Generation and detection of ultrashort pulses

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(1)Generation and dete tion of ultrashort pulses by. Vi toria Onyeka Nwosu Thesis presented in partial fulllment of the requirements for the degree of. Master of S ien e at the University of Stellenbos h. Supervisor: Prof. E. G. Rohwer Co-supervisors: Prof. H. S hwoerer and Dr. G. Arendse Mar h 2009.

(2) De laration By submitting this thesis ele troni ally, I de lare that the entirety of the work ontained therein is my own, original work, that I am the owner of the opyright thereof (unless to the extent expli itly otherwise stated) and that I have not previously in its entirety or in part submitted it for obtaining any quali ation.. .................03/03/2009................... Date. 2008 Stellenbos h University Copyright All rights reserved.

(3) Abstra t The ex iting eld of ultrashort laser opti s has experien ed tremendous growth sin e it's in eption. One of its bran hes that has been of ontinuous interest is the hara terization of ultrashort laser pulses. The need for omplete hara terization of these pulses has in reased with appli ations in various elds of resear h. Opto-ele troni devi es are in apable of resolving the temporal duration of these short opti al pulses and other indire t te hniques have been developed. Most self referen ing te hniques developed for short pulse hara terization exploits the intera tion of the input pulse with a opy of itself in a nonlinear medium. This proje t studies the ultrashort opti al regime.. It explains the theory. on auto orrelation pulse hara terization te hniques in luding the Spe tral Phase Interferometry for Dire t ele tri -eld Re onstru tion (SPIDER) te hnique. SPIDER measures the interferen e between a pair of repli as of the input pulse that has been spe trally sheared. A spe trometer is used to re ord the interferogram and the spe tral phase is retrieved through a non-iterative, and purely algebrai method. A SPIDER setup was onstru ted and used to measure femtose ond laser pulses.. The SPIDER te hnique has the uniqueness of been a single shot,. pre ise, reliable and self referen ing diagnosti s te hnique for ultrashort laser pulse hara terization.. It is also apable of retrieving the amplitude and. phase of the input pulse. The input pulse measured revealed that the pulse duration is about 106 fs with a quadrati spe tral phase implying a linear hirp on the pulse. (Numeri al simulations were used to verify the algorithm).. i.

(4) A knowledgements My sin ere gratitude goes to my supervisors Prof. S hwoerer and Dr.. E.G Rohwer, Prof.. H.. G.J. Arendse for their enormous supervision, support. and motivation. Spe ial thanks to Gurthwin Bosman for all the assistan e he oered during the ourse of this work. I would like to a knowledge all the members of the Laser Resear h Institute (LRI) for inspiration and support. My profound gratitude to the Afri an Institute for Mathemati al S ien es (AIMS) and the Afri an Laser Center for nan ial support. I am indebted to my dear friends Florimond Mpiana, Emmanuel Jonah, Gibson Ikoro, Joy Okonkwo and Justin Mabiala for the love, support and onstru tive riti ism they oered to me during the ourse of the work. I will not forget to appre iate every member of my family, my parents, Lt. Colonel E.O Nwosu and Mrs H.N. Nwosu. My siblings, Marian, Jane, Augustine, Felix, and Daniel for their ontinual love and support. Finally, I thank the almighty God for granting me favor and leading me this far. Thank you all.. ii.

(5) Contents Abstra t. i. List of Figures 1. 2. 3. 4. ix. Introdu tion to ultrashort pulsed lasers. 1. 1.1. Histori al development of the laser. . . . . . . . . . . . . . . .. 1. 1.2. Aim and outline of this work . . . . . . . . . . . . . . . . . . .. 3. 1.3. Generating ultrashort pulses . . . . . . . . . . . . . . . . . . .. 3. 1.3.1. Modelo king . . . . . . . . . . . . . . . . . . . . . . . .. 4. 1.3.2. A tive modelo king . . . . . . . . . . . . . . . . . . . .. 5. 1.3.3. Passive modelo king. 7. . . . . . . . . . . . . . . . . . . .. Intera tion of light with solid materials. 9. 2.1. Signi an e of the generalized wave equation . . . . . . . . . .. 9. 2.2. Origin of dispersion in materials . . . . . . . . . . . . . . . . .. 11. 2.3. Opti al dispersion in materials . . . . . . . . . . . . . . . . . .. 15. 2.4. Ultrashort pulse propagation in a dispersive medium. 16. . . . . .. Des ription of an ultrashort pulse. 20. 3.1. Temporal and spe tral representations. . . . . . . . . . . . . .. 20. 3.2. Time bandwidth produ t . . . . . . . . . . . . . . . . . . . . .. 21. 3.3. Variation of instantaneous frequen y with time . . . . . . . . .. 23. Nonlinear opti al pro esses. 27. 4.1. The origin of nonlinear polarization . . . . . . . . . . . . . . .. 28. 4.2. Se ond harmoni generation (SHG) . . . . . . . . . . . . . . .. 34. 4.3. Phase mat hing . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. iii.

(6) CONTENTS 5. 6. 7. 8. CONTENTS. Correlation te hniques for ultrashort pulse hara terization 39 5.1. Intensity auto orrelation . . . . . . . . . . . . . . . . . . . . .. 40. 5.2. Interferometri Auto orrelation. 41. . . . . . . . . . . . . . . . . .. Spe tral Interferometry. 50. 6.1. Fourier Transform Spe tral Interferometry. . . . . . . . . . . .. 50. 6.2. Prin iple of FTSI . . . . . . . . . . . . . . . . . . . . . . . . .. 51. 6.3. Experimental results of FTSI. . . . . . . . . . . . . . . . . . .. 52. 6.4. Spe tral shearing interferometry . . . . . . . . . . . . . . . . .. 54. Spe tral Phase Interferometry for Dire t Ele tri Field Re onstru tion (SPIDER). 56. 7.1. Optimization parameters . . . . . . . . . . . . . . . . . . . . .. 57. 7.1.1. Generating the repli a. . . . . . . . . . . . . . . . . . .. 57. 7.1.2. Generating the hirped pulse . . . . . . . . . . . . . . .. 58. 7.1.3. Nonlinear Intera tion . . . . . . . . . . . . . . . . . . .. 60. 7.2. SPIDER Theory. . . . . . . . . . . . . . . . . . . . . . . . . .. 7.3. Experimental implementation, results and dis ussion. 7.4. Numeri al simulations. 61. . . . . .. 65. . . . . . . . . . . . . . . . . . . . . . .. 73. Con lusions. 78. Appendix. 79. A. 80 A.1. Gaussian pulse propagation through a medium . . . . . . . . .. 80. A.2. Conversion e ien y of the se ond harmoni wave . . . . . . .. 82. A.3. Matlab ode for the phase retrieval. . . . . . . . . . . . . . . .. 87. A.4. SPIDER veri ation algorithm . . . . . . . . . . . . . . . . . .. 92. iv.

(7) CONTENTS. CONTENTS. Bibliography. 100. v.

(8) List of Figures 1.1. (a) Multimode and (b) Modelo ked laser output . . . . . . . .. 1.2. Frequen y spe trum of an amplitude modulated eld. band at. 1.3. ωn ± Ω. [1℄. 5. Side. . . . . . . . . . . . . . . . . . . . . . . . .. 6. Transmission of a saturable absorber as a fun tion of the in ident intensity. . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.1. Real part of the refra tive index versus frequen y. 2.2. Imaginary part of the refra tive index versus frequen y. 2.3. Dispersion urve for fused quartz. 3.1. Temporal intensity of a Gaussian pulse with onstant phase. .. 21. 3.2. Bandwidth limited Gaussian pulse . . . . . . . . . . . . . . . .. 23. 3.3. (a) Negatively hirped Gaussian pulse (b) Positively hirped Gaussian pulse. 3.4. . . . . . . .. 8. 14. . . . .. 15. . . . . . . . . . . . . . . . .. 16. . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. Illustration of the prin iple of hirped pulse ampli ation te hnique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. 4.1. Potential energy fun tion for a nonlinear medium [2℄. 29. 4.2. S hemati of the prin iple underlying se ond harmoni gener-. . . . . .. ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Phase mat hing fa tor of the onversion e ien y for the se ond harmoni wave . . . . . . . . . . . . . . . . . . . . . . . .. 4.4. 34. 37. Non- ollinear geometry for phase mat hing, energy and momentum is onserved. . . . . . . . . . . . . . . . . . . . . . . .. 5.1. S hemati of an intensity auto orrelator. . . . . . . . . . . . .. 5.2. Simulated intensity auto orrelation fun tion for Gaussian shaped pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi. 38. 41. 42.

