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Under the direction of Dr. William M. Dennis

1. PARAMETERS FROM PARAMETER FILE isteps number of divisions to create in interval mystop mystart when calculating values of density operators and P 2 t mystart time in fs at which your external radiation pulses first hit the sample time at which Greens functions begin taking effect must be greater than or equal to zero mystop time at which external radiation stops hitting sample Greens functions no longer have an effect in sample tau delay between external pulses 2 and 3 should be zero wl frequency of external radiation for all three pulses in wavenumbers weg zero phonon line of first order signal from sample in wavenumbers hbar k Planck Boltzmann constant tl T1 the eigenstate decay rate relaxation of density operator 46 47 diagonal matrix elements t2 T2 dephasing rate relaxation of density operator off diagonal matrix elements fwhm fwhm of external radiation pulse in fs Ur Ur Ur Ur UW Ur Ur Ur Ur Ur Ur nn integer nmax i imax real pi sigma parameter nmax 524288 pi 3 14159 real t signalmax risteps mystart mystop wl weg tau tauprime hbar k rp3 nmax ip3 nmax rp2 nmax ip2 nmax rpl nmax ipl nmax rrgeg nmax igeg nmax tsteps tl t2 fwhm rgegr gegi geer geei real rconv nmax iconv nmax invrco
2. 12 2 3 THE BLOCH MODEL 13 2 4 MODELLING THE POLARIZATION AND PHOTON ECHO 19 2 5 REFERENCES 22 3 EXPERIMENTAL ULTRAFAST DEGENERATE FOUR WAVE MIXING 23 3 1 THE LASER SYSTEM 23 3 2 EXPERIMENTAL CONSIDERATIONS 29 3 3 RESULTS AND DISCUSSION 40 3 4 REFERENCES 42 4 CONCLUSIONS APPENDIX A CALCULATING THE POLARIZATION COMPONENTS B CALCULATING THE PHOTON ECHO C DATA COLLECTION PROGRAM vi 45 46 55 65 2 1 2 2 2 3 2 4 3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 LIST OF FIGURES A Superposition State between the Ground State and First Excited State Feynman Diagrams of the Rotating Wave Approximation Polarization components for five frequencies within the inhomogeneous 1 broadening bandwidth zero phonon line at 3000 cm Photon echoes calculated at three delays between pulses 1 and 2 Four Level Lasing System of the Nd ion Absorption and Emission Spectra of AlpO3 Ti l a ee Autocorrelation trace showing a delay of 310 fs between pulses Bandwidth Characterization Fu Electronic energy levels for trivalent Ce Excitation solid line and emission dotted line spectra for Y2SiO Ce at room temperature us ER
3. Pict urel ScaleLeft xmin Pict ure2 ScaleTop ymax2 Pict Pict ure2 ScaleHeight ymax2 ymin2 ure2 ScaleWidth xmax xmin Pict ure2 ScaleLeft xmin maxcnt xmax xmin 1 Text A For 5 Text Str maxcnt replot icnt xmin To xmax If icnt xmin Then Picturel PSet xmin myxval xmin Picture2 PSet xmin myyval xmin Else Picturel Line icnt myxval icnt Picture2 Line icnt myyval icnt 73 End If Next icnt End Sub c REPLOT button c replots data with different x y axis parameters Private Sub cmdreplot Click define variables Dim deltat As Integer Dim icnt As Long Dim maxcnt As Long Dim n As Integer initialize graph and parameters Picturel Cls Picture2 Cls rename text text include commands for empty textboxes for the case where a file is opened and no parameters exist in the appropriate textboxes If Text7 Text lt gt Then xmin CDbl Text7 Text Else xmin liadata 4 text7 Text CStr xmin End If If Text8 Text lt gt Then xmax CDbl Text8 Text Else xmax liadata 5 text8 Text CStr xmax End If If Text2 Text lt gt Then yminl CDDl Text2 Text Else yminl liadata 6 text2 Text CStr yminl End If If Textl Text lt gt Then ymaxl Else ymaxl liadata 7 textl Text CStr ym
4. initialize the lockin Call ibfind dev8 lia this loop is included after every ib command Do Call ibrsp lia r Loop While r And 2 0 lia send output to gpib Call ibwrt lia outxl Do Call ibrsp lia r Loop While r And 2 0 do a reset all info in lia buffers is lost Call ibwrt lia rst Do Call ibrsp lia r Loop While r And 2 0 set input to 107 6 amps 65 66 Call ibwrt lia isrc 2 Do Call ibrsp lia r Loop While r And 2 0 set the input to read volts Call ibwrt lia isrc 0 Do Call ibrsp lia r Loop While r And 2 0 run the auto gain feature Call ibwrt lia agan Do Call ibrsp lia r Loop While r And 2 0 read previously set default settings from the file defaults txt and place in array defsettings Dim defsettings 10 lines As String Dim i As Integer i 0 Open c deanna defaults txt For Input Access Read Shared As 1 Do Until EOF 1 reads elements of the default file into an array and from that array we give lia the correct settings Do While i lt 4 Line Input 1 lines defsettings i lines If defsettings i X Then Call ibwrt lia ddef 1 0 0 ElseIf defsettings i R Then
5. 56 off diagonal matrix elements fwhm full width half max time length of pulse of external radiation in fs minc number of frequencies to include in the inhomogeneous broadening Ur Ur Ur Ur Ur Ur Ur UU Ur Ur Ur Ur UO integer nmax i m minc isteps real pi sigma imax parameter nmax 524288 pi 3 14159 real t signalmax wegmi wegmf deltaw weight rmystart mystop wl weg weg0 tau tauprime hbar k rp3 nmax ip3 nmax rp nmax ip2 nmax rpl nmax ipl nmax rrgeg nmax igeg nmax tsteps tl t2 fwhm gegr gegi geer geei photonecho nmax real rconv nmax iconv nmax invrconv nmax inviconv nmax invrgeg nmax invigeg nmax rgee nmax igee nmax Signal nmax rrel nmax iel nmax invrel nmax inviel nmax rfunc nmax ifunc nmax re nmax ie2 nmax re3 nmax ie3 nmax rdipge idipge elec complex el nmax e2 nmax e3 nmax invel nmax plinv nmax polar nmax rhototal nmax conv p3 nmax geg gee p2 nmax p1 nmax dipge common values hbar weg tl t2 mystop common times tsteps tau tauprime common elecvals wi sigma character 16 fname rdipge 1 0 idipge 0 0 dipge cmp1x rdipge idipge read isteps mystart mystop tauprime tau rwl weg0 hbar k tl t2 fwhm minc 57 c sigma pulsewidth of external radiation field in fs c sigma fwhm 1 76 in units of param file sigma fwhm 1 76 redefine wavenumber in
6. S cmplx rpl i ipl i enddo cC FT FT G t FT E 2 t p1 t call fft rconv iconv nmax nmax nmax 1 do i 1 nmax enddo i 1 do i 1 nmax t mystart i 1 tsteps if i le isteps 1 then c E3 BEGINS AT T A DELAY OF TAUPRIME LATER THAN EI e3 i elec t cmplx cos wl t sin wl t else e3 i 0 endif p2 1 e3 1 cmplx invrconv i inviconv i rp2 i real p2 i ip2 i imag p2 i enddo c p3 FT G t FT conv t Geg calculated in the c beginning of program call fft rp2 ip2 nmax nmax nmax l call fft rgeg igeg nmax nmax nmax l do i 1 nmax rconv i zreal cmplx rgeg i igeg i cmplx rp2 i ip2 i iconv i imag cmplx rgeg i igeg i cmp1x rp2 i ip2 i enddo call fft rconv iconv nmax nmax nmax 1 a2 do i 1 nmax p3 i 1 cmp1x rconv i iconv i c third order polarization V ge p3 eg c invV ge invp3 eg 2Real V ge p3 eg polar i 2 real dipge p3 i enddo c THIS LOOP IS LIMITED TO RECORDING THE VALUES OF c POLARIZATION OR p3 FROM TIME VALUES OF 20 TO 150 open unit 88 file polarization paw do i 20 mystart tsteps l 150 mystart tstepstl t mystart i 1 tsteps write 88 t real p3 i enddo close 88 end complex function gee t implicit none c green function in time domain real hbar gammae
7. Tr Vp t The third order polarization is a triple integral in time involving Green functions and the interaction Hamiltonian In the Bloch model the triple integral can be written in terms of 17 Gge ti ky SEE 19 gt gi Goglt2 Geglti 10 gi 19 lt gI Figure 2 2 Feynman Diagrams of the Rotating Wave Approximation Diagrams taken from Shaul Mukamel Principles of Nonlinear Optical Spectroscopy Oxford University Press London 1995 p 298 18 three convolutions because the response function can be factored The convolutions can be carried out more efficiently using fast Fourier transform techniques than evaluating the triple integral directly The double sided Feynman diagram for R in Figure 2 2 describes a system which begins in the ground state g gl is then acted on by a field to the left with wave vector k which puts the system in a superposition state g el This state is evolved in time by the matter green function G t until the system is perturbed by a second field acting to the right with wave vector ks The system is now in a population state le el which evolves according to the green function G t2 A third field perturbs the system acting on the left with wave vector ks putting the system back into a superposition state e gl which evolves in time according to the green function Ge tz A signal described by wave vector k is then emitted in the ks k3 kg
8. invrconv i zreal cmplx invrgeg i invigeg i cmplx invrel i inviel i inviconv i imag cmplx invrgeg i invigeg i S cmplx invrel i inviel i end do c pl t obtained from FT FT Ginv t FT Einv t call fft invrconv inviconv isteps isteps isteps l c EXFT FT Ginv t FT Hinv t E pl i 1 do i l isteps t mystart i 1 tsteps if i le istepstl then e2 1 elec t cmplx cos wl t sin wl t else e2 i 0 endif 60 pl 1 e2 1 cmplx invrconv i inviconv i rpl i real pl i ipl4i imag pl i end do c find FT E2 t pl t and FT G t call fft rpl ipl isteps isteps isteps l call fft rgee igee isteps isteps isteps l do i l isteps rconv i 0 rconv i zreal cmplx rgee i igee i 5 cmp1x rp1 i ip1 i iconv i 0 iconv i imag cmplx rgee i igee i cmplx rpi i ipl4i enddo c FT FT G t FT E t pl t call fft rconv iconv isteps isteps isteps l do i l isteps t mystart i 1 tsteps if i le isteps 1 then e3 1 elec tttau S cmplx cos wl t sin wl t else e3 i 0 endif p2 i e3 i cmplx invrconv i inviconv i rp2 1 real p2 i ip2 i imag p2 i enddo C p3 FT G t FT conv t Geg calculated in the c beginning of program call fft rp2 ip2 isteps isteps isteps l call fft rgeg igeg isteps isteps isteps l do i l isteps 61 r
9. isteps 1 then el 1 elec t cmplx cos wl t sin wl t else el i 0 endif invel i zconjg el i invrel i1 real invel i inviel i imag invel i enddo a a a a Green functions must be calculated for t gt 0 only but convoluted with E t lt 0 values determined by value of mystart To ensure that elements E t i begin at mystart and elements G i begin at t 0 separate 97 c loop is needed to redefine t for array elements i c when determining the Green function i 1 do i l ist eps t i l tsteps if i le imax then else endif rgeg i gegr t igeg i gegi t rgee i geer t igee i geei t rgeg i 0 igeg i 0 rgee i 0 igee i 0 invrgeg i real conjg cmplx rgeg i igeg i invigeg i imag conjg cmplx rgeg i igeg i enddo c pl invgeg invel in freq space FT G t FT E t G w E w call call c FT G t i 1 do fft invrgeg invigeg isteps isteps isteps 1 fft invrel inviel isteps isteps isteps 1 FT EL t 1 G w E w i l isteps invrconv i zreal cmplx invrgeg i invigeg i cmplx invrel i inviel i inviconv i zimag cmplx invrgeg i invigeg i cmplx invrel i inviel i end do c pl t obtained from FT FT Ginv t FT Einv t call fft invrconv inviconv isteps isteps isteps l do is1 isteps end
10. 1926 11 Felix Bloch Phys Rev 70 460 1946 12 E M Purcell H C Torrey R V Pound Phys Rev 69 37 1946 13 Theodore H Maiman Nature 187 493 1960 14 P A Franken A E Hill C W Peters and G Weinreich Phys Rev Let 7 118 1961 15 Jagdeep Shah Springer Series in Solid State Sciences 115 Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures Springer Verlag Berlin 1996 16 R L Fork C H Brito Cruz P C Becker C V Shank Opt Lett 12 483 1987 17 Web site listing the winners of the 1999 Nobel Prize http www sciam com 2000 0100issue 0100nobel html 18 Robert W Boyd Nonlinear Optics Academic Press Boston 1992 CHAPTER 2 CALCULATION OF THE NONLINEAR OPTICAL RESPONSE FUNCTIONS The doped inorganic crystal Y2SiO Ce which is the focus of this thesis can be par titioned in the following manner the system which describes a subset of the electronic energy levels of the Ce dopant and the bath which describes the vibrational modes of the host crystal When an external radiation field such as an ultrafast optical pulse resonant with a Ce transition interacts with the material an ensemble of electronic superposition states cy g csle is created Figure 2 1 When the system and bath interact i e by the scattering of phonons off the electronic states the coherence of the ensemble is lost The processes responsible for the loss of c
11. 25 0 25 50 75 t fs Figure 2 3 Polarization components for five frequencies within the inhomogeneous broadening bandwidth zero phonon line at 3000 cm a 2500 cm b 2750 cm c 3000 cm d 3250 cmt e 3500 cm P t arb units a 0 1 0 50 100 150 200 1 b 0 1 0 50 100 150 200 1 c 0 1 0 50 100 150 200 t fs Figure 2 4 Photon echoes calculated at three delays between pulses 1 and 2 a 25 fs delay b 75 fs delay c 125 fs delay 22 Calculations of the photon echo signal as a function of the delay 7 between the pulses will be presented and discussed at the end of Chapter Three 2 5 REFERENCES 1 Mare D Levenson Introduction to Nonlinear Laser Spectroscopy Academic Press New York 1982 2 Shaul Mukamel Principles of Nonlinear Optical Spectroscopy Oxford Univer sity Press London 1995 3 L E Fried S Mukamel Adv Chem Phys LXXXIV 435 1993 4 Eugen Merzbacher Quantum Mechanics Second Edition John Wiley amp Sons New York 1970 5 Stig Stehnolm Foundations of Laser Spectroscopy John Wiley amp Sons New York 1984 6 M Joffre Coherent Effects in Femtosecond Spectroscopy A Simple Picture Using the Bloch Equation in Claude Rulli re Ed Femtosecond Laser Pulses Springer Berlin 1998 7 T K Yee T K Gustafson Phys Rev A 18 1597 1978 8 J D Jackson Classical Electrodynamics John Wiley amp
