time-dependent theoretical model for terahertz wave detector using a parametric process

Time-dependent theoretical model for terahertz wave detector using a parametric process

C. Y. Jiang1, J. S. Liu

1,*, B Sun

1, K. J. Wang

1, S. X. Li

1, and J. Q. Yao

1,2

1Wuhan National Laboratory for Optoelectronics, School of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

2College of Precision Instrument and Opto-electronics Engineering, Tianjin University, Tianjin, 300072, China * [email protected]

Abstract: We have presented a time-dependent theoretical model to describe the time behavior of a quasi-monochromatic nanosecond terahertz detector reported by Guo et. al. (2008 Appl. Phys. Lett. 93, 021106). The temporal input-output characteristic of the detector is investigated numerically by taking the system parameters close to the experimental ones, and the calculated pulse width for the incident terahertz wave agrees well with the experimental one. Our results demonstrate that the energy and width of an output idler wave pulse are proportional to those of the incident terahertz wave pulse. This study provides a strict theoretical basis and could be used to guide the design and optimization for the highly sensitive coherent terahertz detector.

©2010 Optical Society of America

OCIS codes: (040.2235) Far infrared or terahertz; (190.4223) Nonlinear wave mixing; (190.4410) Nonlinear optics, parametric processes; (160.3730) Lithium niobate; (040.1880) Detection

References and links

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#129790 - $15.00 USD Received 9 Jun 2010; revised 2 Aug 2010; accepted 4 Aug 2010; published 9 Aug 2010(C) 2010 OSA 16 August 2010 / Vol. 18, No. 17 / OPTICS EXPRESS 18180

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1. Introduction

Over the past decades the generation and detection of terahertz (THz) pulses have progressed rapidly since these pulses allow many fascinating and important applications, such as medical diagnosis [1,2], and security inspection [3]. Several methods have been developed for the detection of THz pulses, all of which have their specific merits. For instance, it has been demonstrated that a Si bolometer is a very sensitive detector. However, a severe disadvantage of the bolometer is that it must be operated at cryogenic temperatures to reduce noises. Moreover, the bolometer is a square-law detector, which loses all information about the phase of a THz electric field. Thus, there is a critical need for a sensitive detector that works at non-cryogenic temperature. Recently, Guo and the co-operators reported an experiment for coherent detection of narrow-linewidth nanosecond THz radiation at room temperature using a parametric process in nonlinear MgO:LiNbO3 crystals, which demonstrated a new way to realize a kind of THz wave detector that was capable of capturing the temporal profile of THz pulses with nanosecond resolution [4]. Such a detector has several advantages: First, it has at least the sensitivity of an order or more than a typical Si bolometer does for detecting nanosecond THz wave pulses. Second, it could operate at room temperature, unlike a liquid-He-cooled Si bolometer that must work at 4.2 K to reduce noise. And last but not least, room-temperature direct THz wave detectors such as pyroelectrics and Golay cells offer poor sensitivity.

The main optical part of such a detector consisted of two pieces of MgO:LiNbO3 crystals,

as shown in Fig. 1(a). In the first crystal, an intense near-infrared pump beam (p

ω ) was

normally incident on and totally reflected at the inner probe surface of the crystal. The

reflected pump light interacted with an incident THz wave (T

ω ), thus generating a light wave

(namely idler wave) at a difference frequency (id p T

ω ω ω= − ) due to stimulated polariton

scattering [5,6]. Nearly all of the THz wave are absorbed due to the heavy absorption loss or are totally reflected by the right beveled side of the first crystal due to the large refractive index (greater than 5) in THz range [7,8].The pump light and the idler light were then incident to the second crystal, in which the idler light was parametrically amplified. The output idler light was detected with an InGaAs-based photodiode. It was believed that the intensity of the output idler light should be proportional to the intensity of the incident THz wave, which was confirmed by the experiment. The parameters of the incident THz wave could then be determined by studying the measured idler light. Obviously, it needs a strict theoretical model to describe the characteristic of the detector and to guide the design of the detector in the further. Very recently, we established a steady-state model for the detector [9], which can explain the main experimental results and can provide some proposals to improve and optimize the detector to some extent. However, because the detector is operated at unsteady state, the steady-state model does not work well all the time. Therefore, in this paper we establish a time-dependent theoretical model to describe the temporal behavior of the detector.


