AN ABSTRACT OF THE DISSERTATION OF. Naaman Amer for the degree of Doctor of Philosophy in Physics presented on September 20, 2006.

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2 AN ABSTRACT OF THE DISSERTATION OF Naaman Amer for the degree of Doctor of Philosophy in Physics presented on September 0, 006. Title: Generation and manipulation of waves Abstract approved: Yun-Shik Lee In this thesis I studied several components that can be used for potential technology. waves were generated in nonlinear medium via optical rectification of femtosecond optical pulses. Utilizing the phase matching condition between the optical and waves in a ZnTe crystal, single-cycle broad-band pulses were obtained. On the other hand, a tunable narrow-band waves were generated in PPLN structures based on a quasi-phased matching process. waves were detected via free-space electro-optic detection. Several methods of pulse shaping were demonstrated. First, the generated waves replicated the polled lithium niobate (PLN) domain structures. In the nd method, we controlled the pulse shape by adjusting the delay time between two coherent optical pulses impinging on a fanned out periodically poled lithium niobate (FO-PPLN) structure. The 3 rd method is an adaptive pulse shaping technique in which each frequency component of the wave generated in the FO-PPLN crystal was manipulated Controlling the ellipticity of the wave is highly desirable for many applications. Two linearly right-angled and delayed optical pulses incident on a

3 nonlinear medium generated two coherent pulses. The ellipticity of the resultant wave was controlled by adjusting the delay time between the two optical pulses. The other and more efficient technique is to use the wave-plate consisting of a wire-grid polarizer and a mirror. The output of the waveplate is two linearly and perpendicularly polarized waves with a relative phase shift. The ellipticity of the output pulse is controlled by adjusting the distance between the wire-grid and the mirror located behind it. Linear properties of 1D periodic dielectric structure were studied. Distributed-Brag-reflectors with 5, 10, and 15 periods in both transmission and reflection arrangements were investigated. Stop-bands at ( ) and ( ) were shown in the transmission arrangement. High-reflectivity-bands at ( ), ( ), and (.03-.9) in a 45 reflection geometry were observed. A resonant cavity structure was also investigated. A resonant transmission line with frequency in the center of the stop band was demonstrated.

4 Copyright by Naaman Amer September 0, 006 All Rights Reserved

5 Generation and manipulation of waves by Naaman Amer A DISSERTATION submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Presented September 0, 006 Commencement June 007

6 Doctor of Philosophy dissertation of Naaman Amer Presented on September 0, 006 APPROVED: Major Professor, representing Physics Chair of the Department of Physics Dean of the Graduate School I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dissertation to any reader upon request. Naaman Amer, Author

7 TABLE OF CONTENTS Page Introduction generation Wave equation in nonlinear medium Optical Rectification Phase matching versus quasi-phase matching processes detection The Pockels effect and the index ellipsoid Jones Matrix Calculus Response function of the gate crystal Calculation and Measurement of the azimuthal angle Dependence of the emission power in nonlinear crystals Calculation of the azimuthal angle dependence of E...39 Experimental Arrangement time domain spectroscopy (-TDS) setup...4. Laser systems Ti-sapphire oscillator (Coherent Mira 900) Ti-sapphire regenerative amplifier (Legend, Coherent Inc.) output Power pulse shaping Introduction...48

8 TABLE OF CONTENTS (Continued) Page 3. pulse shaping in pre-engineered domain structures generation in Poled Lithium Niobate (PLN) pulse shaping based on PLN domain structures pulse shaping with two pulses in fanned-out PPLN Continuous tuning of frequency pulse shaping using a shaped optical pulse and fanned-out PPLN crystal Adaptive pulse shaping in a fanned-out PPLN Experimental setup (figure 3.9) Experimental data Ongoing project Summary Generation of waves with arbitrary elliptical polarization Two-pulses method Generation of mixed polarization of single-cycle pulses Generation of elliptically polarized multi-cycle pulses Generation of circularly polarized waves Experimental results and analysis...8

9 TABLE OF CONTENTS (Continued) Page 4.3 Summary wave propagation in one-dimensional periodic dielectrics (1D-PD) Experiment Samples and Fabrication Results and Discussion Transmission Reflection resonant Cavity Summary Summary...94 BIBLIOGRAPHY.96

10 LIST OF FIGURES Figure Page 1.1 Optical rectification in a nonlinear crystal E ( z', ω) is the contribution of z position in the nonlinear crystal to the wave (a) Calculation of the generated electric field from a 1mm ZnTe crystal, using n opt = andτ = 10 fs. (b) The corresponding power spectrum calculated by fast Fourier transform (FFT). The detector s response function is not included here () 1.4 PPLN structure: arrows represent the orientation of χ (a) Calculated multi-cycle waveform generated in 0 domain PPLN structure (the thickness of each domain d = 50 μm, n = 5., n opt =. 3).(b) The corresponding narrow band power spectra (a) the (110) cut surface of the ZnTe crystal, (b) the principal axes Pockels effect in ZnTe crystal. (a) The principal axes and the intersection of the index ellipsoid with (110) plane in the absence of electric field. (b) The principal axes and the intersection of the index ellipsoid with the (110) plane in the presence of a field Balanced diode detector geometry. BS: pellicle beam splitter,qwp: quarter wave plate, WP: Wollaston prism Response function of a 1 mm ZnTe crystal Calculated (solid-line), and experimental (dotted-line) of a electric field generated and detected in a 1 mm ZnTe crystal Calculated (solid-line) and experimental (dotted-line) of power spectrum of pulse in figure For comparison, also shown the power spectrum of the pulse before the gate crystal Calculated electric field generated in a 0 layers PPLN structure crystal (the thickness of each domain d = 50 μm, n = 5., n o =. 3) and detected in a 1 mm ZnTe crystal

11 LIST OF FIGURES (Continued) Figure Page 1.13 Calculated power spectrum of the pulse in figure Measurement (line-symbol) and calculation (solid-line) of the azimuthal dependence of power Schematics of pump-probe -TDS setup. BS=Beam splitter, PC=off-axes parabolic mirror, WP=Wollaston prism Schematic diagram of generation of pulse in a PLN structure Diagram of alternating domain structure of 30 μm and 60 μm. (b) The generated waveform. (c) The power spectra (a) Diagram of domain structure. The π phase shift is introduced by introducing a 100 μm domain between sets of 0 domains with 50 μm domain length. (b) The generated waveform. (c) The power spectrum (a) Diagram of chirped domain structure. (a) 71 domains, ranging from 0 μm to 70 μm, with a 1 μm increment. (b) The generated waveform. (c) The power spectra Fanned-out PPLN structure Continuous narrow-band wave generation. Squares represent the experimental data. The solid line represents the calculated frequency Schematics of pulse shaping with to optical pulses in fanned-out PPLN pulse shaping with two phase locked multi-cycle pulses. (a) The generated π π 3π waveforms with phase ΔΦ = 0,,,, andπ. (b) The corresponding 4 4 power spectra Schematics of arbitrary -pulse shaping waveform generated in a fanned-out PPLN crystal without a spatial mask. (b) The corresponding power spectrum... 63

12 LIST OF FIGURES (Continued) Figure Page 3.11 (a) The measured waveform generated in a fanned-out PPLN crystal with a high-pass spatial filter. (b) The solid line is the corresponding power spectrum, while the dotted line corresponds to the no-mask pulse. The inset is the spatial pattern of the high-pass filter waveform generated in a fanned-out PPLN crystal with a low pass spatial filter. (b) The solid-line is the corresponding power spectrum, while the dotted- line corresponds to the no-mask pulse. The inset is the low-pass spatial pattern of the mask (a) waveform generated in a fanned-out PPLN crystal with a double slit spatial filter. (b) The solid-line is the corresponding power spectrum, while the dotted- line corresponds to the no-mask pulse. The inset is the double-slit spatial pattern of the mask (solid-line) Experimental spectrum for the uncovered case. (Dotted-line) The calculated spectrum Schematics of the generation of elliptical waves using two optical pulses technique. This technique was demonstrated using both ZnTe and PPLN crystals as the nonlinear source Diagram of polarization orientations of the incident optical pulses and the generated pulses (a) waveforms of two ( 1 st pulse (dotted-line), nd pulse (solid-line)) singlecycle linearly polarized pulses with delay times -0.4 ps, -0. ps, 0 ps,0. ps, 0.4 ps, and 0.6 ps the measured angle between the two electric fields components is 51±.(b) the corresponding polarization of the superposed field. E x points in the direction of the 1 st pulse polarization. E x an E have the same units y

13 LIST OF FIGURES (Continued) Figure Page 4.4 Diagram of polarization orientations of the incident optical pulses and the generated pulses. γ = 6 is the measured angle between the polarizations of the two pulses E 1 t 4.5 Dotted-line represents the waveform ( ), solid line represents the waveform E ( t τ ). The time between the two pulses is τ = 0.9 ps corresponding to a phase difference of 106. The frequency is ν = Schematics of the wave plate. A combination of a WGP and a mirror separated by a variable distance acts as a variable wave-plate (a) Two linearly and perpendicularly polarized waves with phase difference Φ = 8,58,85,153, and, 35 (dotted-line) reflection from the WGP, and (solid-line) reflection from the mirror. (b) The polarization trajectory in the x-y plane. Where E x, ( E y, ) represents the electric field reflected from the WGP (mirror) Schematic of the transmission (a) and reflection (b) setups. In both setups the emitter and gate crystals are 1 mm (110) cut ZnTe crystals Diagrams of the (a) DBR and (b) the resonant cavity structures. We studied DBR structures with 5, 10, and 15 periods. Each period consists of 75 μm PET and of 15 μm air. The cavity resonant consists of.5 periods of 75 μm DBR s separated by 50 μm layer of air (a) temporal evolution of incident and transmitted waveforms through 5, 10, and 15 periods of PET-air.(b) The corresponding power spectra (solid-line), and the corresponding transmitted calculated power spectra (dotted-line). The measured incident power spectrum was used the input for the calculations (a) temporal evolution of incident and reflected waveforms from 5, 10, and 15 periods of PET-air (b) The corresponding power spectra (solid-line), and the corresponding reflected calculated power spectra (dotted-line). The measured incident power spectra were used the input for the calculations. The incident angle is

14 LIST OF FIGURES (Continued) Figure Page 5.5 (a) temporal evolution of transmitted waveforms through 1D resonant cavity.(b) The corresponding power spectra (solid-line), and the corresponding calculated power spectra (dotted-line). The measured incident power spectra were used the input for the calculations... 9