12. Chr 13 End If If optEPl value True Then Call ibwrt mm IER1 1 Chr 13 ElseIf optEP10 value True Then Call ibwrt mm 1ER1 10 Chr 13 End If If optStep value True Then Call ibwrt mm IFMO1I Chr 13 ElseIf optEnc value True Then Call ibwrt mm IFMOO Chr 13 End If If optLoopOn value True Then Call ibwrt mm ICL ON Chr 13 ElseIf optLoopOff value True Then Call ibwrt mm 1CL OFF Chr 13 End If frmOptionsl Hide End Sub c Determines the type of output c as x y or r theta Private Sub cmdstart Click If optx value True Then Call ibwrt lia ddef 1 0 0 frmMain Text10 Text X channelindex 0 1 ElseIf optr value True Then Call ibwrt lia ddef 1 1 0 frmMain Textl0 Text R channelindex 0 2 End If If opty value True Then Call ibwrt lia ddef 2 0 0 frmMain Textll Text Y channelindex l 1 ElseIf opttheta value True Then Call ibwrt lia ddef 2 1 0 frmMain Textll Text Theta channelindex l 2 End If frmOutput Hide End Sub c CANCEL button hides form Private Sub cmdCancel_Click frmOutput Hide End Sub c saves data to a data file c named by the user 87 Private Sub cmdSave_Click Dim m As Integer Dim count As Long If savel Text lt gt Th
13. Hide Else Label3 Caption This file does not exist tion and End If frmOpen End Sub syntax Hide Private Filel Pa txtPath End Sub Sub Dirl_Change th Dirl Path Text Filel Path Private On Error Dirl Pat Exit Sub Sub Drivel Change GoTo DriveDefault h Drivel Drive 84 Input 10 pnt xval yval CDbl xval CDbl yval al icnt CDbl pnt al icnt xval al icnt yval nt pnt mytval icnt myxval icnt myyval ient Check loca 85 DriveDefault Drivel Drive Dirl Path Exit Sub End Sub Private Sub Filel Click txtOpen Text Filel filename End Sub c allows options available on the c motion stage to be activated Private Sub cmdOK_Click initialize the MM3000 if appropriate If frmMain ckMM3000 value True Then Call ibfind devi mm Do Call ibrsp lia r Loop While r And 2 0 End If If optMotorOn value True Then Call ibwrt mm 1MO Chr 13 ElseIf optMotorOff value True Then Call ibwrt mm IMF Chr 13 End If If optBeepOn value True Then Call ibwrt mm 1FSO Chr 13 ElseIf optBeepOff value True Then Call ibwrt mm 1FS40 Chr 13 End If If optBasel00 value True Then Call ibwrt mm 1VB100 Chr 13 ElseIf optBase4000 value True Then Call ibwrt mm 1VB4000
14. achieved the beams are again directed into the sample 3 2 4 DETECTION After the sample all pump and signal pulses are collimated using a 7 cm focal length convex lens A fast photo diode New Focus Model 1621 having a detection surface of 39 1 mm is placed on a translation stage allowing it to be moved from one signal pulse to the other to select the signal for analysis Laser scatter is reduced using an aperture 3 2 5 DATA ACQUISITION PROGRAM The three motion stages and the lock in amplifier were interfaced through the General Purpose Interface Bus GPIB and controlled by a microcomputer The GPIB interface allows up to fifteen devices to communicate through a central controller regardless of the individual manufacturer or device language A program was written in Visual Basic to simultaneously control the motion stages within the experimental set up and data collec tion by the lock in amplifier This program provides a graphical user interface to enable the user to set the parameters for each experimental scan as listed below Motion Stages e zero point or home position e initial position of scan e starting and stopping acceleration e starting and stopping velocity e velocity during motion e size of each step e resolution of each step e time allotted for settling after each step total travel length in absolute or relative distance Lock In Amplifier e form of output r 0 or x y e triggering function
15. Call ibwrt lia ddef 1 1 0 ElseIf defsettings i Y Then Call ibwrt lia ddef 2 0 0 ElseIf defsettings i Theta Then Call ibwrt lia ddef 2 1 0 ElseIf defsettings i External Then Call ibwrt lia fmod 0 ElseIf defsettings i Internal Then Call ibwrt lia fmod 1 ElseIf defsettings i Sine Then Call ibwrt lia rslp 0 ElseIf defsettings i Positive Edge Then Call ibwrt lia rsip 1 ElseIf defsettings i Negative Edge Then Call ibwrt lia rslp 2 End If is it Loop Line Input 1 lines defsettings 4 lines channelindex 0 CInt defsettings 4 Line Input 1 lines defsettings 5 lines channelindex 1 CInt defsettings 5 Loop Close Read values of array into the correct label box parallel bars after the character Text10 Text defsettings 0 Textll Text defsettings 1 Text9 Text defsettings 2 Text6 Text defsettings 3 txtchannelindex0 Text CStr channelindex 0 txtchannelindexl Text CStr channelindex 1 End Sub c Checks for the presence c of the motion stage Private Sub ckMM3000 Click Call ibfind devl mm Call ibln mm 1 O listen test Text listen End Sub c START button c starts the data collection Private Sub cmdstart Click determine number of data points to be taken minus the linefe
16. End Sub 83 c opens previously saved data c file for viewing Private Sub cmdOpen Click openfile txtOpen Text If openfile lt gt Then m InStr openfile If m lt gt 0 Then datafile MidS openfile 1 m dat liafile MidS openfile 1 m lia Else datafile openfile dat liafile openfile dat End If Open datafile For Input As 10 Open liafile For Input As 11 Do Until EOF 11 reads lst line of headings into junk string lia data into an array Line Input 11 junklin s Line Input 11 Data channell liadata 0 Data Line Input 11 Data channel2 liadata l Data Line Input 11 Data trig liadata 2 Data Line Input 11 Data maxcount liadata 3 Data Line Input 11 Data minx liadata 4 Data Line Input 11 Data maxx liadata 5 Data Line Input 11 Data minyl liadata 6 Data Line Input 11 Data Line Line Loop maxcount icnt 1 Do Until For icnt xval yval mytv myxv myyv LPE Data Data Data Data Data CLng liadata 3 maxyl liadata 7 Input fll miny2 liadata 8 Input 11 maxy2 liadata 9 EOF 10 1 To maxcount Next icnt Loop datalin Close 1 Close 1 frmOpen es Input LOF 10 10 0 1
17. If channelindex 0 1 And channelindex 1l 1 Then Call ibwrt lia snap 1 2 Do Call ibrsp lia r Loop While r And 2 0 for r theta ElseIf channelindex 0 2 And channelindex 1 2 Then Call ibwrt lia snap 3 4 Do Call ibrsp lia r Loop While r And 2 0 for x theta ElseIf channelindex 0 1 And channelindex l 2 Then Call ibwrt lia snap 1 4 Do Call ibrsp lia r Loop While r And 2 0 for ryy ElseIf channelindex 0 2 And channelindex 1 1 Then Call ibwrt lia snap 3 2 Do Call ibrsp lia r Loop While r And 2 0 End If myystr Spaces 255 myxstr Spaces 255 read data from x y coord into myystr Call ibrd lia myystr count position of 1st comma mypos records position mypos InStr myystr place only the x coord into myxstr Mid myystr 1 mypos not in myystr 2nd char actual array elements myxstr 1 myyval icnt A CDb1 myxstr CDbl myystr 71 lose the x coord and the end junk from the y coord myystr will now have only the y coord myystr Mid myystr mypos mytval is an array containing the of the pts read mytval icnt CDbl icnt myxval icnt 1 ibcntl mypos l plot each point connecting line between points If icnt 1 Then Picturel PSet mytval l myxval l Picturel Line mytval icnt myxval icnt If icnt 1 Then Picture2 PSet mytval 1 myyval l P
18. Text2 Text Str p2 ElseIf Text10 Text R Then Picturel ScaleHeight Picturel ScaleTop p2 Picturel ScaleHeight Picturel ScaleTop p2 Format p2 Scientific Text2 Text Str p2 Else Picturel ScaleHeight Picturel ScaleTop End If Picturel ScaleWidth maxcnt Text8 Text Str maxcnt Picturel ScaleLeft 1 Text7 Text Str 1 If channelindex 1 1 Then If Text3 Text lt gt Then Picture2 ScaleTop CDbl Text3 Text Else Picture2 ScaleTop Picturel ScaleTop End If If Text4 Text lt gt Then Picture2 ScaleHeight CDbl Text3 Text CDbl Text4 Text Else Picture2 ScaleHeight Picturel ScaleHeight End If ElseIf channelindex l 2 Then If Text3 Text lt gt Then Picture2 ScaleTop CDbl Text3 Text Else Picture2 ScaleTop 180 End If If Text4 Text lt gt Then Picture2 ScaleHeight CDbl Text3 Text CDbl Text4 Text Else Picture2 ScaleHeight 360 End If End If Text3 Text Str Picture2 ScaleTop Text4 Text Str Picture2 ScaleHeight Picture2 ScaleTop Picture2 ScaleWidth eLeft 1 maxcnt Picture2 Scal start the main loop For icnt 1 To maxcnt Timer 0 08 mytime Do Loop Until Timer gt mytime retrieve correct axes 70 for x y
19. True Then Call ibwrt mm IUU um Chr 13 motionindex 0 1 ElseIf optMillimeter value True Then Call ibwrt mm 1UUmm Chr 13 motionindex 0 2 ElseIf optInch value True Then Call ibwrt mm lUUin Chr 13 motionindex 0 3 ElseIf optMil value True Then Call ibwrt mm IUUmil Chr 13 motionindex 0 4 Else Print Error please choose a value for units End If If opt007 value True Then Call ibwrt mm 1US0 007um Chr 13 motionindex l 1 ElseIf opt050 value True Then Call ibwrt mm 1US0 050um Chr 13 motionindex 1 ElseIf opt055 value True Then Call ibwrt mm 1US0 055um Chr 13 motionindex 1 ElseIf opt074 value True Then Call ibwrt mm 1US0 074um Chr 13 motionindex 1 4 ElseIf opt593 value True Then Call ibwrt mm IUSO 593um Chr 13 motionindex l 5 ElseIf opt100 value True Then Call ibwrt mm 1US0 100um Chr 13 motionindex 1 ElseIf opt500 value True Then Call ibwrt mm 1US0 500um Chr 13 motionindex l 7 ElseIf optl value True Then Call ibwrt mm IUSIum Chr 13 motionindex 1 8 ElseIf opt254 value True Then Call ibwrt mm IUS2 54um Chr 13 motionindex 1 9 ElseIf opt10 value True Then Call ibwrt mm 1US10um Chr 13 motionindex l 10 90 Else Print Error please choose a
20. condition Tr p t 1 for our system forces Tr p t 0 Then pi t p 2 t simplifying p t to PIE 2Geg t D EE Vege2 0 2 13 QlVegl VegGeg t amp TE Gee t amp BG e t Geg t amp E t Since the density matrix is Hermitian p t pl t The external field E t is real and can be expressed in terms of a complex field E t i e E t E t E t 2 14 Applying the rotating wave approximation i e keeping only the terms which are res onant the sixteen terms resulting from substituting Eq 2 14 into p6 t are reduced to four The double sided Feynman diagrams 7 of these four terms are given in Figure 2 2 Because we are studying a two level system in which the population relaxation time is much greater than the experimental time scale we set lee Ig 0 in this case and Ri R4 and R2 R3 Of these two remaining terms only R R3 generates a photon echo it is this term that is explored in this work The appropriate third order matrix element of the density operator described by Rs is py Vaal we AMOS NIE SEMI GAS where Ei t 2 t and 3 t are the contributing components of pulses t E t and E3 t The polarization is the ensemble average of the electric dipole operator By expanding the density matrix perturbatively in the electric field we find the third order polarization as the trace of the dipole operator with the third order density matrix P t
21. k direction having frequency Ws W3 w wi In the two beam degenerate four wave mixing described by this thesis the wave vector and frequency terms simplify to k 2k k and w 2wo w1 wi respectively The evaluation of the polarization using the convolution is greatly simplified for our two level system The 2 x 2 matrix obtained by operating the electric dipole operator on the third order density matrix has the form Verd O eg 0 Vee V p t The third order polarization is the trace of this matrix PO Ve NO Vip 2Re V 0l t 2 16 For a homogeneously broadened system the four wave mixing signal as a function of delay 7 can be calculated using Ssg r x i 2 POG r dt 2 17 19 For an inhomogeneously broadened system the total polarization is the sum of inde pendent two level systems whose transition frequencies we are weighted by g we a Gaussian distribution The four wave mixing photon echo signal can be calculated from Eg 2 17 where P t 7 is now the total polarization Note that the 7 dependence of PO 1 r is implicit in the density matrix 2 4 MODELLING THE POLARIZATION AND PHOTON ECHO The model system is an electronic two level system with 7 gt gt T meaning the popula tion relaxation time T is much longer than the dephasing time 75 A nearly impulsive radiation field with a very short pulsewidth FWHM of 5 fs and low frequency 3000 cm was used
22. of the Graduate School The University of Georgia August 2002 ACKNOWLEDGMENTS I would first like to thank my family for being my faithful cheerleaders and counselors during this long process Their support encouragement and prayers have made it possible to persevere to the end I would like to thank the faculty and staff of the Department of Physics and Astronomy at the University of Georgia for their instruction and help Many excellent professors such as Dr Todd Baker and Dr Kanzo Nakayama endured my lack of knowledge to open my mind to deeper understanding I am also grateful to my committee members Dr Baker Dr Michael Geller and Dr William Dennis for allowing me to take some of their already too short time and their contributions to this thesis Special thanks to Dr Dennis for his patience and long suffering through this process iv TABLE OF CONTENTS Page ACKNOWLEDGMENTS 2 GL IG G GG 2 G2 u FY e onon iV LIST OF FIGURES y vii LIST OE TABLES GE oe eee ke etd A de eten BR dap A TA wi dy A OW Wd AN viii CHAPTER 1 INTRODUCTION ongie vere tan te She el ydd de da MEDd lan a a An 1 1 1 ULTRAFAST APPLICATIONS 3 1 2 INTRODUCTION TO NONLINEAR SPECTROSCOPY 3 123 REFERENCES ars oet RR 404 A O ae ee ke ee de O 4 2 CALCULATION OF THE NONLINEAR OPTICAL RESPONSE FUNCTIONS 7 2 1 FUNDAMENTALS 9 2 2 THE POLARIZATION EXPANSION
