Spectral Approach in the Analysis of Pulsed Terahertz Radiation

Spectral Approach in the Analysis of PulsedTerahertz Radiation

Anna A. Ezerskaya & Dmitry V. Ivanov & Sergey A. Kozlov & Yuri S. Kivshar

Received: 17 January 2011 /Accepted: 30 April 2012# Springer Science+Business Media, LLC 2012

Abstract We derive the spectral analogues of the Maxwell equations for describing thepropagation of electromagnetic waves in linear and weakly nonlinear dielectric media,which can be useful for the THz spectroscopy of short pulses. We discuss the solutions ofthose equations for TM and TE polarized nonlinear waves. We obtain analytical solutions ofthese equations for the case of linear homogeneous isotropic and weakly nonlinear media,and also analyze the patterns of the Fresnel and Fraunhofer diffraction of single-cycleGaussian THz pulses.

Keywords Maxwell’s equations . Terahertz spectroscopy . Ultrashort pulse . Single cyclepulse . Fresnel diffraction . Fraunhofer diffraction

1 Introduction

In the past 20 years terahertz (THz) spectroscopy emerged in radiophysics, as a powerful toolfor the study of the propagation of short sub-millimeter pulses, and also in optics, for theanalysis of waves in the far-infrared spectral range [1, 2]. Because at the THz frequencies wemeasure the wave amplitude rather than its intensity, in THz spectroscopy the signal carriespotentially more information about the properties of the medium where it propagates, incomparison with the conventional optical spectroscopy based on the intensity measurement [3].

The techniques for generating THz radiation as well as the physics behind the propaga-tion of short pulses have a long history [4], however recent advances in novel, highlyefficient generation techniques [5, 6] attracted an additional interest to the physics of few-cycle pulses and opened the door for a variety of new applications including biology andmedicine [7], security control [8], and material characterization [9, 10].

J Infrared Milli Terahz WavesDOI 10.1007/s10762-012-9907-9

A. A. Ezerskaya :D. V. Ivanov (*) : S. A. Kozlov : Y. S. KivsharSt. Petersburg State University of Information Technology, Mechanics and Optics, St. Petersburg 197101,Russiae-mail: [email protected]

Y. S. KivsharNonlinear Physics Center, Research School of Physics and Engineering, Australian National University,Canberra ACT 0200, Australia

In particular, it was shown that THz radiation can be generated by femtosecond lasers, aswell as by using the photoconduction effect in semiconductors [11]. Generated radiationlooks like a burst of the electromagnetic field being represented only by a few completeoscillations of the field [11, 12]. Such short pulses are usually referred to as single-cyclepulses with superwide temporal spectrum. Because the diameter of a light spot generated bya femtosecond laser at a surface of semiconductor can be comparable with the wavelength ofthe generated THz radiation, both temporal and spatial spectra of such single-cycle pulsescan be very broad. In addition, it is expected that the modern sources of THz radiation can bemade powerful enough, so that the corresponding nonlinear effects should be taken intoaccount [13, 14].

In the recent years, several approaches to the analysis of THz wave propagation in linearand nonlinear media were suggested. First, the Gaussian beam optics was used by Ziolkow-ski and Judkins [15] who predicted temporal reshaping of a half-cycle pulse as a result of itsdiffractive propagation. Kaplan [16] extended that approach to the case of pulses with anarbitrary temporal shape. Jepsen et al. [17] as well as You and Bucksbaum [18] used the so-called ABCD matrix formalism to simulate the propagation of THz waves numerically.However, all these approaches are based on the paraxial approximation for THz waves, andthey are valid only for the beams with the Gaussian transverse profile.

In this paper we introduce the spectral method for the analysis of the THz radiationpropagating in both linear and weakly nonlinear media, and for short pulses with arbitrarytemporal and spatial profiles. Our approach does not rely on the paraxial approximation, butit is simple enough in comparison with the well-known methods for the analysis of theelectromagnetic waves propagating in both linear and nonlinear media.

The paper is organized as follows. In Sec. 2 we introduce the spectral equations whichprovide spectral analogue of Maxwell's equations, employed in Sec. 3 for the derivation ofgeneral equations describing the dynamics of spatiotemporal spectra of nonparaxial waves.In Sec. 4 we consider some general methods for solving those nonlinear equations. Thespecial case of linear waves is discussed in Sec. 5 and Sec. 6, where we also consider waveswith the Gaussian transverse profile. Section 7 is devoted to the study of diffraction patternsof single-cycle Gaussian THz waves, whereas Sec. 8 concludes the paper.

2 Spectral analogues of Maxwell's equations for electromagnetic waves

Maxwell's equations describing the propagation of electromagnetic waves in classical opticscan be written in the following form (using the CGSM system) [19],

r� E!¼ � 1

c@ B!@t

r� H!¼ 4p

c j!þ 1

c@D!@t

rD!¼ 4pρ

rB!¼ 0;

8>>>>><>>>>>:ð1Þ

where E!

and H!

are electric and magnetic fields of the electromagnetic radiation,

respectively, D!

- a dielectric flux density, B!

- a magnetic induction, j!

is a density ofan electric current, ρ is a density of an extraneous electrical charge, ∇ - the del operator, t - time;c - velocity of light in vacuum. In this paper we restrict ourselves by the frequently used case

when the external charges and currents are absent (ρ00 and j!¼ 0 ).