15 Introduction The name terahertz waves describes electromagnetic waves with frequency of the order of 1 10 Hz. The region of the electromagnetic spectrum is loosely categorized as 0.1 to 10. The corresponding wavelength range is 3 mm to 30 μm. radiation is therefore located above the frequency range of traditional microwave electronics and below the range of infrared and light sources. Unlike the thoroughly studied IR and microwave regimes, the interaction between waves and materials has remained largely unstudied; mainly because there were no bright sources of waves. With the development of femtosecond lasers many schemes have been employed to generate subpicosecond pulses. Today table-top laser based sources, generating both single-cycle pulses 1 3 and continuous waves (CW) are available. The affordability of these sources has encouraged more research in this field. Many more generation and detection schemes are being developed with continuous improvement in radiation intensity, tunability, and ellipticity control. The following is a brief description of some of these methods:

16 The CO pumped gas laser is one type of CW sources. This type of laser is used mainly for astronomical studies 9. The main limitation of this source is that it only lasses at discrete frequencies. Photo-mixing of two near-ir laser beams in a semiconductor material is another popular method to generate tunable (0.1-3) CW radiation with powers on the order of tens of nanowatts Microwave generation followed by frequency multiplication allows CW tunable generation up to.7. 1 Quantum cascade semiconductor lasers are very promising bright CW sources as well. It contains hundreds of alternating layers of GaAs and AlGaAs. A voltage applied across the device forces the electrons to cascade through each layer and to emit photons with a wavelength determined by the thickness of the layer. The main challenge for this promising source is to raise the operating temperature as high as possible. 13 Sources based on table-top ultrafast lasers, combined with nonlinear optical crystals, are widely available. Recently we have demonstrated the generation of tunable ( ) waves in a fanned-out, periodically-poled lithium niobate (PPLN) structure via optical rectification. 14 Free electron lasers (FELs) are unique sources for generating powerful and narrow-band tunable coherent radiation. Energetic electron bunches (a few tenths of a millimeter long) are sent through a magnetic field, making them emit coherent synchrotron radiation. FELs occupy a huge space, making them impractical for many kinds of experiments and applications. For time domain spectroscopy (-TDS), a wave should be synchronized with a probe pulse. With the use of picosecond and femtosecond lasers it

17 3 became possible to generate broadband subpicosecond pulses via optical rectification in nonlinear media, e.g., ZnTe and LiNbO 3. It was in 1971 when Y.R. Shen s group published their work claiming the generation of pulses via optical rectification in LiNbO Optical rectification is a nd order nonlinear process in which an ultrashort optical pulse creates a time-dependent polarization that radiates electromagnetic waves. In this process the rapid oscillations of the optical pulse are rectified and the polarization resembles the optical pulse envelope. Photoconductive switches are another well-established technique to generate pulsed waves Unlike in optical rectification, which is a nonresonant process, an absorptive medium is used. A broken antenna is built on a GaAs substrate and a DC bias voltage placed across it. Femtosecond optical pulses excite electronhole pairs in the region of the break, making it conductive and completing the antenna. This allows current to flow through the antenna under the influence of the bias voltage. The created time-dependent current radiates pulses. Unlike the optical rectification technique, in a photoconductive switch the range of the generated spectrum is determined by the rising and falling of the current time and not by the optical pulse duration. In addition to the broad-band sources based on ultrafast lasers, there are the electron beam-based sources. Such sources can produce high power radiation with a wide spectrum, and very short pulses Accelerator based sources, table-top sources based on ultrafast lasers, and quantum cascade lasers can be used for studying nonlinear phenomenon in the regime due to the intense radiation generated by such sources.

18 4 radiation poses some unique properties that make it an attractive research tool. waves penetrate many visually opaque nonorganic materials, and do not cause ionizing damage to living cells and tissues. Different materials have different vibrational and rotational resonances in the regime, allowing spectrographic analysis. waves can be focused to produce sharp images. These properties lend themselves to applications in biomedical imaging 1 and medical diagnostics 3. Chemical and biological agents can be detected and identified through their resonance fingerprints. Because water absorbs waves 4, waves can be used to distinguish between materials with different amounts of water. The ability of waves to penetrate dielectric materials (paper, plastics, etc.) opens the way towards better ways of detecting weapons or explosives 5 hidden in boxes or with passengers. CW technology is used in astronomy for solar system studies 6 7, and remote sensing such as air pollution and ozone depletion 8. Electronic devices with speed are under study 9. In purely scientific fields, waves are uniquely suited to study systems in physics, chemistry, medicine, and biology. Many quantum systems in semiconductors and their nanostructures have resonances in the regime. Among the quantum systems with the characteristic energy (1~10 mev) are phonons, hydrogenic states of impurity-bound carriers, internal-exciton and exciton to continuum excitations, intra-band excitations, plasmons and block oscillations, and magnetic level splitting in magnetic fields. spectroscopy is a natural tool to study them. In addition to the resonances, many relaxation processes are in the picosecond and subpicosecond regime, and thus ideally probed by subpicosecond pulses.

19 5 Coherent quantum control in semiconductors is possible in the regime. Different coherent control phenomena such as Rabi oscillations and phonon echoes have been studied using waves 30. Here, pulse-shaping techniques are desirable to generate different types/shapes of waves, according to the experimental needs. Besides being of fundamental interest, studying semiconductors and their nanostructures is critical for applications such as electro-optic devices, sources and detectors 31, quantum information processing 3, and spintronics 33. spectroscopy is also suited to study metals 34, strongly correlated electron systems 35 and insulators. The ability of -TDS to measure both imaginary and real components of the dielectric function in real time has made it a desired method to study superconductors 36 at frequencies. The nondestructive nature of the measurement is another advantage encouraging researchers to use -TDS. spectroscopy in chemistry and biology is another fruitful area of study. Rotational, bending, and torsion dynamics of molecules can be probed by waves. In addition to the gas phase, the dynamics of molecules in condensed phases can be studied. Using subpicosecond waves one can study relaxation processes 39, and high fields can induce interionic motion or molecular orientational motion, changing the local structure. The dynamics of protein folding and unfolding is sensitive to the regime 40 ; and water properties in water/membrane environments have been examined 41. Medical imaging is a very promising field of study and application. radiation can be applied for medical imaging of skin, teeth and breast tissue 4. Unique biological

20 6 resonances are the basis for signature imaging employed to identify disease. Pharmacists can also benefit from spectroscopy. Drug polymorphs (different forms of the same compound with different pharmaceutical activities) can be differentiated and identified using spectroscopy 43. Despite the enormous potential applications of the technology it is still in its infancy. Commercial components are rare, so that scientists have to start almost from the scratch when carrying out their experiments. pulse shapers, waveguides, frequency specific mirrors, waveplates, and spectral filters are very basic and important components for technology. Components and instruments that control and manipulate the propagation of waves in free space or materials are necessary in any lab. This thesis discusses the work I have done while working under the direction of Dr. Yun-Shik Lee at Oregon State University. In chapter 1, I describe generation and detection principles. I discuss two optical rectification techniques to generate waves: The broad-band single-cycle pulses generated in a ZnTe crystal, and the narrow-band multi-cycle wave generated in a poled lithium niobate (PLN) structure. I then describe the principles of the free space electro-optic detection method employed in our lab. Chapter describes the (-TTD) pump-probe setup, and the laser system employed in our lab. In (-TTD) one can obtain the dielectric function (real and imaginary parts) of the studied sample in the regime. The main goal of chapter 3 is to introduce an adaptive pulse shaping technique. In this technique one has the ultimate control over the phase and amplitude

21 7 of each frequency component in the pulse. As an introduction to this end, I describe other techniques for pulse shaping and I compare them to our technique. This chapter was based on reference (44) In chapter 4, I introduce a novel technique to control ellipticity of waves, in which a combination of a wire-grid polarizer and a mirror act as a quarter-wave plate. In addition, other methods to control the ellipticity of waves, based on controlling the time delay between two incident optical pulses on nonlinear crystals, are also described. This chapter is based on reference (45). Chapter 5 is dedicated to the studies of the linear properties of 1D periodic dielectrics in regime using -TDS in both transmission and reflection setups. Such structures can be employed as spectral filters, frequency specific mirrors, and resonant cavities. This chapter was based on reference (46).

22 8 1 generation 1.1 Wave equation in nonlinear medium We use a simple model to calculate the generation and propagation of electro-magnetic (EM) waves in an electro-optic (EO) nonlinear medium. We assume a plane wave model in our calculations. This is justified because of the large Rayleigh length (~1 cm). I therefore consider a one-dimensional wave propagating parallel to the optical axis of the EO crystal. All macroscopic EM phenomena, including propagation of waves, are governed by Maxwell s equations (in cgs units): B = 0 (1.1.1) D = 4πρ (1.1.) 1 B E + = 0 (1.1.3) c t 1 D 4π H = J (1.1.4) c t c B is the magnetic induction, D the displacement field, ρ the free charge, E the electric field, H the magnetic field, and J the free current.

23 9 Assuming no free sources: ρ = 0 (1.1.5a) J = 0 (1.1.5b) Assuming a magnetic permeability of unity, B = H (1.1.6) The response of the medium (including the nonlinearity) is introduced through the relation: D = E + 4πP (1.1.7) In general, the polarization P includes the effects of nonlinear responses of the materials to the electromagnetic field. Taking the curl of (1.1.3) and using equations (1.1.5a), (1.1.5b), and (1.1.6), E 1 + B = 0 c t. Replacing B by 1 D c t and using equation (1.1.7) to eliminate D, one obtains a general expression for the wave equation in media: 1 E + c E t 4π P = c t (1.1.8) To simplify the equation above let us use the identity E = ( E) Assuming that the expression ( E) is negligible compared with E ; equation (1.1.8) takes a simpler form: E E t 1 4 E = c π P. (1.1.9) c t

24 10 Including the nonlinear polarization up to nd order where P P + P D (1) () = (1.1.10) D 4 P (1) () = + π (1.1.11) D (1) E 4 P (1) = + π. (1.1.1) Using equations (1.1.11) and (1.1.1), equation (1.1.9) takes the form D t (1) 1 4 E = c π P c t (). (1.1.13) Next, let us express E, D (1), and () P in the frequency domain: 1 i = ω ω t E( r, ) e dω π E( r, t) (1.1.14) D (1) ( r, t) 1 (1) i = ω ω t D ( r, ) e dω π (1.1.15) P () ( r, t) 1 () i = ω ω t P ( r, ) e dω π (1.1.16) Substituting (1.1.14), (1.1.15), and (1.1.16) into (1.1.13), one can show that the wave equation in frequency domain has the following form: ω (1) 4πω () E( r, ω) + D ( r, ω) = P ( r, ω) (1.1.17) c c For further simplifications we assume a one dimensional propagating wave. (1) Also we eliminate D using the relation (1) D ( z, ω) = ε( ω) E( z, ω) (1.1.18)

