Advances in imaging and electron physics, volume 185

Orthogonal Ray-Centered Coordinate System for Rotating Elliptical Gaussian Beams Propagating Along a Curvilinear Trajectory in a Nonlinear Inhomogeneous Medium 26 10.. Complex Ordinary D

Peter W HawkesCEMES-CNRSToulouse, France

CEMES-CNRS, Toulouse, France

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Academic Press is an imprint of Elsevier

Single-Particle Cryo-Electron Microscopy (Cryo-EM): Progress, Challenges, and Perspectives for Further Improvement

Advances in Imaging and Electron Physics (2014) 185, pp 113–137.

Academic Press is an imprint of Elsevier

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The three chapters in this latest volume span most of the regular themes ofthe series: an ingenious extension of geometrical optics, electron microscopyand mathematical morphology We begin with a very complete account ofcomplex geometrical optics by P Berczynski and S Marczynski Thisvariant of traditional geometrical optics allows diffraction phenomena to bestudied It has two forms, one ray-based, the other eikonal-based, and theauthors describe these fully After presenting the underlying theory, theapproach is used to study the propagation of Gaussian beams in inhomo-geneous media.

This is followed by a summary of the present state of cryo-electronmicroscopy for the study of unstained biological macromolecules by D.Agard, Y Cheng, R.M Glaeser and S Subramaniam The subject is notnew but a large step foward has recently been made with the introduction of

a new type of electron-detection camera I leave the authors to set out theadvantages of this innovation but we can be sure that many valuable newresults can be anticipated

To conclude, we have a long and authoritative account by M Welk and

M Breuß of the image-adaptive structuring elements known as amoebas

It has been shown that for iterated medianfiltering with a fixed element, the process is closely related to a partial differential equationassociated with the image in question Here, the authors examine therelation between discrete amoeba median filtering and their (continuous)counterparts based on partial differential equations I have no doubt thatthis clear and very complete account of the subject will be widelyappreciated

structuring-Peter Hawkes

viij

H.-W Ackermann

Electron micrograph quality

J Andersson and J.-O Str €omberg

Radon transforms and their weighted variants

Femtosecond electron imaging and spectroscopy

C Bobisch and R M€oller

Ballistic electron microscopy

Refelective electron beam lithography

N Chandra and R Ghosh

Quantum entanglement in electron optics

A Cornejo Rodriguez and F Granados Agustin

Ronchigram quantification

N de Jonge and D Peckys

Scanning transmission electron microscopy of whole eukaryotic cells in liquid and in-situ studies of functional materials

ixj

Recent advances in electron holography with point sources

J Grotemeyer and T Muskat

Time-of-flight mass spectrometry

M Haschke

Micro-XRF excitation in the scanning electron microscope

M.I Herrera

The development of electron microscopy in Spain

R Herring and B McMorran

Electron vortex beams

Ultrafast electron microscopy

D Paganin, T Gureyev and K Pavlov

Intensity-linear methods in inverse imaging

M Pap

Hyperbolic wavelets

N Papamarkos and A Kesidis

The inverse Hough transform

S.-C Pei

Linear canonical transforms

P Rocca and M Donelli

Imaging of dielectric objects

J Rodenburg

Lensless imaging

J Rouse, H.-n Liu and E Munro

The role of differential algebra in electron optics

The Rayleigh –Sommerfeld diffraction theory

R Shimizu, T Ikuta and Y Takai

Defocus image modulation processing in real time

T Soma

Focus-deflection systems and their applications

P Sussner and M.E Valle

Fuzzy morphological associative memories

Laboratory for Cell Biology, Center for Cancer Research, National Cancer Institute, National Institutes

of Health (NIH), Bethesda, MD 20892, USA

Martin Welk

UMIT, Biomedical Image Analysis Division, Eduard-Wallnoefer-Zentrum 1, 6060 HALL (Tyrol), Austria

xiiij

Gaussian Beam Propagation in Inhomogeneous Nonlinear Media Description in Ordinary

Differential Equations by Complex Geometrical Optics

Pawel Berczynski1, Slawomir Marczynski2

1 Institute of Physics, West Pomeranian University of Technology, Szczecin 70-310, Poland

2 CGO: Fundamental Equations, Main Assumptions, and Boundary of Applicability 6

3 Gaussian Beam Diffraction in Free Space CGO Method and Classical Diffraction

5 Generalization of the CGO Method for Nonlinear Inhomogeneous Media 18

6 Self-Focusing of an Axially Symmetric Gaussian Beam in a Nonlinear Medium of the Kerr Type The CGO Method and Solutions of the Nonlinear Parabolic Equation 20

7 Self-Focusing of Elliptical GB Propagating in a Nonlinear Medium of the Kerr Type 21

9 Orthogonal Ray-Centered Coordinate System for Rotating Elliptical Gaussian Beams Propagating Along a Curvilinear Trajectory in a Nonlinear Inhomogeneous Medium 26

10 Complex Ordinary Differential Riccati Equations for Elliptical Rotating GB

Propagating Along a Curvilinear Trajectory in a Nonlinear Inhomogeneous Medium 28

11 Ordinary Differential Equation for the Complex Amplitude and Flux Conservation Principle for a Single Rotating Elliptical GB Propagating in a Nonlinear Medium 32