(9) LIST OF FIGURES. LIST OF FIGURES. 5.3. S hemati of an interferometri auto orrelator . . . . . . . . .. 5.4. Theoreti al interferometri auto orrelation tra e for a Gaussian shaped pulse with no hirp. 5.5. β=0. . . . . . . . . . . . . .. b=2. . . . . . . . . .. 49. S hemati of a typi al Fourier transform spe tral interferometry set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.2. 48. Theoreti al interferometri auto orrelation tra e for a Gaussian shaped pulse with hirp parameter,. 6.1. 42. 51. (a) Measured spe tral interferen e for two repli as reated from the ree tion o an un oated glass plate (b) Measured interferen e spe tra of two frequen y doubled repli as . . . . .. 6.3. (a) Inverse Fourier transform of (6.2) (b) Filtered positive side lobe. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Phase transfer fun tion retrieved for about 1.4 using FTSI. µm glass. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Basi omponents of a SPIDER set up. 7.2. S hemati of a grating opti al disperser used for stret hing ultrashort pulses. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. 54. 56. 58. In rease in the stret hed pulse duration with grating pair separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.4. 53. plate. 7.1. 7.3. 52. 60. S hemati of the frequen y shifting pro ess used for the SPIDER te hnique. . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.5. SPIDER inversion routine algorithm. . . . . . . . . . . . . . .. 7.6. Experimental layout of the SPIDER setup: IR-iris, GP-Glass. 61 65. plate, R-ree tor, M-Mirrows, P-Gold oated right angled prism, T-translation stages, G-Gratings, L-Lens, BBO-Beta-barium borate nonlinear rystal 7.7. . . . . . . . . . . . . . . . . . . . . .. 67. Input fundamental spe trum with Gaussian t . . . . . . . . .. 68. vii.

(10) LIST OF FIGURES 7.8. LIST OF FIGURES. Measured SPIDER interferogram (spidergram), phase information of the input pulse is onverted to amplitude information 68. 7.9. Inverse Fourier transform of the measured SPIDER interferogram. Side lobes are entred around the delay of the repli as .. 69. 7.10 Measured alibration tra e used for the inversion routine of the SPIDER interferogram . . . . . . . . . . . . . . . . . . . .. 70. 7.11 Fourier transform of the re orded alibration tra e used in the SPIDER algorithm. . . . . . . . . . . . . . . . . . . . . . . . .. 70. 7.12 Re onstru ted spe tral phase using integration (a) and on atenation methods (b) . . . . . . . . . . . . . . . . . . . . . .. 71. 7.13 Retrieved spe tral intensity and spe tral phase . . . . . . . . .. 72. 7.14 Retrieved temporal prole and time dependent phase. 73. . . . . .. 7.15 Spe tral phase returned by SPIDER (straight line) for an input pulse similar to that shown in gure 7.7, and the same pulse after propagating through a glass of 1 mm thi kness (dots) 73 7.16 A pi ture of the SPIDER setup. . . . . . . . . . . . . . . . . .. 75. 7.17 Simulated SPIDER interferogram . . . . . . . . . . . . . . . .. 76. 7.18 Input (dash line) and re onstru ted quadrati phase (solid line) using the SPIDER algorithm, and. c=0. a = 5 × 10−20. 2 fs ,. b = 0,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76. 7.19 Input and re onstru ted ubi phase using the SPIDER algorithm,. a = 0, b = 5 × 10−56. 2 fs , and. c=0. . . . . . . . . . . . .. 77. 7.20 Input and re onstru ted quarti phase using the SPIDER algorithm,. a = 0, b = 0,. and. c = 5 × 10−71. viii. 2 fs. . . . . . . . . . .. 77.

(11) List of Tables C. for sele ted pulse shapes 23. 3.1. Values of time-bandwidth onstants. 5.1. Mathemati al results for dierent pulse shapes and their orrelation fun tion [3℄ . . . . . . . . . . . . . . . . . . . . . . . .. ix. 41.

(12) 1. Introdu tion to ultrashort pulsed lasers. 1.1 Histori al development of the laser It is interesting to know that as diverse as the eld of ultrashort laser opti s is today, it all sprang up from basi physi s prin iples. In this se tion we will give a brief dis ussion on the history of short pulse lasers and the development of ultrashort pulse hara terization. The prin iple of laser operation was rst dis overed in 1916, when the great physi ist Albert Einstein des ribed the theory of stimulated emission. Before that time, the intera tion of a photon with a mole ule was explained with the prin iple of spontaneous emission and absorption. The former pro ess o urs when a mole ule is ex ited to a higher energy state, but does not remain there, it de ays to a lower energy state and in the pro ess emits a photon that has the energy orresponding to the energy dieren e between the two states. Absorption is the reverse pro ess, where a mole ule already in a lower energy state is raised to a higher energy state if it intera ts with a photon that has the energy orresponding to the energy dieren e between the two states [4℄. The interesting pro ess on whi h the laser operates is stimulated emission where a mole ule in a higher energy state intera ts with a photon and de ays to a lower energy level if the photon has the energy orresponding to the energy dieren e between the two states. The mole ule emits a photon with the same dire tion, energy and phase as the photon that stimulated the pro ess. After the des ription of the theory of stimulated emission by Einstein, it was used for several pra ti al appli ations. However it was not until 1960 that Theodore Maiman utilized this prin iple and invented the rst laser using a lasing medium of ruby rystal of aluminum oxide. (Al2 CO3 ). doped with hromium stimulated by high. energy ashes of intense light. The laser had a u tuating intensity lasting between a mi rose ond and a millise ond [5℄. Shortly after that, in 1961 Hellwart proposed and implemented the on ept of Q-swit hing. A giant pulse. 1.

(13) Se tion 1.1. Histori al development of the laser. Page 2. of about 10 nanose ond was produ ed by Q-swit hing a ruby laser using a Kerr- ell shutter. The introdu tion of these Q-swit hed lasers with the high intensities they produ e gave rise to the studies of various interesting nonlinear opti al ee ts. To obtain shorter pulses, modelo king of a large number of longitudinal modes a tivated by the gain prole of the lasing medium was proposed by Hagrove, Fork and Polla k in 1964 [5℄. Pulses shorter than a nanose ond were obtained by DeMaria et al. in 1966 by passive modelo king of a Nd-glass laser, whi h has a broad gain prole [5℄. Today, modelo king based on transition-metal-doped rystals su h as Ti:Saphire have been optimized to produ e ultrashort pulses of a few femtose onds. As these pulses be ome shorter, their use for fundamental studies and appli ations in reases rapidly. The ability to measure them also be omes in reasingly important [6℄. Measuring these pulses involves, determining the pulse width, the spe tral intensity and phase or in time domain the temporal prole and phase of the ultrashort pulse. There are several important aspe ts of short pulse measurement. First, a pre ise knowledge of the pulse properties is ne essary to verify the theoreti al models of pulse reation. Se ondly, in order to make even shorter pulses, it is ne essary to understand the distortions that limit the length of the urrently available pulses.. Thirdly, in experiments using. these short pulses, it is always important to know at least the pulse duration in order to determine the temporal resolution of a given experiment [6℄. Pulse shaping experiments however requires the omplete hara terization of the shaped pulse [7, 8℄. The te hnology for measuring ultrashort pulses must pa e up with the lasers themselves [9℄.. Traditional orrelation te h-. niques only provide information about the pulse width after an assumption of the pulse shape [10℄. With te hniques su h as Frequen y Resolved Opti al Gating (FROG) and Spe tral Phase Interferometry for Dire t Ele tri eld Re onstru tion (SPIDER) it is possible to measure pulses over a wide range of wavelengths, pulse durations and retrieve the amplitude and phase of the pulse. This leads us to the aim and outline of this work..

(14) Se tion 1.2. Aim and outline of this work. Page 3. 1.2 Aim and outline of this work The main aim of this proje t work is to dis uss the prin iple of the SPIDER te hnique, dene the design onsiderations and experimentally implement the te hnique for laser pulse hara terization. This thesis is split into 8 hapters, in hapter one we introdu e histori al development, motive and on epts used for ultrashort pulse generation. Chapter two explains some interesting phenomena of the intera tion of light with solids. In hapter three we give a theoreti al des ription of an ultrashort pulse. Chapter four fo uses on the origin and onsiderations of nonlinear opti s. In hapter ve, we introdu e the on ept of pulse hara terization and dis ussed some traditional orrelation te hniques for pulse measurement. In hapter six we introdu ed the on ept of spe tral interferometry and explained how the on ept has been previously used to obtain phase information of an ultrashort pulse. Chapter seven overs the main fo us of this work. The SPIDER prin iple is explained, the experimental design onsiderations are explained and the numeri al and experimental results dis ussed.. 1.3 Generating ultrashort pulses There are some basi omponents ne essary to generate an ultrashort pulse. Firstly, it requires a broad gain bandwidth medium to be pumped by a sour e. This requirement is a result of the inverse Fourier relationship between frequen y and time, in order to obtain a short pulse in time, a broad frequen y spe trum is required. Titanium: Saphire (Al2 O3 ) has shown to be an ex ellent gain medium for laser operation in the near infrared region. Its broad gain bandwidth (from 660-1100 nm) allows for a large tuning range and is well suited for ultrashort pulse generation [11℄. Although a broad spe trum is ne essary for ultrashort pulse generation, it is not su ient. One an nd a thermal sour e or an LED that has the same frequen y spe trum as a 10 fs visible pulse. The spe tral intensity only provides information about the relative spe tral intensities or spe tral modes that ompose the light but.

