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МИНОБРНАУКИ РОССИИ

Санкт-Петербургский государственный электротехнический университет «ЛЭТИ» им. В. И. Ульянова (Ленина)

V. Yu. VENEDIKTOV

FUNDAMENTALS OF NONLINEAR OPTICS

Tutorial

Санкт-Петербург Издательство СПбГЭТУ «ЛЭТИ»

2017

УДК 535 (07)

ББК З 86я7 V 38

Venediktov V. Yu.

V 38 Fundamentals of nonlinear optics: Tutorial. SPb.: SPbGETU «LETI» publishers, 2013. 43 p.

ISBN 978-5-7629-2101-5

The fundamentals of nonlinear optics and a number of its most important phenomena used in the development of laser systems are considered. The principles of the device of nonlinear-optical converters of the laser radiation wavelength, systems of dynamic holography and wave front reversal are described, including on the basis of stimulated light scattering. An overview of modern methods for obtaining short and ultrashort pulses of optical radiation is given. The fundamentals of optics and electro-optics of liquid crystals and devices based on them are also presented.

Tutorial is intended for the preparation of masters on the innovative educational program «Laser measuring technology». It can also be useful for engineering and technical workers in this field of knowledge.

УДК 535 (07)

ББК З 86я7 Reviewed by Department of Optical Information Technologies and Materi-

als of ITMO University; Doctor of Science (Physics and Mathematics), professor I.M.Belousova (S.I. Vavilov State Optical Institute).

Утверждено редакционно-издательским советом университета

в качестве учебного пособия

ISBN 978-5-7629-2101-5

INTRODUCTION

The discovery of the high-power sources of coherent light, i.e. of lasers, has resulted in arrival of the new field of investigations, dealing with the interaction of the high-power coherent light flows with the matter. This field of science is known as the nonlinear optics. The optical effects whose character depends upon the radiation intensity are known as the nonlinear ones, and the field of science, investigating the nonlinear optical effects, i.e. the optics of intense radiation, is also known as the nonlinear optics.

Before lasers only few nonlinear-optical effects were known, because in the pre-laser era the light sources were characterized by very low intensity of radiation and hence practically all observed optical effects were linear. Only the use of laser radiation, whose electric field intensity can be as high as 107–

109 V/cm or more, and thus can compete with the intra-atom field intensity, has led to discovery of numerous nonlinear effects. These effects has, in particular, provided the development of numerous practically important optical devices, whose properties seem to be impossible within the frame of conventional (classical) optics. These are, for instance, the devices, providing summation and multiplication of laser radiation frequencies, self-focusing of laser beams, phase conjugation etc. Such devices are now widely used in laser technique. We shall briefly consider some of them.

1. BASIC EQUATIONS OF THE NONLINEAR OPTICS

1.1. Dielectric polarization in the static electric field

Any electromagnetic process in the medium is described by the Maxwell’s equations:

rot E = −	∂B ; div B = 0;	rot H = j +	∂D ; div D = ρ.	(1.1)
	∂t		∂t

Here E(r, t), H(r, t) are the vectors of the electric and magnetic field intensity correspondingly in the point r at the moment of time t; ρ is the charge density; j is the electric current density; and D and B are the vectors of the electric and magnetic induction correspondingly.

This set of equations is not sufficient for solving the electromagnetic problem. One needs also the so-called material equations, which establish the additional mutual correspondence between these vectors:

D = ε0 E + P '; B = µ0 H + M '; j = σE .

(1.2)

The equations (1.2) are establishing the relationships between the vector of

macroscopic polarization of the medium P', of the macroscopic magnetization of the medium M' and the vectors D and B, and also with the current density j and electric field intensity E. Since in the optical spectral range the radiation interaction with matter is usually caused by electric interaction, we shall further take into account the magnetic properties of matter.

In the isotropic case the macroscopic polarization of medium depends upon the electric field intensity E. The proportionality coefficient in such a dependence law is known as the electric susceptibility χ(E). In general case it can also be the function of E. With account for such a dependence, the equation (1.2) transforms

into:

P '= ε0 c(E )× E ® D = (1 + c(E ))× ε0 × E = ε0 × ε(E )× E .