Our results demonstrate that the pulse width and energy of the output idler wave are proportional to those of the incident THz wave for the case of no pump depletion. Such a linear relationship indicates that the system consisted of two pieces of LiNbO3 (LN) crystals can be used as a detector to measure a nanosecond THz wave pulse. Some proposals to improve and optimize the detector are provided.

2. Theoretical analyses

Fig. 1. Schematic diagrams (a) the THz wave detector made of two LN crystals reported in [4],

and (b) wave configuration, boundary conditions and coordinate system.

In our analysis, the wave pulses propagate in the x-z plane with their direction of polarization along y-axis (paralleling to the c-axis of the crystal), as shown in Fig. 1(b). For the case that the effects of higher-order dispersion [10,11] and transverse diffraction are ignored and for quasi-monochromatic pulses, from Maxwell equations we can derive the coupled equation as

( ) ( ) ( )2

0NL 0 02

0 0

1, , exp ,

2z

z

A z t i p z t i k zz u t k c

ωϕ

ε∂ ∂ + = − + ∂ ∂

(1)

where 0ε is the vacuum permittivity, c is the light velocity in the vacuum, ( ),A z t and

( )NL,p z t are the slowly varying field envelope and the nonlinear polarization, respectively,

0ω is the angular frequency,

0zk is the z-component of wave vector at

0ω ,

0ϕ is the initial

phase. More details of the derivation processes can be found in Appendix. For the case that the frequency of the THz wave is lower than or close to the phonon

vibration frequency of the crystal, while the frequencies of the pump and idler waves are much higher than the phonon vibration frequency, following some pioneers’ works [12–14], the coupled wave equations can be written as

( ) ( ) ( ) ( ){

( ) ( ) ( ) ( ) }

20p (3)

p p 0p 0p 0id 0id id p

p 0p

(2)

p 0p 0id 0T id T 0

1, ; , , , ,

2

; , , , exp ,z

A z t i A z t A z tz u t n c

A z t A z t i k z

ωχ ω ω ω ω

χ ω ω ω ϕ

∂ ∂+ = − ∂ ∂

+ − ∆ + ∆

(2a)

( ) ( ){

( ) ( ) ( ) ( ) }

(1)0TT T 0T T

T 0T

(2) *

T 0T 0p 0id p id 0

1, ( ) ,

2

; , , , exp ,z

A z t i A z tz u t n c

A z t A z t i k z

ωχ ω

χ ω ω ω ϕ

∂ ∂+ = ∂ ∂

+ − ∆ + ∆

(2b)

( ) ( ) ( ) ( ){

( ) ( ) ( ) ( ) }

2(3)0id

id id 0id 0p 0p 0id p id

id 0id

(2) *

id 0id 0p 0T p T 0

1, ; , , , ,

2

; , , , exp ,z

A z t i A z t A z tz u t n c

A z t A z t i k z

ωχ ω ω ω ω

χ ω ω ω ϕ

∂ ∂+ = − ∂ ∂

+ − ∆ + ∆

(2c)


where p id T

0 0 0 0z z z zk k k k∆ = − − ;

0p 0id 0Tϕ ϕ ϕ ϕ∆ = − − ;

0mω ,

omn and

mu ( )p,id,Tm = denote the

angular frequencies, real refractive indices and the group velocities of the pump, idler and

THz waves, respectively; the superscript ‘*’ denotes to make conjugation to the fields; (1)

Tχ is

the linear ionic susceptibility at 0T

ω ; (2)χ is the sum of the second order electric and the ionic

susceptibilities; (3)χ is known as Raman susceptibility [15].