(15) Se tion 1.3. Generating ultrashort pulses. Page 4. gives no information about the relative phases between these modes [12℄. For ultrashort pulse generation there must be a well dened phases relation between these modes.. Modelo king is an ee tive way of generating high. intensity short pulsed lasers. Let us explain how this is a hieved.. 1.3.1. Modelo king. A typi al laser avity support the existen e of longitudinal modes separated. ∆ν = c/2ηL,. by the frequen y dieren e of the gain medium,. L. where. η. is the refra tive index. is the length of the avity and. c. the speed of light.. These longitudinal modes generally os illate independently of ea h other and the phases of the modes are randomly distributed. When by some means, many longitudinal modes with xed frequen y dieren e in a laser resonator are for ed to maintain xed phase relationships with ea h other, there is onstru tive interferen e and a short pulse is formed. This is termed modelo king [13℄. The total ele tri eld in the laser avity is the superposition of the elds of the various. N. E(t) =. longitudinal modes [14℄ and an be written as. N X. En ei[(2πν0 +2πn∆ν)t+φn ]. (1.1). n=1. where and. n. φn. ν0 is the fundamental laser enter frequen y nth mode. The output power of su h a system. is the mode number,. is the phase of the. with random phases is proportional to the square of the ele tri eld (and in the time domain, the output onstitutes of series of spikes see gure 1.1(a)). In a modelo ked laser, where all the modes have a xed phase and assuming all modes have the same amplitude, the intensity of su h a laser an then be expressed as [15℄. I(t) = E02. sin2 (N∆νt) . sin2 (∆νt). (1.2). Su h a system allows a well dened pulsed output in time (see gure 1.1(b)), with a repetition rate. 1/T .. Most signi antly, for modelo ked lasers, is the.

(16) Se tion 1.3. Generating ultrashort pulses. Page 5. individual pulse width is then given by [15℄. τp = where. 1 T = N ∆νN. (1.3). 1 is the gain bandwidth of the laser and determines the pulse width. ∆νN. (a). (b). Figure 1.1: (a) Multimode and (b) Modelo ked laser output Pra ti ally modelo king is a hieved either a tively or passively or by the ombination of both methods. A tive modelo king involves pla ing an a ousti opti or ele tro-opti modulator in the avity while passive modelo king involves the use of saturable absorber in the avity [14℄. Let us look at these on epts in detail.. 1.3.2. A tive modelo king. A tive modelo king involves using an external modulator su h as an a ousto opti al rystal or an ele tro opti modulator, a short pulse is generated if the modulation is syn hronized with the avity round trip frequen y amplitude modulation at the angular frequen y. Ω. c/2nL.. An. of the longitudinal modes. ωn ± Ω (see gure arrier frequen y ωn by. results in the generation of side bands of angular frequen y 1.2) su h that the side bands are displa ed from the. Ω.. Of parti ular interest is the ase where the modulation frequen y. equal to the mode frequen y spa ing, that is. Ω. is.

(17) Se tion 1.3. Generating ultrashort pulses. Page 6. Figure 1.2: Frequen y spe trum of an amplitude modulated eld. Side band at. ωn ± Ω. [1℄. Ω = ωn+1 − ωn = 2πc/2nL. (1.4). In su h a situation, the side band of ea h mode exa tly mat hes the frequen ies of the adja ent mode, in this ase ea h mode strongly ouples with its nearest neighbors and this leads to a global phase lo king over the whole spe tral distribution. If the amplitude modulation is periodi , we an write the time dependen e of mode. n. of frequen y. ωn. as [3℄. En (t) = εn cos (ωn t + φn ) [1 − α (1 − cos (Ωt + φ))] where. α. (1.5). is the modulation depth. Using trignometri al identities equation. (1.5) an be rewritten as. α En (t) = εn (1 − α) cos (ωn t + φn ) + εn cos [(ωn − Ω)t + φn − φ] 2 α + εn cos [(ωn + Ω)t + φn − φ] (1.6) 2.

(18) Se tion 1.3. Generating ultrashort pulses. Page 7. So we see from equation (1.6) that an amplitude modulation generates side bands. α times the arrier frequen y amplitude. It is also evident from equa2. tion (1.6) that if the modulation frequen y is driven at the frequen y mode spa ing, that is. Ω = ωn ± ωn+1. then the side bands asso iated with ea h. mode exa tly oin ide with the neighboring modes, resulting in the desired lo king of all the modes.. 1.3.3. Passive modelo king. The rst opti al pulses in the pi ose ond range was generated through passive modelo king. It utilizes the insertion of a saturable absorbing element inside the laser avity.. The absorption oe ient of the material is su h. that it an be saturated by high intensity, in other words it ould rea h a maximum transmission above the saturation intensity. As the gain medium is ontinuously pumped, at rst the avity loss is very large due to the absorber, the transmission of the absorber is almost onstant at. T0. (gure 1.3). and independent of the in ident intensity. Due to the large loss the laser an not os illate and the gain in the medium in reases. However, on e the gain be omes high enough to over ome the losses in the avity, the intensity in the avity grows rapidly..

(19) Se tion 1.3. Generating ultrashort pulses. Page 8. Figure 1.3: Transmission of a saturable absorber as a fun tion of the in ident intensity. At some saturation intensity. Isabs. the absorber be omes saturated, the avity. losses drop rapidly. This lo ks the modes together and the result is a short pulse in time [3℄..

(20) 2. Intera tion of light with solid materials In this hapter we will use the term light when relating to properties or ee ts of light in general, however we will be sure to use the term short pulses, or laser light when referring to ee ts that only the uniqueness of laser light an produ e. Most opti al experiments involve the propagation through or ree tion of light from solid opti al materials. The intera tion of light with these solid materials result in interesting opti al phenomena su h as absorption, dira tion, emission, ree tion, dispersion, and polarization ee ts [16℄. The aim of this hapter will be to understand from a lassi al point of view the mi ros opi origin of some of the afore mentioned opti al properties of solids. This is of relevan e to this study be ause as we would see later, some of these ee ts are useful in ultrashort pulse hara terization.. 2.1 Signi an e of the generalized wave equation If we onsider only nonmagneti , ele tri ally neutral media, then this implies that the magnetization in su h media.. M. and volume density of ele tri harge. ρ. are zero. The ele tromagneti state of the medium an then be de-. s ribed by the polarization. P. des ribing the response of the bound harges. to the applied eld and the urrent density. J. des ribing the response of the. ondu tion ele trons to the applied eld. The polarization. P. is given by the. expression. where spa e.. ǫ. P = (ǫ − ǫ0 ) E = χǫ0 E. (2.1). ǫ0. is the permittivity of free. is the permitivity of the material and. The fa tor. χ =. . ǫ ǫ0. −1. . is alled the ele tri sus eptibility and is. 9.

(21) Se tion 2.1. Signi an e of the generalized wave equation. a unique property of the material.. Page 10. This expression for the polarization is. only valid for isotropi materials su h as glass where. χ. is a s alar quantity. and has the same value for any dire tion of the applied eld. Also the linear relationship between the polarization and ele tri eld is orre t under the approximation that the in ident ele tri eld is not high enough to onsider higher order terms. As we will see later in hapter 4 if the ele tri eld is high enough then the polarization must be expressed as a power series in the ele tri eld. For now, we will treat the linear ase. We begin by deriving the wave equation for an ele tri eld in a medium. Maxwell's equations for the ele tri eld. E. and magneti eld. H. are given by [17℄. ∂H ∂t ∂E ∂P ∇ × H = ǫ0 + +J ∂t ∂t 1 ∇·E = − ∇·P ǫ0 ∇·H = 0 ∇ × E = −µ0. (2.2) (2.3) (2.4) (2.5). The general wave equation for the ele tri eld is derived by taking the url of equation (2.2) and the time derivative of equation (2.3). This gives. 1 ∂2E ∂2P ∂J ∇ × (∇ × E) + 2 2 = −µ0 2 − µ0 c ∂t ∂t ∂t. (2.6). The two terms on the right hand side of equation (2.6) are alled sour e terms and arise from polarization and ondu tion harges within the medium. For a non ondu ting medium only the polarization term is important and is responsible for opti al ee ts su h as dispersion (whi h we will look at in detail), frequen y onversion, and absorption.. In the ase of metals the. ondu tion term is very important and is responsible for the opa ity and high ree tan e of metals. Now that we have an equation that des ribes the propagation of light in a medium.. Our next interest is to understand the. intera tions of the eld with the medium. In the next se tion, we will pay parti ular attention to absorption and dispersion as we would see later these two phenomena are very important in ultrashort opti s and the SPIDER.

(22) Se tion 2.2. Origin of dispersion in materials. Page 11. te hnique in parti ular.. 2.2 Origin of dispersion in materials Considering non ondu ting isotropi materials su h as glass, we assume that the ele trons behave as though the for es binding them to the nu lei are elasti for es des ribed by Hooke's law.. This means we are onsidering a. linear response, where the restoring for e is proportional to the displa ement with opposite dire tion. We an use this idea to obtain an expression for the polarization in terms of the frequen y of the applied eld. The for e on the ele tron due to the ele tri eld is. F = −eE Suppose ea h ele tron with harge. −e. (2.7). is displa ed by a distan e. r. from its. equilibrium position, then the resulting ma ros opi stati polarization of the medium is given by [16℄. P = −Ner = s where. N. Ne2 E K. (2.8). is the number of ele trons per unit volume and. K. is the for e. onstant. If the in ident ele tri eld is time dependent then the expression for polarization in equation (2.8) is in orre t, and the motion of the ele trons must be onsidered.. Considering the bound ele trons as lassi al damped. harmoni os illators, the equation of motion for the ele trons an be written as [16℄. m where. γ. dr d2 r + mγ + Kr = −eE 2 dt dt. (2.9). is the damping onstant. Assume that the motion of the ele trons. has a harmoni time dependen e su h that a solution to the equation of motion in (2.9) would be. r = r0 exp(−iωt) where ω. is the angular os illation.