23. to describing only linear 3 effects until the advent of the laser in 1961 The first laser was built by Theodore H Maiman using a synthetic ruby rod 13 The first nonlinear optical effect was observed by Franken et al in 1961 when harmonic generation was produced by passing the pulse of a ruby laser through a quartz crystal 14 Nonlinear optical effects have since become increasingly important and nonlinear optics has developed into an independent discipline within the field of optics 1 1 ULTRAFAST APPLICATIONS The applicability of nonlinear spectroscopic techniques has been greatly enhanced by the advent of the ultrafast laser The term ultrafast is generally used to describe an event that occurs in a picosecond or less Beginning a decade after the introduction of the laser in the 1960 s ultrafast pulses were generated using organic dye lasers in research laborato ries As lasers using solid state materials capable of emitting a large spectral bandwidth were optimized pulses having a temporal width in the femtosecond 10 s region with some pulses being below 10 fs were produced 12 It can be difficult for the layman to appreciate the shortness of this timescale 1 femtosecond is to 1 second as 1 second is to 31 688 088 years Femtosecond pulses achieved significant notice in the nonscientific media with the awarding of the 1999 Nobel Prize in Chemistry to Ahmed H Zewail for using ultrafast measurements to explore the dynamics
24. 40 e internal or external source for triggering e default values e number of data points collected and averaged per step Data Display e maximum and minimum values of each axis e total data acquisition or portion of data acquisition displayed e step unit desired e total number of scans taken The results were plotted in real time as the data was being taken allowing the user to abort a scan if necessary After data was acquired the user could choose to limit the data shown on the screen and zoom in on regions of interest The values of the axes could be varied both before and after data was taken Data was saved in files named by the user This program is given in Appendix C at the end of this thesis 3 3 RESULTS AND DISCUSSION 3 3 1 EXPERIMENTAL RESULTS The intensity of the four wave mixing signals in the 2k k and 2k k directions was measured as a function of the delay time 7 between the pulse with wave vector k and the pulse with wave vector ks For a sample with a fast dephasing time relative to the pulsewidth the dephasing time of the sample can be studied by examining the shift between the peaks of the 2k k and 2k k signals In the initial experiments the four wave mixing signals exhibited various peak shifts with no discernible pattern It was subsequently discovered that this effect was due to backlash in the translation stage controlling the pathlength of kj After compensating for this probl
25. CH MODEL The treatment in this section closely follows that of Joffre 6 Note that in this section all calculations are done with the reduced density operator to simplify the notation the reduced density operator f t is written as p t In the Bloch model the reduced density operator obeys the following equation no Hs plt Hine PA 4 ih p t dt O Co relax Here Hs is the unperturbed system Hamiltonian having eigenstates E Hs 6n Ele 14 Hing V E t is the electric dipole interaction Hamiltonian describing interaction of the system with the incident pulses E t and V is the electric dipole operator The last term of the equation corresponds to the interaction of the material with the bath which leads to relaxation with matrix elements of the form 2 at 0 p t Dam Pnm t P m relax When n 4 m Pm 1 T5 is denoted the dephasing rate and when n m l an 1 T is denoted the population relaxation rate The density operator at thermal equilibrium is pl where pli 1 since only the ground state is populated Our relaxation term for any matrix element other than simplifies to Pg g relax p O pi nm 8 relax The equation of motion for elements of the density matrix is i Mo Tom Pam 1 a Drood pult Vim O9 where Wim En E h The Green function for the above equation is Com t Oltje ein with the Fourier transform Gon MEES FEE ka W W
26. D amp txtSD Text Call ibwrt mm sd Chr 13 frmMain test Text sd Ene Lt frmVell Hide End Sub Private Sub cmdEnd Click compiles program Call ibwrt mm Chr 13 Call ibwrt mm CP Chr 13 End Sub 92 Private Sub cmdEnter Click com txtCommands Text Call ibwrt mm com Chr 13 Previous txtWindow Text amp Chr 13 txtWindow Text Previous amp com txtCommands Text End Sub c Sets the home position c of the motor stage Private Sub cmdOK_Click initialize the MM3000 if appropriate If frmMain ckMM3000 value True Then Call ibfind devl mm If optFloat value True Then Call ibwrt mm 1OMO Chr 13 ElseIf optSwPulse value True Then Call ibwrt mm 1OM1 Chr 13 ElseIf optSW value True Then Call ibwrt mm 1OM2 Chr 13 End If frmzerol Hide End Sub Private Sub cmdOrigin Click Call ibfind devi mm Call ibln mm 1 O listen Print listen Call ibwrt mm IOR Chr 13 End Sub a a a a a a aaa a a a a a a a a a a a a a a a a a a a a Q APPENDIX D CALCULATING THE FOUR WAVE MIXING SIGNAL program joffres transforms implicit none Uses fast fourier transform method by Joffre to calculate third order polarization signal Equation on pg 277 is one term of the function Each convolution is a separate subroutine convolution from third order expression PARAMETERS FRO
27. DW Y NIW RE dled ae Two Beam Degenerate Four Wave Mixing SS ss Experimental Set Up for Degenerate Four Wave Mixing ancle 0c ur dre rt EE ET EG RS AE DROS MR RE Echo signals as a function of delay calculated at various dephasing times with a laser FWHM pulse of 5 fs SS EE SE SS ES ss se vii CHAPTER 1 INTRODUCTION Until the latter half of the twentieth century the understanding of the interaction of light and matter was limited to the linear response of matter to electromagnetic radiation Beginning in 1666 with the separation of white light into its component colors by Isaac Newton for hundreds of years man has been investigating the linear response of matter to light The apparent continuum observed by Newton was found to contain discrete lines and the field of spectroscopy was born due to the pioneering works of Joseph von Fraunhofer 1778 1826 Robert W Bunsen 1811 1899 and Gustav R Kirchhoff 1824 1887 1 The corpuscular theory of light was strongly adhered to until Thomas Young in 1801 discovered interference shifting belief to a wave description of light The nature of light as electromagnetic radiation travelling in wave form was mathematically described by James Clerk Maxwell first in 1862 and collected into the famous Maxwell s Equations by Oliver Heaviside and Heinrich Hertz in 1885 2 Maxwell believed that electromag netic radiation travelled in a medium known as the ether an assumpt
28. M PARAMETER FILE isteps number of divisions to create in interval mystop mystart when calculating values of density operators and P 2 t Must be an integer mystart time in fs at which your external radiation pulses first hit the sample time at which Greens functions begin taking effect must be greater than or equal to zero mystop time at which external radiation stops hitting sample Greens functions no longer have an effect in sample tau delay between ext pulses 2 and 3 should be zero taustart first value of tauprime incalculations taustop last value of tauprime in calculations tauincrements of tau values to step for each value of tauprime in calculations wl frequency of external radiation laser weg0 zero phonon line of first order signal from sample in wavenumbers hbar k Planck Boltzmann constant ti TI the eigenstate decay rate relaxation of density operator diagonal matrix elements t2 T2 dephasing rate relaxation of density operator off diagonal matrix elements 93 94 c fwhm full width half max time length of pulse of c external radiation in fs integer nmax i j jmax m minc isteps real pi sigma imax parameter nmax 524288 pi 3 14159 real t signalmax wegmi wegmf deltaw weight mystart mystop wl weg weg0 tau tauprime taustart taustop tauincrements hbar k rp3 nmax ip3 nmax S rp2 nmax ip2 nmax rp1 nmax ipl nmax r
29. Sons New York 1962 9 Robert W Boyd Nonlinear Optics Academic Press Boston 1992 CHAPTER 3 EXPERIMENTAL ULTRAFAST DEGENERATE FOUR WAVE MIXING 3 1 THE LASER SYSTEM An actively modelocked Ti Sapphire laser is used to generate ultrashort pulses on the order of 70 femtoseconds The system is composed of 1 an intracavity doubled Nd YVO solid state pump laser and 2 an actively modelocked ultrafast titanium sapphire laser followed by an optional frequency doubler Characterization equipment includes an auto correlator for measuring the pulse width and a rotating grating spectrometer for moni toring the bandwidth Each component is described in detail below 3 1 1 ND YVO PUMP LASER The Nd Y VO laser Spectra Physics Millennia 1 is optically pumped by two fiber optic bundles which propagate the output of two GaAlAs diode laser bars lasing at 809 nm Each diode laser bar is capable of 20 W power output but is operated at only 75 of the maximum power output to increase longevity and to enable stabilization of the Nd YVO laser Operating at less than full power allows for minor misalignment of the Nd YVO laser to be compensated for by increasing or decreasing the electric current to the diodes The usefulness of this feature was observed when the lenses focusing the diode emission into the fiber optic bundles became misaligned resulting in reduction of the pumping efficiency The attempts by the feedback system to co
30. ULTRAFAST DEGENERATE FOUR WAVE MIXING IN Y3SIOs CEST by DEANNA M CLARKSON Under the direction of Dr William M Dennis ABSTRACT Both linear and nonlinear spectroscopic techniques are used to investigate a single crystal of the blue phosphor Y2SiO Ce In particular excitation and emission spectra are measured at room temperature Ultrafast two beam degenerate four wave mixing 1s used to investigate dephasing at room temperature Numerical calculations of optical dephasing on a model system are performed within the optical Bloch model Compar ison of the model calculations with experimental results indicate that the dephasing time of Y2SiO Ce is considerably shorter than the pulsewidth of the laser pulses i e 70 femtoseconds INDEX WORDS Nonlinear spectroscopy Ultrafast optics Four wave mixing Optical Bloch model Dephasing ULTRAFAST DEGENERATE FOUR WAVE MIXING IN Y9SIO CE by DEANNA M CLARKSON B S University of Montevallo 1990 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE ATHENS GEORGIA 2002 2002 Deanna M Clarkson All Rights Reserved ULTRAFAST DEGENERATE FOUR WAVE MIXING IN Y2S1O 5 CE by DEANNA M CLARKSON Approved Major Professor Dr William M Dennis Committee Dr Michael R Geller Dr F Todd Baker Electronic Version Approved Gordhan L Patel Dean
31. UR WAVE MIXING One of the major experimental techniques that takes advantage of the third order suscepti bility is four wave mixing FWM 12 13 This technique which can be used to measure both population decay and electronic dephasing includes a variety of nonlinear optical experiments such as photon echoes Three laser pulses having wave vectors ki k2 and kg interact at the sample to produce a fourth signal with a wave vector in the ky ki ko ks direction The frequencies of the three pulses add with the same sign as the wave vectors The three interacting waves therefore generate a fourth wave with frequency w4 wi w2 ws in the direction specified by the wave vectors There are several variations of the basic FWM experiment FWM signals can be described in either the frequency domain or time domain For ultrashort pulses which have a broad frequency spectrum the time domain description is more appropriate In time domain FWM the behavior of the third pulse as the delay time between the first two pulses is varied provides us with information about population decay time or dephasing 8x10 6x10 arb units 4x10 Intensity 2x10 300 400 500 600 wavelength nm Figure 3 6 Excitation solid line and emission dotted line spectra for Y2SiO Ce at room temperature 35 time By choosing the polarization combination of the incoming pulses it is possible to measure different components of the nonli
32. ator can be used in calcu lations The Hamiltonian which describes the sample can be partitioned into four sections 1 the system Hamiltonian Hs which usually describes the electronic states of interest 2 the interaction Hamiltonian H which describes the interaction of the system with the external electric field 3 the bath Hamiltonian H g which describes the reservoir and 11 4 the system bath Hamiltonian H sg which describes the coupling between the system and the bath For Y2SiO Ce the system is the electronic states of the Ce dopant and the bath is the vibrational modes of the Y2SiO crystalline host The system and bath are described by eigenstates of their respective Hamiltonians We take the eigenstates of Hs to be 77 and the eigenstates of Hg to be After interaction the general state would be ME 2 Crul En n where n from a complete direct product basis in the Hilbert space of the combined system and bath If we wish to Know the ensemble average of an observable which depends only on the system degrees of freedom and not on the bath degrees of freedom we can sum over the eigenstates of the bath and obtain a reduced density operator A t doula Eu L with matrix elements P nn X ma Eulo t Su lv Tre male na L where Trg is a partial trace over only the bath degrees of freedom The ensemble average of an operator depending only on the system eigenstates using