The group velocities of the waves in a LN crystal are p

u = 1.2673 × 108m/s for the pump

wave at 1.064 µm and T

u = 2.7693 × 107 m/s for the THz wave at 1 THz [16]. For a pump

pulse with its width being p

τ = 10ns, the walk-off length will be -1 -1

p p TdL u uτ= − = 135.6

cm, so that the walk-off effect in a LN crystal with a length of no more than twenty centimeters will generally be small. As a result, we can approximately take the pump, idler and THz waves to be of the same group velocity u . Thus we can convert Eq. (2) into a

retarded time frame specified by

and .z z t z uτ′ = = − (3)

Because the pump wave is very strong while the THz wave is relatively much weaker, we can assume that the reduction in the pump wave intensity coming from nonlinear interaction could be ignored and then the amplitude of the pump wave can be regarded as a constant. As a result, the coupled wave Eqs. (2) can be reduced and one can obtain the solutions as

( ) ( ){ ( )

( ) ( ) ( ) ( )} ( )

T T id T 0z T

T id T 0z T 0z

1, (0, ) i 2 (0, ) exp

0, i 2 0, exp exp i 2 ,

A z A g k A g zg g

A g k A g z k z

τ β τ α τ

β τ α τ

∗′− +

+ −

∗′ ′+ −

′ ′ = − − + ∆ −

′ ′ − − − + ∆ ∆

(4a)

( ) ( ) ( ) ( ){ ( )

( ) ( ) ( ) ( )} ( )

id id id T

T z id id T 0z

1, i 2 0, 0, exp

i 2 0, 0, exp exp i 2 ,

T zA z g k A A g zg g

g k A A g z k z

τ α τ β τ

α τ β τ

∗ ∗ ∗′+ +

+ −

∗ ∗′ ′− −

′ ′ = − + ∆ + −

′ ′ − − + ∆ + − ∆

(4b)

(1)0T

T T

0T

,2 cos

in c

ωα χ

ϕ= (5a)

( ) ( )(2)0TT T p

0T

0, exp ,2 cos

i A in c

ωβ χ τ ϕ

ϕ= ∆ (5b)

( )2

(3)0idid id p

0id

0, ,2 cos

i An c

ωα χ τ

θ= (5c)

( ) ( )(2)0idid id p

0id

0, exp ,2 cos

i A in c

ωβ χ τ ϕ

θ= ∆ (5d)

( ) ( )2

T id T id 0z T id

1 1i 4 ,

2 2g kα α α α β β∗ ∗ ∗

′± = + ± − + + ∆ + (5e)

where ( ) ( )p p0, 0,A A zτ τ′= = is the initial complex envelop of the pump wave at the incident

surface of the first crystal; ( )T0,A τ and ( )id

0,A τ are the initial complex envelops of the THz

and idler waves, respectively; θ is the angle between the pump wave and the idler wave; ϕ

denotes the angle between the pump wave and the THz wave as shown in Fig. 1. Using Eq.


(4), we may obtain the information of the idler and the THz waves at arbitrary position in the crystal in terms of the initial conditions.

Based on Eq. (4), we can obtain the expressions for the idler wave at the outgoing end of the two crystals in terms of the different initial conditions of the fields. For the first crystal,

the initial conditions are ( )id0, 0A τ = and ( )T

0, 0A τ ≠ , then we obtain

( ) ( ) ( ) ( ) ( )id id T 0z

exp - exp, 0, exp i 2 .

g z g zA z A k z

g gτ β τ∗ ∗+ −

′+ −

′ ′′ ′= − ∆

− (6)

It should be pointed out that nearly all of the THz waves are absorbed due to the heavy absorption losses or are totally reflected by the right beveled side of the first crystal due to the large refractive index (greater than 5) in the THz range. Therefore, we assume that there are no THz waves propagating into the second crystal from the first crystal. Moreover, the LN

crystals are transparent to the idler wave [7]. Thus, the initial conditions are ( )id 1, 0A l τ ≠ and

