Organization of this thesis

An overview of THz radiation and motivation of this study are presented in chapter 1. In chapter 2, we will introduce the theories of terahertz generation and detection via PC antenna, and the simulation results of THz generation and detection is included. In chapter 3, we will demonstrate the experiment results of THz pulses generation and detection under different chirped pulses. In chapter 4, we utilize this chip-controlled THz-TDS to study the sampling effect on THz-Spectroscopy. In chapter 5, the brief conclusion and future works are presented.

 

Tsunami

P.B.S

4.8mm

LT-GaAs antenna

Computer

Lock-in Amplifier SR830 Function Generator 8cm

8cm LT-GaAs antenna

4.8mm Spatial Light Modulator

20cm

H.W.P.

Q.W.P.

N.D.

N.D.

5 

References 

[1] B. Ferguson, and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Materials, Vol. 1, pp. 26-33, Sep., 2002.

[2] Tonouchi, M. “Cutting-edge terahertz technology,” Nature Photon., Vol.1, pp. 97-105, Feb., 2007.

[3] D.E. Spence, P.N. Kean, and W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett., Vol.16, pp. 42-44, Jan., 1991.

[4] D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi , “Semiconductor saturable-absorber mirror assisted Kerr-lens

mode-locked Ti:sapphire laser producing pulses in the two-cycle regime,” Opt. Lett., Vol. 24, pp.

631-633, May, 1999.

[5] D.H. Auston, K.P. Cheung, and P.R. Smith, “Picosecond photoconducting hertzian dipoles,” App.

Phys. Lett., Vol. 53, pp. 284-286, May, 1984.

[6] X.-C. Zhang, Y. Jin, and X.F. Ma, “Coherent measurement of THz optical rectification from electro-optic crystals,” Appl. Phys. Lett., Vol.61, pp. 2764-2766, Sep., 1992.

[7] A. Rice, Y. Jin, X.F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from <110> zinc-blende crystals,” Appl. Phys. Lett., Vol. 64, pp. 1324-1326, Mar., 1994.

[8] X.-C. Zhang and D.H. Auston, “Optoelectronic measurement of semiconductor surface and interface with femtosecond optics,” J. Appl. Phys., Vol. 71, pp. 326-338, Jan., 1991.

[9] Cing-Jung Chuang, “Femtosecond Pulse Shaping Methodology and its Applications in Tera-Hertz radiation enhancement”, Master Thesis, Dep. Photonics and Inst. Electro-optical and Eng., National Chiao Tung University, 2004.

[10] Sung-Hui Lin, “The Study of Terahertz Radiation Mechanism from Photoconductive Antenna by Femtosecond Pulse Shaping Technology,” Master Thesis, Dep. Photonics and Inst. Electro-optical and Eng., National Chiao Tung University, 2008.

6 

Chapter 2 Principles and Configuration of the  Experimental Apparatus  

In this chapter, we will introduce the basic theory of pulse shaping and principles of THz pulses generation and detection.

2.1 Theory of Pulse Shaping 

What is pulse shaping ?

In a loose definition, it means anything that can change the temporal profile of pulse, but, how to do that ? There are two approaches can be used to modulation the pulses, one is direct modulation the pulses in time. Nanosecond and picoseconds laser pulses can be directly modulation by electronic derived devices [1].

( ) ( ) ( )

out in

E t =h t ⊗E t (2.1)

On a femtosecond time scale, this method is not suitable due to the limitation of modulation speed of electronic devices. The other approach is modulation the pulses in frequency domain.

( ) ( ) ( )

out in

E ω = H ω E ω (2.2)

Where Ein

( )

ω =F E

{

in

( )

t

}

is the spectrum of the input pulse, Eout

( )

ω =F E

{

out

( )

ω

}

is

the spectrum of the modulated pulse and ^H

( )

ω is the frequency response. This approach is the so-called “Fourier synthesis method”, it was first accomplished by Heritage and Weiner [2], and it’s also the mainstream of ultrafast pulse shaping method currently. Figure 2-1 depicts the concepts of pulse shaping in time domain and frequency domain.