(23) Se tion 2.2. Origin of dispersion in materials. Page 12. frequen y. Then solving equation (2.9) we get. −mω 2 − iωmγ + K r = −eE. and. r=. ω02. (2.10). −e/m − ω 2 − iωγ. (2.11). Using equation (2.8) and (2.11), the frequen y dependent polarization an be written as. P=. Ne2 /m E ω0 − ω 2 − iωγ. where we have used. ω0 =. r. (2.12). K m. as the ee tive resonan e frequen y of the bound ele trons.. As expe ted. the polarization expression in equation (2.12) is the amplitude expression for a driven damped harmoni os illator, so we expe t to nd some unique resonan e phenomenon for frequen y omponents lose to the resonant frequen y.. This ee t shows in the hange of the index of refra tion of the. medium and strong absorption of light lose to the resonant frequen y. To show analyti ally how the polarization indu es dispersion and absorption of light pulses, we go ba k to the general wave equation in (2.6), leaving out the ondu tion term be ause we are onsidering diele tri materials and substituting the expression for the polarization. P expressed. in equation (2.12) into. the wave equation. Sin e we are onsidering an isotropi medium, there is no variation of the ee t of ele tri eld on the medium, hen e the term. ∇ × (∇ × E) equals −∇2 E in equation (2.6).. ∇ · E=0. and. Then the polarization. ee t on the propagation of the pulse in the medium an be des ribed by a simpler expression given by. 1 ∇ E= 2 c 2. 2 Ne2 ∂ E 1 1+ · 2 2 mǫ0 ω0 − ω − iγω ∂t2. where we have used the relation. 1/c2 = µ0 ǫ0. and. µ0. (2.13). is the permeability of. free spa e. Already we an see from equation (2.13) that the ele tri eld.

(24) Se tion 2.2. Origin of dispersion in materials. Page 13. in su h a medium is not only time dependent but is ae ted by a material property (terms in bra ket). If we onsider a plane wave of the form. E = E0 ei(kz−ωt) where. k is the wave number.. (2.14). Using this as a trial solution for equation (2.13),. then. −∇2 E = −k 2 E. (2.15). and. ∂2E ∂t2. = −ω 2 E.. (2.16). Substituting equations (2.15) and (2.16) into the wave equation (2.13) gives an expression for. k. whi h is the wave number and relates to the behavior of. the light in the medium as. ω2 k = 2 c 2. . Ne2 1 1+ · 2 mǫ0 ω0 − ω 2 − iγω. . .. (2.17). The imaginary term in the denominator of the expression above means that the wave number. k. must be a omplex number having a real and imaginary. part, meaning that we an write. k. in terms of its real and imaginary parts. as. k = kR + iki. (2.18). Sin e the wave number is related to the refra tive index by the fa tor where. ω. is the angular frequen y of the wave and. an also write the refra tive index. η. c. ω , c. is the speed of light. We. in terms of it real and imaginary parts. as. η = ηR + iηi. (2.19). With equation (2.18) we an rewrite our solution based on equation (2.14) as. E = E0 e−ki z ei(kR z−ωt) The fa tor. e−ki z. (2.20). shows that the amplitude of the wave de reases as it propa-.

(25) Se tion 2.2. Origin of dispersion in materials. Page 14. gates through the medium, this is as a result of the absorption of the energy of the wave by the medium. This absorption is very large at resonant frequen ies. Sin e the intensity of the wave is given by the square of the magnitude of the ele tri eld, the absorption oe ient. α = 2ki .. Using equation (2.17). and (2.19), and equating the real and imaginary parts of the refra tive index, we get. ηi2 and. −. ηR2. Ne2 =1+ mǫ0. Ne2 2ηi ηR = mǫ0. The interesting parameters. . ηR (ω). . ω02 − ω 2 (ω02 − ω 2 ) − γ 2 ω 2. γω 2 (ω0 − ω 2) − γ 2 ω 2. and. ηi (ω). . . (2.21). (2.22). whi h denotes the frequen y. dependent refra tive index (dispersion) of the material and absorption of light by the material respe tively are obtained using equation(2.21) and (2.22). Figures (2.1) and (2.2) shows how these parameters behave and we an make the following dedu tions.. PSfrag repla ements. Figure 2.1: Real part of the refra tive index versus frequen y. The index of refra tion is slightly greater that. 1. for small frequen ies and. in reases rapidly with frequen y as it approa hes the resonant frequen y. This is the normally dispersive region, for most transparent media this.

(26) Se tion 2.3. Opti al dispersion in materials. Page 15. PSfrag repla ements. Figure 2.2: Imaginary part of the refra tive index versus frequen y. overs the visible spe tral region. On the other hand anomalous dispersion involves a de rease in the refra tive index with in reasing frequen y.. 2.3 Opti al dispersion in materials From the previous se tion we have understood that the refra tive index of a material in reases with frequen y in normally dispersive media and this ee t is termed dispersion.. As a result of the wide spe tral width of ultrashort. opti al pulses, and group velo ity dispersion (GVD) in transparent media, dispersion is usually of interest in ultrashort opti s. The propagation of the dierent spe tral omponents in the pulse spe trum results in a spatial and temporal broadening of the pulse. In materials with positive group velo ity dispersion, that is, normally dispersive media, the longer wavelengths travel faster than the shorter ones, and in anomalous dispersive regions the reverse is the ase (see gure 2.1). Figure (2.3) shows the dispersion urve for fused sili a illustrating an in rease in refra tive index with wavelength..

(27) Se tion 2.4. Ultrashort pulse propagation in a dispersive medium Page 16. 1.58. Refractive index. 1.56 1.54 1.52 1.5 1.48 1.46 100. 200. 300. 400 500 Wavelenght(nm). 600. 700. 800. Figure 2.3: Dispersion urve for fused quartz. 2.4 Ultrashort pulse propagation in a dispersive medium A short pulse propagating through a medium su h as an opti al dispersive material, might be ome distorted in shape or broadened in time depending on the spe i material and on the distan e. z. along the dire tion of propa-. gation. In this se tion we try to analyze this phenomena. In pra ti e, an ultrashort pulse is poly hromati that is, it onsist of many frequen y omponents within the visible region of the ele tromagneti spe trum. Ideally the instantaneous frequen y of the pulse should be onstant in time. In su h a situation the pulse is said to be transform limited. However if a broad spe tral width pulse enters a dispersive medium, the dierent frequen y omponents experien es dierent values of the refra tive index, this means they ea h travel at slightly dierent speeds. The result of this phenomena is an in rease in the duration of the pulse, the pulse is thus said to be. hirped.. Now we want to analyze this phenomena in relation to an ultrashort. pulse. Assuming a Gaussian envelope for the ele tri eld of the form.

(28) Se tion 2.4. Ultrashort pulse propagation in a dispersive medium Page 17. E(t, 0) = exp(−a0 t2 + iω0 t),. (2.23). implies an ele tri eld in the frequen y domain given by [18℄. (ω − ω0 )2 E(0, ω) = exp − 4a0 . . (2.24). By hoosing the pulse in the form des ribed in (2.23) note that we have on-. a0 = a − ib with distan e z through a. sidered intially an un hirped pulse with pulse parameter. b = 0.. The output pulse spe trum after propagating a. dispersive medium will be the initial spe trum as expressed in (2.24), multiplied by the frequen y-dependent propagation through the system (we have already established this in se tion (2.1)). This gives. E(z, ω) = E(ω, 0) exp(−ik(ω)z) where. k(ω) = η(ω)ω/c. (2.25). is the wave number of the dispersive medium,. η(ω). is the refra tive index orresponding to a parti ular frequen y omponent or wavelength and. c. is the speed of light in va uum. Considering ases of. relatively narrow band signals having frequen y omponents around some enter frequen y its value at. ω0. ω0 ,. then the propagation onstant an be expanded about. in the form [18℄. 1 k(ω) = k(ω0 ) + k ′ (ω − ω0 ) + k ′′ (ω − ω0 )2 2 where the derivatives. 2. ω0 ) .. Then. E(z, ω). k ′ = dk/dω. and. k ′′ = d2 k/dω 2. an be expressed as [18℄. (2.26). are evaluated at. (ω −. ik ′′ z 1 2 ′ + (ω − ω0 ) E(z, ω) = exp −ik(ω0 )z − ik z(ω − ω0 ) − 4a0 2. (2.27).