(1.3)

The magnitude ε(E)=1+χ(E) is known as the dielectric permittivity. In the

weak field both susceptibility and permittivity are constants, which do not depend upon the field intensity. Hence the reaction of the medium to the external field has the linear nature:

P'= ε0	× c0 × E; D = (1 + c0 ) × ε0 × E = ε0 × ε × E; e = 1 + c0 .	(1.3a)
The nonlinear	effects reveal themselves only when the field is sufficiently

strong and one can no more treat the values of χ and ε as independent upon the field intensity. Let us illustrate the arise of nonlinear behavior of χ and ε by calculation of their values within the simple classical problem. Let us consider some gas, consisting of atoms (two point charges – nucle i and electron) without the constant electric dipole moment. In the absence of external field the positions of both point charges coincide. Let the constant external electric field is now applied to such a gas. For simplicity let the electron displacement coincides with the external electric field direction; in this case one can neglect the vector nature of the problem, and solve it in scalars. Hence the atoms will accumulate the dipole moment of d = e r.

For gas, consisting of N atoms, its macroscopic polarization will be equal to:

P' = N × d = N × e × r .

(1.4)

The electron is subjected to two forces action. The action of the electric field

is described by the relationship F1 = eE, while the elastic force tends to return the electron to its primary position F2 = – k r – q r3; in general case this force reveals

the nonlinear dependence upon the electron displacement. These forces are to be mutually equal, and thus we obtain the equation for the electron displacement in the external field1

		eE = kr + qr3 .				(1.5)
Keeping in mind (1.4) for r, we can get from (1.5) the following nonlinear
equation for polarization:
	e2 N	E = P +	q	P'3 .		(1.6)
		E = P +		P'3 .		(1.6)
	k		ke2 N 2	0
	k		ke2 N 2
Let us solve it for P' in assumption of small P'3 term. Let P′ = P′					+ P′	, cor-
				0	1

responding to two orders of minority. Hence we can get from (1.6) two equations for the terms of zero and first minority orders. We can solve these equations:

P′ =

→ P′ = P′

+ P′

P′ = −

P'3

From comparison with (1.3) we obtain:

χ(E) =

e2N

−

qe4N

k 4

		2	N		qe	4		N
=	e			−					E	2 E .
							4
		k			k

E2 ,

(1.7)

(1.7а)

In other words, the susceptibility is the nonlinear function of the field intensity. If the field is sufficiently weak (it has to be much smaller than the intra-atom field), one can neglect the second term. It means that the displacement r is small and one can neglect the term qr3 in the relationship for F2; in this case the susceptibility is linear.

Till this moment we were considering the case of isotropic medium. For the anisotropic medium the susceptibility and permittivity are scalars no more – now these are the second rank tensors. In this case the relationship of the vectors P', D,

E looks like:

3	3	3
Pi′ = ∑ χij (E)E j ;	Di = ∑ (δij + χij (E))E j = ∑ εij (E)E j .		(1.8)
j =1	j =1	j =1

Here δij is the unit tensor.

For the Cartesian coordinate set is valid:

1 Here and further on the record like r3 means some vector, whose direction coincides with r, while its magnitude is equal to r|3.

We can separate the terms, containing cos ωt and

Dx = εxx Ex Dy = ε yxEx Dz = εzx Ex

+εxy Ey

+ε yy Ey

+εzy E y

+ ε xz Ez ;

+ ε yz Ez ; (1.8а)

+ εzz Ez .

If one knows the external electric field and the susceptibility for the given crystal (it is usually determined in experiment), he can easily calculate the crystal polarization.

1.2. Dielectric polarization in the optical electric field

Let us now consider the polarization of dielectric in a high-frequency field, using the same simple model as considered in 1.1. Now the electric field intensity depends upon time, and thus one has to solve the dynamic problem instead of static one. The equation for the electron movement looks like:

m	d 2r	= F + F + F ;	F = −mγ	dr	,	(1.9)

	dt2	1 2 att	att	dt

Here Fatt is the attenuation force, which is proportional to the electron movement velocity; F1 is the electric force from the external electric field; and F2 is the elastic field. Let us use the linear approximation for the elastic field as is valid for the weak field: F2 = – k r. Thus we obtain from (1.9):

m	d 2r		= eE(t) − kr − mγ			dr	;

		dt 2				dt		(1.10)
d 2P'+ ω2P'+γ				dP' =	e2 N
						E(t).

dt 2		0		dt m

Second equation uses the representation of radius-vector r in terms of po-

larization. The value ϖ02 is equal to k/m.