( )T 1, 0A l τ = for the second crystal. Thus we obtain

( ) ( ) ( ) ( ) ( )

( ) ( )

T T

id 1

id 1 0z

i 2 exp i 2 exp,

, exp i 2 ,

z zg k g z g k g zA z l

g g

A l k z

α ατ

τ

′ ′∗ + + − −

+ −

∗′

′ ′− + ∆ − − + ∆′+ =

−

′× − ∆

(7)

where 1l denotes the effective length of the first crystal. In the processes of deriving Eq. (6)

and Eq. (7), the reflection and transmission losses of the idler wave at the interfaces of the two crystals are neglected. Introducing Eq. (6) into Eq. (7), we obtain the expression for the outgoing idler wave from the second crystal as

( ) ( ) ( ) ( )(2)* *0id 0zid 1 2 id p T 1 2

0id

, 0, 0, exp i + ,2 cos 2

kA l l i A A l l

n c

ωτ κχ τ τ ϕ

θ∗ ′∆ + = − − + ∆

(8)

where

( ) ( ) ( ) ( )( )

( ) ( )

T 0 2 T 0 2

2

1 1

i 2 exp i 2 exp

exp exp ,

z zg k g l g k g l

g g

g l g l

α ακ ′ ′+ + − −

+ −

+ −

− + ∆ − − + ∆=

−

× −

(9)

and 2

l represents the effective length of the second crystal.

The expressions for the electric field of the idler, pump and THz waves are

( ) ( ) ( )id id 0id id, , exp ,E z A z iτ τ ω τ φ′ ′= − + ( ) ( ) ( )p p 0p p, , exp ,E z A z iτ τ ω τ φ ′ ′= − + and

( ) ( ) ( )T T 0T T, , exp ,E z A z iτ τ ω τ φ′ ′= − + respectively, where 0 0

1

2

m

m z mk zφ ϕ′ ′= + ( )p,id,Tm =

is the phase of the corresponding electric filed. From Eq. (8), we can obtain the expression of the electric field for the outgoing idler wave from the second crystal as

( ) ( ) ( )(2)* *0idid 1 2 id p T

0id

, 0, 0, ,2 cos

E l l i E En c

ωτ κχ τ τ

θ∗ + = − (10)

For the nanosecond pulses with slow varying envelopes, the time-dependent intensity I at

position z′ can be approximately written as ( ) ( ) ( )2 2

0 0, 2 , 2 ,I z c n E z c n A zτ ε τ ε τ′ ′ ′≈ =

[17,18]. Therefore, we obtain


( ) ( ) ( )2

2(2)*0id

id 1 2 id p T3 2

0 0p 0id 0T

, 0, 0, .8 cos

I l l I Ic n n n

ωτ κχ τ τ

ε θ+ = (11)

Equations (10) and (11) build a bridge to connect the phase and intensity information between the incident THz wave and the measured idler wave, which are the main results for the time-dependent model presented in this work.

3. Numerical analyses

A LN crystal has four Raman- and infrared-active transverse optical phonon modes called A1-

symmetry modes [19] and has a nonlinear coefficient 33

d = 25.2 µm/V at λ = 1.064 µm [20].

The expressions and data for the susceptibilities (1)χ , (2)χ and (3)χ for the A1-symmetry

modes can be taken from [9] and [21–23]. The refractive indices of the pump and idler waves can be obtained based on the Sellmeier equation for an extraordinary wave [24]. In general, the refractive index of THz waves in a LN crystal is a complex value. We write its real-part as

(1)

0T i 0TRe ( ),n ε χ ω∞= + (12)

where ε∞ is the high frequency dielectric constant.