(29) Se tion 2.4. Ultrashort pulse propagation in a dispersive medium Page 18. The output pulse in time domain after a distan e. z. will then be the inverse. Fourier transform of the output spe trum, that is. E(z, t) ≡ =. Z. ∞. E(z, ω) exp(iωt)d (ω − ω0 ) ik ′′ z 1 2 ′ (ω − ω0 ) exp(iωt)d (ω − ω0 ) + exp −ik(ω0 )z − ik z(ω − ω0 ) − 4a0 2 −∞. Z−∞ ∞. (2.28). The integral in equation (2.28) an be rewritten in the form (see appendix A.1). ei [ω0 t − k(ω0 )z] E(z, t) = 2π where. ∞. (ω − ω0 )2 ′ exp − + i(ω − ω0 )(t − k z) d(ω−ω0 ) 4a(z) −∞. Z. . (2.29). ′′. 1/a(z) ≡ 1/a0 +2ik . The advantage of rearranging in this form is that. the arrier-frequen y time and spa e dependen e have been moved out of the integral, so that the integrand of the expression gives the time and spa e dependen e of the output pulse envelope. From equation (2.29) we an see that the output pulse is still a Gaussian pulse but with an altered Gaussian pulse parameter. a(z).. In order to interprete this mathemati al result, we an. arry out the integration using Siegman's lemma, given by [18℄. Z. ∞. −∞. Ay 2 −2By. e. dy ≡. r. π B2 /A e , A. Re[A] > 0. (2.30). or we an note that the integrand of equation (2.29) is the Fourier transform of a Gaussian pulse in the form. ′. t − k z.. tan e. z. exp(−at2 ),. with a shift in time by. These two approa hes give the output pulse after traveling a disthrough the medium as. i h 2 E(z, t) = exp [i(ω0 t − k(ω0 )z)] exp −a(z) (t − k ′ z) " 2 # z z exp −a(z) t − = exp iω0 t − (2.31) υφ (ω0 ) υg (ω0 ).

(30) Se tion 2.4. Ultrashort pulse propagation in a dispersive medium Page 19. a(z) is the modied Gaussian pulse parameter after traveling a distan e z . This term is usually smaller that the initial parameter a0 showing that the where. pulse width has been broadened. Some interpretations an now be made from the expression for. E(z, t) given in equation (2.31).. From the rst exponential. term it is seen that the phase of the arrier frequen y is delayed by a phase shift. k(ω0 )z. or a midband phase delay in time. tφ =. tφ. given by. k(ω0 ) z = z. υφ (ω0 ) ω0. (2.32). This means that the sinusoidal waves within the pulse envelope will seem to move forward with a phase velo ity of. υφ (ω0 ) =. ω0 z = tφ k(ω0 ). (2.33). The phase velo ity is determined by the propagation frequen y. k(ω0 ). at the arrier. ω0 .. The se ond exponent in (2.31) shows that the pulse envelope remains a Gaus-. a(z),. and is delayed by the group. z = k ′ z. υg (ω0 ). (2.34). sian but with a modied pulse parameter delay time tg given by. tg =. This means the pulse envelope appears to move forward with a group velo ity. 1 = υg (ω0 ) = (dk/dω). . dω dk. . .. (2.35). To summarize this dis ussion, we showed that an ultrashort pulse propagating through a lossless (ignoring absorption) homogeneous media will experien e a delay of the pulse, and an in rease in duration due to a frequen y hirp..

(31) 3. Des ription of an ultrashort pulse. 3.1 Temporal and spe tral representations The fundamental quantity des ribing an individual pulse of light is the real ele tri eld whi h is a fun tion of time and spa e or frequen y and wave ve tor. Consider again a Gaussian pulse shape, and rewrite equation (2.23) and equation (2.24).. The Gaussian pulse time domain ele tri eld with. onstant phase (gure 3.1) an be written as. E(t) = exp(−at2 + iω0 t) where that is. a. is proportional to the inverse of the square of the pulse duration,. a ∝ 1/t2p .. Equation (3.1) des ribes a pulse that has a onstant in-. stantaneous frequen y in time.. |E(t)|. 2. (3.1). The time dependent intensity of the pulse. an be measured and for a onstant phase an be represented as in. gure (3.1) The Fourier transform of equation (3.1) gives the frequen y representation of the light pulse as. (ω − ω0 )2 E(ω) = exp − 4a. and the width of the spe trum is dire tly proportional to. (3.2). a.. We an see that. there is a relationship between the spe tral width of the pulse and the pulse duration. The exa t relationship between these quantities will be shown in the next se tion.. 20.

(32) Se tion 3.2. Time bandwidth produ t. Page 21. Temporal intensity (a.u.). 1. Temporal phase. 0.8. 0.6. 0.4. 0.2. 0 -400. -300. -200. -100. 0 100 Time (fs). 200. 300. 400. Figure 3.1: Temporal intensity of a Gaussian pulse with onstant phase. 3.2 Time bandwidth produ t In general the time-frequen y relationship of a light pulse an be represented as. −∞. 1 E(t) = 2π. Z. E(ω) =. Z. and. E(ω)e−iωt dω. (3.3). ∞ ∞. E(t)eiωt dt. (3.4). −∞. The duration and spe tral width of the pulse are usually al ulated using standard statisti al denitions [3℄. < ∆t > = 2. < ∆ω > =. R −∞. R∞−∞. R∞ −∞ ∞. t|ε(t)|2 dt |ε(t)|2dt. ω 2 |E(ω)|2dω. R −∞ ∞. ,. |E(ω)|2dω. .. (3.5).

(33) Se tion 3.2. Time bandwidth produ t. Page 22. The spe tral width and pulse duration an be shown to be related by the quantities. 1 2. ∆ω∆t ≥ So we note that a short pulse duration. (3.6). ∆t requires a large frequen y range ∆ω. to des ribe the pulse. In order to generate an ultrashort pulse, it is therefore ne essary that the laser is hara terized with a large gain bandwidth and a large number of longitudinal modes.. Equation (3.6) also shows that there. is a limit to the time-bandwidth produ t. The idea of a theoreti al limit to the time-bandwidth produ t of a pulse an be explained better using Fourier transformations. Considering the ase of the Gaussian pulse envelope. The Full Width at Half Maximum (FWHM) of the temporal intensity prole is given by. ∆t =. r. 2ln(2) a. (3.7). and the FWHM of the spe trum is. ∆ω ∆ν = = 2π. r. 2ln(2)a π. (3.8). The time-bandwidth produ t is then. ∆ν∆t =. 2ln(2) = 0.441 π. (3.9). Equation (3.9) gives a theoreti al minimum for the time-bandwidth produ t of the pulse. This value depends on the assumed pulse shape (see table 3.1) and holds for pulses without frequen y modulation or un hirped pulses. We dened the term hirped pulse in the previous se tion. A pulse whose timebandwidth produ t is given by the theoreti al minimum is alled a Fourierlimited or bandwidth limited pulse.. The ele tri eld of a bandwidth. limited Gaussian pulse will be of the form given in gure 3.2.

(34) Se tion 3.3. Variation of instantaneous frequen y with time. Page 23. 1 0.8. Electric field (a.u.). 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -4. -3. -2. -1. 0 1 Time (a.u.). 2. 3. 4. Figure 3.2: Bandwidth limited Gaussian pulse. and is Chara terized by a onstant frequen y with time. In reality, due to inevitable hirp a pulse bandwidth produ t will be greater than the theoreti al minimum. For some appli ations it is important that the pulse is lose to the transform limit. Shape Gaussian fun tion Exponential fun tion Lorentzian Pulse Hyperboli se ant Cardinal sine Re tangle. E(t) C = ∆ν∆t 2 exp(−at ) 0.441 exp(−at) 0.140 [1 + (at)2 ] 0.142 1 0.315 cosh(at) sin2 (at) at2. 0.336. −. Table 3.1: Values of time-bandwidth onstants. 0.892. C. for sele ted pulse shapes. 3.3 Variation of instantaneous frequen y with time As was mentioned earlier, in reality, the time-bandwidth produ t of the pulse will not usually be given by the theoreti al minimum, that is, the pulse will.

(35) Se tion 3.3. Variation of instantaneous frequen y with time. Page 24. originally not be bandwidth limited. This is due to the ee ts of dispersion and nonlinearities.. A pulse that is not bandwidth limited is said to be a. hirped pulse. A hirped pulse is one whose instantaneous frequen y varies during the duration of the pulse.. A brief mathemati al des ription of a. hirped pulse is given below. This will help us understand the omplexities of hirp in a pulse.. The time-varying phase-shift or phase rotation of the. sinusoidal signal within this Gaussian pulse is given by. E(t) ∝ exp[i(ω0 t + φ(t))] where the temporal phase. φ(t). (3.10). ontains the frequen y versus time informa-. tion. For a start we assume. φ(t) = βt2. (3.11). as a rst approximation We will explain shortly what the parameter. β. de-. s ribes. The signal an then be expressed as. E(t) ∝ exp[i(ω0 t + βt2 )] The total instantaneous phase of the signal denoted by. (3.12). φo. φo (t) = ω0 t + βt2. is given by. (3.13). The instantaneous angular frequen y is dened as the rate at whi h the total phase of the sinusoidal signal rotates forward, this an be expressed as. ωinst (t) ≡. dφo(t) . dt. (3.14). Considering the omplex Gaussian with instantaneous angular frequen y given by. ωinst(t) ≡. d (ω0 t + βt2 ) = ω0 + 2βt. dt. (3.15). The omplexity of the phase of the signal determines the omplexity of the frequen y versus time variation. A Gaussian pulse with a non-zero imaginary part. β,. therefore has a linearly time-varying instantaneous frequen y. Su h.