25 where ε (ω) is the linear dielectric function for the nonlinear medium. With these assumptions equation (1.1.17) takes the form 11 E( z, ω) ε ( ω) ω 4πω () + E( z, ω) = P ( z, ω) z c c (1.1.19) There are several nonlinear optical interactions that can be analyzed using (1.1.19) such as second harmonic generation, sum and difference frequency generation. Here I will discuss only the optical rectification process. 1. Optical Rectification Optical rectification is the generation of DC (or low frequency) polarization by the application of optical waves in a non-centrosymmetric medium with large nd order susceptibility. If the driving wave is an ultrashort pulse, the rapid oscillations of the optical pulse are rectified, leaving only the pulse envelope with a carrier bandwidth from DC to few terahertz (Figure 1.1). If the optical pulse duration 1 is τ = 100 fs, then the generated bandwidth is roughly f = 10. τ

26 1 Nonlinear Medium Ultra-short Optical pulse Induced polarization Figure 1.1 Optical rectification in a nonlinear crystal Another way to understand the optical rectification process in the frequency domain is to view it as a special case of difference-frequency generation in which only one optical pulse drives the process. In the time domain, a nonlinear polarization induced by an optical pulse in a medium has the mathematical form: P () () ( z, t) χ Eopt ( z, t = ) (1..1) () Where χ is the nd order nonlinear susceptibility. Assuming a Gaussian optical input pulse

27 13 z t v opt τ E opt ( z, t) = I 0e (1..) Where v opt is the optical group velocity and τ the optical pulse duration. () The Fourier transform of P ( z, t) has the form P () z t vopt ω τ () τ iωt () 4 ω) = χ e e dt = χ I 0 π e ω i z vopt ( z, e (1..3) Substituting the right side of (1..3) at z = z' in (1.1.19), one obtains the equation of electric field with a nonlinear source at z '. E 3, ω τ ω i z ( z, ω) ε ( ω) ω 4π ω () 4 vopt + E( z, ω) = χ I 0e e z c c (1..4) Nonlinear medium E ( z', ω) z ' z 0 L Figure 1. E ( z', ω) is the contribution of z position in the nonlinear crystal to the wave. The wave contribution at the exit surface z = L of the crystal, contributed from a position z inside the crystal (figure 1.), may be expressed as: E ' τ z ( ) ω iω( + 4 vopt ' ( L z ) ε ( ω) ) c ( z', ω ) = A ω e (1..5) 0

28 14 Where A 0 is a constant. I use 1..5 to calculate the far field waveform for two different types of nonlinear crystals, phase matched and quasi-phase matched Phase matching versus quasi-phase matching processes (a) (b) electric field (a.u.) power spectrum (a.u.) Time domain (ps) Frequency () Figure 1.3 (a) Calculation of the generated electric field from a 1mm ZnTe crystal, using n opt = andτ = 10 fs. (b) The corresponding power spectrum calculated by fast Fourier transform (FFT). The detector s response function is not included here.

29 A broad-band single-cycle wave can be generated in a ZnTe crystal. In this method we exploit phase-matching between optical and pulses. In a phase matching process the optical group velocity v opt should be equal (or at least close) to 15 the phase velocity v. ZnTe satisfies this phase-matching condition for optical pulses when the wavelength is around 80 nm. The waves generated in each element within the ZnTe crystal amplified coherently in the forward direction and one obtains a single-cycle pulse (Figure 1.3). Alternatively we can utilize a quasi-phase matching technique with PPLN structure to generate a multi-cycle narrow- band wave. Developed in , quasi-phase matching is a technique for generating a strong nonlinear response in non-phase matched material. In this method one uses a spatially modulated nonlinear polarization medium. The idea is to allow the optical pulse to propagate in a medium for a short distance and to reverse the polarization of the nonlinear medium where the generated waves start canceling each other. This distance is comparable with the walk-off length. Where, the walk-off length is the distance that the optical pulse has to propagate in order to lead the wave by one optical pulse duration and is given by d w = n cτ n opt, where n, n opt are the group refractive indices at and optical frequencies respectively. One engineers a PPLN structure with N alternating domains with thickness in the order of the walk-off length, so that a field consisting of N cycles is generated.

30 A periodic multi-cycle pulse has an intrinsically narrow spectrum, the generated frequency is 16 f = c ( n n d (1..6) opt ) Where d is the domain width, allowing one to tune the generated frequency by adjusting this domain width. The relative spectral bandwidth is given by Δf = f N. (a) Far field waveform from a ZnTe crystal In this case the generated spectrum is very broad, and because there is a strong transverse-optical (TO) phonon resonance at 5.3, the dielectric function is frequency dependent and given by ε stωto ε( ω) = ε el + (1..7) ω ω + iγω TO where ε = 6 represents the electronic contribution to the dielectric el function, ε = 3. 9 represents the strength of the TO phonon to the static dielectric st function, ω TO = π 5.3 is the TO phonon resonance angular frequency, and γ = 1 is the line-width of the resonance. In my calculation I used the values introduced in the reference (48) for all the parameters in the dielectric function. Figure 1.3 shows the calculated waveform generated by a 10 fs optical pulse, with optical refractive index n = 3. 19, in a 1 mm thick ZnTe crystal. The opt corresponding power spectra calculated by the fast Fourier transform is also shown. It s worth mentioning that the optical rectification with 10 fs optical pulses can

31 provide a sampling bandwidth of 10. The phonon resonance of the ZnTe crystal at 5.3, in addition to the not perfect phase matching conditions limits the obtained bandwidth to about 3. In section 1.4, I will discuss the further reduction of the bandwidth as a result of the propagation of the pulse in the ZnTe gate crystal. (b) Far field waveform in PPLN structure The inverse Fourier transform of equation (1..5) yields the time domain of the wave at the output surface contributed from the position z ' inside the crystal (figure 1.): 17 E ' ze vop t τ ' ze z', t) = ± D( 1) e (1..8) v τ ( opt with z ' e vopt = z' + ( L z' ) v v opt t and D is a constant. The ± sign is determined by the orientation of χ (). The total output is obtained by spatially integrating E ( z', t) over the length of the nonlinear crystal L and dividing by L. E L 1 ( z' = L, t) = E ( z', t) dz' (1..9) L 0 Solving for a PPLN crystal with N domains and domain length d (figure 1.4), (1..9) may be written in the form: N 1 i= 0 di + 1 i E ( z' = L, t) = ( 1) E ( z', t) dz' (1..10) di

32 18 where, d is the position of the interface between the domains i and i + 1. Each i domain contributes: d x i + 1 D voptτ E ( z', t) dz' = ( xe v d opt i 1 v ) xi + 1 xi (1..11) to the summation, where x i vopt = ( 1 ) d v i v + v opt L v opt t PPLN structure d 0 domains () Figure 1.4 PPLN structure: arrows represent the orientation of χ. (1..10) is substituted into (1..9) using (1..11) and the summation is performed. The field amplitude at the output surface of the nonlinear crystal contributed from N domains is given by: N 1 i= 0 E ( L, t) = ( H i+ 1 ( t) H i ( t)) (1..1) where

33 19 Dxi H i ( t) = e v opt 1 v xi optτ v (a) (b) electric field (a.u.) Power spectra (a.u.) Time delay (ps) Frequency () Figure 1.5 (a) Calculated multi-cycle waveform generated in 0 domain PPLN structure (the thickness of each domain d = 50 μm, n = 5., n =. 3). (b) The corresponding narrow band power spectra. opt Figure 1.5 shows the calculated multi-cycle waveform generated in a 1 mm (0 domains * 50 μm domain-width) PPLN structure, and the power spectrum is calculated by the FFT. Due to the TO phonon resonance of the lithium niobate

34 material, I introduced an absorption term exp( αω ) where α = 0.01ps and a temporal (or spatial) absorption term exp( βt ), where t is the propagation time 0 in the PPLN structure with 1 β =.0 ps. 1.3 detection The Pockels effect and the index ellipsoid The Pockels effect is an electro-optic effect in which the refractive index of a material changes linearly with the strength of an electric field applied across it. The general relation between the electric field E and the displacement field D in anisotropic medium is given by the following form: with i, j { x, y, z} Di = ε ij E j (1.3.1) j Assuming a lossless, optically inactive medium, one finds six independent elements representing a real symmetric permeability tensorε ij. This tensor can be expressed as a diagonal matrix by relating the principal axis coordinates

35 1 R = ( X, Y, Z) to the general coordinates r = ( x, y, z) through an orthogonal transformation. Equation (1.3.1) may be written as D i = ε E (1.3.) ii i axes. where i { x, y, z} We say that the medium is birefringent if ε ii has different values for different The energy density of the electric field in anisotropic medium is given by W 1 = E D 8π (1.3.3) Using (1.3.) the energy density represented by the principal axis coordinates becomes W 1 = 8π i D i ε ii (1.3.4) This shows that the surfaces of constant energy density are represented by ellipsoid in D space. I define a dimensionless vector R = ( X, Y, Z) in the D direction D R = (1.3.5) 8πW Using (1.3.5), one now can represent (1.3.4) in the simple form X ε xx Y + ε yy Z + ε zz = 1 (1.3.6) The surface defined by the equation (1.3.6) is known as the index ellipsoid.

