Complex geometrical optics of Kerr type nonlinear media

Paweł Berczyński and Yury A. Kravtsov

1) Institute of Physics, West Pomeranian University of Technology, Szczecin 70-310, Poland2) Institute of Physics, Maritime University of Szczecin, Szczecin 70-500, Poland

SEMINAR AT THE INSTITUTE OF PHYSICSWEST POMERANIAN UNIVERSITY OF TECHNOLOGY

Szczecin, 26-th February (2010)

and transport equation for 0

AA

0)( 2 Adiv

Both amplitud and eikonal are complex-valued in framework of CGO.

2. The ray-based and eikonal-based forms of CGO

Complex geometrical optics (CGO) is known to have two equivalent forms:the ray-basedand eikonal-based ones.

Yu.A.Kravtsov. Geometrical Optics in Engineering Physics. Alpha Science International, Harrow, UK, 2005.The ray-based form deals with the complex trajectories,

Surprising feature of CGO is its ability to describe Gaussian beam diffraction.It has been demonstrated analytically (for homogeneous medium)

in frame of the ray-based form still 38 years ago in the paper: Yu.A. Kravtsov, 1967.

)(p is the ray “momentum” and d relates to the elementary arc length d by

/dd

.

It has been shown recently that Eikonal-based form deals directly with complex eikonal and is able to describe Gaussian beam diffraction for arbitrary smoothly inhomogeneous media. Now

CGO is generalized for the case of Gaussian beam (GB) diffraction and self-focusing in

nonlinear media of Kerr type, including nonlinear graded index fiber.

and transport equation for 0

AA

0)( 2 Adiv

Both amplitud and eikonal are complex-valued in framework of CGO.

2. The ray-based and eikonal-based forms of CGO

Complex geometrical optics (CGO) is known to have two equivalent forms:the ray-basedand eikonal-based ones.

Yu.A.Kravtsov. Geometrical Optics in Engineering Physics. Alpha Science International, Harrow, UK, 2005.The ray-based form deals with the complex trajectories,

rp

pr

2

1,

d

d

d

d

Surprising feature of CGO is its ability to describe Gaussian beam diffraction.It has been demonstrated analytically (for homogeneous medium)

in frame of the ray-based form still 42 years ago in the paper: Yu.A. Kravtsov, 1967.

)(p is the ray “momentum” and d relates to the elementary arc length d by

/dd

.

3. CGO for linear media of cylindrical symmetry

3.1. Riccati equation for complex parameter B

For axially symmetric GB in axially symmetric medium CGO solution has the form

is the linear wave number for GB propagating in the z direction

Complex parameter: and is a distance from the axis z

is complex-valued eikonal

The eikonal equation in coordinates takes the form:

Paraxial approximation can be expanded in Taylor series in in the

vicinity of symmetry axis z

2/expexp),( 2 BzikzAikAzu )0(

ck

22 yx IR iBBB

RB2

1

kwBI

),( z

,22

zz

,z

2

||0,2

02

2

0

z

Substitution: into eikonal equation leads to the

ordinary differential equation of Riccati type for complex parameter B:

where

Substituting into eikonal , one obtains Gaussian beam in the form:

Above solution reflects the general feature of CGO, which in fact deals with the Gaussian beams.

2/2 Bz

2Bdz

dB

IR iBBB

2exp

2exp)(exp),(

2

2

2 zikw

zAikAzu

02

2

|21

d

d

3.2. The equation for GB complex amplitude

In frame of paraxial approximation the transport equation for axially symmetric

beam in coordinates takes the form:

For eikonal we obtain that

As a result the transport equation reduces to the ordinary differential equation in the form:

The above equation for GB complex amplitude, as well as the Riccati equation for complex curvature B are the basic CGO equations. Thus, CGO reduces the problem of GB diffraction to the domain of ordinary differential equation. Having calculated the complex parameter B

from Riccati equation one can readily determine complex amplitude A:

),( z0)( 2 Adiv

01 2

2

22

Azzdz

dA

2/2 Bz 1

z

B21

0 BAdz

dA

dzzBAzA exp)0(

3.3. The equation for GB width evolution

The Riccati equation is equivalent to the set of two equations for the real and imaginary parts of the complex parameter B:

Substituting and into above equations, one obtains the known relation between the beam width and the wave front curvature [Kogelnik (1965)]:

