Intrinsic localized modes in nonlinear photonic crystal waveguides: Dispersive modes

13 September 1999

Physics Letters A 260 Ž1999. 314–321 www.elsevier.nlrlocaterphysleta

Intrinsic localized modes in nonlinear photonic crystal waveguides: Dispersive modes Arthur R. McGurn

1

Department of Physics, Western Michigan UniÕersity, Kalamazoo, MI 49008, USA Received 19 April 1999; received in revised form 10 July 1999; accepted 11 July 1999 Communicated by L.J. Sham

Abstract A discussion is given of the theory of intrinsic localized modes ŽILM. for nonlinear waveguide impurities in photonic crystals. Particular attention is made to the determination of the dispersion relation of propagating ILM modes. Both evenand odd-parity ILM modes are considered. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 42.65; 42.65.S; 63.20.P Keywords: Localization; Photonic crystals; Nonlinear optics

1. Introduction There has been considerable interest in intrinsic localized modes ŽILM.. ILM modes have been studied in the vibrational spectra of one-dimensional monatomic w1–11x and diatomic lattices w12,13x, in the excitation spectra of magnetic chains w8,9,14x, and most recently as part of the electromagnetic mode spectra in nonlinear photonic crystal waveguides w15x. In most instances, these studies have employed analytic and computer simulation techniques and, except for the photonic crystal work, both static and dispersive ILM modes w16x have been investigated.

1 Tel.: q1 616-387-4950; fax: q1 616-387-4939; e-mail: [email protected]

Recently, we have given an analytic treatment of the static Žnon propagating. ILM modes in nonlinear photonic crystal waveguides w15x. The conditions needed for such modes to exist and the nature of the localized wavefunctions of the static ILM modes was determined. In the present short note we would like to extend the treatment given in Ref. w15x to treat dispersive Žpropagating. ILM modes. We will determine the conditions necessary for the existence of dispersive ILM modes and will discuss the nature of the localized wavefunctions of these modes. As in Ref. w15x, we consider the E-polarized electromagnetic modes of a two-dimensional photonic crystal formed as a square lattice array of infinitely long, parallel, identical dielectric rods w17,18x. The rods are characterized by a dielectric constant e and are embedded in vacuum w17–22x. The periodic dielectric constant of the system as a

0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 4 7 4 - 0

A.R. McGurnr Physics Letters A 260 (1999) 314–321

function of position, r < < s xiˆq yj,ˆ in the x y y plane is then e Ž r << .s

½

e,

ˆ mac jˆ< F R for n and m integers < r < < y nac iy

1,

otherwise

where E 0 Ž r < < , v ,t . is an envelope function which is slowly varying in time compared to expŽyi v t ., then the wave equation for E 0 Ž r < < , v ,t . becomes

Ž 1. ,2 qe Ž r << .

where a c is the lattice constant of the square lattice, and R - a c is the radius of the dielectric rods. The E-polarized electromagnetic modes of the photonic crystal are solutions of the matrix eigenvalue equation w17–22x

c2

Ý eˆ Ž G < < y GX< < . e Ž k < < < GX< < < v . .

c

Ž 2.

q eT Ž r < < .

X

Here the eigenvalue v 2rc 2 gives the frequency v of the electromagnetic mode, G < < is a reciprocal lattice vector of the square lattice, e Ž r < < . s ÝG < < eˆ Ž G < < .e iG < < P r < < , and eŽ k < < < G < < < v . are related to the electric field, EŽ r < < < v ., of the mode of frequency v by iŽ k <
.

Ž 3.

E 0 Ž r < < , v ,t .

v E E 0 Ž r < < , v ,t . c2

v

G<<

E Ž r << < v . s Ý e Ž k << < G << < v .e

ž /

y de Ž r < < .

Ž k<< qG<< . eŽ k<<
2

v

f y2 i e T Ž r < < .

2

s

315

ž / c

Et 2

E 0 Ž r < < , v ,t .