33. ax t i l tsteps if i le imax then rgeg i gegr t igeg 1 gegi t rgee i geer t igee i geei t else rgeg i 0 igeg i 0 rgee i 0 igee i 0 endif invrgeg i rreal conjg cmplx rgeg i igeg i invigeg i imag conjg cmplx rgeg i igeg i enddo 50 c pl invgeg invel in freq space FT G t FT E t J c G w E w call fft invrgeg invigeg nmax nmax nmax l call fft invrel inviel nmax nmax nmax l C FT G t FT E1 t G w E w i 1 do i 1 nmax invrconv i zreal cmplx invrgeg i invigeg i cmplx invrel i inviel i inviconv i imag cmplx invrgeg i invigeg i S cmplx invrel i inviel i end do c pl t obtained from FT FT Ginv t FT Einv t call fft invrconv inviconv nmax nmax nmax l C E FT FT Ginv t FT Einv t zE pl i 1 do i 1 nmax t mystart i 1 tsteps if i le isteps 1 then c E2 CALCULATED AT T A DELAY OF TAUPRIME LATER THAN El e2 i elec t cmplx cos wl t S sin wl t else e2 i 0 endif p1 i e2 i cmplx invrconv i inviconv 1i rpl i zreal pl i ipl i imag pl i end do c find FT E2 t pl t and FT G t call fft rpl ipl nmax nmax nmax l call fft rgee igee nmax nmax nmax l 51 do i 1 nmax rconv i zreal cmplx rgee i igee i S cmplx rpl i ipl i iconv i rimag cmplx rgee i igee i
34. axl End If If Text3 Text Then ymax2 CDbl Text3 Text Else ymax2 liadata 9 text3 Text CStr ymax2 End If Q Dbl Text1 Text If Text4 Text lt gt Then ymin2 CDbl Text4 Else ymin2 liadata 8 text4 Text CStr ymin2 End If If Texti0 Text Then If liadata 0 1 Then Text10 Text RY ElseIf liadata 0 2 Then Text10 Text X End If End If If Textll Text Then If liadata l 1 Then Textll Text Theta ElseIf liadata l 2 Then Textll Text Y End If End If If Textll Text Then If liadata 2 1 Then Text6 Text Sine ElseIf liadata 2 2 Then Text6 Text Positive Edge ElseIf liadata 2 3 Then Text6 Text Negative Edge End If End If reset graphs using values from text boxes Picturel ScaleTop ymaxl Picturel ScaleHeight ymaxl yminl Picturel ScaleWidth xmax xmin Picturel ScaleLeft xmin Picture2 ScaleTop ymax2 Picture2 ScaleHeight ymax2 ymin2 Picture2 ScaleWidth xmax xmin Picture2 ScaleLeft xmin maxcnt xmax xmin 1 Text5 Text Str maxcnt replot icnt xmin For icnt xmin To xmax If icnt xmin Then Text 74 75 Picturel PSet mytval xmin myxval xmin Picture2 PSet mytval xmin myyval xmin Else Picturel Line mytval icnt
35. conv i 0 rconv i real cmplx rgeg i igeg i cmp1x rp2 i ip2 i iconv i 0 iconv i imag emplx rgeg i igeg i cmp1x 5 rp2 4 pa 4 enddo call fft rconv iconv isteps isteps isteps l i 1 do i l isteps p3 1 rdipge 2 idipge 2 conjg dipge S cmplx rconv i iconv i 4 rhototal i rhototal i weight p3 i c third order polarization V ge p3 eg invV ge invp3 eg 2Real V ge p3 eg Photon echo is found c by adding the polarization terms together Q photonecho i photonecho i S 2 real dipge rhototal i enddo c ending of loop over frequencies in the external c radiation bandwidth m loop enddo c write final values for each element of photonecho i c with respect to real time to a file open unit 89 file photonecho paw do i l isteps t mystart i 1 tsteps write 89 t photonecho i enddo close 89 end complex function gee t c green 62 implicit none function in time domain real hbar gammaee t igee rgee weg t1 t2 mystop common values hbar weg ti t2 mystop gammaee 1 t1 if t gt 0 then gee cmp1x 0 0 exp gammaee t hbar else gee cmplx 0 0 0 O endif return end real function geer t implicit none real t complex gee if t 1t 0 then geer 0 else geer real gee t endif return end real function geei t c gee has only imaginary parts implici
36. d the end junk from the y coord myystr will now have only the y coord myystr MidS myystr mypos 1 ibcntl mypos 1 mytval is an array containing the of the pts read mytval icnt CDbl icnt CDb1 myxstr CDb1 myystr myxval icnt myyval icnt sum data points datasumx myxval icnt datasumx datasumy myyval icnt datasumy Next icnt average data points dataavex q datasumx maxcnt dataavey q datasumy maxcnt save the data from each loop into separate files numeri cally indexed by q If txtFileInd Text lt gt Then m InStr txtFileInd Text If m lt gt 0 Then datafile Mid txtFileInd Text 1 m 6 g amp dat liafile MidS txtFileInd Text 1 m amp g amp lia Else datafile txtFileInd Text amp q amp dat liafile txtFileInd Text amp g amp lia End If Open datafile For Output As 72 Open liafile For Output As 73 cannot save parameters ast Forml textn text becaus variable names are needed in order to pull out data when file is reopened maxcnt CDbl frmMain Text5 Text If CStr frmMain Textl0 Text R Then channell 1 ElseIf CStr frmMain Textl0 Text X Then channell 2 Else Print Error in reading channel 1 channell 0 80 End If If CStr frmMain Textll Text Theta Then channel2 1 ElseIf CStr frmMain Textll Text Y Then c
37. do 98 cC E FT FT Ginv t FT Einv t zE pl i l do i l isteps t mystart i 1 tsteps if i le isteps l then e2 i elec trtauprime S cmplx cos wl t sin wl t else e2 i 0 endif pl i e2 1 cmplx invrconv i inviconv i rpl i real pl i ipl i imag p1 i end do c find FT E2 t pl t and FT G t call fft rpl ipl isteps isteps isteps 1 call fft rgee igee isteps isteps isteps l do i l isteps rconv i O rconv i zreal cmplx rgee i igee i 5 cmplx rpl i ipl i iconv i 0 iconv i imag cmplx rgee i igee i cmplx rpl i ipl i enddo cC FT FT G t FT E t pl t call fft rconv iconv isteps isteps isteps l do i l isteps tzmystartr i l tsteps if i le istepsrl then e3 i elec t tauprime tau S cmplx cos wl t S sin wl t else 99 e3 i 0 endif p2 1 e3 i cmplx invrconv i inviconv 1i rp2 1 real p2 i ip2 i imag p2 1 enddo c p3 FT G t FT conv t Geg calculated in the c beginning of program Q call call do fft rp2 ip2 isteps isteps isteps l fft rgeg igeg isteps isteps isteps l i l isteps rconv i O rconv i zreal cmplx rgeg i igeg i cmplx rp2 i ip2 i iconv i 0 iconv i imag cmplx rgeg i igeg i cmp1x rp2 i ip2 i enddo ca do cmp rhotot enddo ending o radiatio en loop
38. e Li3BO3 non linear mixing crystal to generate a 532 nm continuous wave emission at a power of 5 W it is this laser emission that is used to pump the Ti Sapphire laser 3 1 2 TI SAPPHIRE LASER The ultrashort pulse laser Spectra Physics Tsunami 2 uses a Ti doped sapphire A1 03 crystal as its laser medium The Ti ions are strongly coupled to the vibrational modes of the host resulting in broad emission bands The absorption band extends from 400 nm to 650 nm the peak occurring near 500 nm as shown in Figure 3 2 The emission band extends from 600 nm to 1050 nm but lasing only occurs to the red of 670 nm due to overlap with the absorption band Although the peak of the laser emission occurs at 790 nm the laser has a tuning range from 690 nm to 1080 nm The experiments described in this thesis were performed using pulses with wavelength 790 nm which were then fre quency doubled A 3 66 m pathlength is created in an enclosure smaller than 1 m through the use of a folded cavity A prism is used to spectrally disperse the beam allowing a variable slit to be used to tune the bandwidth Bandwidths smaller than 11 nm cause significant pulse lengthening while bandwidths larger than 27 nm lead to instability of the laser After bandwidth selection the beam is sent through a second prism to reverse the spectral dispersion of the first prism Modelocking of the laser is accomplished using an acousto optic modulator The resulting pulses have a
39. e Click initialize MM3000 Call ibfind devl mm Call ibln mm 1 O listen frmMain test Text listen read sensitivity value from lia Call ibwrt lia sens Do Call ibrsp lia r Loop While r And 2 0 senst Space 255 Call ibrd lia senst remov nd of line character from senst data and convert data to interger senst MidS senst 1 ibcntl l isens CInt senst TI create array of fixed sensitivity values from 1V to 1 10 9 V sensval 26 1 For n 1 To 9 If n lt gt 9 Then sensval 26 3 n 1 10 n sensval 26 3 n 1 2 10 n sensval 26 3 n 2 5 10 n Next set graph parameters and display in appropriate text boxes If frmMain Textl Text lt gt Then pl CDbl frmMain Textl Text Else pl sensval isens for microamp scale Else pl sensval isens 0 000001 End If pl Format pl Scientific If frmMain Text2 Text lt gt Then p2 CDbl frmMain Text2 Text Else p2 O End If p3 Str p2 p4 1 If channelindex 1 1 Then p5 pl p6 p2 ElseIf channelindex 1 2 Then po 180 p6 360 End If variable for MM3000 motion for each loop axis 3 motion 3PR amp txtSteps Text maxcnt txtmaxcount Text marker for ending execution of all motion loops If optAbsLoop value True Then bb txtAbsLoo
40. e Figure 3 7 Two beam DFWM is the technique used in this thesis In this experiment the three pulses can be considered to interact in the following way The first pulse K creates an electronic superposition state creating a time varying polarization A second pulse ko arrives at the sample after a fixed delay time and modulates the polarization created by k This polarization then radiates the signal field If the delay time between the first and second pulse is less than the time required for the sample to lose its phase memory due to system bath interactions termed the dephasing time no nonlinear signal will be observed 36 ul an Figure 3 7 Two Beam Degenerate Four Wave Mixing A particularly simple form of DFWM which is often a useful model for interpreting the experiment is the transient grating Two beams incident on the sample at an angle with respect to each other cause an interference patterns Bright areas where the two beams interfere constructively will cause electronic transitions to a resonant excited state Dark areas where the two beams interfere destructively will result in the electronic population remaining in the ground state resulting in a population grating The third pulse k again then arrives at the sample and is diffracted by the population grating The signal generated by the third pulse provides information on the population lifetimes 3 2 3 DFWM EXPERIMENTAL SET UP A schema
41. e requirements of the experiment these pulses were then frequency doubled to 395 nm Experimental data using a laser pulsewidth of 70 fs revealed no shift between the 2k5 k and 2k k four wave mixing signal peaks In addition the experimentally obtained photon echo signals showed no obvious asymmetry The experimental results are consistent with calculations performed on a model system having a dephasing time significantly shorter than the laser pulsewidth as a result the dephasing time of Y SiO Ce is concluded to be significantly shorter than the pulsewidth of the laser i e 70fs Since the dephasing rate 7 provides a measure of the coupling of the electronic system to the bath this result indicates that the electron lattice coupling is strong in this system as is expected from the large Stokes shift 45 aa a 0 00 0 RO a aannnnnnnnnnnnnnnnn APPENDIX A CALCULATING THE POLARIZATION COMPONENTS program joffres polarization implicit none THIS PROGRAM IS A VARIATION OF MAIN PROGRAM BJOFFRE F AND CALCULATES THE POLARIZATION FOR ONE FREQUENCY COMPONENT OF THE INHOMOGENEOUS BROADENING AT A FIXED TAUPRIME DELAY VALUE BETWEEN PULSE 1 AND PULSE 2 NOTES FOR POLARIZATION PROGRAM WRITTEN IN ALL CAPS THIS PROGRAM IS A MODEL SO WEG IS NOT THE TRUE TRANSITION FREQUENCY OF CERIUM DOPED YTTRIUM SILICATE THE PROGRAM MODELS POLARIZATIONS WHICH OCCUR AT AND SLIGHTLY OFF THE TRANSITION FREQUENCY
42. e t igee rgee weg t1 t2 mystop common values hbar weg tl t2 mystop gammaee l tl if t ge 0 then gee cmplx 0 0 exp gammaee t hbar else gee cmplx 0 0 0 O endif return end real function geer t implicit none real t complex gee if t 1t 0 then geer 0 else 53 geer real gee t endif return end real function geei t c gee has only imaginary parts implicit none real t complex gee if E 1E S0 then geei 0 else endif c green geei imag gee t return end complex function geg t implicit none function in time domain real hbar t weg gammaeg t1 t2 mystop common values hbar weg tl t2 mystop gammaeg 1 t2 if t ge 0 then geg exp gammaeg t cmplx sin weg t cos weg t hbar else gegzcmplx 0 0 0 0 endif return end real function gegr t implicit none complex geg real t if t 1t 0 then gegr 0 else gegr real geg t endif return 54 end real function gegi t implicit none complex geg real t if t 1t 0 then gegi 0 else gegi imag geg t endif return end real function elec t implicit none c real amplitude expression of electric field c E t lec t e iwt real t tsteps sigma wl sech tau tauprime common times tsteps tau tauprime common elecvals wl sigma elec sech t sigma return end real function sech t implicit none real t sech 2 exp t exp t retur
43. ed and end of line characters they print bold 68 If Text5 Text lt gt Then maxcnt CLng Text5 Text Else maxcnt 1000 End If Text5 Text Str maxcnt n 0 initialize graph Picturel Cl Picture2 Cls read sensitivity value from lia Call ibwrt lia sens Do Call ibrsp lia r Loop While r And 2 0 senst Spaces 255 isens 0 Call ibrd lia senst remov nd of line character from senst data and convert data to integer senst MidS senst 1 ibcntl 1 isens CInt senst create array of fixed sensitivity values from 1V to 1 10 9 V sensval 26 1 For n 1 To 9 If n lt gt 9 Then sensval 26 3 n 1 10 n sensval 26 3 n 1 2 10 n sensval 26 3 n 2 5 10 n Next set graph parameters and display in appropriate text boxes If Textl Text lt gt Then Picturel ScaleTop CDbl Texti Text 0 000001 used for microamp scale Else Picturel ScaleTop sensval isens 0 000001 pl Picturel ScaleTop pl Format pl Scientific 69 Textl Text Str pl End If If Text2 Text Then Picturel ScaleHeight CDbl Picturel ScaleTop CDbl Text2 Text ElseIf Texti0 Text X Then Picturel ScaleHeight 2 Picturel ScaleTop p2 Picturel ScaleHeight Picturel ScaleTop p2 Format p2 Scientific