Let the field changes with time in accordance with the harmonic law E(t) =

E0 cos ωt. So we shall search for the solution for polarization in terms of P' = P'0 cos(ωt + ϕ). Let us differentiate this equation. Then (1.10) transforms into:

(ϖ2	− ϖ2 )P (cos ϖt cos ϕ − sin ϖt sin ϕ)−
0	0		2				(1.10a)
− γϖP (cos ϖt sin ϕ − sin ϖt cos ϕ) =		e		N
					E	0	cos ϖt.

0			m

sin ωt:

− (ϖ02 − ϖ2 )sin ϕ − γϖ cos ϕ = 0;

(ϖ2

− ϖ2 )P cos ϕ − γϖP sin ϕ =

e2 NE0

First equation gives to us the polarization phase

tg ϕ = −

γϖ

ϖ02 − ϖ2

and thus second equation gives to us:

e2N

1 + tg

P' =

m E0

= e2N

(ϖ02 − ϖ2 )− γ cosϕ

m (ϖ02 − ϖ2 )2 + (γϖ)2

Hence obviously the solution for the polarization looks like

(1.11)

(1.11a)

(1.11b)

P'0 =	e2N	E0 cos(ϖt + ϕ)	.	(1.12)

	m	(ϖ02 − ϖ2 )2 + (γϖ)2
	m	(ϖ02 − ϖ2 )2 + (γϖ)2

So now we know that the polarization changes with the same frequency ω as the external field, and its amplitude strongly depends upon the ratio of frequencies ω and ω0:

a)in the resonance, when ω = ω0, the amplitude is maximal:

b)far from the resonance conditions, i.e. when |ω−ω0| >> γ,is valid:

γϖ		γϖ		γϖ		γ
	=	(ϖ0 − ϖ)(ϖ0 + ϖ)	≈	2ϖ(ϖ0 + ϖ)	=	2(ϖ0 + ϖ)	<< 1.
ϖ02 − ϖ2

In this case the polarization phase will be close to zero, and the polarization is determined by

P'=

e2N E

cos ϖt

e2 N

E(t)

= χ(ϖ)E(t),

(1.12а)

ϖ02 − ϖ2

In other words, the susceptibility depends upon frequency.

In the constant field limit we obtain for the susceptibility the formula, which is analogous to (1.7a) (ω = 0) → (1.12a)):

χ(0) = e2 N = e2 N . 2mϖ02 k

Till this moment we have supposed that the electron is subjected to the low

intensity field, and thus we have use for the elastic force F2 = – kr, i.e. its linear approximation, which is valid for the case of small displacement of the electron. Now let us suppose that the electric field intensity and thus the electron displacement can be sufficiently high, and thus the elastic force equation looks like:

d 2r

= eE − kr − qr3

− mγ

;

dt2

d 2P'

+ ω2P' + γ

dP'

P'3

e2N

E .

dt2

e2N 2m

(1.13)

(1.14)

Let us once again suppose that the field E(t) obeys the harmonic law, and let us consider the nonresonant case (|ω– ω0| >> γ). Let us neglect the terms, containing γ. Let us once again search for solution the combination of two minority orders

P′ = P′

+ P′

.Hence, collecting in (1.14) the terms of various order of minority, we

get:

′

+ ω0P0 =

E(t);

(1.15)

dt2

2P′

′

′3

= 0.

(1.16)

dt2

+ ω0P1

e2N

We have already solved the first of these equations – it is the solution far from resonance (1.12a). Let us put this solution into (1.16):

	d 2P′	2 ′	qχ3
1			3
		+ ω0P1 = −		E (t) .	(1.17)
	dt2
			e2N 2m
The field intensity varies in accordance with the harmonic law, and thus:
	E 3 (t) = 1 4 × E03 (3 cos vt + 3 cos vt ) .				(1.18)

Equation (1.18) is the harmonic oscillator equation, where the oscillator is

subjected to the external force, comprised by two components. One of these components varies with the frequency ω, and the second one with the frequency 3ω.

Let us thus search for the solution in terms of виде P′						= P′	cos ϖt + P′	cos 3ϖt .
				1		1,ϖ	1,3ϖ
We can obtain from (1.18) the following:
	= −	3 4qχ3(ϖ)E3
P'			0		;			(1.19)
P'			2 (ϖ02 − ϖ2 )		;			(1.19)
1,ϖ		me2N
		me2N
			8

P'	= −	1 4 qχ3(ϖ)E30	.