J Infrared Milli Terahz Waves

We assume that the medium has a selected direction z along which radiation propagates,whereas the directions x and y can be treated as the transverse directions. Certain directionsof propagation of light waves are formalized by the requirements,

Ex;y;z ! 0;x ! �1y ! �1

Hx;y;z ! 0;x ! �1y ! �1

Dx;y ! 0;x ! �1y ! �1

Bx;y ! 0;x ! �1y ! �1

ð2Þ

where Ex,y,z and Hx,y,z are the Cartesian components of the electric and magnetic fields,respectively, Dx,y and Bx,y are the Cartesian components of the dielectric flux density andmagnetic induction.

From the equations for electromagnetic fields (1), we obtain the equations for the spectralcomponents of light radiation by applying the Fourier transform,

gx;y;z w; kx; ky; z� � ¼ Rþ1

�1

Rþ1

�1

Rþ1

�1Ex;y;z t; x; y; zð Þe�i wtþkxxþkyyð Þdtdxdy

hx;y;z w; kx; ky; z� � ¼ Rþ1

�1

Rþ1

�1

Rþ1

�1Hx;y;z t; x; y; zð Þe�i wtþkxxþkyyð Þdtdxdy

dx;y;z w; kx; ky; z� � ¼ Rþ1

�1

Rþ1

�1

Rþ1

�1Dx;y;z t; x; y; zð Þe�i wtþkxxþkyyð Þdtdxdy

bx;y;z w; kx; ky; z� � ¼ Rþ1

�1

Rþ1

�1

Rþ1

�1Bx;y;z t; x; y; zð Þe�i wtþkxxþkyyð Þdtdxdy;

8>>>>>>>>>>><>>>>>>>>>>>:ð3Þ

where gx;y;z; hx;y;z; dx;y;z; bx;y;z are spectral components of the corresponding fields,

whereas ω,kx,ky are the frequencies of temporal and spatial spectra, respectively.

Multiplying all equations of the system (1) by the factor e�i wtþkxxþkyyð Þ , we integrate

them on t, x and y from −∞ up to +∞, taking into account the relations ρ00, j!¼ 0 and

requirements (2), we simply obtain the spectral analogues of the system (1) in the form,

@gx@z � ikxgz ¼ �i wc by@gy@z � ikygz ¼ i wc bxkygx � kxgy ¼ w

c bz

8><>: ð4Þ

@hx@z � ikxhz ¼ i wc dy@hy@z � ikyhz ¼ �i wc dxkyhx � kxhy ¼ � w

c dz

8><>: ð5Þ

@dz@z

þ ikxdx þ ikydy ¼ 0 ð6Þ


@bz@z

þ ikxbx þ ikyby ¼ 0: ð7Þ

Under the conditions (2), the system of equations (4)-(7) has the same solutions, assystem (1), so in this sense these two systems are equivalent.

We restrict our further analysis by the case of nonmagnetic media, for which we have

B!¼ H

!and, hence βx,y,z0hx,y,z. After differentiating the first two equations of the system

(4) in z and expressing hz through gx and gy from the third equation of the system (4), andalso in the third equation of the system (5) having expressed hx and hy (i.e. βx and βy) fromthe system (4), we obtain that the dynamics of the spectral components of the electric field isdescribed by the following equations:

@2gx@z2 � ikx

@gz@z � k2y gx þ kxkygy ¼ � w2

c2 dx@2gy@z2 � iky

@gz@z � k2x gy þ kxkygx ¼ � w2

c2 dy

�k2x � k2y

� �gz � ikx

@gx@z � iky

@gy@z ¼ � w2

c2 dz:

8>><>>: ð8Þ

It is important to mention that solutions of the system (4)-(7) are also solutions of the system(8). The reverse statement, strictly speaking, is not valid. Due to differentiation of the equations(4) with respect to z, the system (8) may possess additional solutions which would be solutionsof the system (4)-(7) with an additional term not depending on z. We notice that the derivedequations (8) under the condition (2) are the spectral analogues of the field equation

r� r� E!� �

¼ � 1

c2@2D!

@t2: ð9Þ

and, depending on the character of response D!

on radiation field E!

, they describe theevolution of electromagnetic radiation in linear homogeneous and nonuniform, isotropic andanisotropic, and in nonlinear dielectric media.

3 General equations for the spatiotemporal spectra of nonparaxial waves

The intensities of the pulsed THz radiations can reach the values of 108 W/cm2 and above. Thismeans that the propagation of such radiation in optical media may be accompanied by self-action effects [13, 14]. Therefore, as the next step we analyze the generalization of the spectralequations to nonlinear propagation of pulses with super-wide temporal and spatial spectra.

To generalize the spectral equations to the case of nonlinear media, we assume that theinteraction of light with a dielectric medium is nonresonant, so that the response of themedium (which we consider as homogeneous and isotropic) to the external field can bepresented in the form [20]

D!¼ D

!lin þ D

!nl ¼

Zþ1

�1" t � t0ð ÞE! t0ð Þdt0 þ "nl E

!� E!� �

� E!; ð10Þ

where the first term stands for the linear part of the field being characterized by dielectricsusceptibility ε, and the second term describes the nonlinear response of the medium with εnlas a nonlinear susceptibility depending on the field intensity. In optics of intensive ultra-


short pulses with super-broad temporal and spatial spectra in dielectrics we may consider thenonlinear contribution as local [20, 21]. However, it is not difficult to consider more generalcase of inertial nonlinear response of the medium [21].