44. em Carl Liebig 4 Normalized DFWM Signal Intensity 0 Delay Time fs Figure 3 9 Two beam degenerate four wave mixing signals measured on Y2SiO Ce at 395 nm Filled circles are 2ky k signal Open circles are 2k k signal Data courtesy of Carl Liebig 42 gathered data on the sample which illustrated no shift between the peaks as shown in Figure 3 9 3 3 2 MODELLING THE RESULTS Four wave mixing signals for the model system described in Chapter Two were calculated from the program given in Appendix D The shape of the four wave mixing signals calculated with dephasing times of 0 01 fs and 2 5 fs 0 002 x pulsewidth and 0 5 x pulsewidth respectively are very similar although with a slight shift illustrating the difficulty in determining an accurate dephasing time when 75 is much shorter than the laser pulsewidth When the dephasing time is equal to or longer than the pulsewidth as shown in the echo signals calculated with dephasing times of 5 0 fs and 10 0 fs equal to and twice the laser pulsewidth respectively a noticeable asymmetry can be seen in the signal Dephasing times which are long relative to the external pulsewidth would give a signal with an exponential tail with characteristic time 75 and would exhibit a peak shifted significantly from zero Dephasing times which are short relative to the external pulsewidth would give a signal shape similar to the external radiation pulseshape obtained by an autoc
45. en m InStr savel Text If m lt gt O Then datafile MidS savel Text 1 m dat liafile MidS savel Text 1 m lia Else datafile savel Text dat liafile savel Text lia End If Open datafile For Output As 2 Open liafile For Output As 3 cannot save parameters ast Forml textn text becaus variable names are needed in order to pull out data when file is reopened maxcount CDbl Forml Text2 Text If CStr Forml Text9 Text R Then channell 1 ElseIf CStr Forml Text9 Text X Then channell 2 Else Print Error in reading channel 1 channell 0 End If If CStr Forml Text10 Text Theta Then channel2 1 ElseIf CStr Forml Textl0 Text Y Then channel2 2 Else Print Error in reading channel 2 channel2 0 End If If CStr Forml Textll Text Sine Then trig 1 ElseIf CStr Forml Textll Text Positive Edge Then trig 2 ElseIf CStr Forml Textll Text Negative Edge Then trig 3 Else 88 Print Error in reading the trigger trig 0 End If minx CDbl Forml text3 Text maxx CDbl Forml text4 Text minyl CDbl Forml text5 Text maxyl CDbl Forml text6 Text miny2 CDb1 Forml text8 Text maxy2 CDbl Forml text7 Text print heading of all info in order then print each piece of i
46. en motion and next loop defined sub procedure Waittime 10 q q 1 Loop While q 1 bb write files of averages of each loop gq 0 tempx 0 tempy 0 fileave ave amp txtFileInd Text amp dat liastuff ave amp txtFileInd Text amp lia Open fileave For Output As 74 For q 1 To bb tempx dataavex q tempy dataavey q Print 74 q Format tempx Scientific Format tempy Scientifi Next q Close 74 Open liastuff For Output As 70 Print 70 Ch 1 Ch 2 Trigger pts minpt maxpt minch1 Print 70 channell Print 70 channel2 Print 70 trig Print 70 maxcnt Print 70 minx Print 70 maxx Print 70 minyl Print 70 maxyl Print 70 miny2 Print 70 maxy2 Close 70 End Sub c Allows the stage to rest c for a specified amount of c time after motion Public Sub Waittime X As Long Dim starttime As Single starttime Timer Do Until Timer starttime 86400 Mod 86400 gt X Loop End Sub c moves the motion stage Private Sub cmdOK Click If txtAbsl Text lt gt Then amove IPA amp txtAbsl Text Call ibwrt mm amove Chr 13 frmMain test Text amove Debug Print amove amove End If If txtRell Text lt gt Then rmove IPR amp txtRell Text Call ibwrt mm rmove Chr 13 frmMain test Text rmove Debug Print rmove rmove End If frmMain test Text frmMovel Hide
47. geg nmax igeg nmax tsteps tl t2 fwhm gegr gegi geer geei real rconv nmax iconv nmax S invrconv nmax inviconv nmax invrgeg nmax invigeg nmax S rgee nmax igee nmax Signal nmax rel nmax iel nmax invrel nmax S inviel nmax rfunc nmax ifunc nmax S re nmax ie2 nmax re3 nmax rdipge idipge elec ie3 nmax complex el nmax e2 nmax e3 nmax invel nmax plinv nmax polar nmax rhototal nmax conv p3 nmax geg gee p2 nmax pl nmax dipge common values hbar weg t1 t2 mystop common times tsteps tau tauprime common elecvals wl sigma character 16 fname rdipge 1 0 idipge 0 0 dipge cmp1x rdipge idipge read isteps mystart mystop tau taustart S taustop tauincrements weg0 wl hbar k tl t2 fwhm minc c sigma fwhm 1 76 in units of param file sigma fwhm 1 76 95 c calculated wegmi wegmf as being 800 cm 1 before c after the zero phonon line weg0 Convert wegmi c wegmf into frequency 1 fs wegmi weg0 800 2 pi 3e8 le 13 wegmf weg0 800 2 pi 3e8 le 13 deltaw wegmf wegmi minc c redefine wavenumber input into correct units of c frequency calcs are done with frequencies in c units of 1 fs lel5 s wl 2 pi 3e8 wl le 13 weg0 2 pi 3e8 weg le l13 c outer loop for calculating the signal at different Q values of the maximum value for tauprime amp i has c to be an integer jma
48. hannel2 2 Else Print Error in reading channel 2 channel2 0 End If If CStr frmMain Text6 Text Sine Then trig 1 ElseIf CStr frmMain Text6 Text Positive Edge Then trig 2 ElseIf CStr frmMain Text6 Text Negative Edge Then trig 3 Else Print Error in reading the trigger trig 0 End If ninx Dbl p4 Db1 p3 C e n maxx minyl CDbl p2 maxyl CDbl pl n n niny2 CDbl p6 naxy2 CDbl p5 print heading of all info in order then print each piece of info on line creating array to be read when file is opened Print 73 Ch 1 Ch 2 Trigger fpts minpt maxpt minchl maxchl minch2 maxch2 Print 73 channell Print 73 channel2 Print 73 trig Print 73 maxcnt Print 73 minx Print 73 maxx Print 73 minyl Print 73 maxyl Print 73 miny2 Print 73 maxy2 For icnt 1 To maxcnt tempx myxval icnt tempy myyval icnt 81 Print 72 icnt Format tempx Scientific Format tempy Scientific Next icnt Close 72 Close 73 Else txtLabel Text You have not specified a filename End If Loop until LIA is done storing data Do Call ibrsp lia r Loop While r And 2 0 move MM3000 Call ibwrt mm motion Chr 13 Do Waittime 1 Call ibwrt mm 3TS Call ibrd mm whereami Loop While Left whereami 1 Chr 68 wait betwe
49. icture2 Line mytval icnt myyval icnt Next icnt End Sub C C C C Private Sub cmdopenplot Cl Dim deltat As Integer Dim icnt As Long Dim maxcnt As Long Dim n As Integer initial displays plots of data saved in a previously saved file liadata contains the saved parameters and dat contains the actual Picturel Cls Picture2 Cls frmMain Cls xmin liadata 4 Text7 Text CStr xmin icnt xmin data Lick ize graph and parameters xmax liadata 5 Text8 Text CStr xmax yminl liadata 6 Text2 Text CStr ymin ymaxl liadata 7 Textl Text CStr ymaxl ymax2 liadata 9 Text3 Text CStr ymax2 ymin2 liadata 8 Text4 Text CStr ymin2 res Pict 12 If liadata 0 1 Then Text10 Text R ElseIf liadata 0 2 Then Text10 Text X Else Text10 Text Error End If If liadata 1 1 Then Text11 Text Theta ElseIf liadata 1 2 Then Text11 Text y Else Text11 Text Error End If If liadata 2 1 Then Text6 Text Sine ElseIf liadata 2 2 Then Text6 Text Positive Edge ElseIf liadata 2 3 Then Text6 Text Negative Edge Else Text6 Text Error End If et graphs using values from text boxes Pict urel ScaleTop ymaxl urel ScaleHeight ymaxl yminl Pict urel ScaleWidth xmax xmin
50. ime interval of the data scan 4 The signal is filtered by an ultraviolet band pass filter and directed into a pho tomultiplier tube housed within the autocorrelator The resulting signal is viewed on an oscilloscope The fused silica block rotates 30 times per second resulting in 60 signals per second being sent to the oscilloscope The pulsewidth of the autocorrelation signal is a t function of the pulse shape and the true FWHM Ac of the sech 55 laser pulse where AX is the pulsewidth is scaled by a factor of s 0 65 of the autocorrelation trace Ao sAo where Ao is the FWHM of the signal as shown on the oscilloscope A fixed delay of 2Ad n c is introduced into one of the pathlengths by placing a cal ibrating etalon within the beampath in a position where the beam will travel through it twice The etalon has a thickness of Ad and an index of refraction of n The temporal FWHM of a pulse Ac is found from the oscilloscope readings by the following calcula tion 3 Ao ZAd m Ac Ap e S where Ap is the distance between adjacent pulses as measured on the oscilloscope and c is the speed of light For the autocorrelator used in this work 2Ad n c 310 fs 29 An example of the autocorrelation trace with and without the etalon delay is given in Figure 3 3 3 1 4 BANDWIDTH CHARACTERIZATION The spectral bandwidth of the ultrashort pulses is measured by directing a small fraction of the laser emission i
51. in the model calculation in order to reduce computer time The polarization was found directly by computing the expression in Eq 2 16 using the program given in Appendix A Polarization components for five frequency compo nents within the inhomogeneous broadening bandwidth are given in Figure 2 3 The delay between pulse one and two was fixed at 50 fs for these calculations with pulse two occur ring at t 0 We note that the five polarization components remain out of phase until 50 fs the time after pulse two which equals the delay between pulse one and pulse two At 50 fs all the polarization components are seen to be in phase creating the photon echo The photon echo was found by summing the calculated polarization components from 200 frequency components contributing to the inhomogeneous broadened line according to the program given in Appendix B Calculations of the photon echo illustrate the behavior of the echo as the delay between pulses k and k kz is varied as shown in Figure 2 4 As expected the photon echoes occur at a time interval after the arrival of ko equal to the delay between k and ks As the delay increases the amplitude of the photon echo signal decreases When the change in amplitude is graphed with respect to delay t T gt time an exponential decay of e results o A 25 0 25 50 75 1 b 0 a 1 DE 0 25 50 75 5 1 C g c S l A 28 25 50 75 1 d 0 1 25 0 25 50 75 1 e 0 1
52. ing the two beams are chopped at different frequencies by a optical chopper wheel Stanford Research Systems Model SRS540 The chopper wheel contains an inner and outer set of blades each set having a different number of apertures One beam is passed through each set of blades ensuring a difference in the chopping frequencies of the two beams The chopper ensures that only a signal generated by a combination of the two frequencies and thus the two beams is detected by the lock in amplifier Stanford Research Systems Model SR380 Any scattering from either of the beams is chopped only at the frequency of the source beam and is therefore discriminated against Only a signal resulting from the interaction of the two beams will be modulated at the sum frequency and therefore amplified The two beams are then focused into the sample using a short focal length 3 cm convex lens Spatial overlap in the sample is achieved by placing one steering mirror in each path after the third iris and before the focusing lens The two beams are aligned parallel over a distance of approximately 5 m ensuring spatial overlap when focused into the sample Temporal overlap is most quickly achieved by using a mirror to divert the aligned and focused beams into an LisBO second harmonic crystal and adjusting the placement of the translation stage until the two focused beams produce a single beam of frequency doubled light visible between them Once temporal overlap is
53. ion proven wrong by Michelson and Morley in 1887 3 but not explained for nearly twenty more years The twentieth century began with radical shifts in the understanding of physics in two areas 1 the nature of the atom and 2 the nature of light time and space The first shift began with the description of a hohlraum or empty container which was to unify electro magnetic theory thermodynamics and classical mechanics by showing that the vibrations of molecules in the wall of the container share energy with the electromagnetic vibra tions of waves within the cavity However theory deviated from experiment inexplicably 2 for high frequency radiation a conundrum termed the ultraviolet catastrophe until 1901 when Max Karl Planck introduced his novel ideas regarding energy distribution within the cavity Planck s equations described energy within the hohlraum as existing not in a continuum ranging from zero to infinity as previously thought but in discrete amounts according to the standing waves allowed by the container i e the energy was quantized Einstein in his explanation of the photoelectric effect in 1905 4 5 and Compton in his observation of the scattering of X rays off a metal foil in 1923 6 provided further evi dence for the quantized nature of energy within the description of atomic behavior The second shift began in 1905 when Einstein explained the anomaly first seen by Michelson and Morley by stating that the speed