12 Generalization of the CGO Method for N-Rotating GB S Propagating Along a Helical

13 Single-Rotating GB Evolution of Beam Cross Section and Wave-Front Cross Section 36

15 Three- and Four-Rotating GB S 75

of a wavefield can be calculated via diffractionless approximation (Kravtsov

& Orlov 1990;Kravtsov, Kravtsov, & Zhu, 2010) Complex generalization

of the classical geometrical optics theory allows one to include diffractionprocesses into the scope of consideration, which characterize wave ratherthan geometrical features of wave beams (by diffraction, we mean diffractionspreading of the wave beam, which results in GB having inhomogeneouswaves) Although thefirst attempts to introduce complex rays and complexincident angles started before World War II, the real understanding of thepotential of complex geometrical optics (CGO) began with the work of

Keller (1958), which contains the consistent definition of a complex ray.Actually, the CGO method took two equivalent forms: the ray-based form,which deals with complex raysdi.e., trajectories in complex space (Kravtsov

et al., 2010; Kravtsov, Forbes, & Asatryan 1999; Chapman et al 1999;

Kravtsov 1967)dand the eikonal-based form, which uses complex eikonalinstead of complex rays (Keller & Streifer 1971; Kravtsov et al., 2010;

Kravtsov, Forbes, & Asatryan 1999;Kravtsov 1967) The ability of the CGOmethod to describe the diffraction of GB on the basis of complex Hamil-tonian ray equations was demonstrated many years ago in the framework ofthe ray-based approach Development of numerical methods in the frame-work of the ray-based CGO in the recent years allowed for the description of

GB diffraction in inhomogeneous media, including GB focusing by localizedinhomogeneities (Deschamps 1971; Egorchenkov & Kravtsov 2000) andreflection from a linear-profile layer (Egorchenkov & Kravtsov 2001) Theevolution of paraxial rays through optical structures also was studied by

Kogelnik and Li (1966), who introduced the concept of a very convenientray-transfer matrix (also seeArnaud 1976) This method of transformation isknown as the ABCD matrix method (Akhmediev 1998; Stegeman & Segev

1999;Chen, Segev, & Christodoulides 2012;Agrawal 1989)

The eikonal-based CGO, which deals with complex eikonal andcomplex amplitude was essentially influenced by quasi-optics (Fox 1964),

which is based on the parabolic wave equation (PWE;Fox 1964; Babi
c &Buldyrev 1991; Kogelnik 1965; Kogelnik & Li 1966; Arnaud 1976;

Akhmanov & Nikitin 1997; Pereverzev 1993) In the case of a spatiallynarrow wave beam concentrated in the vicinity of the central ray, theparabolic equation reduces to the abridged PWE (Vlasov & Talanov 1995;

Permitin & Smirnov 1996), which preserves only quadratic terms in smalldeviations from the central ray The abridged PWE allows for describing theelectromagnetic GB evolution in inhomogeneous and anisotropic plasmas(Pereverzev 1998) and in optically smoothly inhomogeneous media (Permitin

& Smirnov 1996) The description of GB diffraction by the abridged PWE is

an essential feature of quasi-optical model It is a convenient simplification,nevertheless it still requires solving of partial differential equations

The essential step in the development of quasi-optics was done in variousstudies that analyzed laser beams by introducing a quasi-optical complexparameter q (Kogelnik 1965; Kogelnik and Li 1966), which allows forsolving the parabolic equation in a more compact way, taking into accountthe wave nature of the beams The obtained PWE solution enables one todetermine such GB parameters as beam width, amplitude, and wave frontcurvature The quasi-optical approach is very convenient and commonlyused in the framework of beam transmission and transformation throughoptical systems However, modeling GB evolution by means of the quasi-optical parameter q using the ABCD matrix is effective for GB propaga-tion in free space or along axial symmetry in graded-index optics (on axisbeam propagation) when the A,B,C, and D elements of the transformationmatrix are known Thus, the problem of GB evolution along curvilineartrajectories requires the solution of the parabolic equation, which iscomplicated even for inhomogeneous media (Vlasov & Talanov 1995) Infact, the description of GB evolution along curvilinear trajectories by means

of the parabolic equation is limited only to the consideration of linearinhomogeneous media (Pereverzev 1998;Vlasov & Talanov 1995;Permitin

& Smirnov 1996) In our opinion, the eikonal-based form of the paraxialCGO seems to be a more powerful and simpler tool involving wave theory,

as opposed to quasi-optics based on the parabolic equation, and even theCGO ray-based version based on Hamiltonian equations

The problem of Gaussian beam self-focusing in nonlinear media wasusually studied by solving the nonlinear parabolic equation (Akhmanov,Sukhorukov, & Khokhlov 1968; Akhmanov, Khokhlov, & Sukhorukov

1972) The abberrationless approximation enables to reduce the nonlinearparabolic equation to solving the second-order ordinary differential equation

for Gaussian beam width evolution in a nonlinear medium of the Kerr type,but the procedure is complicated Because of the general refraction coeffi-cient, the CGO method presented in this paper deals with ordinary differ-ential equations; it does not ask to reduce diffraction and self-focusingdescriptions starting every time from partial differential equations The well-known approaches of nonlinear optics, such as the variational method andmethod of moments, demand that the nonlinear parabolic equation getssolved by complicated integral procedures of theoretical physics, which can

be unfamiliar to engineers of optoelectronics, computer modeling, andelectron physics It is worthwhile to emphasize that the variational methodand method of moments have been applied to model Gaussian beam evo-lution in nonlinear graded-indexfibers (Manash, Baldeck, & Alfano 1988;

