Chapter 1 Introduction to terahertz radiation
1.6 Organization of this thesis
In chapter 1, an overview of THz radiation and GaSe semiconductors are presented. In chapter 2, the basic principles of theories of terahertz radiation via optical rectification are described, both experimental and simulation results are included. In chapter 3, we demonstrate the superposition of the two terahertz in time domain , and the coherence of the terahertz wave is presented. In chapter 4, we introduce a method to generate higher power terahertz by superposition with high coherence and design a suitable condition to generate higher power terahertz. In chapter 5, the brief conclusion and summary are presented.
Reference
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Chapter 2
Overview of optical rectification
2.1 Background
Optical rectification is the second-order of nonlinear process in which the D.C frequency (comparing to optical frequency range) of electric field is produced by intense optical pump illumination in the nonlinear medium [1]. Due to the development of picosecond and femtosecond pulsed laser, currently, generating ultrashort, single cycle THz electromagnetic pulses via optical rectification is much easier.
Generally, optical rectification refers to the generation of a DC polarization via an intense optical pulse. Most optical electric field strength, even with a laser sources, are small compared with characteristic crystalline chemical bonding. Therefore, the electronic polarization could be expanded in the powers series of the driving electric force.
( ) (2) (3) (4)
0[ ]
Pnl =ε χ E E+χ E E E+χ E E E E+higher order term (1) The nonlinear susceptibility of χ( )n is the tensor of n order of n+1 rank. The second order polarization is the lowest term, which could yield a DC contribution. In general, the individual tensor components will depend on the frequencies fields involved.
Terahertz optical rectification is induced by an intense pump pulses. More accurately, the terahertz optical rectification should be considered as nearly degenerate difference frequency generation with each frequency pair in a single cycle optical pulse. When a pump pulse containing broadband spectrum, which is
dominated by the pulse shape and the pump pulse duration, is incident into the nonlinear EO crystal, the nonlinear interaction between two frequency components will induce a nonlinear polarization at the beating frequency.
Considering the lowest order of nonlinear interaction, the second order polarization for optical rectification is with the form:
(2) (2) *
( T) 0 ( T; p T,- p) ( p T) ( p)
P ω =ε χ ω ω +ω ω E ω +ω E ω (2) χ(2) is the second order susceptibility, third rank tensor which of 3 3 3× × components. Symmetry determines which of the tensor components are vanished and which components are equal. For second order nonlinear interaction, the material must possess non-central symmetry. For broadband pump pulses, the electric field in equation (2) contains the frequencies ωp +ωT and ωp . The ωT is the DC frequency far from optical range. For convenience, we take the effective nonlinearity
1 Under the phase matching condition, the field of the beating frequency radiation has a continuous spectrum with frequency as high as several terahertz and corresponding waveforms.
2.2 Theory of optical rectification
The one-dimensional equation for the Fourier component of the THz field at the angular frequency, ωT , (ET ωT), could be derived directly from the Maxwell’s equations.
• ( , ) 0E z t By double curl operator and constitution relation, the wave equation could be recast into
where ε0 and μ0 are the permittivity and permeability in free space, respectively, and σ is the conductivity. In addition, E and P expresses the time varying electric field and polarization. It is assumed that the medium is with no free current, so that the first term of right handed side is vanished, and we drop the first term of left handed side by the reason of assuming the medium is without free charges.
( , ) 0 D z t
∇ • = (9) Therefore, the wave equation could be simplified as follow:
2 2
Herein, the plane wave analysis is applied, the electric field and the nonlinear polarization are the linear combination of different frequency:
( , ) n( , )n i nt
( , ) ( ) ik zn When the relations above are introduced into Eq.(10), a wave equation analogous to Eq.(10) is obtained. The wave equation valid for each frequency component of the field is shown as follows:
2 2 2 ( )
2.3 Terahertz wave generation by optical rectification
Under the strong pump, the pump pulse is almost constant in propagating.
( , ) ( , ) ~
p p
d E z E z const
dz ω ⇒ ω (16) This implies that the pump is non-depleted. This physical assumption makes us consider terahertz generation only. Under the slowly varying envelope approximation:
2 and in the limit of no absorption α =0, the terahertz generation could be expressed as follow: expressed through the material effective nonlinearity deff as:
* pump field, respectively. The k-vector mismatch Δ is given by the following kT
relation [2][3]:
( ) ( ) ( )
T p T p T
k k ω ω k ω k ω
Δ = + − − (20) and the deff for the crystal of GaSe for type-oee phase matching is given by the handbook as follows [4]:
2
22cos ( ) cos(3 )
deff =d θ ϕ (21) The differential equation Eq.(19) is terahertz generation via optical rectification described the second order nonlinear optical process. The phase matching determine the whether ET(ωT, )z could be generated. The others linear coefficients dominate the amplitude strength of the generated field.