7 

Figure 2-1 Schematic of pulse shaping in (a) time domain (b) frequency domain.

In the approach of Fourier synthesis method, the modulation function we applied is

( )

0

( )

^{( )}

H ω =H ω ej^{φ ω} (2.3)

It contains two parts : (1) H0

( )

ω is the amplitude modulation, and (2) e^{jφ ω( )} is the phase modulation, conventional modulation devices can be used for pulse shaping include

liquid-crystal spatial light modulators [3], acoustic optic modulators [4] and deformable mirrors [5]. In our experiment, we will use LC-SLM as our modulation device, the details are described below.

2.1.1 Measurement of Ultrafast Laser Characteristics 

The laser system we used in experiments is a mode-locked Ti-sapphire laser (Tsunami, Spectra-Physics) with a typical output power of 450mW and pulse repetition rate ~82MHz.

Figure 2-2 shows the spectrum of our laser source, the FWHM spectra bandwidth is about 40nm.

h(t) 

( )

Ein t   Eout

( )

t =h t

( )

⊗Ein

( )

t  

( ) ( )

h t τ E τ τd

∞

=

∫

−∞ − (a) Modulation in the time domain

(b) Modulation in the frequency domain

H(ω) 

in

( )

E ω   Eout

( )

ω = H

( ) ( )

ω Ein ω

 

( )

¹

{ ( ) }

out out

E t =F⁻ E ω  

8 

700 750 800 850 900

-500

Figure 2-2 Spectrum of the mode-locked Ti-sapphire laser.

According to the Fourier theory, the pulse field in the time domain is the inverse Fourier transform of the frequency domain field :

i

( )

^{1 i}

( )

^exp

( )

E t 2 E ω i t dω ω

π

∞

=

∫

−∞ ^(2.4)

The pulse field in the frequency domain can be written as : i

( ) ( )

^exp

( )

E ω = S ω ⎡⎣jϕ ω ⎤⎦ (2.5)

Where ^S

( )

ω is the spectrum and ϕ ω is the spectral phase. We can use this method to

( )

estimate the ideal pulse width direct from the measured spectrum :

2 the speed of light. According to equation (2.6), we can know that the ideal pulse width of our laser pulses is about 23.7 fs, it’s also called the Transform-Limited Pulse (TLP) width.

9 

In order to understand more information about the pulse, we must measure the temporal profile of the pulse, however, due to the measurement limitation of electronic devices, we cannot direct measure the ultrafast laser pulses in time. The only way we can use to measure the ultrafast pulses is use the pulse to measure itself [6], it’s called autocorrelation.

There are several kinds of autocorrelation methods, according to the correlation process we used, autocorrelation can be divided into two classes : (1) Interferometric autocorrelation (2) Intensity autocorrelation. For a complex electric field iE t , the field autocorrelation is

( )

defined by

( )¹

( )

ⁱ

( )

ⁱ

( )

A τ ^∞ E t E t τ dt

=

∫

−∞ − ^(2.7)

Figure 2-3 depicts a experiment setup of intensity autocorrelator using second-harmonic generation. A pulse is divided into two, one is variable delayed with respect to the other, and the two pulses are overlapped into a nonlinear medium, such as second-harmonic-generation (SHG) crystal (BBO or LBO) or third-harmonic-generation (THG) crystal. The SHG intensity will be measured vs. delay, yielding the autocorrelation trace.

( )²

( ) ( ) ( )

A τ ^∞ I t I t τ dt

=

∫

−∞ − ^(2.8)

Where ^{I t}

( )

^{= E ti}

( )

² is the intensity profile of the complex electric field. Figure 2-4 shows the experimental result of intensity autocorrelation.

Figure 2-3 Experimental layout for an intensity autocorrelator.

BBO

Color Filter PMT

Delay Stage

BS

Parabolic mirror

Computer

E(t)

E(t-τ)

10 

1.0 Experimental Result

Theory fitting

Intensity Autocorrelation [a.u.]

Delay [fs]

δt=61fs

Figure 2-4 Experimental result of the intensity autocorrelation trace.