The temporal envelops of the pulses are assumed to have a Gaussian form

( ) ( ) ( )2

0, exp 2ln2

m m mA z A zτ τ τ ′ ′= − ( )p,id,Tm = , where

0mA and

mτ are the real peak

field and the pulse width (full width at half maximum), respectively. The transverse spatial distribution of the pulse intensity is assumed to be uniform, and then the power of the pulse

can be determined by multiplying the intensity by an effective spot areaA

S . Therefore, the

time-dependent power of the pulse at z′ is ( ) ( ) ( )2

0, exp 4ln 2

m m mP z P zτ τ τ ′ ′= − , here, the

peak power ( ) ( ) ( )2

0 0 0 0,0 2m m A m A mP z I z S c n S A zε′ ′ ′≈ = ( )p,id,Tm = . In addition, the energy

of the pulse at a position z′ is simply determined by integrating the powers over time:

( ) ( ) ( ) ( )0, 2 ln 2 .

m m m mW z P z d P zτ τ τ π

∞

−∞′ ′ ′ ′ ′= =∫ (13)

Besides, we also assume that the three waves satisfy the so-called phase matched

condition (0z

0k ′∆ = ). By taking the system parameters close to the experimental ones

reported in [4], we calculate the time-dependent power envelope of the wave pulses, as shown

in Fig. 2. The calculated pulse width of the incident THz wave is Tτ = 4.2 ns, which agrees

well with the value (4.7 ns) of experimental one in [4]. The insert drawings are the initial pump pulse (left) and the measured idler pulse (right), respectively.

Fig. 2. The calculated time-dependent power envelope of the incident THz pulse with regard to an incident pump pulse (left insert drawing) and an output idler pulse (right insert drawing).

The system parameters are p

τ = 15 ns, pλ = 1.064 µm,

idτ = 4.13 ns,

idλ = 1.0697 µm,

pW = 16 mJ,

0idP = 85 mW,

AS = 1.3 × 1.1 mm2,

1l = 5 mm and

2l = 50 mm.


Fig. 3. The curves of the incident THz pulse varying with the output idler wave with p

τ = 15

ns, p

W = 16 mJ, 1l = 5 mm and

2l = 50 mm. The blue one (left coordinate) is the curve

between 0T

P and 0id

P with idτ = 4.13ns, while the red one (right coordinate) is the curve

between Tτ and

idτ with

0idP = 85 mW.

Fig. 4. The curves of the output idler pulse energy id

E varying with the incident THz pulse

energy T

E under (a) different effective lengths of the first crystal, (b) different effective

lengths of the second crystal, (c) different pulse energies of the incident pump beam, and (d)

different pulse widths of the incident pump beam. The other system parameters are A

S = 1.3 ×

1.1 mm2, pλ = 1.064µm,

Tτ = 4 ns and

0T 0T2ν ω π= = 1.5 THz.

The energy input-output characteristics of the detector are investigated in detail. That is,

the curves of the output idler pulse energy id

W varying with the incident THz pulse energy

TW are plotted under different system parameters. Figure 4 (a) shows the curves for different


values of 1l with

2l = 50 mm, while Fig. 4(b) shows the curves for different values of

2l with

1l = 5 mm. Both Figs. 4(a) and 4(b) are corresponding to a given pump pulse with

pτ = 15 ns

and p

W = 16 mJ. As might be expected, id

W increases monotonically as the two effective

lengths grow. Although the influence of the two effective lengths on id

W is very small, it can

be clearly seen from the illustrations within the Figs. 4(a) and 4(b). Furthermore, the impact of

1l on

idW is more significant than that of

2l . Therefore, one should pay more emphasis on

elevating the length effective of the first crystal to achieve higher sensitivity. Figures 4(c) and

4(d) show the curves for different incident pump pulses with 1l = 5 mm and

2l = 50 mm, in

which p

τ = 15 ns for Fig. 4(c), while p

W = 16 mJ for Fig. 4(d). As can be seen, id

W increases

monotonically as p

W increases and p

τ decreases. Therefore, in order to effectively detect a

THz wave, the energy and the duration of the pump pulse should be taken large and short enough. It should be noted that these conclusions are obtained based on the assumption of no pump depletion and out of consideration of other practical factors (e.g. photorefractive effect and noise) [4,25]. In spite of this, these results are still instructive for the parameters selections of an incident pump pulse.