Karlsson, Anderson, & Desaix 1992; Paré & Bélanger 1992; Perez-Garcia

et al 2000;Malomed 2002;Longhi & Janner 2004) Moreover, analogoussolutions can be obtained by the CGO method in a more convenient andillustrative way The CGO method deals with Gaussian beams, which areconvenient and appropriate wave objects to model famous optical solutions(Anderson 1983; Hasegawa 1990; Akhmediev 1998; Stegeman and Segev

1999; Chen, Segev, & Christodoulides 2012) propagating in nonlinearopticalfibers (Agrawal 1989)

The CGO method presented in this paper has been applied in the past todescribe GB evolution in inhomogeneous media (Berczynski and Kravtsov

2004;Berczynski et al 2006), nonlinear media of the Kerr type (Berczynski,Kravtsov, & Sukhorukov 2010), nonlinear inhomogeneous fibers(Berczynski 2011) and nonlinear saturable media (Berczynski 2012,2013a,b)

In Berczynski (2013c), the CGO method was generalized to describespatiotemporal effects for Gaussian wave packets propagating in nonlinearmedia and nonlinear transversely and longitudinally inhomogeneousfibers

InBerczynski (2014), the CGO method was generalized to describe ellipticalGaussian beam evolution in nonlinear inhomogeneousfibers of the Kerr type

To access the accuracy of the CGO method, Berczynski, Kravtsov, andZeglinski (2010)showed that it demonstrated a great ability to describe GBevolution in graded-index optical fibers reducing the time of numericalcalculations by 100 times with a comparable accuracy with the Crank-Nicolson scheme of the beam propagation method (BPM)

The present paper is organized as follows Section 2 presents thefundamental equations of the CGO method and its boundary applicability.Section3presents the analytical CGO solution for paraxial GB propagating

in free space, which demonstrates the advantages of the CGO method over

the standard approach of diffraction theory, specifically from the Fresnel integral Section 4 presents the problem of propagation of theGaussian beam in linear inhomogeneous media by the CGO method, which

Kirchhoff-is a good introduction to the problems of graded-index optics and integratedoptics Section5presents the natural generalization of the CGO method fornonlinear inhomogeneous media Section 6 presents an analytical CGOsolution for an axially symmetric Gaussian beam propagating in a nonlinearmedium of the Kerr type, and it discusses the accuracy of presented CGOresults compared to solutions of the nonlinear parabolic equation withinaberration-less approximations Section 7 discusses the influence of beamellipticity on Gaussian beam propagation in a nonlinear medium of the Kerrtype, in the case when elliptical cross section of the beam conserves itsorientation with respect to a natural trihedral We also discuss the in-terrelations of the CGO method with the standard approaches of nonlinearwave optics In section8, we discuss the sophisticated phenomenon of GBrotation in a nonlinear medium In section9, we present an orthogonal ray-centered coordinate system that is indispensable to describing the problem ofelliptical rotating Gaussian beams propagating along a curvilinear trajectory

in nonlinear inhomogeneous media We also discuss the influence ofnonlinear refraction on the evolution of the central ray of the beam innonlinear inhomogeneous medium, reportedfirst byBerczynski (2013a).For clarity, in section 10, we derive from the eikonal equation thecomplex ordinary differential Riccati equation, which models the problem

of the rotation of elliptical GBs propagating along a curvilinear trajectory in anonlinear inhomogeneous medium In section 11, we derive from thepartial transport equation ordinary differential equations for the evolution ofcomplex amplitude and flux conservation principle for a single ellipticalrotating GB in a nonlinear medium In section 12, we present a general-ization of the CGO method for N-rotating GBs propagating along a helicalray in nonlinear graded-index fiber We also demonstrate here the matrixform of CGO equations that are convenient for numerical simulations Insection 13, we present and discuss the evolution of a single rotating Gaussianbeam propagating along a helical ray in nonlinear graded-index fiber Wealso demonstrate the great ability of the CGO method to model explicitlythe rotation of the beam intensity and wave-front cross section In sections

14 and 15, we discuss the interaction of two, three, and four rotatingGaussian beams in nonlinear graded-indexfiber As mentioned previously,the effect of N-interacting Gaussian beams in a nonlinear inhomogeneousmedium is a new problem, which demands application of a simple, effective

tool and fast and accurate numerical algorithms From a practical point ofview, the rotating Gaussian beams can model the properties of rotatingelliptical solitons Furthermore, the existing state of knowledge of theinteraction of optical wave objects in nonlinear media is limited to thedescription of only two copropagating axially symmetric beams or pulses(Pietrzyk 1999,2001;Jiang et al 2004;Medhekar, Sarkar, & Paltani 2006;