1 E 2 E 0 Ž r < < , v ,t . c2

E t2

.

Ž 6.

Here we make the standard approximation Žsee Section 9.3 of Ref. w23x. of ignoring the time dependence of the small non linear terms of the dielectric constant. This amounts to letting

E2

E2 E t2

e T E 0 Ž r < < , v ,t . s e T

E t2

E 0 Ž r < < , v ,t .

G<<

A waveguide impurity is formed in the system defined in Eq. Ž1. by adding impurity material to a row of rods along one of the directions of the square lattice w15,17,18x. The total dielectric constant of such a waveguide impurity system, e T Ž r < < ., is then given by e T Ž r < < . s e Ž r < < . q de Ž r < < . where de Ž r < < . is the change in the dielectric constant of the photonic crystal upon the addition of impurity dielectric material. For a waveguide composed of an infinite array of single site impurities of square cross sectional area in the x–y plane, de Ž r < < . is

de , de Ž r < < . s 0,

½

in the derivation of the Helmholtz wave equation for E 0 Ž r < < , v ,t .. Using standard techniques w15,19x, Eq. Ž6. can be rewritten as an integral equation given by E 0 Ž r < < , v ,t . s d 2 r X< < G Ž r < < ,rX< < < v .

H

= de Ž rX< < .

q2 i e T Ž

rX< <

< x y nac < , < y < F t for n an integer otherwise

Ž 4. Here 2 t is the length of a side of one of the single site impurities and 2 t < R - a c . If we assume that the electric field of the modes associated with the nonlinear waveguide is of the form E Ž r < < < t . s E 0 Ž r < < , v ,t . exp Ž yi v t .

Ž 5.

ye T Ž rX< < .

v

ž /

.

c

2

E 0 Ž rX< < , v ,t .

v E E 0 Ž rX< < , v ,t . c2

Et

1 E 2 E 0 Ž rX< < , v ,t . c2

E t2

.

Ž 7.

Here GŽ r < < ,rX< < < v . is the Green’s function of the Helmholtz operator on the left hand side of Eq. Ž6.. As in Ref. w15x we assume that t is small enough so that the electromagnetic field at each square cross section rod of impurity material is constant over that volume of the impurity material. The electric field in the nth rod of impurity material of the waveguide,

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316

denoted by En where EnŽ t . s E 0 Ž nac i,ˆ v ,t ., then satisfies the difference equation En Ž t . s Ý Bn ,l de Ž lac iˆ. l

= El Ž t . q

v q2i

c2

Hd

1

y

1 E 2 El Ž t .

v2

E t2

E 0 Ž r < < , v ,t . f

ž na iˆ,r < v /

2 X r<
X <<

c

Ž

2 X r<
. X <<

E 2 E 0 Ž rX< < , v ,t .

=e Ž rX< < .

Et

2

.

Ž 8.

v 2i

Here

v Bn ,l s

2

ž c / Hlth impurityd

2

ž /H c

d 2 r X< < G Ž r < < ,rX< < < v . de Ž rX< < .

Ž 12 .

This is Eq. Ž7. less the small terms involving E E 0rE t and E 2 E 0rE t 2 . Substituting Eq. Ž12. into the integrals in Eq. Ž11. and ignoring the nonlinear terms in Eq. Ž10., we find

ž na iˆ,r < v / c

2

v

=E 0 Ž rX< < , v ,t . .

Et

Hd

c2

Et

v

E E 0 rX< < , v ,t

=e Ž rX< < . y

2 i E El Ž t .