Conclusions

We have presented a time-dependent theoretical model to describe the temporal behavior of a THz wave detector consisted of two pieces of LiNbO3 crystals. This model is suitable for the case that the electromagnetic waves are quasi-monochromatic nanosecond pulses. By taking the system parameters close to the experimental ones reported in [4], the calculated pulse width of the incident THz wave agrees well with the experiment one. The input-output characteristics of the detector are investigated and some conclusions, which could be useful for guiding and optimizing the design of the detector, are obtained. We believe that this study provides a strict theoretical basis for the highly sensitive coherent THz detector.

Appendix: Coupled wave equations for quasi-monochromatic pulses

From Maxwell equations the nonlinear scalar equation describing the propagation of the electric field of plane wave can be written as

2 (1) 2

2 NL

2 2 2 2

0 0

1 1,

D PE

c t c tε ε∂ ∂

∇ = +∂ ∂

(A1)

where (1)

0 LD E Pε= + , and

LP is the linear polarization. We express the quantities in terms of

their Fourier transform as

( ) ( ), , , , / 2 ,i t

E x z t E x z e dωω ω π

∞−

−∞

= ∫ ɶ (A2a)

( ) ( )(1) (1), , , , / 2 ,

i tD x z t D x z e d

ωω ω π∞

−

−∞

= ∫ ɶ (A2b)

( ) ( )NL NL, , , , / 2 ,i t

P x z t P x z e dωω ω π

∞−

−∞

= ∫ ɶ (A2c)

We assume that ( )(1), ,D x z ωɶ and ( ), ,E x z ωɶ are related by the usual linear dispersion relation

as

( ) ( ) ( )(1)

0, , , , .

rD x z E x zω ε ε ω ω=ɶ ɶ (A3)


where ( )rε ω is the relative dielectric constant with respect to ω .

By introducing these forms into Eq. (A1), we can obtain a relation for ω frequency

component of the pulse in the frequency domain and which is given by

( ) ( ) ( ) ( )2 2

2

NL2 2

0

, , , , , , .rE x z E x z P x zc c

ω ε ω ωω ω ω

ε∇ = − −ɶ ɶ ɶ (A4)

Our goal is to derive a wave equation for slowly space and time varying field complex

envelope ( ), ,A x z t defined by

( ) ( ) ( )0 0 0 0, , , , exp . .,x zE x z t A x z t i t k x k z c cω ϕ= − − − − + (A5)

where 0

ω is the angular frequency and 0

k is the linear part of the wave vector at 0

ω , 0

ϕ is

the real constant initial phase. . .c c denotes the complex conjugate. We represent ( ), ,A x z t in

terms of its spectral content as

( ) ( ), , , , / 2 .i t

A x z t A x z e dωω ω π

∞−

−∞

= ∫ ɶ (A6)

From Eqs. (A2)a), (A5) and (A6), we have

( ) ( ) ( )

( ) ( )0 0 0 0

0 0 0 0

, , , , exp

, , exp .

x z

x z

E x z A x z i k x k z

A x z i k x k z

ω ω ω ϕ

ω ω ϕ∗

= − + +

+ + − + +

ɶɶ

ɶ (A7)

By using the quantity ( ), ,A x z ωɶ (the slowly varying field amplitude in the frequency

domain), the wave equation becomes

( ) ( )

( ) ( )

2 22 2 2

0 0 0 0 02 2

2

0 0 0 NL2

0

2 , ,

exp . . , , ,

x z x z

x z

i k k k k k A x zx z x z

i k x k z c c P x zc

ω ω ω

ωϕ ω

ε

∂ ∂ ∂ ∂ + + + + − − − ∂ ∂ ∂ ∂

× + + + =−

ɶ

ɶ

(A8)

where

( ) ( )2 2 2.