Taking into account the equations (10), the system of equations (8) becomes

@2gx@z2 þ k2 � k2y

� �gx � ikx

@gz@z þ kxkygy ¼ � w2

c2 "nlFx

@2gy@z2 þ k2 � k2x

� �gy � iky

@gz@z þ kxkygx ¼ � w2

c2 "nlFy

k2 � k2x � k2y

� �gz � ikx

@gx@z � iky

@gy@z ¼ � w2

c2 "nlFz;

8>>><>>>: ð11Þ

where k wð Þ ¼ wc n wð Þ is the wave number, n(ω) is the frequency-dependent refractive index

of the medium, so that n2(ω) 0 ε(ω), where " wð Þ ¼ Rþ1

�1"ðtÞeiwtdt . Therefore, the terms

Fx;y;z w; kx; ky; z� � ¼ Zþ1

�1

Zþ1

�1

Zþ1

�1E2x þ E2

y þ E2z

� �Ex;y;ze

�i wtþkxxþkyyð Þdtdxdy ¼

¼ 1

2pð Þ6Zþ1

�1

Zþ1

�1

Zþ1

�1

Zþ1

�1

Zþ1

�1

Zþ1

�1

Xi¼x;y;z

gi w � w0; kx � kx0; ky � ky

0; z� �

�gi w0 � w00; kx0 � kx00; ky0 � ky

00; z� �

�

�gx;y;zðw00; kx0 0; ky0 0; zÞdw0dk 0xdky0dw00dkx0 0dky 00

describe the contribution of the medium nonlinear response in the governing equations.It is useful to transform the system of nonlinear equations (11) to some canonical form

that is more convenient in the spectral optics in the linear regime. To do this, we differentiate

the third equation of the system (11) in z and substitute the expressions for the derivatives @2gx@z2

and@2gy@z2 , which follow from the first and second equations of the system, respectively. Next,

we modify the first and second equations by substituting the expression for @gz@z that follows

from the third equation. After this, we obtain the nonlinear spectral equations in the form,

@2gx@z2 þ k2 � k2x � k2y

� �gx ¼ � "nl

" k2 � k2x� �

Fx � kxkyFy þ ikx@Fz@z

� �@2gy@z2 þ k2 � k2x � k2y

� �gy ¼ � "nl

" k2 � k2y

� �Fy � kykxFx þ iky

@Fz@z

� �@gz@z þ ikxgx þ ikygy ¼ � "nl

" ikxFx þ ikyFy þ @Fz@z

� �8>>><>>>: ð12Þ

We notice that the use of higher-order derivative in the third equation does not lead toadditional solutions of Eq. (12) in comparison with Eq. (11) because Eq. (12) is the spectralanalogue of the third equation of the system of Maxwell’s equations (1) for the mediumwithout free charges (ρ00).

Equations (11) and (12) allow analyzing the effect of medium nonlinearities, includingself-action of single-cycle individual unidirectional waves, interaction co- and contra-propagating pulses and collisions of optical waves with extra-wide temporal and spatialspectra. These equations can be applied to the analysis of radiation propagating in nonlineardielectrics, and not only THz radiation.


Below, we consider the simplified versions of the systems (11) and (12) for some particularcases, and also demonstrate some methods of the solution of the nonlinear spectral equations.

4 Nonlinear spectral equations for the evolution of nonparaxial two-dimensional waves

For two-dimensional waves the equations (11) and (12) can be simplified dramatically. Inparticular, for the two-dimensional TM waves polarized in the plane xz, when gy00 and thereis no dependence on ky, the system (11) can be presented in the form

@2gx@z2 þ k2gx � ikx

@gz@z ¼ �k2 "nl

" Fx

k2 � k2x� �

gz � ikx@gx@z ¼ �k2 "nl

" Fz:

(ð13Þ

For weakly nonlinear media this system can be solved by a perturbation theory. Toemploy the iteration procedure, we normalize the system (13) by introducing the dimen-sionless variables and functions,

egx ¼ gxg0; egz ¼ gz

g0; ez ¼ k0z; ew ¼ w

w0; ekx ¼ kx

k0; ek ¼ k

k0; e" ¼ "

"0; ð14Þ

where g0 is the maximum value of the spectral density of radiation at the boundary of thenonlinear medium, ω0 is the central frequency, k0 ¼ w0

c n w0ð Þ; "0 ¼ n2 w0ð Þ . Using thevariables (14) we present the system (13) in the form,

@2egx@ez 2 þ ek2egx � iekx @ egz

@ez ¼ � ek 2e" μ eFxek2 � ek2x� �egz � iekx @egx

@ez ¼ � ek 2e" μ eFz;

8<: ð15Þ

Where we consider μ ¼ "nl"0g20k

20w

20 ¼ "nlg20w

40

c2 as a small parameter

eFx;z egx;egzð Þ ¼ 1

2pð Þ4Z1�1

Z1�1

Z1�1

Z11

Xi¼x;z

egi ew � ew0;ekx � ekx0;ez� �� egi ew0 � ew00;ekx0 � ekx 00;ez� �

�egx;z ew00;ekx00; z� �dew0dekx0dew00dekx00:

We look for solutions of the system (15) in the form

egx ¼ gð0Þx þ μgð1Þx þ μ2gð2Þx þ . . .egz ¼ gð0Þz þ μgð1Þz þ μ2gð2Þz þ . . . ;