Consider an optical pump, broad bandwidth-limited ultrashort (e. g, femtosecond) laser pulses propagating along z in the form of infinite plane waves with the Gaussian time envelope profile of the electric field. In simulation, we assume that the incident pump light pulse has a Gaussian temporal shape as:
2 which is defined by the full width at half maximum of the intensity of pulse shape.
Under the Fourier transform limit, the pump light pulse could be expressed as:
( , ) ( , ) ik z0 i t
p p
e z t =
∫
E ω z e e−ωdω (24) The Fourier transform of the incident optical wave at z=0 becomes:2 2
Similar one-sided Fourier representation will be used for other fields below in the text.
2.4 Parameters of GaSe
We use the parameters of GaSe crystal as the electro-optical medium. The central wavelength of the pump pulse is treated as 800 nm. The dielectric function in terahertz range and Sellmeier equation in optical range are introduced by [5].
Dispersion of the GaSe crystal in terahertz region is described by a complex dielectric function at angular frequency, ω :
2 In optical region, the refractive index ( )n λ for ordinary and extraordinary optical wave are shown as derived from Sellmeier equation, respectively:
2 2
here λ represents the wavelength in μm. The parameters of each equation are list in the table 2-1.
Ordinary εo
Table 2-1 Parameters of Sellmeier equation dielectric function.
2.5 Simulation result
T 0
Δ =k and αT =0
First, here we conduct numerical simulation under conditions of perfect phase matching (Δ = ) and without absorption (kT 0 αT = ). On the other hand, we neglect 0 the pump-depleted effect. It is assumed that the pump pulse is not changed with propagating, in order to understand the basic physical nature of the optical rectification process.
The power spectrum of terahertz and corresponding waveform obtained by the calculation are shown in Fig.2-1(a) as a function of the propagation length inside GaSe crystal from 0 to 2mm. the terahertz spectrum grows almost linearly as a function of the propagation length, and a little change in the shape change in the shape is observed.
0 1 2 3 4 5
Fig.2-1 (a) Power spectra of THz under perfect phase matching and no absorption condition at propagation distances in the GaSe crystal form 0 to 2mm. (b) Temporal waveforms of THz field obtained corresponding to the same
condition.
The temporal waveform of the terahertz pulses are displayed in Fig. 2-1(b). The waveforms are almost proportional to the second order time derivative of the pump pulse shape in this figure. Although the assumption of no change in the pump field is the simulation condition, in very low conversion efficiency the results we obtain from numerical calculation study is still adoptable and predictable.
T 0
Δ ≠k and αT =0
Now we examine the effects of real phase condition on the terahertz optical rectification processes by finite phase mismatching, Δ ≠ . Similarly, we keep the kT 0 absorption as zero. Parameters of the pump pulses are the same as that in the previous case. Terahertz spectrum obtained by simulations is shown in Fig.2-2(a) as a function of propagation length. The peak of the terahertz spectrum is not linearly growing, and the spectral bandwidth becomes narrower as the propagation length increases due to the fast varying phase matching condition. As the propagating length increasing, the spectrum peaks shift to lower frequency. It is noticed that the spectrum with a deep around 0.58 THz is due to the absorption of transverse optical phonon [2].
Temporal waveform of the terahertz radiation is shown in the Fig.2-2(b). It could be observed that the dispersion relation causes the oscillation tailoring and make the terahertz waveform broadened.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
Fig.2-2 (a) Power spectra of THz under realistic phase mismatches and no absorption conditions at propagation distances in the GaSe crystal from 0 to 2mm. (b) Temporal waveforms of THz pulses corresponding to same conditions.
T 0
Δ ≠ and k αT ≠ 0
The frequency dependent absorption effect is included in this case. The absorption coefficient αT in terahertz range is as a function of frequency. The absorption coefficient is defined as:
T
=4π ( )
α κ ω
λ (30) where κ is the imaginary part of refractive index derived from dielectric function in terahertz range.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
Fig.2-3 (a) Power spectra of THz under realistic phase mismatches and absorption along propagating distances in the GaSe crystal from 0 to 2mm. (b)Temporal waveforms of THz pulses corresponding to same conditions.