The electromagnetic modes of the waveguide are now determined as solutions of Eq. Ž11.. To evaluate the two integrals on the left hand side of Eq. Ž11., we approximate E 0 Ž rX< < , v ,t . in both integrands by

r < < G nac iˆ,r < < < v ,

ž

l runs over the integers, and v is the frequency of the impurity mode. The terms in Eq. Ž8. having integrands involving E E 0rE t and E 2 E 0rE t 2 arise as part of the time dependence of the envelope function and will be discussed shortly. For the moment, we will look at the sum on the right hand side of Eq. Ž8.. For a Kerr waveguide impurity media, the field and frequency dependent change in the dielectric constant at r < < s lac iˆ is given by

de Ž lac iˆ. s A Ž 1 q D < El < 2 .

Ž 10 .

where A and D are constants. Using Eq. Ž10. in Eq. Ž8., our recursion relation becomes

H

ž nac iˆ,rX< < < v / e Ž rX< < .

f yi Ý ACn0,l

Ž 9.

/

c2

d 2 r X< < G

E El Ž t . Et

l

E E 0 Ž rX< < , v ,t . Et

,

Ž 13 .

where 3

1 v

Cn0,l s y2

c

ž /H c

=

Hlthimpurityd

d 2 r X< < G nac iˆ,rX< < < v e Ž rX< < .

2

ž

/

r < < G Ž rX< < ,r < < < v . ,

Ž 14 .

and 1 c2

Hd

2 X r<
fy Ý

En Ž t . s A Ý Bn ,l Ž 1 q D < El Ž t . < 2 .

ž

nac iˆ,rX< < < v e Ž rX< < .

/

ADn0,l

E 2 El Ž t .

l

E t2

,

E 2 E 0 Ž rX< < , v ,t . E t2

Ž 15 .

l

= El Ž t . q

v q2i

Hd

c2

=e Ž rX< < . 1 y

c2

=e Ž

H

rX< <

2 i E El Ž t .

Et

v

2 X r<
y

v2

Dn0,l s Cn0,lr Ž 2 v . .

E t2

c

X <<

E E 0 Ž rX< < , v ,t . Et

ž

E 2 E 0 Ž rX< < , v ,t . E t2

.

Ž 16 .

In neglecting E E 0rE t and E 2 E 0rE t 2 in Eq. Ž12. we are treating the envelop function, E 0 , in Eq. Ž5. in the slowly varying envelop approximation w24x. This assumes that

ž na iˆ,r < v /

d 2 r X< < G nac iˆ,rX< < < v

.

where

1 E 2 El Ž t .

1 E 2E0

/

v

Ž 11 .

E t2

E E0 <

Et


and is essentially a search for pulse type solutions for EŽ r < < < t . with envelops which move slowly in

A.R. McGurnr Physics Letters A 260 (1999) 314–321

space and are modulated by the rapidly varying eyi v t time dependence shown explicitly in Eq. Ž5.. In doing this we assume that the integral in Eq. Ž12. not only dominates the terms involving 1 E E0

and

1 E 2E0

v Et v2 E t2 but that its derivative gives the major contributions to E E 0rE t and E 2 E 0rE t 2 . These arguments can be refined a bit by retaining in Eq. Ž12. the first two terms of the three terms in the integrand in Eq. Ž7. instead of just the first term retained in Eq. Ž12.. Using this to evaluate Eqs. Ž13. and Ž15. gives an additional contribution to Eq. Ž13.. This additional term can be treated, in the sum of Eqs. Ž13. and Ž15. which enters into Eq. Ž11., as a renormalization of 0 0 Dn,l . In particular, Dn,l in the sum of Eqs. Ž13. and Ž15., when renormalized in this way, becomes v4

d 2 r < < d 2 r X< < G nac iˆ,r < < c6 =e Ž r < < . G Ž r < < ,rX< < . e Ž rX< < .