�ð16Þ

where for each component we take only two first terms into account. Then, following theperturbation theory [22] the system of nonlinear equations (15) is reduced to the system oflinear homogeneous and inhomogeneous equations,

@2gð0Þx

@ez 2 þ ek2gð0Þx � iekx @gð0Þz

@ez ¼ 0ek2 � ek2x� �gð0Þz � iekx @gð0Þx

@ez ¼ 0

8<: ð17aÞ


@2gð1Þx

@ez 2 þek 2gð1Þx � iekx @gð1Þz

@ ez ¼ � ek 2e" eFx gð0Þx ; gð0Þz

� �ek2 � ek2x� �

gð1Þz � iekx @gð1Þx

@ez ¼ � ek 2e" eFz gð0Þx ; gð0Þz

� �:

8<: ð17bÞ

The system (17a) appears in the zero-order approximation as linearization of the system(13). Its general solution can be written in the following form (provided it has no spatial

components ekx ! ek ),

gð0Þx ew;ekx;ez� �¼ C ew;ekx� �

e�i

ffiffiffiffiffiffiffiffiffiffiffiek2�ek 2x

p ez þ D ew;ekx� �ei

ffiffiffiffiffiffiffiffiffiffiek2�ek2xp ezgð0Þz ew;ekx;ez� �

¼ ekxC ew;ekx� �ffiffiffiffiffiffiffiffiffiffiek2� ek2xp e�i

ffiffiffiffiffiffiffiffiffiffiffiffiek2�ek 2x

q ez � ekxD ew;ekx� �ffiffiffiffiffiffiffiffiffiffiek2� ek2xp eiffiffiffiffiffiffiffiffiffiffiffiek2�ek2xp ez;

8>>>>>><>>>>>>:ð18Þ

where C и D are two independent constants (in z) describing the amplitudes of the wavespropagating in the forward and backward directions of the z axes. If the spectrum has the

spatial components ekx ! ek , from the second equation (17a) it follows @egx@ez ��ekx!ek ! 0 . To

have the solutions of Eq. (18) to be bounded, we assume that the following additional

condition holds, D ew;ekx� ��ekx!ek ! C ew;ekx� ��ekx!ek .In the next approximation the system (17b) takes the medium nonlinearity into account

and it has the general solution

gð1Þx ew;ekx;ez� �¼ C1 ew;ekx� �

e�i

ffiffiffiffiffiffiffiffiffiffiffiek 2

�ek2x eqz þ D1 ew;ekx� �

eiffiffiffiffiffiffiffiffiffiffiffiek2�ek 2

xeq

z�� 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiek2 � ek2xq Zezez0

sin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiek2 � ek2xq� ez�ez0ð Þ

�ek2 � ek2xe" eFx gð0Þx ez0ð Þ; gð0Þz ez0ð Þ

h i(þþiekxe" � @

eFz gð0Þx ez0ð Þ; gð0Þz ez0ð Þ� �@ez0

)dez0

gð1Þz ew;ekx;ez� �¼

ekxC1 ew;ekx� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiek2 � ek2xq e�i

ffiffiffiffiffiffiffiffiffiffiek2�ek2xq ez � ekxD1 ew;ekx� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiek2 � ek2xq ei

ffiffiffiffiffiffiffiffiffiek2�ek2xq ez �

�ek2ek2 � ek2x� �e" eFz gð0Þx ez0ð Þ; gð0Þz ez0ð Þ

h i� iekxek2 � ek2x

Zezez0

cos

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiek2 � ek2xq ez�ez0ð Þ

�ek2 � ek2xe" eFx gð0Þx ez 0ð Þ; gð0Þz ez 0ð Þ

h iþ iekxe" � @

eFz gð0Þx ez 0ð Þ; gð0Þz ez 0ð Þ� �@ ez 0

( )dez 0;

ð19ÞFirst two terms in the solution (19) describe a general solution of the homogeneous

ordinary differential equations, whereas the third term is a particular solution of thecorresponding inhomogeneous equation (17b) [23].

Approximate solution of the system (13) in the form (16) where the terms of the zero-order are given by the expressions (18), and the first-order terms are presented by Eq. (19),can describe for example, the wave interaction in weakly nonlinear media which for the THz


frequencies are generated in plasmas [24, 25], as well as self-action of strong THz wavesgenerated from subpicosecond relativistic electron bunches [26]. We notice that the appear-

ance of the spectral components ekx ! ek in the unidirectional wave due to the mediumnonlinearity describes self-reflection of radiation.

The simplicity of the perturbation theory is based on the fact that the solutions of thenonlinear spectral equations can be obtained in a simple form. For the field equations (seebelow), this is not so and the situation becomes more complicated.