2.6 Experimental results
The experimental setup for terahertz generation by optical rectification is shown in Fig.2-4. The optical pump source utilizes a Ti: sapphire laser with a 1-kHz repetition rate. The pulse energy is about 700 Jμ , central wavelength at 800 nm, with 280 fs pulse duration. The beam splitter reflects 10% of input power as THz-TDS gating pulse, transmits 90% as terahertz pump source.
Terahertz radiation is generated from 2 mm GaSe crystal. Fig.2-5 shows the geometry of terahertz radiation via GaSe crystal by optical pump pulse. The residual laser beam are blocked by Teflon and terahertz wave are passed through wire-grid polarizer to make sure the polarization, then terahertz wave are guided by four parabolic mirrors to ZnTe crystal [6]. The indium-tin-oxide (ITO) glass plate, which transmit the optical pump pulse while partially reflects the terahertz pulse, is used as the terahertz dichroic mirror reflecting terahertz between second and third parabolic mirror. A pellicle beam splitter which is transparent to the terahertz beam and has a reflectivity of 5% for optical pump pulse is used to combine terahertz pulse and gating pulse and make the gating pulse collinear with the terahertz pulse into ZnTe crystal.
For monitoring the time-domain waveform of terahertz radiation, we employed the electro-optical sampling technique with a 1 mm thick ZnTe crystal. The terahertz pulse and gating pulse co-propagate through ZnTe crystal. The linear polarization of the gating beam is perpendicular to the polarization of the terahertz pulse. The azimuthal angle of ZnTe could be adjusted to obtain maximum E-O signal. The linear polarization of the gating pulse without been modulated by the terahertz pulse is converted to circular polarization by quarter wave plate. Polarization of the gating pulse modulated by the terahertz pulse is converted to ellipsoid polarization by a quarter wave plate. Behind the quarter wave plate, the Wollaston prism follows, and
separates the modulated ellipsoid polarized gating pulse into two orthogonal polarizations, s- and p-polarization. Both polarizations are guided to balance detector.
The electrical signal of balanced detector is connected to a lock-in amplifier in order to increase the signal to noise ratio (S/N ratio). An optical chopper is used in this experimental arrangement to modulate the optical pump beams for optical rectification of terahertz radiation.
Fig.2-4 The experimental setup of electro-optic THz system.
Fig.2-5 The scheme of THz generation via optical rectification in GaSe crystal.
K-vector C-axis
Horizontal axis Vertical axis
θ
ψ
Ti:Sapphire 1 kHz 270 fs
BS
Teflon+ Wire grid
Wollaston ZnTe
Pellicle QWP chopper
ND-filter
ND-filter
probe stage
GaSe ITO
Balanced Detector
0 10 20 -4
-3 -2 -1 0 1 2 3 4
THz field (a.u)
time delay (ps) (a)
0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5
power (a.u)
frequency (THz) (b)
Fig. 2-6 (a) THz electric field profile in the time domain via EO sampling detection.
(b) Power spectrum versus frequency of the THz pulse in (a)
Fig.2-6(a) shows the terahertz radiation time domain waveform and corresponding spectrum from GaSe crystal. The terahertz spectrum is also with a deep around 0.58 THz agreed well with simulation result, also corresponds to the absorption of transverse optical phonon of dielectric function.
2.7 Effective nonlinearity
Optical rectification is nonlinear optical behavior related to the second-order susceptibility. THz output from the OR is dependent on the value of the nonlinearity deff. The deff of the GaSe crystal is expressed as follows:
2
22cos cos 3
THz eff
E ∝d =d θ ϕ (31) By selecting the suitable φ angles, THz peak electric field and the deff could be optimized. Fig.2-7 shows the THz time domain waveforms at different GaSe azimuthal φ angle. This relationship is attributed to the hexagonal crystal symmetry in the uniaxial crystal. It has been shown that the maximum signal is achieved when the GaSe emitter has its azimuthal angle set at cos 3ϕ =1, since the effective nonlinear coefficient of type-II phase matching is proportional to the value of cos3φ. Fig.2-8 presents the THz wave peak amplitude versus azimuthal φ angle for GaSe emitter.
5 10 15 20 25
Fig.2-7 THz time domain waveforms at different azimuthal angle.
-50 0 50 100 150 200 250 300 350 400
Fig.2-8 THz wave peak amplitude versus azimuthal angle for GaSe emitter.
2.8 Summary
We demonstrated the numerical simulations based on one dimensional terahertz generation by optical rectification in GaSe crystal which possesses the large nonlinearity suitable for terahertz generation. Effects of perfect phase matching and no absorption, and also realistic phase matching and absorption conditions are also discussed by use of the parameter of GaSe crystal. The numerical simulation results of several cases show that the phase matching condition could have a significant effect on the optical rectification processes for terahertz pulse generation despite the dispersion of the EO medium.