Sarkar & Medhekar 2009) Thus, the CGO method presented in this paper is

a convenient tool that easily and effectively generalizes the results of vious research (e.g., Pietrzyk 1999, 2001; Jiang et al 2004; Medhekar,Sarkar, & Paltani 2006;Sarkar & Medhekar 2009) on N-rotating Gaussianbeams interacting during propagation along a curvilinear trajectory In ouropinion, CGO can be recognized as the simplest method of nonlinear waveoptics, which can make it applicable not only to theoretical physicists butalso to engineers of electron physics

pre-2 CGO: FUNDAMENTAL EQUATIONS, MAIN

ASSUMPTIONS, AND BOUNDARY OF APPLICABILITY

As is well known, classical geometrical optics represents diffractionlessbehavior of the wavefield (i.e., behavior that does not take into accountwave phenomena) In classical geometrical optics, we deal with a quasi-plane wave uðrÞ ¼ AðrÞexpðiJðrÞÞ, where the real amplitude AðrÞ andthe real local wave vectors kðrÞ ¼ VJðrÞ vary insignificantly over thewavelengthlðrÞ in the medium The wave fronts of the quasi-plane waveexperience geometrical transformations because of wave focusing or defo-cusing in inhomogeneous media The preservation of the quasi-plane waveform of the wave front is the necessary condition of classical geometricaloptics applicability Thus, classical geometrical optics becomes invalid nearfocal points, where the wave front loses its quasi-planar form CGO is thegeneralization of classical geometrical optics Unlike classical geometricaloptics, which deals with real rays and quasi-plane homogeneous waves, theCGO method is involved with quasi-plane inhomogeneous waves (i.e.,evanescent waves) in the form

In contrast to classical geometrical optics, the eikonal (optical path)jðrÞ andamplitude AðrÞ are complex values in the framework of CGO The di-rection of wave propagation is determined by the gradient of the real part ofthe complex eikonalVjRðrÞ ¼ VRefjðrÞg The direction of exponential

decay of the field’s magnitude is given principally by the gradient of theimaginary part VjIðrÞ ¼ VImfjðrÞg The gradient of complex eikonaldetermines the ray momentump ¼ VjðrÞ, which satisfies the ray equations

where n is the refractive index and ds is the elementary arc length In Eq.(2),

we can use also the parameter of relative permittivity εðrÞ, which for anisotropic nonmagnetic medium, is related with refractive index by the formula

Based on the assumption of a quasi-plane inhomogeneous wave structure ofthe CGO wave field, it is natural to require (analogously, as in classicalgeometrical optics) that the real and imaginary parts of the complexamplitude AðrÞ and local wave vector k ¼ k0p ¼ k0VjðrÞ do not changesignificantly on the scale of l0ðrÞ ¼ lðrÞ=2p ¼ 1=jkðrÞj That is, theinequalities

l0jVQRj << jQRj; l0jVQIj << jQIj; (4)where Q stands for kR¼ k0pR¼ k0VjRðrÞ; kI ¼ k0pI ¼ k0VjIðrÞ, and

AðrÞ, together with the expression

determine the necessary conditions for the validity of CGO in weakly

l0ðrÞ ¼ l0

0=nðrÞ, where l0

0¼ l0=2p and l0is the wavelength in free space

By using inequalities in Eqs.(4) and (5), we can introduce the parameters ofcharacteristic scales: LRi (i¼ 1; 2; 3Þ for quantity εðrÞ, real parts of

nRðrÞ<<

jQIjjVQIjhLiI: (6)Alternatively, these conditions, stating thatεðrÞ; kðrÞ, and AðrÞ vary insig-nificantly within the region of the order of l0ðrÞ may be united in a singleinequality:

k0 ffiffiffiffiffiffiffiffiεðrÞ

p

L¼ l0ðrÞ

wheremCGO is the parameter of smallness in the method of CGO, and L isthe smallest of the characteristic lengths of εðrÞ; kðrÞ, and AðrÞ; i.e.,

L¼ minðLR

i ; LI

iÞ To derive the basic equations of the CGO method, let ustake advantage of the Rytov expansion (Kravtsov et al., 2010) of thefield in

a small dimensionless parameter m ¼ 1=k0L, where we assume that

mCGOym Within dimensionless variables xI ¼ k0x; yI ¼ k0y, and

zI ¼ k0z, the Helmholtz equation for an inhomogeneous medium takes theform

kL¼ 1=mCGO We can introduce this parameter into Eq.(8)by transforming

rII ¼ mCGOrI ¼ r=L; nðrIIÞ ¼ nðmCGOrIÞ ¼ nðr=LÞ into the following:

DIIuðrIIÞ þ εðrIIÞ

m2 CGO

II The amplitude is assumed to vary slowly, so

A¼ AðmCGOrIÞ ¼ AðrIIÞ It is also convenient to write the phase inthe form JðrÞ ¼ J1ðmCGOrIÞ=mCGO ¼ J1ðrIIÞ=mCGO, so that the localwave vector as a gradient of phasek ¼ VJ ¼ k0VIJ1ðmCGOrIÞ=mCGO ¼

k0V2J1ðrIIÞ would also be assumed to be a function that changes slowlywith the coordinates, whereVI ¼ v=vrI andVII ¼ v=vrII As a result, thewavefield in coordinates rI andrII has the form

u¼ AðmCGOrIÞexpðiJ1ðmCGOrIÞ=mCGOÞ ¼ AðrIIÞexpðiJ1ðrIIÞ=mCGOÞ:

(10)Substituting Eq.(10)into Eq.(9), we obtain

m2 CGO

(11)Thus, comparison of the components with 1=m2CGO in Eq.(11)leads tothe eikonal equation