Below we discuss another useful method for the analysis of nonlinear spectral equationsof nonparaxial waves for the case of two-dimensional TE polarized waves. For such waves,we can have gx0gx00 and the dependence on ky disappears. Then, Eqs. (11) and (12) can bereduced to the simple form

@2gy@z2

þ k2 � k2x� �

gy ¼ �k2"nl"Fy: ð20Þ

We transform Eq. (20), by using the substitution

gy w; kx; zð Þ ¼ G w; kx; zð Þe�iffiffiffiffiffiffiffiffiffik2�k2x

pz; ð21Þ

to the form

@2G

@z2� 2i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � k2x

q@G

@z

�e�i

ffiffiffiffiffiffiffiffiffik2�k2x

pz ¼ �k2

"nl"Fy: ð22Þ

Multiplying Eq. (22) by e�iffiffiffiffiffiffiffiffiffik2�k2x

pz , we obtain the following equation

@

@z

@G

@ze�2i


pz

�¼ �k2

"nl"e�i


pzFy: ð23Þ

Integrating Eq. (23) from z0, the boundary of a nonlinear medium, to the current coordinate z,we obtain the integro-differential equation with the first-order derivative in z of the function Gin the form,

@G@z e

�2iffiffiffiffiffiffiffiffiffik2�k2x

pz � @G

@z e�2i


pz

� �z0¼

¼ �k2 "nl"

Rzz0

e�iffiffiffiffiffiffiffiffiffik2�k2x

pz�

� 12pð Þ4

R1�1

R1�1

R1�1

R1�1

G w � w0; kx � kx0; zð Þ � G w0 � w00; kx0 � kx

00; zð Þ � G w00; kx00; zð Þ�

�e�iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w�w0ð Þ� kx�kx

0ð Þ2p

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w0�w0 0ð Þ� kx

0�kx0 0ð Þ2

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w0 0ð Þ�kx

0 02p� �

zdw0dkx0dw00dkx00dzð24Þ

Taking the integral in the r.h.s of Eq. (24) by parts, we assume

@G@z e

�2iffiffiffiffiffiffiffiffiffik2�k2x

pz

� �z0¼ � i

2pð Þ4 k2 "nl

"

n�

� R1�1

R1�1

R1�1

R1�1

e�i

ffiffiffiffiffiffiffiffik2�k2x

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w�w0ð Þ� kx�kx 0ð Þ2

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w0�w0 0ð Þ� kx 0�kx 0 0ð Þ2

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w0 0ð Þ�k2x

p� �zffiffiffiffiffiffiffiffiffi

k2�k2xp

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w�w0ð Þ� kx�kx

0ð Þ2p


0�kx0 0ð Þ2

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w0 0ð Þ�k2x

p ��G w � w0; kx � kx

0; zð Þ � G w0 � w00; kx0 � kx00; zð Þ � G w00; kx00; zð Þdw0dkx0dw00dkx00dz

ð25Þ


valid for the case when the wave propagates in one direction, and small terms up tothe order "2nlG

5 and higher can be neglected (since the original equation (20) is derived with thesame accuracy). Finally, for the new variable G we obtain

@G@z ¼ � i

2pð Þ4 k2 "nl

" �

� R1�1

R1�1

R1�1

R1�1

e�i

ffiffiffiffiffiffiffiffik2�k2x

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w�w0ð Þ� kx�kx 0ð Þ2

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w0�w0 0ð Þ� kx 0�kx 0 0ð Þ2

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w0 0ð Þ�k2x

p� �zffiffiffiffiffiffiffiffiffi

k2�k2xp

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w�w0ð Þ� kx�kx

0ð Þ2p

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2ðw0�w0 0Þ�ðkx 0�kx

0 0Þ2p

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2ðw0 0Þ�k2x

p ��G w � w0; kx � kx

0; zð Þ � G w0 � w00; kx0 � kx00; zð Þ � G w00; kx00; zð Þdw0dkx0dw00dkx00dz

ð26Þ

For the initial evolution (short propagation distances), the solution of this equation can beevaluated by employing the perturbation theory, similar to the TM waves, moreover in thecase of Eq. (26) the zero-order term does not depend on z.

The approach presented above allows to reduce the nonlinear spectral equations substan-tially when the backward waves are neglected. In this case we can analyse theoretically the self-focusing of nonparaxial beams by solving simpler equations (with the first but not secondderivatives).

We notice that using Eq. (21) we can return to the analysis of the dynamics of gy, andfrom Eq. (26) obtain the equation

@gy@z þ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � k2x

qgy þ i

2pð Þ4 k2 "nl

" �� R1�1

R1�1

R1�1

R1�1

g w�w0;kx�kx0 ;zð Þ�g w0�w0 0;kx 0�kx

0 0;zð Þ�g w0 0;kx 0 0 ;zð Þffiffiffiffiffiffiffiffiffik2�k2x

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w�w0ð Þ� kx�kx

0ð Þ2p


0�kx0 0ð Þ2

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 w0 0ð Þ�k2x

p ��dw0dkx0dw00dkx00dz ¼ 0

ð27Þ

This equation was obtained earlier in Ref. [27] (see also Ref. [28]) by selecting anonlinear term to satisfy the condition when solution of Eq. (27) are the solutions of moregeneral Eq. (20).

Finally, we notice that nonlinear optics of pulsed THz radiations is at the initial stage ofits development. More practical applications today deal with the analysis of linear propaga-tion that we consider below for specific case by applying our spectral approach.

5 Linear spectral equations in the nonparaxial and paraxial approximation

In the case of weak electromagnetic fields, the system (12) can be linearized and reduced tothe form [28]:

@2gx@z2 þ k2 � k2x � k2y

� �gx ¼ 0

@2gy@z2 þ k2 � k2x � k2y

� �gy ¼ 0

@gz@z þ ikxgx þ ikygy ¼ 0:

8>>><>>>: ð28Þ

General solution of the system of ordinary differential equations (28) can bewritten in the form

gx;y w; kx; ky; z� � ¼ Cx;y w; kx; ky

� �e�i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2�k2x�k2y

pz þ Dx;y w; kx; ky

� �ei


pz

gz w; kx; ky; z� � ¼ kxCx w;kx;kyð ÞþkyCy w;kx;kyð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2�k2x�k2y

p e�iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2�k2x�k2y

pz�

� kxDx w;kx;kyð ÞþkyDy w;kx;kyð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2�k2x�k2y

p eiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2�k2x�k2y

pz;

8>>>>><>>>>>:ð29Þ

where the constants Cx, Cy, Dx and Dy are determined from the boundary conditions.