36 The optical properties of anisotropic media can be described using the index ellipsoid. In the plane perpendicular to the propagation direction, the index ellipsoid traces an ellipse. The refractive indices associated with this plane are nothing but the semimajor and semiminor axes of this ellipse. In a general coordinate system, one can conveniently describe the index ellipsoid by the relation x n x y + n y z + n z xy + n xy xz + n xz yz + n yz = 1 The n coefficients describe the index ellipsoid in the r = ( x, y, z) coordinate system and can be expressed in terms ofε ii. The impermeability tensor η ij is defined as the inverse matrix of ε ij η = ( ε 1 ij ) ij In terms ofη ij, equation (1.3.6) may be written Riη ij R j = 1 (1.3.7) ij The Index Ellipsoid in a Static Electric Field Consider an impermeability tensor that is dependent on an applied electric field. Assume the change to the tensor vanishes as field vanishes. Under such conditions η ij can be expanded in a Taylor series, (0) 1 μ ij = ηij + r ijk Ek +... = ( ) ij + rijk E k (1.3.8) n k k

37 (0) Here, η ij is a diagonal matrix representing the impermeability with no applied electric field. r ijk is the tensor describing the linear optic effect. The Kerr quadratic electro-optic effect is neglected here. Substituting the right hand side of equation the (1.3.8) in the equation (1.3.7), 1 R i ( ) = 1 ij + rijk Ek R (1.3.9) n k j ij Equation (1.3.9) represents the modified index ellipsoid under the application of the electric field. Since the tensor ε ij is symmetric, η ij is also symmetric. r ijk must therefore be symmetric under interchange of the first two indices 3 i, j. To simplify the notation we express the different combinations of i, j by a single index h, so that rijk r hk according to the following notation r 11k r k r 33k r 3k r 13k r 1k r = r = r = r 1k r r k 3k 3k 31k 1k r r r 4k 5k 6k (1.3.10) For EO detection, we use a (110) oriented ZnTe crystal as the gate crystal (see figure 1.6). ZnTe has a zincblende structure; such a structure is optically symmetric when no field is applied. One can write n = 1 = n = n3 n o (1.3.11) ZnTe has only one independent electro-optic coefficient

38 4 r = = (1.3.1) 41 r5 r63 The other r hk components are all zero. Substituting (1.3.11) and (1.3.1) into (1.3.9), and using the notation in (1.3.10), one gets a simple form of the index ellipsoid after the influence of a static electric field with components E, E, E ) ( x y z x n o y z r 41 ( E yz + E zx + E xy) = 1 x y z n n o o (1.3.13) Where ( x, y, z) are defined as (figure 6a): x : Points along the [1,0,0] direction. y : Points along the [0,1,0] direction. z : Points along the [0,0,1] direction. (a) (b) [ 1 10] Z Figure 1. 6 (a) the (110) cut surface of the ZnTe crystal, (b) the principal axes.

39 5 Let us introduce a new coordinate system defined as (figure 1.6(b)) X points along the [-1,1,0]. Y points along the [0,0,1]. Z points along the [-1,-1,0]. In the absence of the electric field the material is isotropic, and one can choose ),, ( Z Y X as the principal axes. Let E make an angle α with the X axis and propagate perpendicular to the [110] plane (See figure 1.7). Under such conditions it can be shown that the new principal axes point in the following directions 49 : = ) sin( ) ( 3cos 1 ) cos( 1 1 ) ( 3cos 1 ) sin( α α α α α U = ) sin( ) ( 3cos 1 ) cos( 1 1 ) ( 3cos 1 ) sin( 1 1 α α α α α U = U Where 3 U is the direction of propagation of the and optical pulses. The three axes are orthogonal, so U 1 and U lie in the (1,-1,0)*(0,0,1) plane. U 1 makes an angle ψ with the X axis given by

40 6 cos(ψ ) = sin( α) 1+ 3cos ( α) 1 Assuming r 41E << it can also be shown that the main refractive indices n o are approximated by 3 nor41e n 1 = no + [sin( α ) cos ( α)] 4 3 nor41e n = no + [sin( α ) 1+ 3cos ( α)] 4 3 nor41e n3 = no sin( α) Figure 1.7 illustrates the principal axes and the intersection of the index ellipsoid with the (110) plane. Figure 1.7 Pockels effect in ZnTe crystal. (a) The principal axes and the intersection of the index ellipsoid with (110) plane in the absence of electric field. (b) The principal axes and the intersection of the index ellipsoid with the (110) plane in the presence of a field.

41 7 A probe beam propagates in the U 3 direction with electric field E p polarized in the (110) plane. As the probe beam propagates through a ZnTe crystal of thickness d, the two perpendicular components of E p along the U 1, and U axes receive a relative phase shift ωo ωd 3 Γ( α ) = ( n n1 ) d = no r41e 1 + 3cos ( α) c c where ω o is the angular frequency of the probe beam and, n 1 n are the refractive indices corresponding to the two orthogonal components of the probe beam along the principal axis of the index ellipsoid. One gets maximum phase shift for α = 0, ψ (α = 0) = 45, and maximum rotation of the probe polarization ( hence, maximum signal) is obtained when the angles between the probe and polarizations are 0 or Jones Matrix Calculus The propagation of a polarized electromagnetic beam through polarizers and birefringent plates is conveniently described using the Jones calculus. In this technique: 1) Rotating a dimensional system by Ψ is represented by: cos Ψ R ( Ψ) sin Ψ sin Ψ cos Ψ

42 ) The phase retardation caused by the ZnTe crystal subject to field is expressed as 8 in1ω od exp( ) ZnTe c 0 0 inωod exp( ) c 3) A quarter wave-plate is described by iπ exp( ) QWP iπ exp( ) 4 For simplicity let us assume a horizontally polarized probe beam incident on the ZnTe crystal. E p 1 0 First I should rotate E p into the index ellipsoid system (rotate by ψ). Then I apply the phase retardation on the two orthogonal components of the rotated electric field, then rotate back to the lab frame system (rotate by -ψ). In Jones matrix language, 1 R( ψ ) ZnTe R( ψ ) 0 This calculation is used to interpret the polarization of the probe beam after passing through the ZnTe gate crystal. (a) Crossed polarizer geometry E p

43 9 In this geometry a polarizer is placed after the ZnTe crystal. The polarizer orientation should be perpendicular to the input polarization of the probe beam, so that the transmitted probe beam vanishes in the absence of the waves. In this case the measured intensity in a detector placed after the crossed polarizer is given by: S cp ( α ) = E sin (ψ ( α))sin Γ( ) ( ) x b (b) Balanced diode detector geometry Here a more advanced geometry is employed to measure the change of the probe beam polarization caused by the field (see figure 8). In this case the difference signal at diodes 1 and is S bd = D1 D sin( Γ( α)) Where, D 1 and D are the amplitudes at diodes 1, and respectively. In general, the generated electric field ( E retardation Γ (α ) is very small ( Γ << 1). Hence, sin Γ ( α) >> sin Γ( α). Consequently, it is more sensitive to use the balanced detectors. ) is very weak so that the In the lab we are employing the balanced diode detector geometry to measure the electric field. Principles of operation of balanced diode detector geometry (figure 8): After passing through the ZnTe crystal, the probe beam passes through a quarter wave plate (QWP) rotated by 45 with respect to the horizontal axis (lab frame). Next,

44 30 a Wollaston prism (WP) separates the two orthogonally polarized components of the electric field of the probe pulse. A pair of mirrors guides them to the balanced detectors. After passing the EO crystal the probe beam is in one of the following polarization states: a - Horizontally polarized if E = b - Elliptically polarized if E 0 After passing the quarter wave plate the probe beam polarization can be described by the following states a - Circularly polarized if E = 0 b - Elliptically polarized if E 0 A circularly polarized light leaves the Wollaston prism in two equal components. The two diodes measure the same light intensity, so that the difference signal is zero. In the other case where E 0, the two orthogonal components of the elliptically polarized light emerging from the Wollaston prism have different amplitudes and will be recorded on the two diodes with different intensities. Hence a non-zero difference signal will be obtained.

45 31 E p Balanced diode detector ZnTe QWP WP E BS Figure 1.8 Balanced diode detector geometry. BS: pellicle beam splitter, QWP: quarter wave plate, WP: Wollaston prism. 1.4 Response function of the gate crystal Several papers have discussed the characterization of waveforms after propagation through electro-optic gate or emitter crystal In my calculations I consider the following factors that deform the shape of the detected electric field in the ZnTe gate crystal: a) dispersion - High frequency components move more slowly than low frequency components due to the TO phonon resonance at 5.3. This leads to an oscillating tail, especially when the damping constant is small.

46 3 Where absorption due to the TO phonon resonance is stronger, the oscillating tail becomes shorter. b) Absorption of high frequency components of the waves due to the TO resonance. Upon propagating a distance z in the ZnTe crystal (in the generating crystal z = L z', where z ' is the position where E (ω) was generated. In the gate crystal z = L, where L is the crystal thickness.) each frequency component s amplitude E (ω) receives a complex phase factor (real phase and absorption) given by exp( ik( ω) z) where w w k( w) = ε ( w) = ( n( w) + iβ ( w)) c c where n (w) and β (w) are the refractive index and absorption coefficients of the ZnTe crystal, respectively. c) Frequency dependent reflection at the boundaries between air and ZnTe. Because the refractive index and absorption coefficient increase monotonically with frequency as frequency approaches the TO resonance, the reflected portion of the wave also grows monotonically with frequency. One may express this as E E ref inc ( w) = ( w) β ( w) + ( n( w) 1) β ( w) + ( n( w) + 1) Inverse Fourier transform yields a time dependent electric field E ( z, t). Up to this point E ( z, t) includes reflection, absorption and dispersion effects in both emitting and detecting crystals.

47 d) The phase mismatch between the optical group velocity v opt and the phase velocityv. The electro-optic detection process can be easily and precisely analyzed as the generation of optical frequency components as a result of sum and difference frequency generation of optical and waves 51. The phase matching equation for sum-frequency generation is k w ) + k( w w ) = k( w ) (1.4.1) ( opt opt 33 where k is the wavevector corresponding to various frequency components. Here k = w ( w ), v and k( w opt dk w w ) k( wopt ) w = k( wopt ) dw w v opt opt Substituting in equation (1.4.1), one finds v = v. The phase matching opt condition is satisfied when the phase velocity equals the optical group velocity. To calculate the effect of the frequency dependent phase mismatch, I present the calculation from (51). I begin from the integral equation I L + EO ( τ ) r41 dz I E z t p (, ) 0 where, r 41 is the electro-optic coefficient, and dt I p ( z v exp ( v ( t τ )) opt d = optτ )

48 34 where, τ d is the delay time between the and optical pulses,τ is the probe pulse duration, and E ( z, t) is the electric field calculated after including the effects a, b, and c in both the emitter and gate crystals. Figure 1.9 shows the response function of a 1 mm thick ZnTe gate crystal. The E(0, ω) response function is defined as the ratio R ( ω) =, where E ( 0, ω) is the E( L, ω) amplitude of the frequency ω before entering the gate crystal, and E ( L, ω) is the amplitude at the output surface. This response function acts as a filter for frequencies higher than.3. Response functionr(ω)- (a.u) frequency() Figure 1.9 Response function of a 1 mm ZnTe crystal

49 35 Figure 1.10 shows the calculated (solid-line) and experimental data (dotted-line) of a electric field waveform generated in a 1 mm ZnTe crystal by a 10 fs optical pulse centered around 800 nm. It is detected using a 1 mm ZnTe crystal with probing pulse duration of 10 fs. The corresponding power spectra are shown in figure Figure 1.1 shows a calculation of a multi-cycle THZ electric field generated in a 0 layer PPLN structure. The domain width is d = 50 μm, n = 5., n =. 3. The wave was detected using the electro-optic method with the same parameters used to detect the pulse in figure Figure 1.13 shows the narrowband power spectrum corresponding to the waveform in the figure 1.1. o cal exp E (a.u) Time (ps) Figure 1.10 Calculated (solid-line), and experimental (dotted-line) of a electric field generated and detected in a 1 mm ZnTe crystal.