From the experiment result, we can know that the intensity FWHM pulse width is about 61fs, the TL-pulse width is about 21fs. This difference is due to the 2^nd order spectra phase, which will broaden the temporal profile of the pulse. A Gaussian type frequency domain field with spectral phase can be written as : following discussion, because b0 wouldn’t change the temporal profile of the pulse, and according to the Fourier theory, the 1^st order phase b₁ in frequency domain will only cause a linear shift in time, the shape of the pulse also remains the same. The temporal electric field is :

11 

From the equation above, we can know that if there exist 2^nd order spectra phase b2, it will causes two effects : (1) Broaden the temporal pulse duration (2) Instantaneous frequency will change with time, it depend on the sign of b2. If b2 is greater than zero, the carrier frequency will increase with time, we will say the pulse is up-chirped (or positive chirp).

(a) Up-chirped pulse (b) Down-chirped pulse

Figure 2-5 (a) An example of up-chirped pulse, it means the instantaneous frequency is increase with time. (b) down-chirped pulse, the instantaneous frequency is decrease with time.

The relationship between 2^nd order spectra phase b2 and temporal pulse width wis :

where τ is the transform-limited pulse width. ₀

From equation(2.11) we can know that, if two pulses with the same 2^nd order spectra phase b₂, they will have the same pulse width, however, the physical meaning are quite different. A pulse is called positive-chirped if lower frequency light travels ahead of higher frequency light and negative if opposite holds.

We will discuss the role of chirp in the processes of THz pulses excitation and detection in the next chapter.

-60 -40 -20 0 20 40 60

12 

2.1.2 Pulse Measurement in Time­Frequency Domain 

Although autocorrelation is a convenient method to measure the temporal profile of the pulses, however, the phase of the pulse is an unsolvable problem, in many applications of ultrafast laser, the phase of the pulse plays very important role. In additional, there exist ambiguities of the temporal shape, two different temporal profile may lead to the same autocorrelation trace.

Frequency-Resolved-Optical-Gating (FROG) [7] is developed to measure the amplitude

and phase of the ultrafast laser pulse. The basic concept of FROG is similar to autocorrelation, it measures spectrum of the correlation signal vs. delay :

(

^,

) ( ) ( )

^{j t}

IFROG τ ω ^∞ P t G t τ e^−ωdt

=

∫

−∞ − ^(2.12)

Where ^{P t}

( )

is the complex electric field we want to measure, ^{G t}

( )

is the gating function, it depends on what kind of nonlinear process we used, and will have different forms. After measuring FROG trace, an iterative method is used to reconstruct the original complex pulse.

And then we can get the information of the pulse amplitude and phase. Figure 2-6 shows an example of FROG trace.

Figure 2-6 An example of SHG-FROG trace [8], (a) amplitude and phase of the pulse in time-domain, and (b) the corresponding SHG-FROG trace. 

(a)  (b) 

13 

2.1.3 Dispersion Free Pulse Shaping System   

Figure 2-7 depicts the dispersion free pulse shaping system, this system consists of a pair of gratings (600 grooves/mm), two spherical reflectors with a focal length f=20cm and two reflection mirrors. Optical pulse will be diffracted by first grating and then focus by the spherical reflector, the first two components performs a Fourier transform which converts the angular dispersion from the grating to a spatial separation. LC-SLM is placed in the

Frequency domain in order to manipulate the spatially dispersed optical Fourier components.

After a second spherical reflector and grating recombine all the frequencies into a single collimated beam, a shaped output pulse is obtained.

Optical pulse shaping is accomplished utilizing a programmable spatial light modulator [9]

(SLM Cambridge Research and Instrumentation Inc. (CRI) Woburn, MA, SLM-128). The SLM which induce a individual phase retardation on the pulse spectrum consists of 128 pixels, each pixel with 5 mm high and 100 μm width. We can tailor and design different pulse

waveforms using GS algorithm [10].

Figure 2-7 Schematic of dispersion-free pulse shaping system.