The first terms in the right-hand sides of the equations (29) describe the diffraction of thewave propagating in the positive direction z. Second terms describe the diffraction of thebackward waves, i.e. those propagating in the opposite direction (negative z). In linear media,the propagation of such waves can be studied independently. Accordingly the diffractiondynamics of the spatiotemporal spectral components of a unidirectional wave (at Dx0Dy00)in the homogeneous, isotropic dielectric medium is described by the following relations


� �e�i


pz

gz w; kx; ky; z� � ¼ kxCx w;kx;kyð ÞþkyCy w;kx;kyð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2�k2x�k2y

p e�iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2�k2x�k2y

pz:

8><>: ð30Þ

In the equations (30) Cx and Cy are the spectral components in plane z00 which are notdefined yet. We notice that the boundary conditions for the z-components of the spectrum, asfollows from (30), aren’t arbitrary but it is linked to the integration parameters Cx and Cy.

Existence of general solutions of the linearized equations (28) in the form of elementaryfunctions (29) and (30) allows simple finding of various analytical solutions of linearequations in quadratures. Such solutions, as was shown in Sec. 4, can also be useful forsolving weakly nonlinear equations by employing the perturbation theory, where in the zeroapproximation we can use the solutions (29) and (30).

We mention that this procedure is not simple in the direct approach dealing with the fieldsrather than their spectra. To explain this statement, we notice that the Fourier transform ofthe first function of the system (30) describing the diffraction of the transverse component ofthe optical field, the expressions are rather cumbersome even for the monochromaticradiation,

E x; y; zð Þ ¼ �i kz2p

R1�1

R1�1

E x0; y0; 0ð Þ eikffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix�x0ð Þ2þ y�y0ð Þ2þz2

px�x0ð Þ2þ y�y0ð Þ2þz2

�

1� 1

ikffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix�x0ð Þ2þ y�y0ð Þ2þz2

p �

dx0dy0;ð31Þ

where x′, y′ are the coordinates in the plane z00. For kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

p� 1 the expression

(15) is nothing but a heuristic Huygens-Fresnel principle of the diffraction theory [19].Thus, a simple algebraic relation obtained in the spectral approach (30) corresponds to

rather complicated expressions such as (31) in the field approach even for monochromaticradiation. Taking into account that the Fourier method is very fast for numerical calculations[29], we expect that our spectral approach will be useful for analyzing various types ofdiffraction phenomena in both linear and nonlinear media. This approach seems mostsuitable for nonparaxial wave packets with a wide spectrum.

When the paraxial approximation is valid, i.e. for the waves with a narrow spatialspectrum, the field integral (31) becomes simpler [30] and the advantages of the spectralapproach are not so obvious. Nevertheless, the analysis based on elementary functions is stillvery convenient.

As a specific example, we consider the radiation with narrow spatial spectrum,

k2x� �

; k2y

n o� w2

� �c2

n2 wð Þ: ð32Þ

In an inequality (32) k2x� �

, k2y

n o, {ω2} are the squares of spatial and temporal

frequencies where the energy of the wave packet is mainly concentrated.


As follows from (30), in this case the longitudinal component of the field can beneglected, and the expressions for spectra of the transverse components can be presentedin a simpler form,


� �e�ikz 1�k2x þk2y

2k2

� �: ð33Þ

In what follows, we apply the general formulism outlined above to the case of Gaussiantransverse spatial distribution of the field generated at the medium boundary (at z00).

6 Evolution of the spatiotemporal spectra of paraxial Gaussian waves

Now we assume that the radiation is linearly polarized along x and its spectrum at z00 hasthe form

Cx w; kx; ky� � ¼ pρ2e�

ρ2 k2x þk2yð Þ4 G0 wð Þ: ð34Þ

i.e. the field is symmetric being described at the surface of the emitter as

Ex t; x; yð Þ ¼ e�x2þy2

ρ2 F0ðtÞ; ð35Þ

where ρ is the transverse size of the wave packet, F0(t) is its temporal profile, which is to bedetermined yet, and G0(ω) is the Fourier transform of the function F0(t).

According to (33), the spatiotemporal spectrum of a wave at an arbitrary distance z can bedescribed by the following relation

g w; kx; ky; z� � ¼ pρ2e

�ρ2 k2x þk2yð Þ

4 � 1�i 2czρ2n wð Þw

� �� e�in wð Þwz

c � G0 wð Þ: ð36ÞUsing the Fourier transform formula,

Ex;y;z t; x; y; zð Þ ¼ 1

2pð Þ3Z1�1

Z1�1

Z1�1

gx;y;z w; kx; ky; z� �

ei wtþkxxþkyyð Þdwdkxdky ð37Þ

and the relation (36), we can present the field evolution in the form

E t; x; y; zð Þ ¼ 1

2p

Z1�1

G w; x; y; zð Þeiwtdw; ð38Þ

where the spatial dependence of the temporal spectrum of radiation is

G w; x; y; zð Þ ¼1þ i 2cz

ρ2n wð Þw

1þ 2czρ2n wð Þw

� �2 e

�x2þy2

ρ2�

1þi 2czρ2n wð Þw

1þ 2czρ2n wð Þw

� �2

e�in wð Þwzc G0 wð Þ: ð39Þ

In the expressions (36)-(39) and in what follows, the index x should be used to mark thelinearly polarized waves along the axis x, which we omit for simplicity.