50 36 exp cal Power spectrum (a.u) frequency () Figure 1.11 Calculated (solid-line) and experimental (dotted-line) of power spectrum of pulse in figure For comparison, also shown the power spectrum of the pulse before the gate crystal. In the reference (55) the group velocity mismatch of the ZnTe was measured as 0.4 ps/mm, which corresponds to a cutoff frequency around.5 for a 1 mm thick ZnTe crystal, in good agreement with our observations. In figure 1.10 we see that the calculated and experimental data are almost identical in the main pulse. The discrepancy between them starts just after ps in the tail portion of the pulse. Analysis of the tail reveals that the calculated tail spectrum contains high frequency components (-3 ), in agreement with the calculations in (51). The experimental tail spectrum also contains a peak around 1.7, in addition to several weaker peaks. These peaks correspond to free induction decay (FID) of the absorption resonances of water vapor. These FID lines were not taken into account in my calculations.

51 37 It is worth mentioning that after 3 ps the calculated spectrum almost vanishes, while the experimental spectrum contains only the FID lines. Further analysis reveals that the main source of the oscillations in the tail (except the FID) is the wave s dispersion in the ZnTe crystal. The most important factor for determining the frequency components of the spectrum is the mis-match between the optical group velocity and the phase velocity in both the generation and the detection crystals. Absorption due to the TO phonon resonance and frequency dependent reflection are important near the TO resonance. For frequency components lower than.5, the strongest filtering effect in the response function is the phase mis-matching. In order to generate and detect higher frequency components, researchers have begun using electro-optic crystals with high TO phonon resonance frequencies (GaAs, InP, and GaP), thin gate crystals, and short pulse duration 54. Using thin crystals, however, reduces the detection sensitivity due to the reduction of the interaction length. To minimize vapor absorption in air, I compared my calculation with an experimental data in which all the setup was purged with dry nitrogen. Even with the dry nitrogen purge, the absorption line around 1.7 is still observed.

52 38 Electric field (a.u) tim e (ps) Figure 1.1 Calculated electric field generated in a 0 layers PPLN structure crystal (the thickness of each domain d = 50 μm, n = 5., n o =. 3) and detected in a 1 mm ZnTe crystal. Power Spectra (a.u.) Frequency () Figure 1.13 Calculated power spectrum of the pulse in figure 1.1.

53 Calculation and Measurement of the azimuthal angle Dependence of the emission power in nonlinear crystals Let θ be the angle between the [001] direction of the (110) cut crystal and the pump field, pump E. In the crystal frame one can express E pump as E pump = E ) ˆ sin( ) ˆ 0 ( sin( θ x θ y + cos( θ ) zˆ) (1.5.1) Where xˆ, yˆ, zˆ describe the [100], [010], and [001] axes, respectively (figure 1.6) Calculation of the azimuthal angle dependence of E The radiated electric field is proportional to the second order time derivative of the induced second order nonlinear polarization, () P : E P t () Hence E and () P have the same spatial dependence. The induced second order polarization () P by an optical field ( z, t) E opt maybe written as P () () ( z, t) χ Eopt ( z, t = ) (1.5.)

54 40 To simplify the notation I write ), ( () t z P as P, ), ( t z E opt as E, and () χ as χ. For a ZnTe crystal one can express (1.5.) in the following matrix form: = y x z x z y z y x z y x E E E E E E E E E P P P χ χ χ (1.5.3) Substituting (1.5.1) into (1.5.3) I obtain = ) ( sin ) sin( ) sin( θ χ θ χ θ χ E E E P P P E E E z y x z y x (1.5.4) The azimuthal dependence of the emitted power from a ZnTe crystal is shown in figure Calculation (solid-line) agrees well with measurement (line-symbol).

55 41 measurement calculation Power (a.u.) azimuthal angle θ Figure 1.14 Measurement (line-symbol) and calculation (solid-line) of the azimuthal dependence of power.

56 4 Experimental Arrangement.1 time domain spectroscopy (-TDS) setup -TDS bridges the gap between the microwave and infrared spectroscopy. It is a technique in which the shape of the pulse is measured in time, giving complete pulse information. The amplitude and phase of the electric field are determined with high precision. To measure the time evolution of the waveform, the generated pulse must be synchronized with a probe pulse. By scanning the delay time between the pulse and the probe pulse one obtains the waveform. We are using two-beam (pump-probe) interferometry in the time domain (Figure.1). In this technique the main optical pulse is separated into an intense pump pulse (95%), and a weak probe pulse (5%) using a beam splitter. The pump pulse induces a nd order polarization in the nonlinear crystal (ZnTe, or PPLN) via optical rectification. The generated waves are collimated and then focused in a (110) cut ZnTe crystal by a pair of off-axis parabolic mirrors. A μm pellicle BS directs the probe pulse to propagate collinearly with the beam through the ZnTe gate crystal for free space electro-optic detection (chapter 1.3). To resolve waveforms in the time domain

57 we introduce a time delay between the pump and probe the pulses. The pump pulse is chopped at 1 khz for lock-in detection. 43 ZnTe, PPLN Pump Optical chopper Femtosecond laser PC BS Polyethylene Filter Sample Polarizer Translational stage (Delay Line) Probe BS EO sampling λ/4-plate Balanced Detector PC Lock-In amplifier ZnTe WP Figure.1 Schematics of the pump-probe -TDS setup. BS=Beam splitter, PC= off-axes parabolic mirror, WP=Wollaston prism.

58 44. Laser systems..1 Ti-sapphire oscillator (Coherent Mira 900) The source for our optical pulses is a mode-locked Ti: Sapphire femtosecond laser (Coherent Mira 900). Specifications: 1) The laser is tunable between nm (we use mode locked pulses of nm, a dry nitrogen purge is needed to go further) ) Pulse duration is about 90 fs. One can use the short pulse option to shorten the pulse duration. 3) The repetition rate of the pulse is 76 MHz corresponding to a period of 13. ns. 4) The average output power is about 1.5 W, corresponding to optical pulse energy of 6 nj. 5) The Kerr lens mode-locking is applied to produce femtosecond pulses. This technique uses the fact that the index of refraction of the lasing medium depends on the intensity of pulse. With a Gaussian beam profile, it acts as a converging lens. After passing through this lens, the beam diameter of the CW wave will be larger than that of the mode-locked wave (as it is much more intense). A mechanical slit is placed at an appropriate location (after the lens). When adjusting the slit width, the

59 45 large diameter CW wave will be blocked by the slit, while the modelocked pulse passes uninterrupted. The pump laser: We are using a diode-pumped solid-state laser that utilizes a neodymium vanadate (Nd: YVO4) crystal. A laser diode system located in the power supply pumps the (Nd: YVO4) via fiber delivery. The pump laser provides a single frequency (53 nm) output. Specifications: Output power =10 W Wave length = 53 nm Polarization = vertical.. Ti-sapphire regenerative amplifier (Legend, Coherent Inc.) The Ti-sapphire amplifier contains the following main components: 1) Ti-sapphire Oscillator provides the seed pulses. ) Pump laser: A diode pumped, Q-switched, intra-cavity doubler with Nd: YLF as the gain medium and a 1 khz repetition rate. 3) Stretcher: In order to avoid severe damage during the pulse amplification the seed pulse of 100 fs is stretched to about 3 ns. 4) Amplifier: Amplifies the stretched pulses with Ti-sapphire as a gain medium of the cavity. After being amplified, the stretched pulses leave the cavity.

60 46 5) Compressor: After being amplified the pulses are stretched back to 100 fs duration. The repetition rate of the output pulse is 1 khz corresponding to a period of 1 ms. The average output power is about 1 W. This corresponds to optical pulse energy of 1mJ, and a peak power of 5 GW..3 output Power Optical power Optical pulse energy Peak optical power Generated power pulse energy Peak power Peak Thz Field Ti-sapphire oscillator 1 W 13 nj 65 kw 10 μ W.13 pj. W 50 V/cm Ti-sapphire regenerative amplifier 1 W 1 mj 5 GW 10 μ W 10 nj 10 kw 10 kv/cm Table: Typical optically pumped powers/energies and the corresponding calculated 5 powers/energies/electric fields generated with conversion efficiency of 10.

61 47 Optical-to- conversion efficiency is P P opt 10 5 where P is the generated power via optical rectification process as a result of an input optical power P opt. The table summarizes typical input optical powers/energies and the corresponding Powers/energies/electric fields. In the generation calculations, I assumed a mm beam, and pulse duration of 0.4 ps.

62 48 3 pulse shaping 3.1 Introduction technology fills the gap between the thoroughly studied infrared and microwave regimes. Several generation and detection schemes are being developed with a continual improvement in radiation intensity, tunability, polarizability, and detection control. In recent years, researchers in the community are concentrating more on applications of the technology. The ability to generate and manipulate pulses in a variety of forms and shapes is required for many applications of the technology. Having full control over waveforms means having the ability to control both the amplitude and phase of pulses. This gives us great flexibility to coherently control and manipulate the system under study, not just to observe it. For example, using temporally shaped pulses one can drive a quantum system into a specific state, which is very difficult to do with other kinds of pulses. Quantum coherent phenomena such as Rabi oscillations and photon echoes can be observed and studied by applying strong and shaped waves to molecules or semiconductor nanostructures. These waves can be used for coherent control of intra-band transitions of semiconductor nanostructures, phonon

63 49 modes in solids, vibrational and rotational excitations in molecules, and Rydberg wavepackets in atoms 3, among other applications. One method of generating shaped pulses is a technique based on optical pulse shaping. Stepanov et al. 57 have demonstrated the idea of pulse shaping by spatial optical pulse shaping. pulse shaping is obtained by manipulating the intensity distribution created by two femtosecond pump pulses on a LiNbO 3 crystal. Sohn et al. demonstrated temporal optical pulse shaping 58. The pulse shaping is obtained by controlling the spectral amplitudes of femtosecond pulses incident on semi-insulator GaAs emitter. The main advantage of these pulse-shaping methods, based on optical pulse shaping, is their adaptability. One can alter the pulse as needed while taking data. On the other hand, the quality of this pulse shaping is limited because the available time window is 10-0 ps at best. The corresponding spectral resolution is about 100 GHz. In this chapter I will discuss different methods employed in our lab to improve the quality of pulse shaping. These are pulse shaping in poled LiNbO 3 structures (section 3.), and pulse shaping with two optical pulses (section 3.3). I also describe our adaptive pulse shaping technique (section 3.4).