Comparing also the components of i=mCGO, we obtain the transportequation in the form

Strictly speaking, Eq.(13)is the transport equation derived in a zeroth-ordergeometrical optics approximation (Kravtsov et al., 2010), which enablessatisfactory accuracy of wave analysis on the example of Gaussian beamevolution in inhomogeneous media (Berczynski & Kravtsov 2004;

Berczynski et al 2006) and nonlinear media (Berczynski, Kravtsov, &Sukhorukov 2010), including optical fibers (Berczynski 2011) In moresophisticated problems, such as wavefield reflection from a weak interface(Kravtsov et al., 2010), which requires the description of the entire wavephenomena in the framework of CGO, one can use the expansion of the

or a Debye expansion of thefield in inverse powers of wave number 1=k0inthe form

The CGO method deals with small-angle (paraxial) beams, which arelocalized in the vicinity of the central ray (beam trajectory) satisfying the ray

equations in Eq (2) To satisfy the condition of the CGO method cability, we introduce the small paraxial parameter, which takes the form

appli-mParax ¼ 1=k0w0¼ l00=w0<< 1: (18)The CGO paraxial parameter defined in Eq.(18)appears usually in explicitform in problems of nonlinear optics, where the optical description bymeans of the nonlinear Helmholtz equation reduces to the nonlinearparabolic equation The propagation of linearly polarized, continuous wavebeams in an isotropic Kerr is governed by the scalar nonlinear Helmholtzequation, where variabler has the form

(19), we obtain the equation for complex envelope evolution in the form

m2Parax4

inhomoge-a nonlineinhomoge-ar medium of the Kerr type, which is essentiinhomoge-al for the ment of the use of the CGO method for nonlinear saturable media

develop-Berczynski (2011)demonstrated that the CGO method supplies solutionsfor GB evolution in inhomogeneous and nonlinear Kerr-typefibers in a

much simpler way than standard methods of nonlinear optics such as thevariational method and the method of moments The CGO method alsoreduces essentially the time spent on numerical calculations compared tothe beam propagation method (BPM), which was shown in the example of

GB propagation in optical graded-indexfibers inBerczynski, Kravtsov, andZeglinski (2010) The Gaussian beam is a self-sustained solution in theframework of the CGO method, and we noticed from our numericalcalculations that the necessary condition for GB to preserve its“Gaussian”profile in a nonlinear medium of the Kerr type is that GB width be smallenough with respect to the characteristic nonlinear scale To satisfy thiscondition, let us introduce the next small, nonlinear parameter of thefollowing form:

where LNL¼ w0=qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiεNLjA0j2

(with w0being the initial beam width and A0being the initial complex amplitude), limiting ourselves to low intensities ofthe beams One can deduce from numerical calculations that the smallparameter in Eq (22)determines the condition of the applicability of theCGO method in nonlinear saturable media To generalize the description bythe CGO method for nonlinear inhomogeneous media, we should take intoaccount the effect of linear refraction Thus, introducing the refractionparameter mREF, we notice that the beam preserves its “Gaussian” formduring propagation in an inhomogeneous medium when this refractionparameter is small enough Thus, we obtain that

where L ¼ jεj=jVεj is the inhomogeneity scale of smoothly inhomogeneousmedia Small parameters in Eqs.(22) and (23)determine also the boundaryapplicability of the abridged PWE (Vlasov & Talanov 1995; Permitin &Smirnov 1996)

3 GAUSSIAN BEAM DIFFRACTION IN FREE SPACE CGOMETHOD AND CLASSICAL DIFFRACTION THEORY

For an axially symmetric wave beam propagating along the z-axis in freespace, the CGO method suggests a solution of the form

uðr; zÞ ¼ A expðik0jÞ ¼ AðzÞexp

ik0BðzÞr2=2  z; (24)

wherej is a complex-valued eikonal, which takes the form

vjvr

dA2dz

As a result, the complex amplitude of the axially symmetric GB takes theform

ZBðzÞdz

where Að0Þ is the initial amplitude Putting Eq.(28)into Eq.(33), we obtainthe connection between the amplitude of GB and the complex wave frontcurvature in the form

Let us compare the obtained CGO results presented in Eq.(28)and Eq.(35)

for GB diffraction in free space with solutions of the diffraction theorywithin Fresnel approximation, for which the diffraction integral has the form



ðx  x0Þ2

þðy  y0Þ2

dx0dy0; (36)whereðx0; y0Þ are the coordinates in the plane of the screen and ðx; yÞ are thecoordinates in the observation plane Eðx; y; zÞ is the envelope of the field inthe observation plane, E0ðx0; y0Þ denotes the field envelope in the plane ofthe screen with the aperture, and z is the distance between the screen planeand the observation plane The integral in Eq.(36)is the approximation ofthe diffraction integral written in standard form:

is the distance between two points

on the aperture and observation plane Remember that Fresnel’s mation describes the diffraction of the paraxial (weakly diverging) opticalbeams for the inequalities, where z>> x; y; x0; y0 are well satisfied The

approxi-inequalities shown here allow one to write the following approximateexpression for parameter r: r ¼ z þ ððx  x0Þ2þ ðy  y0Þ2Þ=2z In thisway, we can disregard the difference between parameters r and z in thedenominator of the integrand in Eq.(37) From a physical point of view, theformula r ¼ z þ ððx  x0Þ2þ ðy  y0Þ2Þ=2z obtained within the Fresnelapproximation implies substitution of the parabolic surfaces for the sphericalwave fronts of the Huygens secondary wavelets By virtue of the axialsymmetry of the GB with initially circular cross sections, we perform furthercalculations of diffraction by a round aperture, making the transition to thepolar coordinates via the following formula:

x¼ r cos 4; y ¼ r sin 4 x0¼ r0cos40; y0 ¼ r0sin40: (38)Writing the surface area in the form ds0 ¼ r0dr0d40, the integral in