Spatial Light Modulator, CRi -128 pixels Grating 600 grooves/mm

Curve mirror, f=20cm Mode-Locked Ti-Sapphire Laser

35fs BBO

Color Filter Lens Spectrometer Computer

FROG system

Feedback control

14 

2.1.4 Freezing Phase Algorithm 

In the previous discussion, we know that if there exist 2^nd order spectra phase b₂, it will broaden the temporal profile of the pulse, and induce linear chirp. If we can compensate this phase in frequency domain, then we can get transform-limited pulse with shorter pulse duration and higher peak intensity than the chirped pulse.

Traditional phase compensation algorithms including simulated annealing algorithm [11], and genetic algorithm [12], we adopt freezing phase algorithm [13] to compensate the spectra phase, which is very rapid compare with other algorithms. The concept of freezing phase algorithm (FPA) is illustrated below.

The mode-locked pulse can be expanded by its spectra components in frequency domain :

i

( ) ( ) [ ]

0 all this phase compensation algorithms are attempt to compensate the spectra phase φ . _n

Figure 2-7 depicts the experiment apparatus of freezing phase algorithm, the laser pulse is expanded into frequency domain, each frequency components will be turning by SLM, after recombination by the 2^nd grating, the laser pulse is focused by a lens into a nonlinear crystal (type-I BBO), the SHG signal will be measured by a spectrometer, the total power of the pulses will be return to computer as feedback control.

We can use phasor to represent each spectra phase components vn =^exp

[ ]

jφn , here we only use 3 phasors v₁ ~v₃to simplify the freezing process. Figure 2-8 (b) show the

transformation process of each phasors during freezing phase algorithm. First we pick up one of the phasors v₃ and make the other phasors v₁ and v₂ as one group, then turning v₃ from 0~2π by SLM, during the turning process, we measuring the power of the SHG signal. If

15 

v3 are in the same direction of v₁+v₂, it will have maximum SHG intensity, we can use this concept to determine the spectra phase of that component. Second, we pick another phasor v₂ and regroup the other SLM pixels, then determine the phase of v₂. Finally, all the spectra phase will be compensated after freezing, figure 2-8(a) shows the flowchart of freezing phase algorithm.

(a) (b)

Figure 2-8 (a) Flow chart of Freezing Phase Algorithm (b) Schematic of the freezing process.

 

Figure 2-9 (a) shows the autocorrelation trace of the laser pulse, we can see that the pulse width is about 100fs before freezing procedure. After freezing phase, the pulse width is about 38fs. The compensated phase of SLM is shown in Figure 2-9 (b).

  spectral components from 0 to 2π

Measure the SHG intensity

Determine the spectral phase of the chosen components

16 

(a)   

-200 -100 0 100 200

0 2 4 6 8 10

δt'=38fs

Intensity (a.u.)

Time (fs)

before Freezng after Freezing

δt=93fs

 

(b)   

360 370 380 390 400

-8 -6 -4 -2 0 2 4

Phase (rad)

Frequency (THz)

SLM phase Fitting

Figure 2-9 (a) autocorrelation trace of the pulse : circle - before freezing phase procedure, triangle – after freezing phase procedure, and (b) the compensated phase in the SLM.

   

17 

2.2 Terahertz­Pulse Generation from Photoconductive  Antenna 

Figure 2-10 Schematic view of photoconductive dipole antenna.

The schematic view of photoconductive antenna we used in experiment is shown in Figure 2-10, the antenna with dipole structure is fabricated on low-temperature grown (LT) GaAs, which has very high breakdown field and fast carrier life time (~ps) along with good mobility (~200 cm²V^-1s^-1). The LT-GaAs substrate is attached with a silicon lens, which is used to efficiently coupling the THz radiation out of the LT-GaAs substrate.