64 50 3. pulse shaping in pre-engineered domain structures generation in Poled Lithium Niobate (PLN) In LiNbO 3 an optical pulse moves faster than the generated pulse ( n = 5. 3, n =. 9 ). Due to the group velocity mismatch between the optical and op pulses, in a bulk LiNbO 3 the only contribution to radiation by a femtosecond optical pulse is from the front and back surfaces of the sample. Here, we exploit a quasi-phase matching technique to generate multicycle waves in PLN structures. As an optical pulse propagates through PLN, it generates a wave via optical rectification. Because of the group velocity mismatch between the optical and pulses, the generated wave falls behind the optical pulse as they travel through the crystal. The polarization (sign of the nd order nonlinear susceptibility χ flipped in the next domain when the waves are about to cancel each other completely. By doing so, the waves start to build up again and to construct a new half cycle. By repeating this process N times (fabricating N domain structures () where χ reverses sign between neighboring domains (figure 3.1)), one can generate N / cycles of waves. In addition, the length of a cycle is determined by the length of the corresponding domain, so the generated waveform replicates the domain structure of the PLN. () ) is

65 51 () χ () χ () χ () χ () χ v opt v pulse Femtosecond optical pulse Figure 3.1 Schematic diagram of generation of pulse in a PLN structure 3.. pulse shaping based on PLN domain structures One way to generate arbitrary waveforms is to design and fabricate PLN structures corresponding to the desired waveforms. We have demonstrated this idea using three different types of PLN structures (a) Alternating waveform The domain structure contains multiple structure of alternating domain lengths (30 μm, and 60 μm) (figure 3.(a)). The pulse contains alternating narrow and broad half cycles (figure 3.(b)). In the power spectrum (figure 3.(c)) we see a sharp spectrum corresponding to the period of 90 μm. The generated pulse decays in time, meaning the pulse is suppressed as it passes through the medium.

66 5 (b ) E (a.u.) Tim e delay (ps) (c ) E (ω) Frequency () Figure 3. Diagram of alternating domain structure of 30 μm and 60 μm. (b) The generated waveform. (c) The power spectra.

67 53 (b) E (t) (a.u.) Time Delay (ps) (c) E(ω) (a.u.) Frequency () Figure 3.3 (a) Diagram of domain structure. The π phase shift is introduced by introducing a 100 μm domain between sets of 0 domains with 50 μm domain length. (b) The generated waveform. (c) The power spectrum. (b) Zero-area double pulse This contains two coherent pulses with a π phase shift between them (figure 3.3(b)). Such a pulse is generated from a domain structure in which a single domain of 100 μm is placed between two sets of multiple domains of 50 μm

68 54 (figure 3.3(a)). In the power spectrum (figure 3.3(c)) we see the interference fringes of the two coherent pulses. Because the pulse decays in time, the second pulse that propagates more in the medium gets suppressed more. Hence the interference pattern is asymmetric. (c) Chirped Thz waveform (figure 3.4(b)) The domain structure includes multiple domain lengths ranging from 0 μm to 90 μm (figure 3.4(a)). In the power spectra (figure 3.4(c)) we see the broadband spectra with a lot of water absorption lines. Low frequency components propagate a farther distance and get suppressed more than high frequency components. At frequency f = 7.6 there is a strong TO phonon resonance in LiNbO 3 and this is the main source for the absorption. As a result, high frequency components have higher decay constants than low frequency components. This technique covers a broad time window (several hundred picoseconds) and broad spectral window (0.1~5 ). Simultaneously, the spectral resolution is less than 10 GHz and the temporal resolution is less than 10 fs. The main limitation of this method is that the pulse shape is fixed; the pulse shape is predetermined by the domain structure. In other words, for each desirable waveform you need to fabricate a new structure.

69 55 (b ) E (t) (a.u.) Tim e Delay (ps) (c) E(ω) (a.u.) Frequency () Figure 3.4 (a) Diagram of chirped domain structure. (a) 71 domains, ranging from 0 μm to 70 μm, with a 1 μm increment. (b) The generated waveform. (c) The power spectra.

70 pulse shaping with two pulses in fanned-out PPLN I begin by introducing our source for generating continuously tunable multi-cycle narrowband waves Continuous tuning of frequency We can generate narrow band waves using PPLN structure. As discussed above, each domain contributes a half-cycle to the total wave. This generates a sine wave in the limit N. The frequency of the generated wave is inversely proportional to the domain length (equation (1..6)), so one can tune the frequency by adjusting the domain width. A continuously tunable source of narrowband waves is accomplished by fabricating the fanned-out PPLN structure (figure 3.5). We used a 5 mm long, 10 mm wide, and 0.5 mm high crystal. By scanning the structure laterally with femtosecond optical pulses, we demonstrated the generation of continuously tunable frequency ( ) of multi-cycle waves (figure 3.6) 64.

71 57 Figure 3.5 Fanned-out PPLN structure experiment theory 1. Frequency () thickness (μm) Figure 3.6 Continuous narrow-band wave generation. Squares represent the experimental data. The solid line represents the calculated frequency.

72 pulse shaping using a shaped optical pulse and fanned-out PPLN crystal To combine this pulse shaping technique (based on pre-engineered structures) with optical pulse shaping, a pair of temporally separated optical pulses generates two phase-locked pulses in PPLN (figure 3.7) 64.. The shape of the generated waveform can be controlled by adjusting the intensity of the second pulse, the delay between them and the generated frequency by lateral translation of the Fanned-out crystal (figure 3.7). Figure 3.7 Schematics of pulse shaping with to optical pulses in fanned-out PPLN. In figure 3.8 we show experimental data when the relative phase between the two pulses is varied between 0 and π. Figure 3.8(a) shows the generated waveforms and figure 3.8(b) shows the corresponding power spectra. Here, the interference between the two pulses took place after the 9 th period (the delay time between the two pulses varies between periods ( Δφ = 0 π )).

73 This corresponds to a central frequency of At Δφ = 0 we have a constructive interference and it becomes more and more destructive as the relative phase approachesπ. 59 Field (a.u.) (a) Δφ= Δφ=π/ Δφ=π/ Δφ=3π/4 Power spectra (a.u.) (b) Δφ=0 π/4 π/ 3π/4 π Δφ=π Time delay (ps) Frequency () Figure 3.8 pulse shaping with two phase locked multi-cycle pulses. (a) The π π 3π generated waveforms with phase ΔΦ = 0,,,, andπ. (b) The corresponding 4 4 power spectra. This method covers a broad time window and the pulse shaping is adaptable. The limitation of this technique is that the spectral window is narrow because the PPLN generates a narrow spectrum.

74 Adaptive pulse shaping in a fanned-out PPLN Here we demonstrate novel technique to eliminate the weak points of the previously mentioned methods. It is an adaptive pulse shaping technique with broad temporal and spectral windows, and a spectral resolution of less than 10 GHz. In this technique we have the ultimate control over the amplitude and phase of each frequency components in a pulse. The major components of the pulse shaping setup (figure 3.9) are a cylindrical lens, a mask (or MEMS mirrors), a FO-PPLN crystal and a spherical mirror. The first step to generate spatially separated frequency components. The optical pulses are linefocused on the FO-PPLN crystal using a cylindrical lens. A mask is placed in front the FO-PPLN crystal to manipulate the intensity distribution of the optical pulse across the surface. This modulates the amplitudes of the various narrow-frequency components emerging from the FO-PPLN crystal. A spherical mirror assembles the various frequencies at a point on a ZnTe crystal for EO detection.

75 61 Cylindrical lens Mask Fanned-out PPLN Spherical mirror φ Optical beam EO sampling Figure 3.9 Schematics of arbitrary -pulse shaping. The spectral resolution of this method is determined by the fanned-out PPLN bandwidth and the mask resolution: The estimated practical resolution is ~ 0.01 for a 1 bandwidth pulse Experimental setup (figure 3.9) The fanned-out PPLN crystal (figure 3.5) was pumped by 100 fs optical pulses centered at 800 nm with 0.1 mj pulse. The optical pulses were generated from a 1 khz

76 6 Ti: Sapphire regenerative amplifier (Legend, Coherent Inc.). A cylindrical lens of 10 cm focal length transfers the optical beam profile into an elliptical profile (3 mm 0.3 mm). We used a 5 mm long, 10 mm wide, and 0.5 mm high fanned-out PPLN crystal as a source for generating continuously tunable waves ranging between ( ), as was demonstrated in chapter 3.3. The spectral bandwidth is determined by the major axis optical beam size and the center frequency of generated pulse. Here the central frequency is 0.7 and the optical beam size is ~3 mm, corresponding to ~ 0. bandwidth. In this stage of the experiment we manipulate and control the generated waveform by manipulating the intensity distribution of the optical beam by introducing spatial metal masks in front of the fanned-out crystal and therefore control the intensity profile of the optical beam in the crystal. The line image of the beam was transferred to a point image by a spherical mirror (R =15 cm), here we exploited the discrepancy between the vertical focal length ( R f v = ) and horizontal focal length ( cosθ f h = Rcosθ ) of the spherical mirror, where θ = 45. The shape of the waveform is detected using the free space electro-optic detection described in chapter Experimental data We demonstrated the pulse shaping technique using three different spatial mask patterns: high-pass filter, low-pass filter, and double slit. The corresponding spatial

77 63 patterns of the masks are shown in the insets of Figures 11(b), 1(b), and 13(b) respectively, where blocked regions are in black. Each waveform was measured with an 80 ps time window corresponding to a spectral resolution of 1.5 GHz. (a ) experim ent (b ) Uncovered field (a.u.) Power spectra (a.u) Tim e delay (ps) Frequency () Figure 3.10 waveform generated in a fanned-out PPLN crystal without a spatial mask. (b) The corresponding power spectrum. Figure 3.10 shows the time domain behavior of a multi-cycle wave without a mask and the corresponding power spectrum. The spectrum ranges from 0.55 to 0.85, with a central frequency of 0.7. The waveform is positively chirped. At early times (time delay (t) < 0 ps) the pulse contains low frequency components centered around 0.6. For later times the main peak is composed mainly from

78 64 components centered around 0.7. The appearance of frequency components centered around 0.6 for t > 60 ps imply a multiple reflection. Figure 3.11 shows (a) the waveform and (b) the corresponding power spectrum as we use a high pass filter. Using the mask, we blocked all frequency components lower than The signal is still strong because the main peak of the pulse around 0.7 was not filtered out. Careful analysis of the spectrum (figure 10(b)) also shows positive chirping behavior. The earlier part of the pulse (t < 7 ps) carries the low frequency components of the spectrum centered around The main part of the pulse (7 ps < t < 60 ps) carries mainly frequency components centered on 0.7. Frequency components centered on 0.66 are again dominant for (60 ps < t < 70 ps) as a result of multiple reflections. (a) (b) experiment calculation High pass uncovered field (a.u.) Power spectra (a.u) Time delay (ps) Frequency () Figure 3.11 (a) The measured waveform generated in a fanned-out PPLN crystal with a high-pass spatial filter. (b) The solid line is the corresponding power spectrum, while the dotted line corresponds to the no-mask pulse. The inset is the spatial pattern of the high-pass filter.