Dn0,l s Cn0,lr Ž 2 v . y 4

H

2

=

X <<

H

ž

/

Hlthimpurityd s G Ž r , s . . <<

<<

As an additional simplification in Eq. Ž13. through Ž16., we assume that e 4 1 ŽThis facilitates the formation of large photonic stop bands and assures that the integrands in Eqs. Ž13. and Ž15. are dominated by their behavior in the e rods.. and that the localized modes of the system are confined to a small enough segment of the waveguide so that the sum over l in Eqs. Ž13. and Ž15. can be restricted to l s n. This can alway be arranged by choosing the waveguide channel to be along directions in the square lattice which have high Miller indices. For such directions the distance between nearest neighbor impurity sites is large, and for frequencies in the stop band the values of the Green’s functions are also small between such nearest neighbor impurity sites. We then have from Eq. Ž11. through Ž16. En Ž t . s A Ý Bn ,l Ž 1 q D < El Ž t . < 2 . El Ž t . l

y iACn , n

E En Ž t .

q ADn , n

E 2 En Ž t .

, Ž 17 . Et E t2 where Cn, n s Cn,0 n y v2 Bn, n , Dn, n s Dn,0 n y v12 Bn, n , and in evaluating Cn, n and Dn, n we have taken

317

de Ž nac iˆ. f A. This equation determines the field in the impurity rods forming the photonic crystal waveguide. Following the discussion in Ref. w15x, we simplify the mathematics of our treatment by restricting the recursion relation in Eq. Ž17. to consider only same site and nearest neighbor site couplings of the Bn,l . With this provision, Eq. Ž17. can be written as En s g a 0 En q b 0 < En < 2 En q a 1 Ž Enq1 q Eny1 . qb 1 Ž < Enq1 < 2 Enq1 q < Eny1 < 2 Eny1 . yi d 0

E En Et

qj0

E 2 En Ž t . E t2

.

Ž 18 .

Here a 0 s Bn, nrŽ4 t 2 ., a 1 s Bn, nq1rŽ4 t 2 ., b 0 s DBn, nrŽ4 t 2 ., b 1 s DBn, nq1rŽ4 t 2 ., g s 4 t 2A, d 0 s Cn, nrŽ4 t 2 ., and j 0 s Dn, nrŽ4 t 2 .. The electromagnetic mode solutions of Eq. Ž18. which are bound to the waveguide are obtained by first choosing v to be a frequency in the stop band of the photonic crystal and computing a 0 , a 1 , b 0 , b 1 , d 0 , and j 0 . For fixed values of D, the resulting Eq. Ž18. can then be solved for  En4 and g Ži.e., A. characterizing these modes. We note that in going from Eq. Ž17. to Eq. Ž18. we retain only terms involving Bn, n and Bn, nq1 in the sum over l. In doing this, it is assumed that < Bn, n < 4 < Bn, nq1 < 4 < Bn, nqq < for q ) 1 because of the general rapid decay of GŽ r < < ,rX< < < v . with increasing < r < < y rX< < < for frequencies well within the stop band gaps. This is similar to the argument we used in Eq. Ž17. for retaining only the Cn, n and Dn, n terms in the sums over E ElrE t and E 2 ElrE t 2 . We have retained Bn, n and Bn, nq1 terms in the parts of Eq. Ž18. involving  En4 , < En < 2 En 4 but have only retained Cn, n and Dn, n terms in the parts of Eq. Ž18. involving E EnrE t and E 2 EnrE t 2 . This is done as terms proportional to the product of Cn, nq1 or Dn, nq1 with  E EnrE t 4 or with  E 2 EnrE t 2 4 are doubly small due to the smallness of both Cn, nq1 or Dn, nq1 and the field time derivatives whereas terms involving products of Bn, nq1 and  En4 or < En < 2 < En <4 are only small due to Bn, nq1. The recursion relation in Eq. Ž18. is very similar to that addressed by Bickham et al. w16x in their

A.R. McGurnr Physics Letters A 260 (1999) 314–321

318

study of ILM modes in one-dimensional anharmonic lattices and in Ref. w15x we have used their methods to study the static Žnon propagating., non dispersive, ILM modes. The methods of Bickham et al. have been extended by them in Ref. w16x to consider propagating ILM modes in the anharmonic lattice. These methods will now be used below to treat the propagating ILM modes of Eq. Ž18..