When the photoconductive gap is irradiated by femtosecond laser pulses with energy greater than the band gap of the semiconductor, electrons and holes pairs are generated in the conduction band and valance band respectively. The carriers are then accelerated by the bias field and decay with a time constant τ , resulting in a pulsed photocurrent _C J t in the

( )

photoconductive antenna, this transient current will induce electromagnetic transient (THz radiation), the radiated electric field is proportional to the first-order time derivate of the transient current :

( ) ( )

THz

E t J t

t

∝∂

∂ (2.14)

LS

dS

Lm/2

vb

silicon lens LT-GaAs

18 

2.2.1 One Dimensional Drude­Lorentz Model   

In order to understand the carrier transport mechanism in the THz radiation generation process, one dimensional Drude-Lorentz model [14-16] is used in this study. When the biased semiconductor is irradiated by the femtosecond laser pulses with energy greater than its bandgap, the electron-hole pairs are generated, the time dependence of the carrier density

( )

laser pulse, assume the generation rate of the carrier is direct proportional to the laser pulse, it can be written as

Where n₀ represents the carrier generation density at t=0 ps, t_pis the intensity FWHM of the laser pulse. The generated carriers will be accelerated in the bias electric field, the acceleration of electrons and holes is given by

( ) ( ) ( )

E t is the local electric field.

The local electric field E t is smaller than the applied bias field,

( )

E_bias, due to the

19 

screening effects[17] of the space charge,

( ) ( )

bias

E t E P t

= − αε (2.18)

Where P t is the polarization induced by the spatial separation of the electron and hole,

( )

ε is dielectric constant of the substrate, and α is the geometrical factor of the semiconductor [18]. The time dependence of polarization P t can be written as

( )

( ) ( ) ( )

r

dP t P t

dt ^{= −} τ ⁺J t ^(2.19)

Where τ is the recombination time between an electron and hole, _r J t is the current

( )

density contributed by an electron and hole,

( ) ( ) ( )

h e

( ) ( ) ( )

J t =en t ⎡⎣v t −v t ⎤⎦=en t v t (2.20) Where e is the charge of a proton, and the change of electric currents leads to

electromagnetic radiation according to Maxwell’s equations. The far field of the radiation

( )

ETHz t is given by

( ) ( ) ( ) ( ) ( ) ( )

THz

dJ t dn t dv t

E t ev t en t

dt dt dt

∝ = + (2.21)

In the following section, we will show the simulation results of THz radiation generation from the Drude-Lorentz model of photoconductive antenna.

20 

2.2.2 Simulation Results and Discussion 

The coupled differential equations (2.15)-(2.20) can be numerically solved by fourth order Runge-Kutta method [19], the parameters we used in the simulation are listed in Table 2-1.

Table 2-1 Parameters used in the THz radiation generation simulation

parameters parameters

Carrier life time τ_c 0.5 [ps] Carrier density n₀ 10²⁴ [m^-3] Momentum relaxation time τ_s 0.2 [ps] Effective mass of electron m_e 0.067 m₀

Recombination time τ_r 10 [ps] Effective mass of hole m_h 0.37 m₀ Pulse duration t_{p 100 [fs]} Applied bias field E_bias 10⁶ [V/m]

Charge of electron q_e 1.6*10^-19 [Q] Geometrical factor α 900 Charge of electron q_h 1.6*10^-19 [Q] Relative Static Permittivity ε_r 12.96

From the 1D Drude-Lorentz model, we can know that there are several factors will influence the THz generation, such as A. Excitation pulse width t_p, B. Carrier life/trapping time τ , C. Momentum relaxation time _c τ , D. Bias field _s E_b, and E. Excitation density

n0.We will show the simulation results and discuss its impact below.

   

21 

Figure 2-11 Simulated THz field at excitation pulse width t of 25fs, 50fs, 100fs, 200fs, and _p 500fs, (a) time domain profile and (b) power spectrum.

0 100 200 300 400 500 Figure 2-12 Excitation pulse width dependence of THz bandwidth.

Figure 2-11 shows (a) the simulated THz pulses generated from different excitation pulse width, and (b) the corresponding power spectrum. From the simulation results we can know that the more shorter excitation pulse width, the broader bandwidth and higher power THz pulse we can get, the relationship between excitation pulse width and efficient THz bandwidth is shown in Figure 2-12.