54. ll come tauprime fs before pulses e2 and e3 e2 and e3 will start at O The photon echo will occur tauprime fs after e2 and e3 have hit the sample temystart i l tsteps if i le isteps 1 then el i elec t tauprime cmp1x S cos wl t sin wl t else el i O endif invel i zconjg el i invrel i real invel i inviel i imag invel i enddo a aa a a a c C Green functions must be calculated for t gt 0 only but convoluted with E t lt 0 values determined by value of mystart To ensure that elements E t i begin at mystart and elements G i begin at t 0 separate loop is needed to redefine t for array elements i when determining the Green function GF exists only for 1st half of total increments so imax is set at 5 of total isteps imax int isteps l 2 J 1 do i l isteps te TE i 1 tsteps i le imax then rgeg i gegr t 59 igeg i gegi t rgee 1 geer t igee i geei t else rgeg i 0 igeg i 0 rgee i 0 igee i 0 endif invrgeg i real conjg cmplx rgeg i igeg i invigeg i imag conjg cmplx rgeg i igeg i enddo c pl invgeg invel in freq space FT G t FT E t zG w E w call fft invrgeg invigeg isteps isteps isteps 1 call fft invrel inviel isteps isteps isteps l C FT G t FT E1 t G w E w i 1 do i l isteps
55. ment described in this thesis is performed in the time domain and thus the third order polarization will also be calculated in the time domain All relevant material properties can be described by Eq 2 5 The first order term gives the strength of the induced polarization within the material and is responsible for the 13 refractive index and absorption The second order term is responsible for second harmonic generation or frequency doubling as used in the Nd YVO laser used in this experi ment and also sum and difference frequency generation Materials such as liquids gases amorphous solids and many crystals do not exhibit any second order effects due to the presence of inversion symmetry In materials with inversion symmetry the third order term is the leading nonlinear term in the polarization The third order term is responsible for third harmonic generation the generation of an intensity dependent refractive index four wave mixing effects and photon echoes which are closely related to this experiment The polarization is the sum effect of the molecular dipoles contained in the material In quantum mechanics the dipole operator for a member a of a dilute ensemble of chro mophores in the Schr dinger picture is V r ga FE ra 2 6 where g is the electronic charge and r is the position operator The polarization is described by the ensemble average of the dipole operator Prt Tr V r p t 2 7 2 3 THE BLO
56. mystart isteps establish initial arrays from which all convolutions will be composed i 1 do i 1 nmax The characteristics of how convolutions are programmed require that the response functions are calculated so G i I is evaluated at t 0 and elec funcs are calculated so E i l is evaluated at mystart The GF t gt 0 are convoluted however with the E t lt 0 Do not wait for t gt 0 to begin convoluting GF with E t t mystart i 1 tsteps if i le istepst l then EI T IS CHANGED TO E1 T TAUPRIME CAUSING THE TIME AT c WHICH El OCCURS TO BE SHIFTED TO THE 49 LEFT EARL ER c BY TAUPRIME SO THAT E2 E3 AND GF WILL BE CALCULATED c FROM T 0 el i elec t tauprime cmplx cos w1 t tauprime 0 S sin wl t tauprime 0 else el i 0 endif invel i zconjg el i invrel i real invel 1i inviel i imag invel i enddo c Green functions must be calculated for t gt 0 only but c convoluted with E t lt 0 values determined by value of c mystart To ensure that elements E t i begin at c mystart and elements G i begin at t 0 separate loop c is needed to redefine t for array elements i when c determining the Green function GF do not care about c time values they care only that they are divided into c the same number of intervals as the pulses regardless c of time values i 1 do i 1 nm
57. myxval icnt Picture2 Line mytval icnt myyval icnt End If Next icnt End Sub c plots upper of two data windows Private Sub cmdUpperPlot Click dataplot e tempplot dat Open dataplot For Output As 20 maxcount Text5 Text For icnt 1 To maxcount plotx myxval icnt ploty myyval icnt Write 2 icnt templ temp2 Print 20 icnt Format plotx Scientific Format ploty Scientific Next icnt Shell e reprsh32 rsh elderberry plot tempplot dat Close 20 End Sub c sets accelerations values for the c motion stages Private Sub mAccell Click frmAccell Show End Sub c OK button sets acceleration c options as the default c values to be used on c all runs Private Sub cmdOK_Click Dim jog home accel As String jog IJA amp txtJogAc Text home 1OA amp txtHomeAc Text accel JAC amp txtAc Text Call ibwrt mm jog Chr 13 76 Call ibwrt mm home Chr 13 Call ibwrt mm accel Chr 13 If optAccelDefl value True Then Open c deanna acceldefaultsl txt For Output Access Write Shared As 35 Print 35 jog Print 35 home Print 35 accel Close End If frmAccell Hide End Sub c executes movement of the motion stages Private Sub cmdExe Click ex EX amp txtExePro Text Call ibwrt mm ex Chr 13 frmExePro Hide End Sub c executes data collection Private Sub cmdExecut
58. n end O A AWA OO TOWO AA a aaa a a a a a aa a a a a a a a a APPENDIX B CALCULATING THE PHOTON ECHO program joffres photonecho implicit none Variation of main program bjoffre f Uses fast fourier transform method by Joffre to calculate photon echo signal at a fixed delay between pulses 1 and 2 Polarization signals are calculated for frequencies within the inhomogeneous broadening of the material The polarization signals are then added together to generate a photon echo occuring at a time after the 3rd pulse equal to the delay between pulses 1 and 2 PARAMETERS FROM PARAMETER FILE isteps number of divisions to create in interval mystop mystart when calculating values of density operators and P 2 t Must be an integer mystart time in fs at which your external radiation pulses first hit the sample time at which Greens functions begin taking effect must be greater than or equal to zero mystop time at which external radiation stops hitting sample Greens functions no longer have an effect in sample tauprime fixed delay between pulses 1 and 2 tau delay between ext pulses 2 and 3 should be zero weg0 zero phonon line for transition freq in sample wl frequency of external radiation pulse hbar k Planck Boltzmann constant tl T1 the eigenstate decay rate relaxation of density operator diagonal matrix elements t2 T2 dephasing rate relaxation of density operator 55 a aa a 0
59. near susceptibility tensor and create different types of electronic polarizations within the material In addition the direction of the signal chosen for observation can be varied according to the characteristics of the system The diffracted signal can be measured in the forward direction transmission geometry or in the backward direction reflection geometry The reflection geometry is valuable for thin films where the absorption of the substrate would make the forward signal weak and thus difficult to measure 12 By varying the propagation directions of the incoming beams different experiments can be performed For example having two pump beams approach the sample from opposite sides followed by a third beam incident on the medium in a different direction results in the phase conjugate of the third beam When the phase con jugated beam passes through the same aberrating medium the third wave passed through aberrations in the third beam are removed In this type of experiment the two pump beams can be used to remove aberrations of waves passing through an external medium 11 When all three pulses originate from the same laser then they are of equal frequencies leading to the simplest type of FWM termed degenerate four wave mixing DFWM In addition ky and k3 can originate from the same beam simplifying the geometry to two beam DFWM The resulting signal wave vector is 2k2 k and the resulting signal frequency is 2wo wi wi se
60. nfo on line creating array to be read when file is opened Print 3 Ch 1 Ch 2 Trigger pts minpt maxpt minchl maxchl minch2 maxch2 Print 3 channell Print 3 channel2 Print 3 trig Print 3 maxcount Print 3 minx Print 3 maxx Print 3 minyl Print 3 maxyl Print 3 miny2 Print 3 maxy2 For icnt 1 To maxcount tempx myxval icnt tempy myyval icnt Write 2 icnt templ temp2 Print 2 icnt Format tempx Scientific Format tempy Scientific Next icnt Close 2 Close 3 savel Text Form5 Hide Else Label5 Caption You have not specified a filename End If End Sub 89 c determines settings for lock in amplifier Private Sub cmdOK Click If optExternal value True Then Call ibwrt lia fmod 0 frmMain Text9 Text External ElseIf optInternal value True Then Call ibwrt lia fmod 1 frmMain Text9 Text Internal End If If optSine value True Then Call ibwrt lia rslp 0 frmMain Text6 Text Sine ElseIf optPositive value True Then Call ibwrt lia rslp 1 frmMain Text6 Text Positive Edge ElseIf optNegative value True Then Call ibwrt lia rslp 2 frmMain Text6 Text Negative Edge End If frmSettings Hide End Sub c sets the stepsize and resolution c of the motion stages Private Sub cmdSS1OK_Click If optMicron value
61. nm an 1E nin Using Fourier transform techniques the solution of Eq 2 9 is Pam 8 Gam 8 amp E t SVupa 8 Pni t Vin 2 10 l where represents convolution 15 2 3 1 THE DENSITY OPERATOR EXPANSION Because the external electric field is weak the density operator is expanded perturbatively in the electric field p t p OE p t O 2 11 where ps 1 and pl 0 for all other states By substituting Eg 2 11 into Eg 2 10 and collecting terms of different orders the nth term of the expansion is be found to be p t Gr amp EE Vupi t oh Viml which can be iterated 2 3 2 THE TWO LEVEL SYSTEM Y SiO Ce can be described by a two level system with the ground state g and the excited state e Since Y2SiO possesses a center of inversion the dipole operator is of the form 0 Vag V 0 where Vye V The first order term in the expansion of the density matrix is PRO VegGeg t 8 E t py 8 where pi pl 0 The second order terms are limited to containing only the off diagonal first order terms and therefore only the two diagonal second order terms survive PDE lVPGe t EG Goel Goult SEM C19 POE MG E t Geg t GO amp E t The third order terms needed to describe the system can now be found and will contain only the p t and p t terms p t Geg t BO Veqp t p2 t Vaa 16 Because Tr p t 1 the initial
62. nto a laser spectrum analyzer IST REES E200 series This instrument enables the spectrum of the laser to be monitored in real time facilitating optimization of the laser A diffraction grating inside the instrument spins at a rate of 18 revolutions per second continually scanning the spectrum The output is viewed on an oscilloscope in real time The bandwidth is measured by calculating the FWHM of the spectrum displayed on the oscilloscope to an accuracy of 0 3 nm as shown in Figure 3 4 The laser is adjusted so that the bandwidth is in the range of 12 to 23 nm 3 2 EXPERIMENTAL CONSIDERATIONS 3 2 1 Y9S10 CE Y SiO is a monoclinic crystal that belongs to the C space group 5 Each unit cell contains eight molecules and the Ce cations occupy two inequivalent crystallographic sites within the cell It is not known which site is preferred by the cations which are considered to be distributed randomly in the host Trivalent cerium has the electronic configuration Xe 4f The energy gap between the ground state and the lowest 5d orbital is large ranging from 20 000 35 000 cm The ground state in free radical form is split by spin orbit effects When surrounded by the crystal field of a host the two levels are split further Excitation occurs from the 4f ground level to the 5d level An energy level diagram is given in Figure 3 5 Y SiO Ce is a rapid response blue phosphor 6 7 8 having a decay to 10 time of 120 ns and i
63. nv nmax inviconv nmax invrgeg nmax invigeg nmax rgee nmax igee nmax signal nmax rel nmax iel nmax invrel nmax rinviel nmax rfunc nmax ifunc nmax re nmax ie2 nmax re3 nmax ie3 nmax rdipge idipge elec complex el nmax e2 nmax e3 nmax invel nmax plinv nmax polar nmax conv p3 nmax geg gee p2 nmax pl nmax dipge common values hbar weg ti t2 mystop common times tsteps tau tauprime common elecvals w1 sigma character 16 fname rdipge 1 0 idipge 0 0 dipge cmp1x rdipge idipge read isteps mystart mystop tau wl weg hbar k rtl t2 fwhm redefine wavenumber input into correct units Q a 48 of frequency calcs are done with frequencies in units of 1 fs lel5 s wl 2 pi 3e8 wl le 13 weg 2 pi 3e8 weg le 13 sigma pulsewidth of external radiation field in fs fwhm 1 76 sigma fwhm 1 76 GF exists only for 1st half of total increments so imax is set at 5 of total isteps imax int isteps 1 2 Or Or Ot ea SOL lt o FIRST PULSE El OCCURS AT 50FS COMPARED TO E2 AND E3 FOR POLARIZATION CALC FOR THE POLARIZATION CALC THE SECOND PULSE OCCURS AT A DELAY OF 50 FS POLARIZATION WILL BEGIN FROM POINT WHEN SECOND AND THIRD PULSE SET UP AND DIFFRACT THROUGH TRANSIENT GRATING tauprime 50 a aa a OO tsteps mystop
64. of chemical reactions involving very short lived transition species 17 As commercially available lasers approach the theoret ical limit of pulse duration 5 fs for an 800 nm pulse 12 16 the body of experimental research using ultrafast technology grows accordingly 1 2 INTRODUCTION TO NONLINEAR SPECTROSCOPY When the strength of an external radiation field such as an ultrafast pulse incident on a material is much smaller than the electric fields within the material itself then the polar 4 ization induced in the material by the pulse can be expanded as a power series in the electric field E 11 gt P FB 4 9 E x EEE where y x and y6 are the linear second order and third order susceptibilities and are second third and fourth rank tensors respectively For a nonmagnetic material all relevant material properties are encapsulated in the polarization The second order term is responsible for second harmonic generation or frequency doubling and sum and difference frequency generation Materials such as liq uids gases amorphous solids and many crystals do not exhibit any second order effects due to inversion symmetry In materials with inversion symmetry the third order term is the leading nonlinear term in the polarization The third order term is responsible for third harmonic generation generating an intensity dependent refractive index and photon echoes One of the major experimental techniques