Eqs. Ž21. and Ž22., describing the envelope functions of the modes in the photonic waveguide, can be rewritten in a more convenient form as w 2fn y

d 2fn dt 2

s J 2 fn q b 1 Ž fnq1 q fny1 . cos k 3L q 4

3 qfny 1 . cos k

2. Odd parity modes Following the treatment of Bickham et al. w16x for propagating ILM modes, we assume that the traveling ILM mode solution in the waveguide system is given in terms of the real valued envelope function fnŽ t . by En Ž t . exp Ž yi v t . s fn Ž t . expi Ž kn y v t . .

Ž 19 .

Here fnŽ t . changes slowly in time compared with the expŽyi v t . term. From Eq. Ž18. we find 2

fn Ž t . s g 0 Ž 1 q a 0 fn Ž t . . fn Ž t . 2

q Ž b 0 q c 0 fnq1 Ž t . . fnq1 Ž t . cos k 2

q Ž b 0 q c 0 fny1 Ž t . . fny1 Ž t . cos k qe0

E 2fn E t2

Ž b0 q c0 fnq1Ž t . 2 . fnq1Ž t . 2 y Ž b 0 q c 0 fny1 Ž t . . fny1 Ž t . sin k q ig 0

y ig 0 d 0

Efn

, Ž 20 . Et where g 0 s ga 0 , a 0 s b 0ra 0 , b 0 s a 1ra 0 , c 0 s b 1ra 0 s a0 b 0 , d 0 s d 0ra 0 , e 0 s j 0ra 0 . The real and imaginary parts of Eq. Ž20. separate into two equations

fn y g 0 e 0

E 2fn E t2

s g 0 Ž 1 q Hn . fn q Jnq1 fnq1cos k qJny 1 fny1cos k ,

Efn

s Jnq1 fnq1 sin k y Jny1 fny1 sin k , Et where Hn s a0 fn2 and Jn s b 0 q a 0 b 0 fn2 . d0

3 2 fn3 q b 1 Ž fnq1

Ž 21 . Ž 22 .

Ž 23 .

where w 2 s Žg 0 e 0 .y1 , J s 1rŽ2 e 0 ., 3 Lr4 s a0 , b 1 s 2 b 0 , and d fn dt

s n k Ž fnq1 y fny1 . = 1q

3L 4

2 2 Ž fnq 1 q f nq1 f ny1 q f ny1 .

,

Ž 24 . where n k s bd00 sin k. Eqs. Ž23. and Ž24. are essentially the same as the equations studied by Bickham et al. in Ref. w16x. In the case that b 1 s y1, Eqs. Ž23. and Ž24. are identical to Eqs. Ž5. and Ž6. of Ref. w16x, and exhibit the same intrinsic localized mode solutions as were demonstrated by analytical and computer simulation methods in Ref. w16x. Our Eqs. Ž23. and Ž24. are a little more general than Eqs. Ž5. and Ž6. in Ref. w16x Ži.e., b 1 is not restricted to be y1., and we will now give an analytical discussion of the more general form of the solution of Eqs. Ž23. and Ž24. in the case that b 1 / y1. Our interest in showing that the envelop function equations in Eq. Ž23. and Ž24. under certain circumstances reduce to those studied by Bickham et al. w16x are two-fold. First, Bickham et al. studied Eqs. Ž23. and Ž24. for b 1 s y1 by computer simulations and showed that ILM modes exist for this equation. Secondly, Bickham et al. developed an analytical treatment which effectively approximated the dispersion relation of the ILM modes determined from computer simulations studies for this case. Our expectation is that the analytical techniques which were effective in approximating the dispersion relation of the b 1 s y1 case of Eqs. Ž23. and Ž24. will also be effective in approximating the general b 1 / y1 case of Eqs. Ž23. and Ž24..