22  time domain profile and (b) power spectrum.

0 1 2 3 4 5 6 7 8 9 10

THz Positive Peak Amplitude THz Negative Peak Amplitude

Carrier Life Time (ps)

THz Positive Peak Amplitude (a.u.) THz Negative Peak Amplitude (a.u.)

(a)

Carrier Life Time (ps)

Bandwidth

(b)

Figure 2-14 Relationship between carrier life time and (a) THz positive & negative amplitude (b) bandwidth.

Figure 2-13 shows (a) the simulated THz pulses generated with different carrier life time, and (b) the corresponding power spectrum. From the simulation results we can know that the faster the carrier life time, the broader bandwidth we can get, however, it will sacrifice the total power of THz radiation. From figure 2-14, we can know that suitable carrier life time will have both higher THz radiation power and broader bandwidth.

23 

Carrier Life Time 0.1ps Carrier Life Time 100ps

 

Figure 2-15 Comparison of THz waveforms generated with shorter and longer carrier time, the peak value is normalized to 1 respectively.

From Figure 2-15, we can compare the two different waveforms calculated with shorter carrier life time (0.1ps) and longer carrier life time(100ps), it shows two different kinds waveform – bipolar waveform(0.1ps) and unipolar(100ps) waveform, it’s agree with the experiment results [21].

C. Momentum relaxation time dependence 

Figure 2-16 Simulated THz field at momentum relaxation time τ   of 25fs, 50fs, 100fs, 200fs, _s and 500fs, (a) time domain profile and (b) power spectrum. 

24 

    The momentum relaxation time it’s also very important in the THz field generation process,  from Figure 2‐16, we can find out the amplitude of THz radiation increases as the 

momentum relaxation time increases. The momentum relaxation time is corresponding to  the saturation velocity of the carriers, the higher the saturation velocity, the higher THz peak  amplitude we can get.   

 

D. Bias Field Dependence 

4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8

Amplitude (a.u.)

Time (ps)

Bias Field

10⁴[V/m]

2*10⁴[V/m]

5*10⁴[V/m]

10⁵[V/m]

2*10⁵[V/m]

(a) Time Domain 

0 5 10 15 20 25 30

Bias Field

10⁴[V/m]

2*10⁴[V/m]

5*10⁴[V/m]

10⁵[V/m]

2*10⁵[V/m]

Power (a.u.)

Frequency (THz)

(b) Power Spectrum 

Figure 2-17 Simulated THz field at bias field  E_b  of  10⁴[V/m],2 10× ⁴ [V/m],  5 10× ⁴[V/m],  105[V/m], and  2 10× ⁵ [V/m] (a) time domain profile and (b) power spectrum. 

The bias electric field provides the acceleration of electrons and holes, when we increase bias field, it can also increase the saturation velocity, so we can get higher peak amplitude of THz field, however, the efficient bandwidth is remains unchanged under higher bias field.

25 

[m^‐3], 10²⁷[m^‐3], and  10²⁸ [m^‐3] (a) time domain profile and (b) power spectrum. 

When the electrons and holes are generated by the optical pulses, it will be accelerated in opposite directions in the local electric field. This induces a polarization, which acts as a restoring force for the motion of electrons and holes, when the carrier density increase, the local electric field can be screened to a comparable magnitude of the restoring force. In that case, the electrons and holes will serves as an oscillator, and induces an oscillation electric field. From the simulation results in Figure 2-18, it shows different THz fields generated at various carrier densities. We can see the amplitude of the THz field increases and tends to oscillate as the increase of the carrier density. This results shows the possibility to get THz fields with higher frequency using screening effect.

 

   

26 

2.3 Terahertz­Pulse Detection with Photoconductive Antenna 

The pulse duration of the THz radiated from the PC antenna is typically several

picoseconds, in order to obtain the temporal profile of the THz waveform, cross-correlation measurement is used. According to the measurement mechanism, it can be majorly divided

picoseconds, in order to obtain the temporal profile of the THz waveform, cross-correlation measurement is used. According to the measurement mechanism, it can be majorly divided

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