(36) Se tion 3.3. Variation of instantaneous frequen y with time. a signal is said to be hirped, with the parameter hirp.. β. Page 25. being a measure of this. In the ase we illustrated above, we have been onsidering pulses. whose instantaneous frequen y in reases linearly with time, this is alled a positively hirped pulse. On the other hand, if the instantaneous frequen y de reases with time we get a negatively hirped Gaussian pulse. We simply. β. negative.. 1. 1. 0.8. 0.8. 0.6. 0.6. 0.4. 0.4. Electric field (a.u.). Electric field (a.u.). make. 0.2 0 -0.2 -0.4. 0.2 0 -0.2 -0.4. -0.6. -0.6. -0.8. -0.8. -1. -1 -4. -3. -2. -1. 0. 1. 2. 3. 4. -4. Time (a.u.). -2. -1. 0. 1. 2. Time (a.u.). (a). Figure 3.3:. -3. (b). (a) Negatively hirped Gaussian pulse (b) Positively hirped. Gaussian pulse. The frequen y of the light wave an also vary nonlinearly with time. Consider the ase where it varies quadrati ally with time. ωinst = ω0 + 3γt2 where. γ. (3.16). is a measure of the quadrati hirp. The phase of eld is given by. E(t) ∝ exp[i(ω0 t + γt3 )]. (3.17). Dispersion has played an important role in ultrafast laser physi s. A typi al example is in the. hirped pulse ampli ation system (CPA) whi h is the main. omponent in ultrashort pulse ampliers.. CPA requires proper dispersion. management to produ e ultrashort, high energy pulses [19℄. In this system the short pulse to be amplied is rst stret hed temporarily, by passing it. 3. 4.

(37) Se tion 3.3. Variation of instantaneous frequen y with time. Page 26. through a dispersive delay line typi ally one with a positive group delay dispersion. This redu es the peak intensity and the pulse is now safely amplied by several passes through a broad band gain medium. After ampli ation the pulse is re ompressed in a dispersive delay line, with opposite sign to that of the stret her. The ompressor orre ts the dispersion indu ed by the gain medium, resulting in an ultrashort high energy pulse. See gure 3.4.. Figure 3.4: Illustration of the prin iple of hirped pulse ampli ation te hnique.

(38) 4. Nonlinear opti al pro esses The nonlinear opti al ee t has been a useful tool in the hara terization of ultrashort pulses. It is interesting to note that while there exist a problem of ultrashort pulse hara terization due to the very short duration of these pulses, the high intensities of these pulses allow for the use of nonlinear opti al pro ess to hara terize the pulse prole either in the temporal or spatial domain.. Nonlinear opti al pro esses be ame measureable after the. demonstration of the ruby laser by Theodore Harold Maiman in July, 1960. It was however rst implemented by Franken and his olleagues in 1961 [20℄. A 694.3 nm light from a ruby laser was fo used into a quartz rystal and a very low energy beam of wavelength 347.15 nm was dete ted. Though with a poor onversion e ien y, the demonstration of this experiment brought great enthusiasm and opened a new world for signi ant nonlinear opti al intera tions. Pre eding this time, the polarization of a material was assumed to be linear with the indu ed ele tri eld. E. P = ǫ0 χE where. ǫ0. is the permittivity of free spa e and. (4.1). χ. is the linear sus eptibility. whi h is a property of the material. However, if the applied ele tri eld is high enough as in the ase of lasers, the nonlinear terms in the polarization of a material an no longer be ignored. The nonlinear polarization an then be des ribed as a power series expansion of the ele tri eld [2℄. PN L = ǫ0 χ(1) E + χ(2) E 2 + · · · Here. χ(2). . (4.2). is the se ond order non-linear sus eptibility of the material and the. nonlinear polarizability PN L is seen to depend nonlinearly on the fundamental applied eld. In entrosymetri materials the even terms vanish. Assuming a linear relationship between the polarization of a material and the input ele tri eld implies that the intensity of the applied eld is small enough to ignore the higher order ontributions of the eld to the polarization of the. 27.

(39) Se tion 4.1. The origin of nonlinear polarization. Page 28. material. Nonlinear opti al pro ess have found a variety of appli ations in ultrashort laser studies and of interest to this proje t is its appli ation in opti al short pulse hara terization.. Most self referen ing te hniques used. for pulse hara terization requires a nonlinear intera tion of the test pulse in a medium. Se ond harmoni generation whi h relates to the se ond term in the nonlinear polarization is often utilized.. 4.1 The origin of nonlinear polarization In se tion 2.2 we had modelled the linear response of the ele trons in a medium using the ele tron os illator model.. Similarly in this se tion we. will model the nonlinear atomi response of a non entrosymmetri medium, (mediums la king inversion symmetry) by allowing for nonlinearity in the restoring for e exerted on the ele trons. In this ase, the restoring for e on the ele trons is not limited to Hooke's law. F = −K r but higher order terms be omes signi ant (larger), onsidering only the rst two terms. F = −K1 r − K2 r2 + · · · and we have negle ted higher order terms.. The Equation of motion of an. ele tron then be omes [2℄. ¨r + γ r˙ + ω02 r + r · a˜ · r = −eE (t) /m where we have introdu ed a quadrati term,. ˜ · r, a ˜ r·a. (4.3). being a third-rank. tensor. For the sake of simpli ity we will only onsider the one dimensional ase so that equation (4.3) an now be written as. r¨ + γ r˙ + ω02 r + ar 2 = −eE (t) /m.. (4.4).

(40) Se tion 4.1. The origin of nonlinear polarization. Page 29. We obtain this form by assuming that the restoring for e is a nonlinear fun tion of the displa ement of the ele tron. Considering only the rst and se ond term of the nonlinear expansion of the restoring for e, we an understand the nature of the for e by noting that it orresponds to a potential energy fun tion of the form. U =−. Z. 1 1 F dr = mω02 r 2 + mar 3 2 3. (4.5). Here the rst term orresponds to a harmoni potential and the se ond term orresponds to an anharmoni orre tion term. This model orresponds to the physi al situation of ele trons in real materials, be ause the a tual potential well that the atomi ele tron feels is not perfe tly paraboli [2℄.. PSfrag repla ements U Figure 4.1: Potential energy fun tion for a nonlinear medium [2℄. In this ase, let us assume that the applied opti al eld is of the form. E (t) = E1 eiω1 t + E2 eiω2 t + c.c.. No general analyti solution to equation (4.4) exist. However, if the applied.

(41) Se tion 4.1. The origin of nonlinear polarization. eld is su iently weak, the nonlinear term. 2 the linear term ω0 r for any displa ement. r.. ar 2. Page 30. will be mu h smaller than. In order to solve equation (4.4). we use the idea of perturbation expansion [2℄. This means we now repla e. E (t). by. λE (t). , where. λ. is a parameter that ranges ontinuously between. zero and one. The expansion parameter. λ. thus hara terizes the strength of. the perturbation. Equation (4.4) then be omes. r¨ + γ r˙ + ω02r + ar 2 = −λeE (t) /m Note here that. λ. (4.6). in this Se tion has nothing to do with wavelength, and we. just employ this method be ause we need to solve the equation (4.4). We now seek a solution to Equation (4.6) in the form of a power series expansion, that is. r = λr (1) + λ2 r (2) + λ3 r (3) + · · · .. (4.7). Using only the rst and se ond term, substituting into equation (4.6), and equating the oe ients of the rst and se ond powers of following equations for the oe ients of. λ1. and. λ2. λ,. we obtain the. respe tively as. r¨(1) + γ r˙ (1) + ω02 r (1) = −eE (t) /m 2 r¨(2) + γ r˙ (2) + ω02 r (2) + a r (1) = 0. (4.8). (4.9). Noti e that equation (4.8) is the same as the equation of motion des ribing the linear response of the atoms to the ele tri eld we obtained in equation (2.9), and we have already found the solution in equation (2.11). In order to solve equation (4.9), we hoose a solution of the form. r (1) (t) = r (1) (ω1 )e−iω1 t + r (1) (ω2 )e−iω2 t + c.c. (4.10).

(42) Se tion 4.1. The origin of nonlinear polarization. where the amplitudes. r (1) (ωj ). Page 31. have the form. r (1) (ωj ) = −. e Ej m D(ωj ). (4.11). and we introdu e the omplex denominator fun tion. D(ω) = ω02 − ω 2 − iωγ.. (4.12). Squaring equation (4.10) gives. . r (1) (t). 2. =. (1) 2 2 r (ω1 ) e−2iω1 t + 2r (1) (ω1 )r (1) (ω2 )e−i(ω2 +ω1 )t + r (1) (ω2 ) e−2iω2 t. + 2r (1) (ω1 )r (1)∗ (ω2 )e−i(ω1 −ω2 )t + 2r (1)∗ (ω1 )r (1) (ω2 )e−i(ω2 −ω1 )t 2 2 + 2r (1)∗ (ω1 )r (1)∗ (ω2 )e−i(−ω1 −ω2 )t + r (1)∗ (ω1 ) e−i(−2ω1 )t + r (1)∗ (ω2 ) e−i(2ω2 )t + 2r (1) (ω1 )r (1)∗ (ω1 ) + 2r (1) (ω2 )r (1)∗ (ω2 ). Noti e that the square of. ± (ω1 − ω2 ). 0.. and. r (1) (t) ontains the frequen ies ±2ω1 , ±2ω2 , ± (ω1 + ω2 ),. We see here, and for the rst time, the sour e of the. se ond-order opti al ee ts. When we onsidered the se ond order term in the nonlinear displa ement of the ele trons by the applied for e, and solved the equation of motion for the ele trons, we noti e the se ond order nonlinear ee ts su h as the frequen y doubled beam (2ω1 and. 2ω2 ). and the sum. frequen y beam (ω1 + ω2 ). In order to determine the response of the material at frequen y. 2ω1 ,. for instan e, we must solve the equation. 2 r¨(2) + γ r˙ (2) + ω02 r (2) = −a r (1) (ω1 ). (4.13). and this implies solving. (2). r¨. + γ r˙. (2). +. ω02 r (2). e2 E12 e−2iω1 t = −a m2 D 2 (ω1 ) . . (4.14). We seek a solution of the form. r (2) (t) = r (2) (2ω1 )e−i2ω1 t. (4.15).