A.R. McGurnr Physics Letters A 260 (1999) 314–321

For odd parity mode solutions of Eq. Ž23. and Ž24., we follow the analytical method of Bickham et al. in determining the parameters of the dispersive intrinsic localized mode solutions from Eq. Ž23.. Ignoring E 2fnrE t 2 compared to v 2fn ŽThis is the slowly varying envelop approximation discussed below Eq. Ž16.. in Eq. Ž23., we look for a solution of Eq. Ž23. of the form

f0 s a ,

Ž 25 . n

fn s a A 0 Ž y1 . eyn q ,

Ž 26 .

and

fyn s fn

Ž 27 .

for n ) 0. For n s 0,1 and n ™ ` we find three self-consistent equations w15,16x: 1

g0

s 1 q a y 2 bD y 2 c D 3 ,

1

1 s

g0

D

Ž 28 .

D q a D3 y b Ž 1 q D K . y c Ž 1 q D3K 3 . ,

Ž 29 . and 1

g0

319

crystal with a channel in the Ž1,0. direction and with t s 0.01a c . For a static Ži.e., cos k s 1. waveguide mode to exist in this system it was shown in Ref. w15x that at the mid stop gap frequency v a cr2p c s 0.440 the ratio Bn, nq1rBn, n s 0.0869. For a s 1.0 a static mode with v a cr2p c s 0.440 was found to exist if g 0 s 0.492 and for a s 0.1 a static mode with v a cr2p c s 0.440 was found to exist if g 0 s 0.815. To obtain the dispersion relation of the dispersive waveguide modes by the method discussed in the previous paragraph, let us first ignore the mild frequency dependence of a 0 in the definition of g 0 . In this approximation, we fix the value of g 0 at g 0 s 0.492 for a s 1.0 modes or g 0 s 0.815 for a s 0.1 modes. From Eq. Ž25. through Ž30. above this requires that the renormalized b s Bn, nq1r Bn, n cos k s 0.0869. If we know Bn, nq1rBn, n as a function of frequency, then we can solve for v as a function of k. From the numerical results in Figure 4 of Ref. w20x we obtain an approximation v ac Bn , nq1 s 0.88 y 0.3003 Ž 31 . Bn , n 2p c for this ratio, which gives in the small k limit v ac 0.0988 s q 0.3412 f 0.44 Ž 1 q 0.11k 2 . 2p c cos k

Ž 32 . s 1 y 2 bcosh q,

Ž 30 .

for g 0 , D s A 0 eyq and K s eyq . Here a s a 0 a 2 , b s b 0 cos k, and c s c 0 a 2 cos k. Eq. Ž28. through Ž30. are essentially the self-consistent equations for the static odd parity modes of the nonlinear photonic crystal waveguide as given in Eq. Ž14. through Ž17. of Ref. w15x. The k dependence of the propagating modes enters into the renormalized b s b 0 cos k and c s c 0 a 2 cos k coefficients in Eq. Ž28. through Ž30. and for k s 0 these equations reduce to the static equations in Ref. w15x. Figs. Ž1. and Ž2. of Ref. w15x give solutions for g 0 , D s A 0 eyq , K s eyq , in terms of a and b as redefined in this paper. As an example of how to extract dispersive ILM mode solutions from Figs. Ž1. and Ž2. of Ref. w15x, let us consider the system used as an example in Section 3.1 of Ref. w15x. The photonic crystal examined there is a system with R s 0.37796 a c and e s 9. A waveguide impurity was formed in this

as the dispersion relation of our dispersive ILM modes. The envelop solutions for D and K for the initial state of these modes can be taken directly from Ref. w15x. If we take into account the frequency dependence of a 0 in the definition of g 0 , we can make the rough approximation a 0 Ž v a cr2p c .r a 0 Ž0.44. f 10.3Ž v acr2p c . y 3.53 from the numerical data in Ref. w20x. Using this with a linear interpolation of the dependence of g 0 on b, we find the dispersion relation v a c r2 p c f 0.44Ž1 q 0.0045k 2 . for a s 1.0 and v a cr2p c f 0.44Ž1 q 0.014 k 2 . for a s 0.1. 3. Even parity modes To study even parity modes we look for a solution of Eqs. Ž23. and Ž24. of the form f 1 s yfy1 s a , Ž 33 .

fn s a A 0 Ž y1 .

ny 1 yŽ ny1. q

e

,

Ž 34 .