rk cω ε ω ω= (A9)

We then expand ( )k ω as a power series with respect to 0

ω as

( ) ( ) ( )00 0| .

dkk k high order dispersion

dω

ωω ω ω

ω= + − + (A10)

We shall ignore the high order dispersion, which is permissible in the case of nanosecond pulses (the dispersion length is much greater than the distance of the nonlinear pulse interaction [10,11]). So we have

( ) ( ) ( )22 2 00 0 02

12 ,

kk k

u uω ω ω ω ω= + − + − (A11)

where ( )u d dkω ω= is the group velocity of wave packet.

We now introduce ( )2k ω into the wave equation, which then becomes


( ) ( ) ( )

( ) ( )

2 220

0 0 0 0 02 2 2

2

0 0 0 NL2

0

12 2 , ,

exp . . , , .

x z

x z

ki k k A x z

x z x z u u

i k x k z c c P x zc

ω ω ω ω ω ω

ωϕ ω

ε

∂ ∂ ∂ ∂ + + + + − + − − ∂ ∂ ∂ ∂

× + + + = −

ɶ

ɶ

(A12)

We convert this equation back to the time domain and obtain

( )

( ) ( )

2 2 2

00 02 2 2 2

2

NL

0 0 0 0 2 2

0

12 2 , ,

, ,1exp . . .

x z

x z

ki k k i A x z t

x z x z u t u t

P x z ti k x k z t c c

c tϕ ω

ε

∂ ∂ ∂ ∂ ∂ ∂ + + + + − ∂ ∂ ∂ ∂ ∂ ∂

∂× − + + − + = ∂

(A13)

In our model, the directions of energy flows are the same to those of wave vectors. Because of the small angular between the pump wave and the idler wave, the x-direction component of energy flow for the idler wave is much smaller than that along z-direction (we choose the propagation direction of pump wave along z-direction). Thus we drop the quantities along x-direction. We also adopt the so-called slowly varying envelope approximation to Eq. (A13) and simplify this expression to obtain

( ) ( ) ( )2

NL

0 0 0 0 2 2

0

,1 12 , exp . . .z z

P z tik A z t i k z t c c

z u t c tϕ ω

ε∂∂ ∂ + + − + = ∂ ∂ ∂

(A14)

We next express the polarization in term of its slowly varying envelope ( )NL,p z t as

( ) ( ) ( )NL NL 0, , exp . .P z t p z t i t c cω= − + (A15)

Then we obtain

( ) ( ) ( )

2 22NL

0 0 NL 02 2

,2 , exp . ..

P z ti p z t i t c c

t t tω ω ω

∂ ∂ ∂= − − − + ∂ ∂ ∂

(A16)

In the case of the pulse whose duration is much larger than its optical period, which can be satisfied for a nanosecond pulse in optical or THz range, we may neglect the first two terms on the right-hand side of Eq. (A16), and then this expression becomes

( ) ( ) ( )

2

2NL

0 NL 02

,, exp . ..

P z tp z t i t c c

tω ω

∂= − − +

∂ (A17)

By introducing Eq. (A17) into the wave equation in the form of Eq. (A14), we obtain

( ) ( ) ( )2

0NL 0 02

0 0

1, , exp .

2z

z

A z t i p z t i k zz u t k c

ωϕ

ε∂ ∂ + = − + ∂ ∂

(A18)

Acknowledgements

The National Natural Science Foundation of China under grant No. 10974063, the Research Foundation of Wuhan National Laboratory under Grant No. P080008, and the National “973” Project under Grant No. 2007CB310403 have supported this research. The authors thank the reviewers for their valuable comments and helpful suggestions on the quality improvement of our present work.


time-dependent theoretical model for terahertz wave detector using a parametric process

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