65. of light is independent of the motion of its source 7 8 He stated that rather than the speed of light changing within inertial frames the speed of light is constant and it is time previously thought to be absolute in all frames which is relevant and changing Though the theory of the wave nature of light had become predominant his work on the photoelectric effect that same year gave greater credence to the particulate nature of light It was principally for his explanation of the photoelectric effect that he received the 1921 Nobel Prize in Physics In 1924 the duality of wave and particle by then accepted for electromagnetic radia tion was applied by de Broglie in his doctoral dissertation to electrons and alpha par ticles which at that time were assumed to exist only as particles establishing the de Broglie wavelength of the electron 9 The waves of de Broglie s theory were shown to propagate according to the equations developed by Schr dinger in 1925 resulting in the famous Schr dinger equation and the study of quantum mechanics was given structure and definition 10 All nonrelativistic quantum effects can be described by solving the Schr dinger equation a task that is in general easier said than done Nonlinear effects in quantum systems were an inaccessible realm until the discovery of nuclear magnetic resonance NMR in 1946 by Felix Bloch 11 and Edward Purcell 12 working independently However spectroscopy was limited
66. oherence are termed dephasing processes Optical dephasing is often described in terms of the optical Bloch equations which are optical analogues of the nuclear magnetic resonance Bloch equations The Bloch equations were originally derived to describe the evolution of a nuclear spin 1 system interacting with an external radiation field and a thermal bath Extensions to the Bloch eguations enabled additional levels to be taken into account using the same formalism In the optical Bloch eguations population relaxation mechanisms such as nonradiative decay result in a longitudinal relaxation time denoted 7 The lifetime of a coherent superpo sition state or rather the time during which coherence is maintained between states is called the transverse relaxation time and is denoted 75 1 When the fluctuation time of the bath is slow compared to the characteristic timescale of the experiment the dephasing is said to be inhomogeneous When the fluctuation of the bath is very fast compared to the System Interaction Bath Electronic degrees of between system nuclear degrees freedom interact with and bath of freedom the electromagnetic field Figure 2 1 A Superposition State between the Ground State and First Excited State 9 characteristic timescale of the experiment the dephasing leads to homogeneous broad ening Under some experimental conditions inhomogeneous dephasing can be reve
67. orrelation measurement Both the 0 01 fs and 2 5 fs dephasing time calculations and the experimental data see Figure 3 9 exhibit minimal shift and the pulseshapes of both are identical to that of the laser as shown in Figure 3 3 It is therefore concluded that the dephasing time of Y2SiO Ce is significantly shorter than the laser pulsewidth of 70 fs 3 4 REFERENCES 1 Millennia Diode pumped CW Visible Laser User s Manual Instruction Manual Spectra Physics Mountain View 1997 0 5 gt 2 D CG Q 25 25 50 c 7 b an o 0 5 U 2 i Z 25 0 25 50 La 1 Q ze Do5 Go o Os 0 25 50 Z 1 y JNO 0 mmm 25 25 50 0 Delay Time fs Figure 3 10 Echo signals as a function of delay calculated at various dephasing times with a laser FWHM pulse of 5 fs a T 0 01 fs b 75 2 5 fs c T 5 0 fs d T gt 10 0 fs 44 2 Tsunami Mode Locked Ti Sapphire Laser User s Manual Instruction Manual Spectra Physics Mountain View 1995 3 Model 409 Autocorrelator User s Manual Instruction Manual Spectra Physics Mountain View 1995 4 Gregory H Wannier Statistical Physics John Wiley amp Sons Inc New York 1966 5 Rufus L Cone Phys Rev B 52 6 1995 6 P J Marsh J Silver A Vecht A Newport J Lumin 97 229 2002 7 S H Shin D Y Jeon K S Suh Jap J App Phys I 40 4715 2001 8 Y Liu C N Xu K Nonaka H Tate
68. ove ll fft rconv iconv isteps isteps isteps l is1 isteps p3 i rdipge 2tidipge 2 conjg dipge lx rconv i iconv i 4 al i rhototal i weight p3 i f loop over frequencies in the external n bandwidth m loop ddo r i after all frequency contributions have been included in each rhototal i matrix element to calculate the polarization term and the four wave mixing signal do i 1 i third or steps der polarization V ge p3 eg 100 c invV ge invp3 eg 2Real V ge p3 eg polar i 2 real dipge rhototal i signal j signal j polar i 2 enddo c ending of loop over delays tauprime j loop enddo c normalize data signalmax 0 0 do j l jmax l1l if j le 1 then signalmax signal 1 else if signalmax lt signal j then signalmax signal j endif endif enddo c loop to write normalized signal data to file with c respect to delay j 1 do j l jmax 1l tauprime taustart j l tauincrements write 89 tauprime signal j signalmax enddo close 89 end c SUBROUTINES complex function gee t implicit none c green function in time domain real hbar gammaee t igee rgee weg ti t2 mystop common values hbar weg ti t2 mystop 101 gammaee l tl if t gt 0 then gee cmplx 0 0 exp gammaee t hbar else geezrcmplx 0 0 0 0 endif return end real function geer t implicit none real t complex gee Tf
69. p Text CDbl txtSteps Text ElseIf optRelLoop value True Then bb txtRelLoop Text CDbl txtSteps Text End If counter for saving files of loops gq 0 start main loop take data then move MM3000 the required number of steps Do 78 ident 1 datasumx 0 datasumy 0 start the data acquisition per loop For icnt 1 To maxcnt mytime Timer 0 08 Do Loop Until Timer gt mytime retrieve correct axes d for x y If channelindex 0 1 And channelindex 1 1 Then Call ibwrt lia snap 1 2 Do Call ibrsp lia r Loop While r And 2 0 for r theta ElseIf channelindex 0 2 And channelindex 1 2 Then Call ibwrt lia snap 3 4 Do Call ibrsp lia r Loop While r And 2 0 for x theta ElseIf channelindex 0 1 And channelindex 1 2 Then Call ibwrt lia snap 1 4 Do Call ibrsp lia r Loop While r And 2 0 PEO EV ElseIf channelindex 0 2 And channelindex 1 1 Then Call ibwrt lia snap 3 2 Do Call ibrsp lia r Loop While r And 2 0 End If myystr Spaces 255 read data from x y coord into myystr Call ibrd lia myystr count position of 1st comma in myystr 2nd char 79 mypos records position not actual array elements mypos InStr myystr place only the x coord into myxstr myxstr MidS myystr 1 mypos 1 lose the x coord an
70. put into correct units of frequency calcs are done with frequencies in units of 1 fs lel5 s wegmi wegmf is the initial final frequency from which inhomogeneous broadening is calculated a a a a a wl 2 pi 3e8 wl le 13 weg0 2 pi 3e8 weg0 le l3 wegmizweg0 2 pi 3e8 800 1e 13 wegmfzweg0r2 pi 3e8 800 1e 13 deltaw wegmf wegmi minc c loop over i before beginning m loop to initialize c each element of the total density matrix and photon c echo signal do i l isteps photonecho i O O rhototal i 0 0 enddo loop over different frequency contributions weg to the inhomogeneous broadening bandwidth Weighting factor weight will be multiplied to each frequency s contribution to the density matrix element for each value of real time a a a a a do m 1 minctl weg 0 0 weg wegmi m 1 deltaw weight exp weg0 weg a APIE IEBFSLDFIERI 3 ERA tsteps mystop mystart isteps c establish initial arrays from which all convolutions c will be composed a aa a a a 2 9 58 i 1 do i l isteps The characteristics of how convolutions are programmed require that the response functions are calculated so G i 1 is evaluated at t 0 and elec funcs are calculated so E i l is evaluated at mystart The GF t gt 0 are convoluted however with the E t 0O Do not wait for t gt 0 to begin convoluting GF with E t el is calculated at el t rtauprime meaning el wi
71. rrect the power deficiency resulted in dramatic current increases recorded to the diodes and provided a clue as to the source 23 20 000 18 000 16 000 14 000 12 000 cm 10 000 8 000 6 000 4 000 2 000 0 San Fan Hop F sp Fan 4 Iisn Tis Tue 4 Ion 24 Pump Bands Lasing Transition Figure 3 1 Four Level Lasing System of the Nd ion of the problem The increase in current was capable of counterracting the misalignment problem until the misalignment became severe The neodymium yttrium vanadate YVO Nd 37 laser is operated in a continuous working CW mode YVO Nd is a four level laser as can be seen from Figure 3 1 A four level system enables a population inversion to be maintained thus allowing con tinuous working operation Nd has a strong absorption band at 860 nm which is over lapped by the diode laser emission The optically excited electrons rapidly nonradiatively 25 decay from the S 8 4F z 2H 2 and F 5 levels to the 4F 3 level and radiatively decay to the Tu state lasing at 1064 nm From the I u state they relax rapidly to the 1I 9 ground state The combination of the relatively long lifetime at the F 3 Storage level 60 us and the rapid relaxation to the ground level creates a population inversion and therefore an ideal lasing transition The 1064 nm emission is intracavity doubled using a lithium triborat
72. rsed resulting in a photon echo The Bloch equations have several limitations i Only the electronic degrees of freedom the system that interact with the external radiation field are included requiring the system bath interactions which are responsible for dephasing to be treated phe nomenologically Systems where the broadening is intermediate between inhomogeneous and homogeneous cannot be treated accurately with the Bloch equations 11 Effects due to vibrational quantization of the bath degrees of freedom i e zero point motion are not included iii Identical ground state and excited state potential surfaces are assumed Despite their limitations the Bloch equations provide a useful starting point for the description of Y2SiO Ce More general approaches include the Brownian oscillator model 2 and semi classical simulations 3 In this chapter the Bloch model is described following the treatment by Joffre 6 2 1 FUNDAMENTALS In quantum mechanics a system can be described by a state vector w t the time evolution of which is described by the time dependent Schr dinger equation EO EED 21 dt n l l It is often convenient to expand the state vector in a basis set which consists of eigenstates of the Hamiltonian H En Le MO YO lon Onl 2 2 where gt gt on Onl La On W t and e is the probability of finding w t in state 10 The density operator p t w t w
73. s mainly used for electron detection in scientific instruments such as mass At 310 fs Normalized Autocorrelation Signal 0 1 2 3 4 5 6 t arbitrary units Figure 3 3 Autocorrelation trace showing a delay of 310 fs between pulses Unscaled autocorrelator signal pulsewidth 110 fs Actual laser pulsewidth 71 fs Normalized intensity arb units gt Un O ka 770 T15 780 785 790 795 800 Wavelength nm Figure 3 4 Bandwidth Characterization 805 aedd 810 32 Energy x 1000 cm 30 5d 5 2 Fin 2 Af 0 Psp Figure 3 5 Electronic energy levels for trivalent Ce 33 spectrometers and electron microscopes Its resistance to ultraviolet and fluid damage also makes it useful in high energy scientific instruments 9 The sample used in this research was a small plate that was optically polished on the two largest faces using 5 um diamond grit The dimensions of the polished sample were 2 5mm x 1 5 mm x 0 1 mm The excitation and emission spectra for the sample were taken using a spectrofluo rometer system Jobin Yvon SPEX Fluoromax 2 at room temperature and are given in Figure 3 6 Peak excitation occurs at 350 nm and peak emission at 400 nm Emission is broadband extending from 400 to 420 nm The shift between the excitation and emission spectrum is called the Stokes shift and is a measure of the electron lattice coupling 3 2 2 INTRODUCTION TO NONLINEAR SPECTROSCOPY AND DEGENERATE FO
74. t can be expressed in the same basis p t Mm Cn n Pr and obeys the Liouville equation d GP EIE pli 03 h The expectation value of an arbitrary operator O O v t O w t can be calcu lated using the density operator as O Tr Op t A system comprised of many identical particles is denoted an ensemble If all of the particles in the system are in the same guantum state then the system is said to be in a pure ensemble or pure state A pure state can be represented by a single ket If some fraction of the particles are characterized by a different ket it is said to be in a mixed ensemble or mixed state 5 If a particular state w t occurs N times in an ensemble comprised of particles in N total states then the density matrix for that mixed state is p t X willt Vi where w N N is the weighting factor for w t and X w 1 If the states w t are orthogonal the weight w N N vi t o t wi t is also the probability of finding the ensemble in the state w The ensemble average of an arbitrary operator O for a system in a mixed ensemble is also given by Tr Op t For a large ensemble the Schr dinger eguation becomes highly complex and unwieldy because all the details of the particles within the ensemble must be known An advantage of the density operator is i complete knowledge of the wavefunction is not needed and ii a reduced density operator rather than the full density oper