Eq.(36)takes the form



r02þ r2 2rr0cosð4  40Þ

By virtue of the axial symmetry of the field distribution, where

E0ðr0; 40Þ ¼ E0ðr0Þ, and expressing the integral

Putting Eq.(42)into Eq.(41), we obtain the diffractionfield, which onthe observation plane has the form

Eðr; zÞ ¼E0expðik0zÞ

1 þ z

ik0w02 exp  r2

2w20

as applied to Gaussian beam propagating and diffracting in free space, weobtain in a simple and illustrative way the same result as can be obtained inthe standard way within a Fresnel approximation to the Kirchoff integral,taking into account the fact that the wave function uðr; zÞ used in the CGOmethod plays the same role as thefield envelope Eðr; zÞ used within classicaldiffraction theory, and the CGO quantity Að0Þ is equivalent to parameter

E0 used in Eq.(43)

4 ON-AXIS PROPAGATION OF AN AXIALLY

SYMMETRIC GAUSSIAN BEAM IN SMOOTHLY

vjvr

relative permittivity in Eq.(47)can be expanded in the Taylor series in r inthe vicinity of symmetry axis z:

r¼0describes the linear refraction When we

describe GB diffraction in a homogeneous medium where ε ¼ const, thisparameter equals zero At this point, let us determine the physical meaning

of complex parameter B The real and imaginary parts of parameter

B¼ ReB þ iImB determine the real curvature k of the wave front and thebeam width w correspondingly:

Substituting Eq.(51)into Eq.(53), we obtain the expression

ddz

4.3 The First-Order Ordinary Differential Equation

for the GB Complex Amplitude

As previously discussed, now we describe within the CGO method paraxialGBs that are now localized in the vicinity of symmetry axis z Thus, in theframework of paraxial approximation, radius r is a small parameter andamplitude A¼ A(z) is complex-valued It satisfies the transport equation in

Eq.(17), which for an axially symmetric beam in cylindrical coordinates (r, z)takes the following form:

dA2dz

Eq.(59)for GB complex amplitude and the Riccati equation for complexcurvature parameter B are the basic CGO equations, which in fact reducethe problem of GB diffraction to the domain of an ordinary differentialequation Having calculated the complex parameter B from the Riccatiequation in Eq (50), we can readily find complex amplitude A by inte-grating Eq.(59) As a result, the complex amplitude of axially symmetric GBpropagating in inhomogeneous medium takes the form

AðzÞ ¼ Að0Þexp 

ZBðzÞdz

where Að0Þ is the initial amplitude

4.4 The Energy Flux Conservation Principle in GB CrossSection

The absolute value of the complex amplitude in Eq.(60)equals

jAðzÞj ¼ jAð0Þjexp 

ZReBðzÞdz

5 GENERALIZATION OF THE CGO METHOD FOR

NONLINEAR INHOMOGENEOUS MEDIA

For clarity, let us start our analysis with the case of on-axis beam propagation

in the simplest nonlinear mediumdnamely, a medium with (cubic) type nonlinearity In such a medium, the relative permittivity depends onthe beam intensityjuj2 in the form

where coefficient εNLis assumed to be positive (εNL > 0) when we considerthe self-focusing nonlinear effect Putting the wavefield in Eq.(52)into Eq.

by GB parameters w and A This is the simplest explanation whythe CGO method presented earlier in this chapter for the linearinhomogeneous case also can be applicable for nonlinear media ofthe Kerr type In accordance with relation in Eq (63), we can present

we obtain the Riccati equation, generalized now for the case of on-axis

GB propagation in a nonlinear inhomogeneous medium of the Kerrtype:

6 SELF-FOCUSING OF AN AXIALLY SYMMETRIC

GAUSSIAN BEAM IN A NONLINEAR MEDIUM OF THEKERR TYPE THE CGO METHOD AND SOLUTIONS OFTHE NONLINEAR PARABOLIC EQUATION

In this section, let us describe the classical example of on-axis GB diffractionand self-focusing in a nonlinear medium of the Kerr type without any in-fluence of linear refraction In conditions when the contribution of thelinear term in Eq.(68)is negligibly small, the Riccati equation in Eq.(70)

takes the form

where LD ¼ k0w2ð0Þ is the diffraction length and LNL¼ wð0Þ=qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiεNLjAð0Þj2

is the characteristic nonlinear scale Eq.(73)also can be presented as

p

k 2 ε NListhe critical power As a result, the equation for GB width evolution in anonlinear medium of the Kerr type takes the form

Integrating once Eq(76)and assuming thatdfdz

z ¼0¼ 0, which corresponds tothe GB with an initial wave front [see Eq.(55)], we obtain the following solution:

1 Under-critical power: P < Pcrit In this case, the beam width increaseswithout limits, and in accordance with Eq (63), the beam amplitudetends to zero at z/N

2 Critical power: P ¼ Pcrit In this case, we obtain a stationary solution

3 Over-critical power: P > Pcrit, and the beam width decreases to zero at afinite propagation distance In accordance with Eq (63), the waveamplitude increases to infinity over such a distance In this case, thecollapse phenomenon takes place (Akhmanov, Sukhorukov, &Khokhlov 1968;Akhmanov, Khokhlov, & Sukhorukov 1972)

7 SELF-FOCUSING OF ELLIPTICAL GB PROPAGATING IN

A NONLINEAR MEDIUM OF THE KERR TYPE

In realistic optical systems of integrated nonlinear optics, we should includebeam ellipticity in the description To analyze the problem of rotatingelliptical beam propagating along a curvilinear trajectory in an inhomoge-neous nonlinear medium, let us consider first the case where an ellipticalbeam conserves its orientation in transverse Cartesian coordinates x, y when

it propagates along the symmetry axis in a nonlinear medium of the Kerrtype AsBerczynski (2014)did, we model by the CGO method the elliptical

GB propagating along the z-axis in the form

an axially symmetric GB We can present Eqs.(79) and (80)in the followingforms:



and has the following solution:

w12þ w2

2 ¼ 1

k2

0w2 01

þ 1

k2

0w2 02

k02ð0Þ ¼ 0] In the solution presented in Eq.(89), we can distinguish threesubcases:

1 When P <P crit

2



q þ1q, whereq denotes the ratio of the GB width alongthe large axis to the GB width along the small axis of the elliptical crosssection q ¼ w02=w01, the combination of principal widths w2

1þ w2

2 crease like parabola

in-2 When P ¼P crit

2



q þ1 q

, the self-trapping effect takes place

3 When P >P crit

2



q þ1 q

, the combination of the widths w12þ w2

2decreases

to zero at a finite (self-focusing) distance, and the GB collapses

The accuracy of the CGO method as compared to solutions of the nonlinearparabolic equations for elliptical GB propagating in a nonlinear medium ofthe Kerr type, as well as the problem of the influence of initial beamdivergence and convergence, were discussed byBerczynski (2014)

8 ROTATING ELLIPTICAL GAUSSIAN BEAMS IN

NONLINEAR MEDIA

In the previous section, we considered the problem of elliptical GB agating along the symmetry axis in a nonlinear medium of the Kerr type.The form of a complex eikonal,

sym-Eq.(52)for two-dimensional (2-D) GB for the case of a three-dimensional(3-D) beam, one can notice that real parts of the functions Rij¼ ReBij

determine the principal wave front curvatures and imaginary parts

Iij¼ ImBij determine the widths of elliptical cross sections of the beam.Thus, the complex eikonal in Eq.(91)can be written as

in the following way:

h1¼ x cos 4 þ y sin 4; h2 ¼ x cos 4  y sin 4; (94)where

tg24 ¼ 2I12

The parameter Iijof a beam cross section ellipse I11x2þ 2I12xyþ I22y2 ¼ 1

is connected with principal widths w1and w2of a rotating coordinate system

by the following relations:

Following analogously, as inBerczynski (2014), we can derive the set ofequations for principal widths w1and w2 in the form

0 12

where Poe is the self-trapping power for an ordinary elliptical beam (i.e., anelliptical beam that conserves its orientation relative to transverse Cartesiancoordinates x, y), as analyzed previously Strictly speaking, the resultpresented in Eq (103) is one of the most important conclusions on theevolution of rotating GB propagating along a rectilinear trajectory inCartesian coordinates Namely, as the degree of symmetry of GB decreases(GB starts to rotate with respect to transverse Cartesian coordinates x, y), theself-trapping power for self-focusing increases A much more complexproblem is the elliptical GB rotating in a nonlinear saturable mediumpropagating along a curvilinear trajectory in an inhomogeneous nonlinearmedium This indicates the need for a simple method effectively describingthe wave motion in curvilinear differential geometry, computationally

efficient and convenient for the implementation in typical environments fornumerical computations (eg Matlab or MathCAD) The CGO methodmeets these requirements It need solving only ordinary differential equationsinstead much more complicated partial differential ones Therefore,computational algorithms are relatively simple Suitable implementations areavailable and can be run in software environments well known to engineers.Therefore CGO method should be preferred way to treat the self-focussingand the diffraction of Gaussian beams, especially in nonlinear systems.

9 ORTHOGONAL RAY-CENTERED COORDINATE SYSTEMFOR ROTATING ELLIPTICAL GAUSSIAN BEAMS

PROPAGATING ALONG A CURVILINEAR TRAJECTORY IN

A NONLINEAR INHOMOGENEOUS MEDIUM

CGO is a method based on geometrical optics, nevertheless it describesaccurately wave phenomena related to Gaussian beam propagation (alongthe central ray of the beam) Because CGO is a paraxial method, we describespatially narrow beams, which are localized in the vicinity of central rays Ininhomogeneous media, such central rays are curvilinear due to a nonzerogradient of the relative medium permittivity and can be described usingHamiltonian equations in the following form:

ds¼ ffiffiffiε

p

As mentioned previously, the CGO method deals with paraxial (spatiallynarrow) beams that are localized along central rays described by the Hamil-tonian equations shown in Eq.(104) Thus, the natural choice of the 3-Dreference system is when the longitudinal axis is located on the central ray

of the beam, which is curvilinear for inhomogeneous media, and tworemaining transverse axis are orthogonal to one another In this way, weconstructed a ray-centered coordinate system From fundamental differentialgeometry, we know that with each curve, we can associate three characteristicunit vectors,l; n and b Vector l is tangent to the curve and vectors n and bare perpendicular to the tangent vectorl The normal vector is n, and the

binormal one is b In accordance with Serret-Frenet formulas, we candescribe the evolution of such vectors in 3-D geometry in the following form:

to construct an orthogonal ray-centered coordinate system that allows fordescribing beam wave motion in an unequivocal way So, the unit vectors

e1¼ n cos 4 þ b sin 4; e2¼ b cos 4  n sin 4 (109)undergo rotation relative to the basen and b, with angular velocity equal to

1982; Popov and P
sen
cik 1978a, b) is now widely applied both ingeophysics and in optics In Babi
c and Buldyrev (1991) and Babi
c andKirpichnikova (1980), such an orthogonal base was applied to the problem

of paraxial beams described by the abridged PWE Such a base is consideredessential for the description of seismic rays (
Cervený 2001) One alsofindsapplications in quasi-optics (Permitin & Smirnov 1996), CGO of inho-mogeneous media (Berczynski et al 2006), and CGO of nonlinear

inhomogeneous media (Berczynski 2013a) Ifx denotes a vector lying in theplane perpendicular to the central rayrcðsÞ in the form

As was shown byBerczynski (2013a), the central ray of a symmetric GB

is not subjected to nonlinear refraction caused by a nonlinear part of relativepermittivity Therefore, the trajectory of the central ray in inhomogeneousnonlinear media coincides with the central ray in linear inhomogeneousmedia This means that low-powered and high-powered beams propagatealong the same trajectory

10 COMPLEX ORDINARY DIFFERENTIAL RICCATI

EQUATIONS FOR ELLIPTICAL ROTATING GB

PROPAGATING ALONG A CURVILINEAR TRAJECTORY IN

A NONLINEAR INHOMOGENEOUS MEDIUM

Now let us consider the propagation of a monochromatic scalar Gaussianbeam in a smoothly inhomogeneous isotropic and nonlinear saturablemedium, with a permittivity profile in the form

In above equation, g is an arbitrary function of the beam intensity IðrÞ and

εSis saturating permittivity, andεðsÞ ¼ εðrcÞ and rc ¼ ðs; 0; 0Þ is the vector for the central ray inðs; x1; x2Þ coordinates In the framework of theCGO method, the eikonal consists of two summands:jcðsÞ is the eikonal onthe central ray, while4ðs; x1; x2Þ is a small deviation from jc in the form

radius-jðs; x1; x2Þ ¼ jcðsÞ þ 4ðs; x1; x2Þ; (116)

where s is the parameter along the central ray and x1;2 are coordinatesorthogonal to the ray in a ray centered-reference system (Figure 1) Thecomplex eikonal4 describes both the curvature of the phase front of the beamand its intensity profile Within paraxial approximation, the deviation 4 for

GB can be presented as a quadratic form:

4ðs; x1; x2Þ ¼ 1

In what follows, i¼ 1; 2 and summation over repeated indices isimplied; Bij¼ Rijþ iIij, where RijhReBij and IijhImBij are complex-valued functions, which constitute a symmetric tensor with B12hB21.The real parts of these functions characterize the curvatures of the GB phasefront, whereas the imaginary parts determine the elliptical cross section ofthe GB In view of the extreme properties of the central ray, the linear inxi

terms does not contribute to Eq.(117)in an isotropic medium The totalcomplex eikonal j satisfies Eq (16), which, for the case of nonlinearsaturable medium, has the form

whereεðrÞ is defined in Eq.(115) For the paraxial beam, we can expand thenext permittivityε in Eq.(120)in a Taylor series in small deviation x:

εðrÞ ¼ εLINðrcÞ þ εNLgðIðrcÞÞ þ ððxVtÞεLINÞr¼rc

Eq.(123)is the quadratic form inxi In order to satisfy this equation, oneshould equal all the coefficients in the left and right sides of this equation As

a result, we obtain a tensor Riccati-type equation for BijðsÞ:

cijðsÞ ¼ εNL

2

vgvI

If we denote the eigenvalues of tensors RijðsÞ and IijðsÞ as RiðsÞ and

IiðsÞ, thus principal curvatures of the wave front, ki, and the principal beamwidths, wi, can be presented in the form:

In the ray-centered coordinatesðs; x1; x2Þ, the transport equation in Eq.(17)

takes the form

1h

vjvs

A NONLINEAR MEDIUM OF THE KERR TYPE

In realistic optical systems of integrated nonlinear optics, we should includebeam ellipticity in the... Cartesian coordinates x, y), theself-trapping power for self-focusing increases A much more complexproblem is the elliptical GB rotating in a nonlinear saturable mediumpropagating along a curvilinear... the basen and b, with angular velocity equal to

1982; Popov and P
sen
cik 1978a, b) is now widely applied both ingeophysics and in optics In Babi
c and Buldyrev (1991) and Babi
c andKirpichnikova