We would like to mention that in the expressions (36) and (39) the refractive index n(ω)can be a complex function, n wð Þ ¼ n wð Þ þ ik wð Þ , therefore these relations describe thediffraction dynamics of radiation spectra not only in transparent media but also in other typesof dispersive and lossy media with κ(ω). However, below we assume that the medium istransparent and κ(ω)00.

From (36) and (39), we can introduce the following characteristic distances,

z1 ¼ ρ2

2cn wð Þwf gmin; ð40Þ

z2 ¼ ρ2

2cn wð Þwf gmax: ð41Þ

Here n wð Þwf gminmax are the maximum and minimum values of the quantity n(ω)ω in the

frequency band where the majority of the radiation energy is trapped.For

z � z1 ð42Þthe relations (36) and (38) become,

g w; kx; ky; z� � ¼ pρ2e

�ρ2 k2xþk2yð Þ4 � e�in wð Þwz

c � G0 wð Þ; ð43Þ

E t; x; y; zð Þ ¼ e�x2þy2

ρ2 � 1

2p

Z1�1

G0 wð Þ � eiw t�n wð Þzcð Þdw: ð44Þ

Inequality (42) is usually termed as a "shadow approximation" [31]. It corresponds to thedistances near the surface of the radiation emitter. As follows from (43)-(44), for shortdistances, the transverse profile of the field distribution does not change, however we shouldtake into account a change of the phase of the wave during its propagation. For

z � z2 ð45Þthe relation (39) becomes

G w; x; y; zð Þ ¼ iρ2n wð Þw

2cz� e

� x2þy2

2czρn wð Þw

� �2

� e�i x2þy2

2czn wð Þw

� �� e�in wð Þwz

c � G0 wð Þ: ð46Þ

Inequality (42) defines the area of the Fraunhofer diffraction for all spectral parts of theradiation. For each of these parts the expression (46) describes the well-known dynamics ofGaussian beams in the far field region [32].

We now analyze a change of the temporal profile of the wave packet that accompanies itsdiffraction spreading. We restrict ourselves by the case of dispersionless media when we canpresent the refractive index in the form, n(ω)0n00const. For such media, the dynamics ofthe spectrum (46) can be presented in the form

G w; x; y; zð Þ ¼ iTðzÞw � e�T2ðzÞw2�x2þy2

ρ2 � e�iwn0c zþx2þy2

2z

� �� G0 wð Þ: ð47Þ


Here TðzÞ ¼ ρ2n02c � 1z , and the Fourier transform (38) is given by the relation

E t0; x; y; zð Þ ¼ 1

2p

Z1�1

TðzÞe�T2ðzÞw2�x2þy2

ρ2 � iw � G0 wð Þeiwt0dw; ð48Þ

where we introduce a "delay time",

t0 ¼ t � n0c

z� x2 þ y2

2z

�: ð49Þ

As follows from the relation (48), the temporal spectrum of the radiation field can be writtenin new variables t′,x,y,z as

G w; x; y; zð Þ ¼ TðzÞe�T2ðzÞw2�x2þy2

ρ2 � iw � G0 wð Þ: ð50Þ

At the center of the wave packet (at x00), the spectrum (50) takes a simple form

G w; 0; 0; zð Þ ¼ TðzÞ � iw � G0 wð Þ; ð51Þ

from which it follows that for any pulse shape the temporal structure of the field at themedium edge E0(t) in the far field diffraction region is defined by its derivative [15, 16, 30]

E t0; 0; 0; zð Þ ¼ TðzÞ @E0 t0ð Þ@t0

: ð52Þ

As follows from (50) and (51), in the Fraunhofer diffraction region for small x and y, thetemporal spectrum of the radiation field is shifted towards higher frequencies in comparisonwith the spectrum at the input G0(ω); however for larger x and y it is shifted downwards, tothe low frequencies. The energy conservation lawZ1

�1

Z1�1

G w; x; y; zð Þj j2dwdxdy¼Z1�1

Z1�1

G w; x; y; 0ð Þj j2dwdxdy ð53Þ

is valid for the dependence (46) , so the integral in (53) does not depend on the coordinate z.

7 Diffraction of a single-cycle Gaussian beam in the paraxial approximation

To give an example of the temporal evolution of a single-cycle wavepacket in the far fieldregion, we consider an input pulse of the form [see a dashed line in Fig. 1(a) below]

E0ðtÞ ¼ E0t

te�

t2

t2 ; ð54Þ

that approximates a THz pulse generated from sub-picosecond relativistic electron bunches[26] through optical rectification of femtosecond pulses in crystals [33], by a photoconduc-tion semiconductor emitter irradiated by pulsed femtosecond lasers [6]. In the latter case, thefocusing of a single-mode Gaussian beam leads to the spatial spectrum of the THz radiationin the form of Eq. (35).