A.R. McGurnr Physics Letters A 260 (1999) 314–321

320

and

fyn s yfn

Ž 35 .

for n ) 1. ŽHere, following Ref. w15x, we have relabeled the waveguide sites so that sites that were labeled by n s 0," 1," 2, . . . are now labeled by n s "1," 2," 3, . . . .. Ignoring E 2fnrE t 2 compared to v 2fn , we find upon substitution of Eq. Ž33. through Ž35. in Eq. Ž23. that for n s 1,2 and n ™ ` w15x: 1

g0

s 1 q a y b Ž D q 1. y c Ž D 3 q 1. ,

1

1 s

g0

D

Ž 36 .

D q a D 3 y b Ž D K q 1. y c Ž D 3 K 3 q 1. ,

Ž 37 . and 1

g0

s 1 y 2 bcosh q.

Ž 38 .

for g 0 , D s A 0 eyq and K s eyq . Here a s a 0 a 2 , b s b 0 cos k, and c s c 0 a 2 cos k. Eq. Ž36. through Ž38. are essentially the self-consistent equations for the static even parity modes of the nonlinear photonic crystal waveguide as given in Eq. Ž24. through Ž26. of Ref. w15x. The k dependence of the propagating modes enters, again, into the renormalized b s b 0 cos k and c s c 0 a 2 cos k coefficients and the results in Figs. Ž3. and Ž4. of Ref. w15x can be taken over with these renormalizations. The validity of these results require that v 2fn 4 vEfnrE t 4 E 2fnrE t 2 . This will be the case for waveguides with small nearest neighbor couplings and nonlinearities which give large contributions in Eqs. Ž23. and Ž24. compared to the nearest neighbor couplings. These conditions are met for waveguides along directions of high Miller indices because along these directions the Green’s functions between nearest neighbor sites will be small.

4. Conclusions We have extended our treatment of ILM modes in nonlinear photonic crystal waveguides to consider propagating ILM modes. The conditions needed for

the existence of propagating modes are found to depend on the k-wavenumber in Eq. Ž19. and these conditions reduce in the limit that k goes to zero to the conditions obtained for static ILM modes. For some particular sets of values of the parameters in our equations ŽEq. Ž23. and Ž24.., our equations are shown to reduce to those of Bickham et al.. These cases have been investigated by Bickham et al. and shown by them in numerical simulations to exhibit sustained ILM mode solutions. Bickham et al. also showed that a certain analytical treatment was effective in obtaining an approximation of the dispersion relations of the simulated ILM modes. We have applied the analytical treatment developed by Bickham, et al. to our more general equations. Our equations, Eq. Ž23. and Ž24., are more general that those of Bickham et al. and should exhibit a wider range of solutions. We hope that the results presented here will be helpful in developing computer simulations of photonic crystal waveguides which exhibit intrinsic localized modes and in the experimental investigation of these modes. In addition, we would make two points. First, we note that the methods developed in Ref. w15x and in this paper can be applied to acoustic and magnetic superlattice systems which have descriptions based on Helmholtz type wave equations. Second, we note that the nearest neighbor and same site approximations made in Eq. Ž17. and Ž18. can be relaxed so that farther neighbor couplings are included. We have considered systems in which only same site and nearest neighbor site couplings are important so that we could make contact with the simulation studies of Bickham et al. We expect that systems with important far neighbor couplings will exhibit ILM mode solutions which are qualitatively similar to those studied here and in Ref. w15x.

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