Substituting above relation into the first equation of the system, we obtain the ordinary

differential equation of the second order for GB width evolution

The absolute value of complex amplitude: leads to

energy flux conservation principle in GB cross-section

02

22

IRI

IRR

BBdz

dB

BBdz

dB

RB 2/1 kwBI

dz

dw

w

1

322

2 1

wkw

dz

wd

dzzBAzA Reexp0

2222 00 AwAw

4. Generalization of CGO for nonlinear media of Kerr type

In nonlinear media of Kerr type the permittivity depends on the beam intensity

where coefficient is assumed to be positive: . Substituting Gaussian wave field into

above equation one obtains

Thus, in the framework of CGO method the nonlinear medium of Kerr type can be formally treated as a smoothly inhomogeneous medium, whose profile is additionally modulated by GB parameters and . The alpha parameter can be given as:

As a result the Riccati equation, generalized for nonlinear media of Kerr type has form:

where describes the linear refraction and accounts for self-focusing process

in nonlinear medium.

2u

2rrr uNLLIN

NL 0NL

2

22

exp)(,,zw

zAzz NLLIN

.NLLIN

zw

wA

d

dB

dzdB NLLIN

NLLIN 4

22

02

22 0)0(

|21

w A

LIN NL

4.1. CGO solution for nonlinear medium of Kerr type without contribution of linear refraction

When the contribution of linear term is negligibly small: the Riccati equation has the form:

and the equation for the beam width has the form (where ):

where is diffraction length and is the nonlinear scale.Following the papers of Akhmanov, Sukhorukov and Khokhlov (1967,1968,1976) we prove that:

where is the total beam power and is the critical power.

0LIN

zw

wAB

dz

dB NLNL 4

222 0)0(

0wwf

2232

2 111

DNL LLfdz

fd

02kwLD 20/0 AwL NLNL

critNL

D

P

P

L

L

2

2

220 00

8

1AwcP

NLcrit

k

cP

20

8

1

As a result, equation for GB width evolution takes the form:

The above CGO result is in total agreement with the solution of nonlinear parabolic equation (NLS) [Akhmanov, Sukhorukov and Khokhlov (1967), (1968), (1976)].

Thus, the PCGO method reproduces the classical results of nonlinear optics but in a more simple and illustrative way

[P.Berczynski, Yu.A.Kravtsov, A.P.Sukhorukov, Physica D, 239, p. 241-247, (2010)].

The three partial cases deserve to be distinguished in the above solution:

1. Under-critical power: , where the beam width increases.

2. Critical power: , Gaussian soliton. It was shown by Desaix, Anderson, Lisak, [Phys.

Rev. A. 40(5), 2441-2445, (1989)] that Gaussian soliton can approximate exact soliton

solution (hyperbolic secant) with relative error not exceeding 6%.

3. Over-critical power: , the beam width decreases to zero at a finite propagation distance and GB amplitude increases to infinity: collapse phenomenon.

011

32

2

critP

P

fd

fd

0| 0zdz

df

11

2

22

critD P

P

L

zf

critPP

Solution

critPP

critPP

DLz /

0 2000 4000 60000

0.5

1

1.5

2

trace 1trace 2trace 3

The distance kz

The

bea

m w

idth

FFig. 1. Evolution of the squared relative beam width in nonlinear medium of Kerr type for the beam power: P=1.5Pcrit (curve 1), P=Pcrit (curve 2), P=0.5Pcrit (curve 3)

It follows from CGO solution for nonlinear medium of Kerr type that the

self-focusing distance is given by:

2fF

1

crit

Df

P

P

Lz

4.2. Influence of the initial phase front curvature on the beam evolution in nonlinear medium of Kerr type

The first integral of eq. for GB width takes the form

The initial condition for at presents the squared initial wave front curvature:

As a result: , where

Taking advantage of differential relation one obtains:

CfLLdz

df

NLD

222

2111

2

1

2

1

dzdf / 0z

0|

0

1| 2

0

2

20

2

zz dz

dw

wdz

df

101 2222

2

critD

crit P

PfL

P

P

d

dff

2222 4)( fff

critD P

PL

d

fd 222

22

012

Solution

00 10210 22

2

22

z

PP

LL

zf

critD

D

DLz /

00

00

As in the previous case, it is worth analyzing the following partial cases:

1) A. Sub-critical regime:

a)For a divergent beam (positive initial curvature) : diffraction widening prevails over self-focusing effect and GB width at once increases (curve 1 in Fig. 2). For a convergent

beam, corresponding to , the beam width initially decreases reaching minimum

value

1) Next, diffraction widening dominates and the beam width starts increasing (curve 2 in Fig.