75. t none real t complex gee TE Belt 0 then geei 0 else endif geei imag gee t return end complex function geg t c green 63 implicit none function in time domain real hbar t weg gammaeg tl t2 mystop common values hbar weg tl t2 mystop gammaeg 1 t2 if t gt O then geg exp gammaeg t cmplx sin weg t cos weg t hbar else geg cmp1x 0 0 0 0 endif return end real function gegr t implicit none complex geg real t if t 1t 0 then gegr 0 else gegr real geg t endif return end real function gegi t implicit none complex geg real t if t 1t 0 then gegi 0 else gegi imag geg t endif return end real function elec t implicit none 64 c real amplitude expression of electric field c E t elec t e iwt real t tsteps sigma wl sech tau tauprime common times tsteps tau tauprime common elecvals wl sigma elec sech t sigma return end real function sech t implicit none real t sech 2 exp t texp t return end APPENDIX C DATA COLLECTION PROGRAM c All forms have CANCEL or c EXIT button which hides c the form Program for c all forms is as follows Private Sub cmdCancel_Click frmname Hide End Sub loads the Main window and initializes of program assigns pretermined defaults to the settings of the lock in amplifier Private Sub Form_Load c c the amplifier upon start up c c
76. temporal width of nominally 70 femtoseconds with a repetition rate of 82 MHz yielding an average power of 800 mW Though the energy of each pulse is very small on the order of 1078 J the peak power Intensity arb units 26 Absorption Emission 400 500 600 700 800 900 1000 Wavelength nm Figure 3 2 Absorption and Emission Spectra of A1 03 Ti From Ref 2 27 is high 140 kW due to the short temporal width The ultrafast laser emission is ana lyzed using a fast photo diode a real time spectrometer and an autocorrelator The laser emission can be frequency doubled to create ultraviolet light at 395 nm Laser power is monitored by a power meter placed at the output coupler Usually adjustments are made only to the prisms and tuning slit to obtain the highest power read ings Very low laser power and instability are corrected by adjusting the placement of the internal steering mirrors of the laser A beam splitter directs a small portion of the pulsing output into a fast photodiode The readings of the photodiode are shown on an oscilloscope and reveal the stability of the pulsetrain If a strong and stable pulse is not consistently seen then the Ti Sapphire laser should be optimized to increase both power and stability Once optimized the design of the laser provides stable operation under consistent environmental conditions for over the period of a day and requires only prism and slit adjustments to maintain po
77. teoEE O then geer 0 else geer real gee t endif return end real function geei t c gee has only imaginary parts implicit none real t complex gee if t It O then geei 0 else geei imag gee t endif return end complex function geg t implicit none c green function in time domain real hbar t weg gammaeg tl t2 mystop common values hbar weg ti t2 mystop 102 gammaeg 1 t2 if t gt 0 then geg exp gammaeg t cmplx sin weg t cos weg t hbar else geg cmp1x 0 0 0 0 endif return end real function gegr t implicit none complex geg real t if t 1t 0 then gegr 0 else gegr real geg t endif return end real function gegi t implicit none complex geg real t if t 1t 0 then gegi 0 else gegi imag geg t endif return end real function elec t implicit none c real amplitude expression of electric field real t tsteps sigma wl sech tau tauprime common times tsteps tau tauprime common elecvals wi sigma elec sech t sigma return end real function sech t implicit none real t sech 2 exp t texp t return end 103
78. that uses the third order polarization is four wave mixing FWM 12 FWM gets it name from three waves having frequencies wy wo and wg interacting to generate a fourth wave with frequency w4 wi two w3 Four wave mixing will be discussed further in Chapter Three An outline of this thesis is as follows The second chapter of this thesis will provide a method for numerically calculating the third order polarization signal and will describe results calculated on a model system The third chapter will present the experimental set up sample description experimental results and a comparison to the model calculation The final chapter will present the conclusions of this thesis 1 3 REFERENCES 1 Loyd S Swenson The Ethereal Aether a history of the Michelson Morley Miller aether drift experiments 1880 1930 University of Texas Press Austin 1972 2 Kenneth W Ford Basic Physics Blaisdell Publishing Co Waltham 1968 3 A A Michelson E W Morley Am J Sci 134 333 1887 4 Albert Einstein Annalen der Physik 17 132 1905 5 Albert Einstein Annalen der Physik 20 199 1906 6 Arthur H Compton Phys Rev 21 483 22 409 1923 7 Robert M Eisberg Fundamental of Modern Physics John Wiley amp Sons New York 1961 8 Albert Einstein Annalen der Physik 17 891 1905 9 L de Broglie Comptes Rendus Acad Sci Paris 183 447 1924 10 E Schr dinger Annalen der Physik 79
79. the reduced density operator can be shown to be equivalent to the expectation value using the full density operator Tr p t O X nnkEulo Olnn lEu 2 4 25 ml Gul Ew rw HE OIE IN DD a Sul CE IE hare mw lOl nl inr Mw On nn 12 5 Pnn HO TO Thus the observables of the operators acting on the system can be described entirely by the trace over the system eigenstates of the reduced density operator p t 5 2 2 THE POLARIZATION EXPANSION In the Maxwell equations which describe electromagnetic phenomena the polarization is the only observable quantity that contains information on the nonmagnetic material i e all processes in a nonmagnetic material can be studied according to how they affect the polarization When the strength of an external radiation field incident on a material is much smaller than the electric fields within the material itself then the polarization induced in the mate rial by the pulse can be expanded as a power series in the electric field 11 gt P x9 B 0 BE OERE pe 2 5 pH po po i ae where x is the nth order susceptibility The susceptibility a frequency domain func tion is related to the Fourier transform of the response function a time domain function denoted t t2 t1 In this case the nth order polarization is given by P t dt f dini a di S tp tnis ads ti 0 0 0 BOZA Let eta tn tn 1 tek The experi
80. tic of the experimental setup is shown in Figure 3 8 The output of the Ti Sapphire laser is divided by a 50 beamsplitter into two paths One path contains a high precision translation stage Newport UTMPP 1 allowing for a delay in the optical path of up to 4 inches with a precision of 0 1 um An iris is placed immediately in front of Nd YVO Laser MI 37 Diode Laser Optical Fibers M3 Ti Sapphire Laser M2 BS2 BS3 BS1 Autocorrelator FM1 M5 Fast Photodiode Spectrum Analyzer Optional Frequency Doubler M6 M7 L Sa n 13 Ms M9 EE M10 Na FM2 BS4 k Optical e Chopper focusing Retro reflector lens Sample Translation Stage focusing ens 2k k 2k k Y yY Photodiode on translation stage Lock In Amplifier Figure 3 8 Experimental Set Up for Degenerate Four Wave Mixing BS beam splitter FM flip mirror I Iris M mirror 38 the beam splitter In addition a second and third iris are placed at the end of each pathway immediately in front of the focusing lens Because even a small change in pathlength of one path can drastically effect the temporal overlap area of 70 fs pulses the use of these irises enables the beam paths to be reproduced from day to day counteracting the effects of mirror drift and changes resulting from laser alignment Before focus
81. value for resolution put int statement to oder program to beginning of loop End If If optDef value True Then Open c deanna stepdefaultsl txt For Output Access Write Shared As 33 Print 33 CStr motionindex 0 Print 33 CStr motionindex l Close End If frmStepSizel Hide End Sub 91 c OK button sets velocity c options for stage at various c points of travel c jog velocity for manually moving c stage outside of program c homelo homehi min max velocity as stage c approaches home position c vel velocity in middle of motion Private Sub cmdOK_Click If txtHiJog Text lt gt Then joghi IJH amp txtHiJog Text If txtLoJog Text lt gt Then joglo IJW amp txtLoJog Text If txtHiHome Text lt gt Then homehi 1OH amp txtHiHome Text If txtLoHome Text lt gt Then homelo IOL amp txtLoHome Text If txtVel Text lt gt Then vel IVA amp txtVel Text Call ibwrt mm joghi Chr 13 Cal ibwrt mm joglo Chr 13 Cal ibwrt mm homehi Chr 13 Cal ibwrt mm homelo Chr 13 Cal ibwrt mm vel Chr 13 If optDef value True Then Open c deanna VelDefl txt For Output Access Write Shared As 34 Print 34 joghi Print 34 joglo Print 34 homehi Print 34 homelo Print 34 vel Close End If Tf ckSD value 1 Then sd 1S
82. wer and stability beyond that time period 3 1 3 PULSEWIDTH MEASUREMENT An indirect measurement of the pulsewidth is obtained using an autocorrelator Spectra Physics Model 409 3 The autocorrelator splits the incoming beam and uses a rotating block of fused silica placed in the path of both beams to create a variable difference in the optical pathlength The difference in pathlength AL of one beam is given by the expression AL 2d n sin 0 cos0 1 n where d and n are the thickness and index of refraction of the block respectively The beams are aligned onto the rotating block at complementary angles resulting in a differ ence in pathlength between the two given by AL 2d y n sin 6 cos 9 Vn cos sin 0 28 As the block rotates the beams travel in opposite directions alternating between max imum and minimum pathlength distances When one beam is at a maximum the other is at a minimum When both beams are incident on the block at an angle of 45 the two path lengths are equal and the pulses are temporally overlapped The beams are then focused by a lens and spatially overlapped inside a frequency doubling crystal The autocorrelator signal is seen when the beams are both spatially and temporally overlapped within the frequency doubling crystal The autocorrelator signal is of the form 1 pt r C r lim x t a t r dt where 7 is the difference in temporal pathlengths and T is the t
83. x int taustop taustart tauincrements c GF exists only for lst half of total increments so c imax is set at 5 of total isteps imax int isteps 1 2 open unit 89 file signal paw do j l jmax l1l signal j 0 0 c loop over all i s to initialize rhototal i polar i c before any frequency contributions are made do i l isteps rhototal i 0 0 polar i 0 0 enddo loop over different frequency contributions to the external radiation pulse bandwidth Weighting factor weight will be multiplied to each frequency s contribution to the density matrix element for each value of real time a aa a a 96 m 1 do m 1 minctl weg 0 0 weight 0 0 a aa a a a weg wegmit m 1 deltaw weight exp weg0 weg S 2 pi 3e8 500 1e 13 2 tauprime delay between external pulses 1 and 2 varied as program progresses tauprime delay value calculated tauprime taustart j 1 tauincrements tsteps mystop mystart isteps establish initial arrays from which all convolutions will be composed i 1 do i l isteps The characteristics of how convolutions are programmed require that the response functions are calculated so G i 1 is evaluated at t 0 and elec funcs are calculated so E i l is evaluated at mystart The GF t gt 0 are convoluted however with the E t 0O Do not wait for t gt 0 to begin convoluting GF with E t temystart i l tsteps if i le
84. yama J Mat Sci 36 4361 2001 9 Phospor Scintillator Data Sheet 24 yttrium silicate cerium doped Applied Scintillation Technologies London 2000 10 Madis Raukas Ph D dissertation Luminescence Efficiency and Electronic Properties of Cerium Doped Insulating Oxides 1997 11 Robert W Boyd Nonlinear Optics Academic Press Boston 1992 12 Jagdeep Shah Springer Series in Solid State Sciences 115 Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures Springer Verlag Berlin 1996 13 A M Weiner S De Silvestri E P Ippen J Opt Soc Am B 2 654 1985 CHAPTER 4 CONCLUSIONS The spectroscopic properties of Y2SiO Ce were investigated at room temperature The sample was found to have a peak excitation at 350 nm and a broadband emission between 400 and 420 nm An ultrafast two beam degenerate four wave mixing experiment was set up and utilized to establish an lower limit on the dephasing time of Y SiO Ce at room temperature at an excitation frequency of 395 nm For these experiments a Ti sapphire laser was used to produce 790 nm pulses at a repetition rate of 82 MHz Laser pulses were characterized using a rotating grating spectrometer to determine the spectral bandwidth and using a rotating block autocorrelator to determine the temporal pulsewidth The laser pulses had a temporal pulsewidth of 60 to 80 fs and a spectral bandwidth ranging from 12 to 23 nm depending on th

Under the direction of Dr. William M. Dennis

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