The input wave (54) possesses a spectrum [see a dashed curve in Fig. 1(b)],

G0 wð Þ ¼ �ffiffiffip

p2

t2E0iwe� tw

2ð Þ2 : ð55Þ

The Fourier transform (38) of the function (50) with a specific form of the radiation spectrumof the emitter (55) can be performed in an explicit form,

E t0; x; y; zð Þ ¼ E0 � A3 x; y; zð Þ � TðzÞt

1� 2 A x; y; zð Þ t0

t

�2" #

� e� A x;y;zð Þt0tð Þ2 ; ð56Þ

where A x; y; zð Þ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2TðzÞ

tð Þ2�x2þy2

ρ2

q , TðzÞ ¼ n0ρ2

2cz , and the delay time t′ is defined by the

expression (49).In spite of the seemingly simple derivation and the final expression (56) describing the

Fraunhofer diffraction of a single-cycle pulse (54) with the transverse Gaussian spectrum(35), this result is obtained, for the best of our knowledge, for the first time. However, it isuseful to remember the analytical results obtained for the description of the waves along thebeam axes [16] in the focal plane at their focusing [34].

Near the beam axis

x2 þ y2 � ρ2t2

4T2ðzÞ ¼ a2z2; ð57Þ

where

a ¼ ctn0ρ

; ð58Þ

the expression for the field (56) becomes simpler, and it can be written in the form

E t0; zð Þ ¼ E0TðzÞt

1� 2t0

t

�2" #

e�t0tð Þ2 ; ð59Þ

which, as a matter of fact, is the derivative of the field E0(t) multiplied by the function T(z).

Fig. 1 (a) Normalized electric field E vs. time t, and (b) spectrum Gj j vs. frequency υ of a THz wave in theregime of the Fraunhofer diffraction near the axis of the wavepacket. Dashed curves show the same functionsat the input.


Next, we estimate the typical diffraction distances for one period on the emitted THzwaves (54) with the Gaussian transverse profile (36), assuming that the pulses of theduration τ00,2 ps have the transverse spatial size of ρ03 mm. The temporal profile of sucha pulse and its spectrum are shown in Fig. 1(a,b) by dashed curves. As follows from thisfigure, the main part of the radiation energy occupies the frequency interval from νmin00.25THz up to νmax 02.25 THz (e−1 level).

At the distances much smaller than z1025 mm (40), for the wavepacket under consider-ation the condition (41) is satisfied, so that the wave structure does not change substantially.However, for the distances much larger than z2022,5 cm (41), the regime of the Fraunhoferdiffraction is realized, and the THz radiation transforms into a spherical wave (56). Thetransverse size of the light spot in this diffraction regime grows in t

2T times and, forexample, at the distance of 1 m the light spot is 10 cm. In the corners smaller than α00.1

-0,5 0 0,5

0,5 0 -0,5

t, ps

x/ 0

-0,5 0 0,5

0,5 0 -0,5

-0,5 0 0,5

0,5 0 -0,5

-0,5 0 0,5

0,5 0 -0,5

t, ps

x/ 0

t, ps

x/ 0

t, ps

x/ 0

-0,5 0 0,5

0,5 0 -0,5

t, ps

x/ 0

-0,5 0 0,5

0,5 0 -0,5

t, ps

x/ 0

-0,5 0 0,5

0,5 0 -0,5

t, ps

x/ 0

a

e f g

b c d

Fig. 2 Spatiotemporal evolution of the electric field of the Gaussian THz wave with the input parameters λ000,3 mm, ρ010λ0, τ00,2 ps. This example corresponds to the propagation in air for the distances (a) 0, (b) 40,(c) 75, (d) 125, (e) 200, (f) 300, and (g) 400 mm.


(58), we may neglect the field dependence on the transverse coordinate, so that the temporalprofile of the wavepacket can be presented in the form (59). In Fig. 1(a,b) we show the pulseprofile and its spectrum after some propagation, where for comparison the input profile isgiven by a dashed line. As follows from this figure, in far field region a single-period wavebecomes 1,5 period, and its spectrum is shifted towards higher frequencies.

Figure 2 shows the spatial structure of the wavepackets with the Gaussian transverseprofile in both near and far diffraction regions, for several moments of time. Light-gray anddark-gray areas correspond to the positive and negative values of the field amplitude,respectively. As follows from the figure, in the near field area (the so-called Fresneldiffraction) we observe additional oscillations of the optical field, whereas in the far fieldarea (the so-called Fraunhofer diffraction), a single-cycle pulse is transformed into a 1,5cycle pulse for the whole wavepacket. Near the optical axis the frequency maximum isshifted to high frequencies, whereas the frequency is shifted downwards far from the opticalaxis.

8 Conclusions

Using the general form of the Maxwell equations, we have derived the dynamical systemdescribing the nonparaxial propagation of electromagnetic waves in terms of the evolution ofthe spatiotemporal spectra of the THz radiation. Our derivation is valid for homogeneousisotropic dielectric media with an arbitrary linear refractive index and weak Kerr nonline-arity. We have demonstrated that, for a broad class of problems involving the propagation ofshort pulses with broad spectra, it is possible to find the corresponding solutions by theiterative methods, because the solution of the zero-order approximation is expressed inelementary functions. We have demonstrated the simplicity of the spectral method applied tothe analysis of broadband THz radiation for a classical problem of Gaussian diffraction of asingle-cycle THz pulse.

This work was supported by grants SS-5707.2010.2, SC-16.513.11.3070 and SC-16.740.11.0459 in Russia,and the Australian Research Council in Australia.

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Spectral Approach in the Analysis of Pulsed Terahertz Radiation