Reference
[1] A. Nahata, A.S. Weling, T.F. Heinz, “A wideband coherent terahertz
spectroscopy system using optical rectification and electro-optic sampling,” Appl.
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[5] Ching-Wei Chen, Tsung-Ta Tang, Sung-Hui Lin, Jung Y.Huang, Chen-Shiung Chang, Ci-Ling Pan, Pei-Kang Chung, Shun-Tung Yen, “Optical properties and potential applications of ε-GaSe crystal in terahertz frequencies,” prepared to submit.
[6] Q. Wu, and X. -C. Zhang, “Free-space electro-optic sampling of terahertz beams,” Appl. Phys. Lett. 67, 3523-3525 (1995).
Chapter 3
Superposition and coherence in multiple stages of optical rectification
3.1 Theoretical model
The nonlinear interacting processes between the optical and terahertz pulses in cascaded GaSe multiple stages optical rectification could be properly described with the coupled wave equations. Under the slowly varying envelope approximation, the one dimensional coupled propagation equations [1] are derived as
2
Here z is the propagation distance in GaSe crystal, E and P E denote the terahertz T radiation field and the optical pump pulse in the second stage, respectively; ε0 and
μ0 are permittivity and permeability in free space, respectively; deff is effective nonlinearity; ωP and ωT are angular frequencies of optical pump pulse and terahertz pulse; αPand αT,2are the linear absorption coefficients of optical pump pulse and terahertz pulse in second GaSe; the Δ and kP ΔkT,2 denote the wave vector phase mismatches.
The total terahertz field in time domain could be described by
,1
,1 ,2
( ) ( ) i T ( )
T T T T T T
E ω =E ω e−ω τ +E ω (3)
where ET,1and ET,2are the terahertz radiation fields from the first and second optical rectification stage, respectively; τ expresses the propagation time delay of the two stage optical rectification stage.
3.2 The minimum variance distortionless response
Here we used a method of minimum variance distortionless response (MVDR), it is used in spectral estimation. And we used this approach to estimate the cross- spectrum and magnitude squared coherence function (MSC) function. First, we calculated cross spectrum. By using the cross spectrum, we analyzed the coherence of each frequency between the two terahertz radiations.
Let ( )x n be a time varying signal, which is the input of K filters of length L, where { }E i expresses the expectation value, and superscript H expresses the transpose conjugate of a vector, and R is defined as: xx
( -1)
The F matrix is called the Fourier matrix and be unitary. This implies
H H
F F =FF = . In the MVDR spectrum, the filter coefficients are chosen so as to I minimize the variance of the filter output, subject to the constraint:
1
H H
k k k k
g f = f g = (8) Under the Eq. (8), the signal x n is passed through the filter ( ) g without distortion k at frequency ωk, and signals at other frequencies tend to be attenuated. This expression could be described by mathematical model:
[1 ]
Replacing Eq. (10), we obtain
xx k xx( )k k
which is a diagonal matrix.
Assuming here that the two signal x n and 1( ) x n with respective spectra 2( ) Here the super script * denote the complex conjugate.
For the same method, we could get
2 1 1 2
To obtain the cross-correlation spectrum, we apply Eq. (15) into Eq. (20):
1 1 1 2 2 2
3.3 Magnitude squared coherence (MSC) function
According cross-correlation spectrum, we define the magnitude squared coherence (MSC) function between two signals x n and 1( ) x n as [3], 2( )
So from Eq. (21), the magnitude squared cross correlation spectrum is:
1 1 1 2 2 2
Using the expressions of Eq. (16) and Eq. (23) into Eq. (22), The MSC function becomes:
3.4 Experiment setup and result
The used laser system is a 1k-Hz amplified Ti:sapphire laser with pulse energy of 700 Jμ and pulse duration of 280 femtosecond. The experimental setup for multi- stage optical rectification is shown in Fig. 3-1. The first beam splitter reflects 10% of input power as gating beam, and the second beam splitter reflects 60% for first terahertz generation and transmitted 40% for second terahertz generation. The typical average pump power on the GaSe crystals is about 130mW and 150mW for the first and second stage respectively. The pump beam diameter for both stages is adjusted to be about 3 mm. Both GaSe crystals are configured for non-phase matched optical rectification. The terahertz radiation field generated from the first optical rectification stage of 2-mm thick GaSe crystal is guided into the second stage of 3-mm thick GaSe.
Again we block the residual optical pump laser with Teflon plates terahertz wave are
Again we block the residual optical pump laser with Teflon plates terahertz wave are