79 65 (a ) (b ) experim ent c a lc u la tio n lo w p a s s uncovered field (a.u.) Power spectra (a.u) Tim e delay (ps) Frequency () Figure 3.1 waveform generated in a fanned-out PPLN crystal with a low pass spatial filter. (b) The solid-line is the corresponding power spectrum, while the dottedline corresponds to the no-mask pulse. The inset is the low-pass spatial pattern of the mask. (a) experiment ca lcu la tio n (b) Double slit uncovered field (a.u.) Power spectra (a.u) Time delay (ps) Frequency () Figure 3.13(a) waveform generated in a fanned-out PPLN crystal with a double slit spatial filter. (b) The solid-line is the corresponding power spectrum, while the dotted- line corresponds to the no-mask pulse. The inset is the double-slit spatial pattern of the mask.

80 66 Figure 3.1(a) shows the generated waveform as we are blocking frequency components higher than 0.65 (low pass filter); the waveform is a long multicycle -pulse. Consequently, the corresponding spectrum centered at 0.6 is narrow (Figure 3.1(b)): the bandwidth is ~0.0. In figure 3.13(a) the double slit mask blocks the middle part of the spectrum, especially the main peak of the spectrum (figure 3.13(b)) around 0.7. The interference between high and low frequency components is evident by the beating waveform pattern with about 10 ps period. It is worth noting here that dispersion behavior is not manifested in the low-pass filter and double-slit pulses. I think that the dispersion of the pulse is an alignment error. This problem will be addressed by introducing a MEMS mirrors into the system. The dotted-line in figure 3.11(a), 3.1(a), and 3.13(a) show the calculated waveform for high-pass filter, low pass filter and double slit correspondingly. Here we consider the uncovered spectrum as the reference spectrum, apply the appropriate spectrum filter (frequency (f) f < 0.65 for high pass, f > 0.65 for low pass, and 0.67 < f < 0.73 for the double slit), then recover the time domain waveform by applying the inverse FFT. In figure 3.11(a) there is almost perfect agreement between experiment and calculation. In figure 3.1(a) and 3.13(a) the agreement is not quite as good. The reason for this discrepancy, maybe the positive chirp seen in both the uncovered and high-pass filter waveforms. The uncovered and high-pass filter spectra (for f > 0.67 ) have the same phase because both waveforms have similar chirping as the unmasked wave. On the other hand the lowpass filter and double slit waveforms are not chirped, so even though the calculated

81 spectrum and the experimental spectrum are the same the phase is different than in the reference spectrum. 67 calculation experiment Power spectra Frequency () Figure 3.14 (solid-line) Experimental spectrum for the uncovered case. (Dotted-line) The calculated spectrum. Figure 3.14 shows a comparison between the experimental spectrum in the uncovered case and the calculated spectrum based on a Gaussian input optical beam. In the calculation I assumed a beam size (length) of 5 mm, central frequency 0.67 (corresponding to a domain width of 77 μm, and a spectrum band Δ f = 0.3 ).

82 68 Notice that the power spectrum is not symmetric (higher spectrum density for higher frequencies) and this is because the generated frequency is inversely proportional to the domain width (equation (1..6)) Ongoing project We are making improvements on the setup (figure 3.9): (a) Replacing the mask by a MEMS mirror A mask controls the components amplitudes, while a MEMS mirror controls their phases (besides the amplitudes). This should also address the chirping behavior discussed above. Phase control is preferable to amplitude control because it doesn t reduce the pulse power. (b) Introducing a grating A spherical mirror focuses a line image into a point image (on the ZnTe detection crystal), with different k -vector directions for the different frequency components. In other words, spatial dispersion is compensated by the spherical mirror, but angular dispersion is still present. This is manifested in the fact that the detected spectrum is narrower than the estimated one based on the optical beam size. Introducing a grating into the setup should collimate the different components.

83 Summary Briefly I described the generation of arbitrary waveforms with pre-engineered structures of PLN. Then I demonstrated pulse shaping based on PLN structures with optical pulse shaping. In the main part of the chapter I demonstrated a novel technique for an arbitrary control of -pulse shape. Preliminary results were demonstrated. Three types of spatial filters modulated the spectral intensity: high pass filter, low pass filter, and double slit. The next two steps are to introduce a MEMS mirror and a grating in the pulse shaper. The goal is to have the ultimate control over the different frequency components of the pulse. Introducing a grating solves the problem of angular dispersion on the detection crystal.

84 70 4 Generation of waves with arbitrary elliptical polarization Most coherent sources generate linearly polarized or mixed-polarization waves. However, some applications require circularly polarized waves. As an example, ionization of Na Rydberg atoms by circularly polarized quarter-cycle pulses has been reported 65. Some excitation states of excitons in semiconductors and their nanostructures, and some rotational states in molecules are accessible only with circularly polarized pulses. Having control over the ellipticity of the waves allows better coherent control of excitation states. In spite of this, there are few methods to generate circularly polarized radiation, making it a relatively unexplored area. In this chapter I describe two different techniques to control the ellipticity of waves. In the first, two linearly polarized and time delayed optical pulses with perpendicular polarization generate pulses via optical rectification in a nonlinear medium. The polarization of the resultant pulse controlled by adjusting the relative time delay between the two pumped optical pulses. In another, more efficient technique, the generated wave is incident on a waveplate consisting of a wire grid polarizer (WGP) and a mirror. The output of the waveplate is two linear and perpendicularly polarized waves with a relative phase shift. The ellipticity of the resultant pulse controlled by adjusting the distance between the WGP and the mirror located behind it.

85 71 sources We performed this experiment with two different sources. Single-cycle pulses was generated in a (110) cut ZnTe crystal and multi-cycle waves were generated in a PPLN structure. In both cases 100 fs pulses centered at 80 nm, with 76 MHz repetition rate from a Ti: sapphire oscillator (coherent Mira 900 F) were focused to a spot of 0 μm diameter in the emitter crystals. The radiation was generated () via optical rectification of the femtosecond optical pulses in the χ medium. detection As in the previous experiments, we measured the actual electric field using the free space electro-optic sampling described in chapter 1.3. Because our experimental setup is not capable of determining the instantaneous polarization of the beam, we determined the polarization by measuring the two linearly polarized electric fields separately using a WGP. 4.1 Two-pulses method In this method, two separated femtosecond optical pulses are incident on a nonlinear crystal (ZnTe or PPLN) to generate two linearly polarized pulses (figure 4.1). The polarization of the combined wave controlled by adjusting the relative time delay, intensity, and the polarization angle between the two incident optical pulses. Because the ultimate goal is to achieve a circularly polarized

86 7 wave, the intensity of the two pulses should be equal. This is achieved when the two incident optical pulses make the same angle relative to the optical axis of the nonlinear emitter crystal. E 1 op ZnTe Or PPLN E 1 E E op τ Pellicle BS Laser detection beam ZnTe Figure 4.1 Schematics of the generation of elliptical waves using two optical pulses technique. This technique was demonstrated using both ZnTe and PPLN crystals as the nonlinear source Generation of mixed polarization of single-cycle pulses A ZnTe crystal generates single-cycle pulses. The ellipticity the resultant pulse of two delayed pulses is not well defined, and one can talk only about generating and controlling mixed polarization of broadband waves.

87 73 (a) Simulations In chapter 1.5 I calculated the azimuthal angle dependence of the generated electric field (equation (1.5.4)). There I assumed θ to be the angle between the [001] direction (figure 1.6) and the electric field of the incident optical pulse. There are two optical pulses with different polarizations. If we assign θ 1 ( θ ) to be the polarization angle of the 1 st ( nd ) pulse, and if θ1 = θ, symmetric with respect to [001] direction, then the angle between the generated fields α increases monotonically with θ 1 so that α max = α θ = 45 ) = 53. I substitute θ = θ 1 = 45 ( θ = θ = 45 ) in equation ( 1 (1.5.1) for the 1 st ( nd ) optical pulse and repeat the same process in equation (1.5.4) for the generated fields. The normalized electric fields are summarized in figure 4.. (Note that I rotated the z-component of the (b) Experimental results and analysis E by 180 ). [001] opt E opt E = ( 1,1, ) E E = ( 5,,1) E E1 = (, 5,1) opt E opt E1 = (1, 1, ) θ = 53 (110) plane Figure 4. Diagram of polarization orientations of the incident optical pulses and the generated pulses.

88 74 (a) 1st pulse nd pulse τ=-0.4 ps (b) τ=-0. ps electric field (a.u.) τ=0 ps τ=0. ps τ=0.4 ps E y (a.u.) 0 τ=0.6 ps Time delay (ps).5 3 E x (a.u.) Figure 4. 3 (a) waveforms of two ( 1 st pulse (dotted-line), nd pulse (solid-line)) single-cycle linearly polarized pulses with delay times -0.4 ps, -0. ps, 0 ps, 0. ps, 0.4 ps, and 0.6 ps the measured angle between the two electric fields components is 51±.(b) the corresponding polarization of the superposed field. E x points in the direction of the 1 st pulse polarization. E x and E have the same units. y

89 75 Figure 4.3(a) shows the measured waveforms of two single-cycle pulses. The time delay (τ) between the 1 st pulse (dotted-line), and the nd pulse (solid-line), ranges between -0.4ps and 0.6ps with a step size of 0. ps. The corresponding polarizations (projection of the superposed electric field in the (110) plane are shown in Figure 4.3(b). We measured the angle between the generated electric fields by measuring the polarization of each pulse separately and found it to be 51 ±.This value is consistent with the calculated value ( 53 ). Figure 4.3(b) shows that whenτ = 0, the polarization of the superposed pulse is linear. For other delay times the polarization is mixed and takes complicated forms. Still, when τ = ±0. ps the polarization trajectories are close to an ellipse. The spectrum of the Pulse is centered around 1., so a delay time τ = ±0. ps corresponds to a relative phase of π ± at this center frequency. We also noticed that the polarization patterns for the same delay times but magnitudes with opposite signs (e.g. τ = ±0. ps) are very similar, and the projection trajectories are mirror images Generation of elliptically polarized multi-cycle pulses To generate waves with elliptical polarization, we use the same scheme (figure 4.1) of two optical pulses incident on a PPLN structure instead of ZnTe crystal.