1,3ϖ		me2N 2 (ϖ02 − 9ϖ2 )

Uniting (1.19) and (1.20) we can get the general solution:

P' = P'0 + P'1 = c(v, E0 )× E0 cos vt + c(3v, E0 )× E0 cos 3vt ;

χ(ϖ, E0 ) = χ(ϖ0 )−

3 4 qχ3 (ϖ)E30

;

χ(ϖ) =

e2 N

;

ϖ02 − ϖ2

me2 N 2 (ϖ02 − ϖ2 )

χ(3ϖ, E0 ) = −

1 4 qχ3

(ϖ)E30

me2 N 2 (ϖ02 − 9ϖ2 )

(1.20)

(1.21)

(1.22)

Hence in the intense light field the polarization is function not only of the input radiation frequency, but also of its third harmonics. It is well known that the charge, which is harmonically moving with some frequency, is emitting the monochrome electromagnetic wave of the same frequency. Hence in the considered problem we get two waves – one with the frequency ω and the second one with the frequency 3ω. In other words, we have shown within the simplest model how the nonlinear properties of medium lead to excitation of the higher harmonics in the intense light field.

1.3. Nonlinear susceptibility tensor

Let us consider the nonlinear interaction of two electromagnetic fields. Let one of them, polarized along the axis j, is described by the relationship:

Eϖ 1	(t ) = Re (Eϖ 1	exp iϖ t )= 1 2	(Eϖ 1	exp iϖ t + c.c.) 1,	(1.23)
j	j	1	j	1

and the polarized along k is described by Eϖk 2 (t) = Re(Eϖk 2 exp iϖ 2t ). . If the medium is a nonlinear one, the presence of these two fields can lead to arise of polarization at the frequencies nω1 + mω2, where n and m are the integer numbers. Let us write out the i-the component of polarization at the frequency ω3 = ω1 + ω2 as follows:

Piϖ 3 =ϖ1 +ϖ 2 (t) = Re(Piϖ 3 exp iϖ 3t ),

1 The abbreviation “c.c.” means “complex conjugate”.

and thus we can determine the tensor of nonlinear susceptibility (earlier we have

introduced the linear susceptibility tensor cik) d	ϖ3 =ϖ1 +ϖ2	, using the following re-
	ijk
lationship for the complex amplitudes:
Piϖ3 = ∑∑dijkϖ3 =ϖ1 +ϖ2 Eϖj 1 Ekϖ2 .			(1.24)
j k

The susceptibility tensor dijkϖ4 =ϖ1 −ϖ2 can be introduced in a similar manner:

Piϖ4 = ∑∑dijkϖ4 =ϖ1−ϖ2 Eϖj 1 Ek−ϖ2 ,	(1.25)
j k

Here, in accordance with (1.23), Ek−ϖ 2 = (Eϖk 2 )* .

We shall start consideration of interaction of the electromagnetic from the Maxwell’s equations, where the polarization P is written out:

rot H = j +	∂D	= j +	∂(ε0E + P)	; rot E = -	∂(μ0H)
	¶t		¶t		¶t .

waves

(1.26)

Let us consider the polarization as the sum of linear (“lin”) and nonlinear (“nl”) terms. Then the first equation will look lik e:

P = e	0	c	lin	E + P ;,	rot H = sE + ∂(εE) + ∂Pnl .			(1.27)
	0		lin	nl		¶t	¶t
						¶t	¶t
Let us calculate the rotor of both parts. Keeping in mind the relationship for
rot H (1.27) and also that rot rot E = grad div E –						Ñ2E and that div E = 0, we ob-
tain:
Ñ2E = m0s ¶E + m0e					¶2E + m0	¶2P .		(1.28)
				¶t	¶t2	¶t2

Let us carry out the further consideration only for the single dimension case (¶/¶x = ¶/¶y = 0). We shall limit ourselves by consideration of interaction of three frequencies. The interacting fields will be presented as the plane waves:

Eϖ 1 (z, t) = 1 2[E		(z) × exp i(ϖ t - k z) + c.c.];
i	1i	1 1
Eϖk	2 (z, t) = 1 2[E2k (z) × exp i(ϖ 2t - k2 z) + c.c.];				(1.29)
Eϖ 3 (z, t) = 1 2 [E		(z) × exp i(ϖ t - k	3	z) + c.c.],
j	3 j	3	3

Here i, j, k are the Cartesian coordinates. Note that for Pnl = 0 the solution of the equation (1.28) corresponds to relationships (1.29), where the amplitudes do not

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