(43) Se tion 4.1. The origin of nonlinear polarization. Page 32. Substituting Equation (4.15) into Equation (4.14) we get that. −4ω12. − 2iγω1 +. ω02. . (2). −2iω1 t. r (2ω1 )e. e2 E12 e−2iωt = −a m2 D 2 (ω1 ) . . (4.16). and this implies. E12 e2 r (2ω1 ) = −a m2 D(2ω1 )D 2 (ω1 ) (2). . . (4.17). where. D(2ω1 ) = −4ω12 − 2iω1 γ + ω02 Similarly, the amplitudes of the response at frequen y. e2 E12 r (2ω2) = −a m2 D(2ω2) D 2 (ω1 ) (2). . (4.18). 2ω2. is found to be. . (4.19). The linear ontribution to polarization from Equation (2.8) is given in a more general form by. P (1) (ω1 ) = −Ner (1) (ωj ) then using equation (4.11), we an now write. P (1) (ω1 ) =. P (1) (ω1 ). as. Ne2 Ej mD(ωj ). (4.20). From equation (4.20), we dene the linear sus eptibility in general as. χ(1) (ωi ) =. Ne2 mD(ωj ). so that the linear polarization in a more general form is dened by. P (1) (ω1 ) = χ(1) (ωj )E(ωi ). (4.21). The nonlinear polarization and sus eptibilities are al ulated in an analogous manner. The nonlinear sus eptibility des ribing se ond-harmoni generation.

(44) Se tion 4.1. The origin of nonlinear polarization. χ(2). Page 33. is dened by the relation. P (2) (2ω1 ) = χ(2) (2ω1 , ω1 , ω1 )E(ω1 )2. (4.22). Here ,. χ(2) (2ω1 , ω1 , ω1 ) = and. P (2) (2ω1). N(e3 /m2 )a D(2ω1 )D 2 (ω1 ). (4.23). is the nonlinear polarization sour e for the se ond-harmoni. generation. From equation (4.22), we see that the nonlinear polarization sour e for the se ond-harmoni generation depends on the square of the in ident ele tri eld. The invention of lasers with high intensity light brought to the fore the nonlinear opti al ee ts in materials. Also, the nonlinear polarization sour e for the se ond-harmoni generation depends on the nonlinear sus eptibility. χ(2) (2ω1 , ω1 , ω1). expressed in equation (4.23).. In the denominator of this. sus eptibility, we see from equation (4.12) and (4.18), that the se ond order nonlinear sus eptibility frequen ies (ω0. ≈ ω1. χ(2) (2ω1 , ω1 , ω1 ). and. ω0 ≈ 2ω1 ).. may be resonantly enhan ed at the. In resonant enhan ement pro esses the. frequen y of the lasers is tuned, so that it is lose to the resonant frequen y of the material. This in reases the sus eptibility. This se tion helps to understand that if we onsider the appli ation of an intense ele tri eld to a medium, the expression for a linear polarization obtained in equation (4.1) does not des ribe the polarization of a medium ompletely. From the ele tron model we would say that the displa ement. r. of the ele trons is no longer small sin e the eld is intense and so we must onsider at least the quadrati term in the nonlinear expansion of the elasti restoring for e. Doing this, we dis overed that the polarization of a nonlinear medium is expressed as a power series in the applied ele tri eld. Sin e the se ond order nonlinear pro esses, in parti ular se ond harmoni generation is mainly utilized in pulse hara terization let us look briey at these on ept..

(45) Se tion 4.2. Se ond harmoni generation (SHG). Page 34. 4.2 Se ond harmoni generation (SHG) Se ond harmoni generation o urs when two photons of frequen y onverted by a nonlinear pro ess in a medium to a frequen y of. 2ω .. ω. are. Let us. Figure 4.2: S hemati of the prin iple underlying se ond harmoni generation. onsider the ase in whi h two input photons have dierent frequen ies and. ω2 ,. and. ω1 6= ω2 .. ω1. The total in ident ele tri eld an be written as. E = E1 e−iω1 t + E1∗ eiω1 t + E2 e−iω2 t + E2∗ eiω2 t. (4.24). The se ond order nonlinear polarizability from equation (4.2) is then. P (2) = ǫ0 χ(2) E 2 = ǫ0 χ(2) [E12 e−i(2ω1 )t + (E1∗ )2 ei(2ω1 )t + E22 e−i(2ω2 )t + (E2∗ )2 ei(2ω2 )t +2E1 E1∗ + 2E2 E2∗ + 2E1 E2 e−i(ω1 +ω2 )t + 2E1∗ E2∗ ei(ω1 +ω2 )t +2E1 E2∗ e−i(ω1 −ω2 )t + 2E1∗ E2 ei(ω1 −ω2 )t ] Note that the indu ed se ond order polarization ontains the se ond-harmoni terms with 2ω1 and 2ω2 , the dire t urrent terms representing a time independent fa tor that produ es no os illating ele tromagneti radiation, the sumfrequen y generation (SFG) orresponding to the addition of the two fundamental frequen ies. ω1 − ω2 .. ω1 + ω2. and the dieren e frequen y generation(DFG). For e ient se ond harmoni generation, it is desirable that the. nonlinear opti al medium possess properties su h as high opti al power damage threshold, transparen y over a large range of frequen ies, large ee tive se ond order sus eptibility (see appendix A.2) and it is essential that the.

(46) Se tion 4.3. Phase mat hing. Page 35. medium is a non- entrosymmetri medium.. Symmetry onstraints applies. only to even order polarization of a material as they vanish in entrosymetri materials. A entrosymetri material is one whose stru ture remains un hanged upon inversion whi h implies repla ing ea h oordinate. r. with. -r. [20℄. For su h a material the ele tri eld experien es the same polarization when the dire tion is reversed that is for the ele tri eld. E,. the polarization. is. P2n = ǫ0. X. χ(2n) E2n. (4.25). n. -E. and for a reverse dire tion. −P2n = ǫ0. X. Sin e the ontributions from. χ(2n) (-E)2n = ǫ0. n. E. X. χ(2n) E2n. (4.26). n. and. −E are. the same, the total indu ed even. (2n) order nonlinear polarization must be zero (χ. = 0).. Another very impor-. tant ondition ne essary for se ond harmoni generation is phase mat hing.. 4.3 Phase mat hing Even when the other riteria ne essary for se ond harmoni generation have been satised, there is still a need for phase mat hing. Phase mat hing in simple terms involves getting the fundamental wave and the se ond harmoni waves to be in phase. This implies traveling at the same speeds in the medium if the beams are ollinear. This is espe ially riti al when the length of the medium ex eeds the oheren e length. fra tion of the length. L of. The oheren e length is a ertain. the medium over whi h there is signi ant se ond. harmoni generation and is usually very small, typi ally in the order of 10. µm. [1℄. The fundamental wave indu es a nonlinear linear polarization that. will normally travel slower than the fundamental wave through the length of the medium, due to normal dispersion. In this ase, the e ien y of the se ond harmoni wave redu es again through the se ond oheren e length in the medium be ause of onstru tive interferen e.. The se ond harmoni.

(47) Se tion 4.3. Phase mat hing. Page 36. wave thus os illates between a maximum intensity and zero intensity. Phase mat hing in rystals is typi ally a hieved by using anisotropi birefringent materials whi h oer ele tromagneti waves dierent refra tive indi es for dierent polarization dire tions in the material. Generally the e ien y of the se ond harmoni wave depends on (see appendix A.2, equation (A.36)). I2ω (L) Iω (0) 2 1 3/2 sin ∆kL ω 2 d2 µ0 2 2 Iω (0)L = 2 1 ǫ0 η 2 (ω)η(2ω) ∆kL 2. CSHG =. (4.27). (4.28). We an see that the e ien y of the se ond harmoni wave depends on the nonlinear sus eptibility tensor, the intensity of the fundamental beam and the square of the total length of the rystal if phase mat hing has been a hieved. To maximize this e ien y, phase mat hing is targetted at making. ∆k. whi h. orresponds to the dieren e in the refra tive indi es of the fundamental and se ond harmoni beam almost equal to zero. The phase mat hing fa tor for se ond harmoni wave is quantitatively given by the fun tion. F (Lc ) =. . sin. 2 1 ∆kL 2. 1 ∆kL 2. (4.29). This phase mat hing fa tor drops to about 40 per ent after the distan e. 1 π ∆kL = 2 2 This leads to the denition of the oheren e length as.

(48) π

(49)

(50) Lc =

(51) ∆k. From the expression for the onversion e ien y of the SHG in appendix A.2, (equation (A.36)) it an be seen that in reasing the length of the rystal without phase mat hing does not improve the e ien y of the se ond.