90 The PPLN crystal generated multi-cycle narrow- band pulses with linear polarization. (a) Calculation Let θ = ±45 be the angle between the [001] direction of the z-cut PPLN crystal and the polarization of the two incident optical pulses ( E 1, direction is in [100] direction. In the crystal frame one can express pump 76 ). The propagation E 1, pump as E pump 1 = Eo ( t) (0,1,1) E E ( t τ ) (0, pump o = 1,1) Where, τ is the delay time between the two optical pump pulses. The crystal class () of LiNbO 3 is 3m with three independent elements in its χ tensor. Using equation (1.5.) for the case of a PPLN crystal (note χ ij = dij ) one can express the generated electric field as where x E E E x y z E, E, and y E x E y P x d 31 d Ez Py = d d 0 d E y Ez P z d31 d31 d E x Ez E x E y E z on the right hand side of the equation are the x, y, and z components of the incident pump pulses can be expressed as pump E 1 and E pump. The generated pulses

91 77 1 E x ( t) 0 1 E1 ( t) = E y ( t) = E ( t) d + d 1 E z ( t) d 31 + d E x ( t) 0 E ( t τ ) = E y ( t) = E ( t τ ) d d Ez ( t) d 31 + d The angle between the generated fields E ( ) and E ( t ) 1 t τ is given by: γ = tan 1 d ( d 31 + d + d ) tan 1 d ( d 31 d + d The values of the nonlinear susceptibility components in the regime are not available. In the optical regime these values are given by d 33 = 98, d 31 = 14, and d = 7.4 ( 9 10 esu) 66, then γ = 8. (b) Experimental results and analysis Figure 4.4 summarizes polarization orientations of the incident optical pulses and the generated pulses in a z-cut PPLN crystal. The measured angle between the polarizations of the two pulses is γ = 6. This means that the polarization of the superposed pulse is almost linear in the [001] direction. This weak ellipticity indicates that the generated nonlinear polarization in the [001] direction is much stronger than the generated polarization in the [010] direction. Comparison between the measured γ (in the regime), and the calculated γ (in the optical regime) indicates that the relative nonlinearity in the [001] direction is stronger than the relative optical nonlinearity. Therefore, in the regime one can make the )

92 78 approximation d 33 >> d,d31. With this approximation, the angle between the generated fields E ( ), and E ( t ) 1 t τ takes the approximate form γ 4d d and we obtain the relation d d pump E E [001] E 1 γ = 6 pump E 1 [010] PPLN Figure 4.4 Diagram of polarization orientations of the incident optical pulses and the generated pulses. γ = 6 is the measured angle between the polarizations of the two pulses. Figure 4.5 shows the generated waveforms E ( ) (dots), and E ( t ) (solid line). 1 t τ

93 79 E (a.u.) Time Delay (ps) Figure 4.5 Dotted-line represents the waveform E 1 ( t), solid line represents the waveform E ( t ). The time between the two pulses is τ τ = 0.9 ps corresponding to a phase difference of 106. The frequency isν = The generated frequency is ω = 1.015, and the delay is τ = 0.9 ps, π corresponding to a phase difference of 106 between the two pulses. The superposed pulse can be expressed mathematically as E Total yˆ E0 cos( ω ) ˆ y kx t + z E0z cos( kx ωt + ϕ) = where, y ˆ points in E0z and = E 0 y E direction and z ˆ is perpendicular to y ˆ. ϕ = 50,

94 80 4. Generation of circularly polarized waves One conclusion of the previous section is that the two optical pulse methods is not an efficient way to generate circularly polarized waves. In this section, I describe a wave plate, which can produce any polarization in a broad spectrum. With this technique we have complete control over the polarization state of waves, and, in particular one can generate circularly polarized pulses. Figure 4.6 shows the schematics of the wave plate. A linearly polarized narrowband multi-cycle wave is generated when a 100 fs optical pulse is incident in a PPLN crystal. A parabolic mirror collimates the wave into the wave plate which consists of a WGP and a high reflection mirror separated by a variable distance (d). If the electric field of the pulse makes a 45 with respect to the WGP wires, then two delayed, linearly and perpendicularly polarized pulses with equal amplitudes will be reflected by the two components. The first is reflected from the WGP and the nd from the mirror located behind the WGP. The ellipticity of the superposed wave is controlled by adjusting the distance between the mirror and the WG polarizer, or by rotating the polarizer. The pulses are collected and focused by a nd parabolic mirror in a ZnTe crystal for electro-optic detection.

95 81 Optical pulse PPLN Parabolic Mirror Wire-Grid Mirror d Pellicle BS Laser detection beam ZnTe Figure 4.6 Schematics of the wave plate. A combination of a WGP and a mirror separated by a variable distance acts as a variable wave-plate.

96 Experimental results and analysis (a) reflection from WG reflection from mirror (b) a 1 b 1 Φ=8 o a b Φ=58 o E (a.u.) a 3 a 4 b 3 Φ=85 o b 4 E y, (a.u.) Φ=153 o a 5 b Time Delay (ps) Φ=-35 o E x, (a.u.) Figure 4.7 (a) Two linearly and perpendicularly polarized waves with phase difference Φ = 8,58,85,153, and, 35 (dotted-line) reflection from the WGP, and (solid-line) reflection from the mirror. (b) The polarization trajectory in the x-y plane. Where E E ) represents the electric field reflected from the WGP (mirror). x, ( y, Figure 4.7 shows the experimental data that demonstrate our ability to control the ellipticity. Figure 4.7 (a) shows the time evolution of the two linearly and

97 perpendicularly polarized waves. The dotted-line represents the fixed pulse reflected from the WGP, and the solid line represents the delayed pulse reflected from the mirror placed behind the WGP. Figure 4.7(b) shows the corresponding x-y polarization trajectory. Where E x,, E y, represents the fields reflected from the WGP and mirror respectively. We can summarize the results mathematically and write the resultant electric field in the following form 83 E Total = xe ˆ cos( ) ˆ 0xˆ kx ω t + ye0 yˆ cos( kx ωt + Φ) where Φ = 8,58,85,153, and 35 for the waveforms in the figures a 1, a, a3, a4, and 5 E0xˆ a respectively. = 1 E 0 yˆ except for the case of figure a. The ω carrier frequency is ν = π = The phase resolution is ΔΦ = 0.5. In the first periods the resultant waveform contains mainly the pulse reflected from the WG (dotted-line in figure 4.7(a), and E x, in figure 4.7(b)) so that the x-y polarization trajectory shows a linearly polarized field oriented in the x- axis. After the nd period the two pulses are fully overlapped. Figure 4.7(b) shows how the polarization of the resultant waves various: a 45 linearly polarized state ( Φ = 8 ), elliptically polarized wave ( Φ = 58, 153, and 35 ) and circularly polarized wave ( Φ = 85 ).

98 Summary In this chapter we have demonstrated two methods to control the ellipticity of the generated waves. In the first method, two linearly and perpendicularly polarized optical pulses generate two coherent pulses in the nonlinear medium. A 1 mm ZnTe crystal was employed to generate the broad-band pulses, and a PPLN structure was employed to generate the narrow band pulses. In this method, the ellipticity of the resultant pulse were controlled by controlling the time delay between the two optical pulses. In the ZnTe case we obtained a mixed polarization states because of the broad-band of the generated pulses. In the PPLN case, the strong nonlinearity in the [001] direction compared with the [010] direction limits our ability to control the ellipticity of the resultant pulse. In the second method a WGP and a mirror act as a wave plate. If a pulse impinges on the WGP, where the electric field makes a 45 with respect to the WGP wires, then two delayed linearly and perpendicularly polarized pulses with equal amplitudes will be reflected. The first is reflected from the WGP and the nd from the mirror located behind the WGP. Adjusting the distance between the WGP and the mirror controlled the ellipticity of the resultant pulse.

99 85 5 wave propagation in one-dimensional periodic dielectrics (1D-PD) Compact devices are essential for the development of applications, and basic components such as waveguides, mirrors, spectral filters and cavities are indispensable to built compact devices. Waveguide propagation of waves has been investigated by several research groups, such as the coupling of freely propagating waves into a waveguides of metal circular and rectangular 68 shapes; dielectric fibers 69 and plastic ribbon 70 also were studied. Guided waves in plastic photonic crystal fiber have been demonstrated 71. Other groups have investigated the behavior of pulses after passing through submillimeter and subcentimeter conductive apertures. As a preliminary contribution to the effort of facilitating the lab with the desired wave components, we investigated the reflection and transmission properties of 1D-PD structures. Two forms of the structures were investigated, distributed brag reflectors (DBRs), and a resonant cavity.

100 Experiment Figure 5.1 Schematic of the transmission (a) and reflection (b) setups. In both setups the emitter and gate crystals are 1 mm (110) cut ZnTe crystals. Figure 5.1 shows the schematics of the transmission (figure 5.1(a)) and reflection (figure 5.1(b)) measurements. The experiment was performed using 100 fs optical pulses, centered at 80 nm with 76 MHz repetition rate from a Ti: sapphire oscillator (Coherent Mira 900). Average power of 300 mw is focused to a 30 μm spot on a 1 mm thickness (110) cut ZnTe crystal. The emitted single-cycle pulses were collected and focused by an off-axis parabolic mirror and incident upon the 1D-PD samples. A second off-axis parabolic mirror collected and focused the transmitted (or reflected) pulses onto a nd (110) cut ZnTe gate crystal for electro-optic detection of the waves. A Pellicle beam splitter is used to allow the gating beam to be overlapped with the beam on the ZnTe gate crystal. The pump beam was chopped

101 at 760 Hz for lock-in amplification. The transmission setup was purged with dry nitrogen to eliminate water vapor absorption Samples and Fabrication Figure 5. Diagrams of the (a) DBR and (b) the resonant cavity structures. We studied DBR structures with 5, 10, and 15 periods. Each period consists of 75 μm PET and of 15 μm air. The cavity resonant consists of.5 periods of 75 μm DBR s separated by 50 μm layer of air. In this experiment we investigated 5, 10, and 15 layers of (DBRs) and a 1D resonant cavity. Figure 5.(a) shows a diagram of the DBR structure studied in the experiment. The geometry involves a periodic structure of dielectric ( 75 μm )-air ( 15 μm ) layers clamped together. Figure.b shows the diagram of the resonant cavity structure. The geometry involves two sets of.5 DBR layers separated by two air sub-layers ( 50 μm ). The dielectric material used here is a dupont melinex 453 polyethylene terephthalate (PET), which is commercially available. We used a fourth