2).

critPP

1/0

/10

22min

critD

crit

PPL

PPww

at

1/0

022

2

min

critD

D

PPL

Lz

Fig. 2. Presented for and (trace1) (trace2) critPP 5.0 DL/10 DL/10

B. Critical regime:

In this case diffraction divergence and self-focusing effects compensate each other and depends only on the value of . As a result, the beam width w changes as a linear function of distance z:

a) In above solution factor leads to a growth of the GB width, whereas for one obtains Gaussian soliton with w=w(0).

b) For the beam is focused at a distance .

critPP NLD LL

2f 0

10)0(/ zwwf

00 00

00 0/11 fz

C. Over-critical regime:

Fig. 3. Evolution of for and (trace1), (trace 2)

One can notice in Fig. 3 that the positive value of the initial wave front curvature can eliminate collapse in over critical regime.

In fact one can distinguish the characteristic beam power for above which GB is always

focused regardless of the sign and value of the initial wave front curvature:

But when GB power is greater than the critical power and smaller than the characteristic power , a positive value of the initial wave front curvature eliminates the collapse effect

whereas a negative value enhances it.

critPP

2fF critPP 5.1 DL/10 DL/10

22 01 Dcrit LPP

P

C.1. When the total power is equal to characteristic power ,the beam width squared is a linear function of z:

For the beam width increases and for the beam is focused at a distance

C.2. When the GB total power is greater than the characteristic power , the GB is focused at a distance:

Fig. 4: illustrates the case

P P2f

102)0(/ 222 zwwf

00 00

02/12 fz 2

12

ff

zz critPP

PP

Dcrit

Df

LP

P

Lz

01

0 1000 2000 3000 40000

0.5

1

1.5

2

trace 1trace 2

The distance kz

The

bea

m w

idth

FcritPP 4

DL/10

5. Gaussian beam propagation in nonlinear inhomogeneous medium of cylindrical symmetry

The CGO method is applied for the beam propagation in a nonlinear and inhomogeneous medium with electrical permittivity of the form

The case when and corresponds to graded-index nonlinear optical fiber,with distance from the fiber axis. The Riccati equation takes now the form:

, where

leads to

Integration above equation with the initial conditions:

0NL

220 uNL

01 2 L

NLLINBdzdB 2

2

1

LLIN

4

2200

w

wANLNL

322

2 1

wkw

dz

wdNLLIN 0

11132222

2

fLLL

f

dz

fd

DNL)0(/ wwf

10 f 00|/ 0 zdzdf

222

22222

22111

0111

NLDNLD LLLfLLL

f

dz

df

Taking advantage of differential relation we have

and differentiating it we obtain the equation:

which has this time a simple physical interpretation in the form of a harmonic oscillator with a

constant force, where ,

This analog can let us easily find the solution of above equation in the form

2222 4)( fff

2222

2222

422 1110

11

4

1f

LLLLLL

f

dz

df

NLDNLD

2222

2

2

2

22 11102

4

NLD LLLL

f

dz

fd

2222

0111

02NLD LLL

F 0202

2

Fxdt

xd

L

zL

L

z

L

L

L

LLf

NLD

2sin0sin101 2

2

2

2

2222

zt 02 xf

6. Conclusions

1. CGO is developed to describe the diffraction and self-focusing of axially symmetric Gaussian beams in nonlinear media of Kerr type.

2. This paraxial method reduces the GB diffraction problem to the ordinary differential equations for complex curvature of the wave front, amplitude and for GB width evolution.

3. This method supplies results, which happen to be identical with the solutions of nonlinear parabolic equation (NLS) for a nonlinear medium of Kerr type.

4. CGO allows easily to include into analysis the initial curvature of the wave front and to study its influence on GB dynamics.

5. It is shown for nonlinear Kerr medium that in an over-critical regime one can distinguish the characteristic beam power for above which GB is always focused regardless of the sign and value of the initial wave front curvature. It is also shown that, when GB power is greater than the critical power and smaller than the above mentioned characteristic power, a positive value of the initial wave front curvature eliminates the collapse effect whereas a negative value enhances it.

6. CGO method is also applied for the case of inhomogeneous and nonlinear medium. The solution for GB diffraction in a graded-index Kerr nonlinear optical fiber is obtained and influence of the initial curvature of the wave front is taken into account.

7. Thereby, the complex geometrical optics greatly simplifies description of Gaussian beam diffraction and self-focusing as compared with the traditional methods of nonlinear optics based-on nonlinear parabolic equation.