(52) PSfrag repla ements. Se tion 4.3. Phase mat hing. Page 37. 1. F(Lc ). Lc. F (Lc ). 0.8. 0.6. 0.4. 0.2. 0 -8. -6. -4. -2. 0. 2. 4. 6. 8. 1 ∆kL 2 Figure 4.3: Phase mat hing fa tor of the onversion e ien y for the se ond harmoni wave. harmoni generated. However, in reasing the oheren e length does in rease the e ien y [21℄.. The method of phase mat hing des ribed above holds. for the ollinear geometry where the individual input photons having different or the same frequen y propagates in the same dire tion.. However,. the individual photons an propagate in dierent dire tions and when su h a geometry is utilized for phase mat hing it is referred to as non- ollinear phase mat hing.. Both ollinear and non- ollinear phase mat hing utilizes. the prin iple of onservation of energy and onservation of momentum. The SPIDER experimental set up des ribed in hapter 7 of this proje ts implements a non ollinear phase mat hing geometry for sum frequen y generation. One advantage of the non ollinear geometry over the ollinear one, is that it allows easy separation between the individual input photons and their se ond harmoni s from the sum frequen y beam as a result of the sum of the two input photons. Figure (4.4) shows how the beams emerging from the doubling rystal using non- ollinear geometry are separated spatially [3℄. Two beams with angular frequen y. ω. and momenta. k1. and. k2. are in ident on the.

(53) Se tion 4.3. Phase mat hing. Page 38. PSfrag repla ements. Figure 4.4: Non- ollinear geometry for phase mat hing, energy and momentum is onserved. nonlinear rystal at an angle say frequen y. ω2. and momentum. k.. β. and produ e one photon with angular. Energy onservation gives. ω2 = 2ω ,. k = k1 + k2 , k = 2k1 , or k = 2k2 .. momentum onservation requires. and. For su h. a onguration in gure (4.4), ve beams exit the rystal, two of them are at the fundamental frequen y. ω. and in the dire tions. k1. and. k2. of the in ident. rays. Another two of the rays are the doubled beams of frequen y momentum ve tors. 2k1. and. k2 .. along the bise tor of the angle. k1 + k2 .. The last beam of frequen y. β. 2ω. 2ω. and. propagates. and the momentum from the geometry is.

(54) 5. Correlation te hniques for ultrashort pulse hara terization In se tion 1.1 of this proje t, we gave motivations of why ultrashort pulse hara terization is ne essary. Unfortunately photoele tri response times are very slow and therefore limited when dealing with femtose ond laser pulses. For laser optimization and other signi ant reasons in luding those outlined in se tion 1.1 one wants to at least measure the pulse duration of femtose ond laser pulses and indire t te hniques have to be used. Sin e these te hniques are indire t, it is important to understand not only how the te hnique works, but also the model implemented to retrieve the desired information from the experimental data. Typi ally orrelation te hniques have been useful for measuring pulse duration of ultrashort pulses. fun tions, the probe pulse. g(t). Given two time-dependent. and the test pulse. f (t),. where. g(t). is known.. The measurement of the ross orrelation fun tion [22℄. G (t) =. Z. −∞. ∞. gives the test fun tion. f (t).. g(t)f (t − τ )dt. (5.1). Higher order ross orrelation fun tions an also. be dened. The limitation of using the ross orrelation measurement is that the probe or known pulse has to be shorter in time than the test pulse. Even when a temporally shorter pulse is available, like all orrelation methods, the ross orrelation fun tion does not provide any information on the phase ontent of the test pulse [23℄. Self referen ing diagnosti te hniques, where the unknown pulse is orrelated with itself has been su essfully implemented for measuring pulse durations and hirp parameters of short opti al pulses. This is termed. auto orrelation. and pra ti ally involves delaying a opy of. the test pulse with itself and measuring the orrelation signal.. Two main. te hniques used to implement auto orrelation measurements are the intensity auto orrelation and the interferometri auto orrelation te hniques.. 39.

(55) Se tion 5.1. Intensity auto orrelation. Page 40. 5.1 Intensity auto orrelation The intensity auto orrelator uses a Mi helson interferometer arrangement (gure (5.1)). The input beam is split into two and one of them undergoes a variable delay. The beams are re ombined and dire ted into a nonlinear rystal for frequen y doubling. A spatial lter is used to reje t the un onverted fundamental beams as well as any se ond harmoni that is ollinear to the fundamental beams whi h is dete ted by a photodiode. A signal an only be dete ted if the two pulses overlap in time sin e the intensity of the light generated by se ond harmoni generation s ales as. ISHG ∝ I(t)I(t − τ ) and is maximum at. τ = 0.. Sin e the pro esses that o ur in time. (5.2). t are. mu h. faster than the response time of the dete tor what is re orded is a tually the intensity auto orrelation fun tion given by [22℄. 2. G (τ ) =. 1+. R −∞. I(t)I(t − τ )dt. −∞ R −∞ ∞. I(t)2 dt. (5.3). A simulated intensity auto orrelation fun tion assuming a Gaussian shaped pulse is shown in gure 5.2.. The width of the auto orrelation fun tion is. related to the input pulse width, and depending on the shape assumed, the pulse width an be extra ted from the auto orrelation signal (see Table 5.1). The intensity auto orrelation annot provide information about the exa t pulse shape due to the symmetri nature of the auto orrelation tra e. Also be ause the pulse is used to measure it self, the intensity auto orrelation annot provide information about the temporal or spe tral phase of the pulse and information about the evolution of the pulse prole is lost. Another draw ba k of the intensity auto orrelation is that there exist pulses with dierent hara teristi s but having the same auto orrelation signal [22℄..

(56) Se tion 5.2. Interferometri Auto orrelation. Page 41. Figure 5.1: S hemati of an intensity auto orrelator. I (t) −t2. e. se h. 2. (t). ∆t. I (ω). 1.665. −ω 2. 1.763. e se h. 2. ∆ω. 1.665 π ω 1.122 2. ∆ω∆t 2.772 1.978. G2 (τ ) e. τ2 2. 3[τ cosh(τ )−sinh(τ )] sinh3 (τ ). ∆τ. ∆τ /∆t. 2.355 2.720. 1.414 1.543. Table 5.1: Mathemati al results for dierent pulse shapes and their orrelation fun tion [3℄. 5.2 Interferometri Auto orrelation Interferometri auto orrelation is a variant of the intensity auto orrelator. While the intensity auto orrelator lters out any signal ontributed by a single pulse repli a, the interferometri auto orrelator uses a ollinear geometry as shown the gure 5.3 The input laser beam is split into two and allowed to travel dierent paths by mounting one of the mirrors (on whi h the beams will strike) to a loud speaker.. This moves the mirror ba k and forth through a distan e. responding to a delay time. τ.. x. or-. The two pulses are sent ollinearly into a. photodiode whi h a ts as a se ond order pro ess. A se ond harmoni rystal an also be used. A photodiode has the advantage that it is heaper, and does not have the problem of phase mat hing [24℄. The ele tri al signal generated is then dire ted to the os illos ope. We an al ulate the interferometri auto orrelation fun tion. An expression for the interferometri auto orrelation.

(57) Se tion 5.2. Interferometri Auto orrelation. Page 42. 3. Intensity AC. 2.5 2 1.5 1 0.5 0 -400. -200. 0 Delay time (fs). 200. 400. Figure 5.2: Simulated intensity auto orrelation fun tion for Gaussian shaped pulse. Figure 5.3: S hemati of an interferometri auto orrelator. fun tion will in lude the pulse width and hirp parameter with the assumption of a pulse shape. The total eld after the Mi helson interferometer is.

(58) Se tion 5.2. Interferometri Auto orrelation. the sum of the two identi al pulses delayed by. Page 43. τ. with respe t to ea h other. E(t, τ ) = E(t + τ ) + E(t) = A(t + τ )eiω(t+τ ) + A(t)eiω(t) (5.4). where A(t) is the omplex amplitude, and. ω.. the arrier frequen y. eiωt. des ribes the os illation with. What we desire to measure is the ele tri eld. in time, but sin e this pro ess is very fast to apture we rather measure the intensity as a fun tion of the delay. The dete tor integrates over the envelope of the light pulses. After undergoing a se ond order pro ess, the total signal produ ed by the interferometri auto orrelator an be written as [25℄. Z. I(τ ) ∝. −∞. Z∞−∞. ∝. Z∞−∞. +. Z∞−∞. +. ∞. | A(t + τ )eiω(t+τ ) + A(t)eiω(t) |A2 (t + τ )ei2ω(t+τ ). 2. |2 dt. 2A(t + τ )A(t)eiω(t+τ ) eiω(t) A2 (t)e2iω(t) |2 dt.. (5.5). Taking the absolute square we get. I(τ ) ∝. Z. −∞. ∞. [|A(t + τ )|4 + 4|A(t + τ )|2 |A(t)|2 + |A(t)|4. + 2A(t + τ )|A(t)|2 A∗ (t)eiωτ + c.c. + 2A(t)|A(t + τ )|2 A∗ (t + τ )e−iωτ + c.c. + A2 (t + τ ) (A∗ (t))2 e2iωτ + c.c.]dt (5.6). From (5.6), the interferometri auto orrelation fun tion onsist of the follow-.

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