5
PHYSICAL REVIE% B VOLUME 50, NUMBER 6 1 AUGUST 1994-II Bragg difFraction of waves in one-dimensional doubly periodic media F. G. Bass Institute of Radiophysics and Electronics, Ukraine Academy of Sciences, 310085, Proskura, 12, Kharkov, Ukraine G. Ya. Slepyan, S. T. Zavtrak, and A. V. Gurevich Institute of Nuclear Problems, Belarus State University, Bobruiskaya, 11, 220050, Minsk, Belarus (Received 28 January 1994) The Bragg diffraction of waves in one-dimensional doubly periodic media is analyzed by means of Kogelnik's coupled-waves technique. The spectrum problem and the problem of reflection from a half- space and from a layer are considered. It is shown that a devil' s-staircase type of spectrum causes characteristic peaks and valleys in the frequency dependence of the reflection coeScient. I. INTRODUCTION The energy spectrum of periodic media is investigated in detail in connection with different problems of solid- state physics, mechanics, radiophysics, optics, etc. ' Al- though the physical nature of periodic media and of periodic configurations can vary greatly, the main quali- tative feature of the spectrum its band structure— remains in all cases. This band structure determines the qualitative features of wave reflection and transmission characteristics. Of considerable interest is the wave process described by the equation d A +k [I+ f(x)]A =0, The analysis of this question is the main subject of this work. At the same time we also consider the spectral problem for the case when both periods are approximate- ly equal. Our approach for solving this problem is different from that used in Refs. 6 and 9-11. It permits solving the spectral and boundary problems in a unified way. This approach is based on Kogelnik's theory of coupled waves. The function f(x } for simplicity is tak- en to be f(x)= tttc po2 sqx+ls, cos2q, x, where pp „q, and q, are given real numbers. The case of a general form for the doubly periodic function f (x ) can be analyzed in a similar way, and does not possess any qualitatively difFerent features. where f(x ) is a given function representing the superpo- sition of two functions with diff'erent incommensurate periods. Equation (1) describes different physical processes: wave propagation in quasicrystals, light diffraction in crystals from two ultrasonic waves with nearly equal fre- quencies, monoatomic film on a crystal substrate with a different period, mercury chains, etc. Artificially creat- ed multilayered doubly periodic structures and crystal su- perlattices are promising as highly efficient x-ray reflectors and high-Q x-ray resonators. ' Two types of problems can be studied on the basis of Eq. (1): spectral problems (k is the eigenvalue sought) and "reflection-transmission" problems (in this case, k is given and Eq. (1) must be solved simultaneously with cor- responding boundary conditions). The spectral problems for (1) are considered in Refs. 6 and 9 11. The effects of Anderson localization and features of a devil*s-staircase type of spectrum are re- vealed. However, in these works very specific models and techniques are used: In Ref. 9, f (x) is defined by the Kronig-Penny approximation, while in Refs. 6, 10, and 11 the tight-binding model with periodic level modula- tion is used (this is valid when the periods differ greatly). But the main thing is that from these results it is not clear how the features revealed in the spectrum appear in the reflection and transmission coefficients. II. THE SPECTRUM Let us assume that qi=q+c, where ~c~ &&q, and represent f (x}in the following form: f (x ) =is(x )cos2qx o (x )sin2qx, where p(x ) =pp+picos2cx and o'(x ) =lsisin2cx. Then, for ~f(x)~ &&1, Eq. (1) with the coefficient (2) can be solved by Kogelnik's method of coupled waves. In accordance with Ref. 2, solution (1) can be represented as A(x)=A+(x)e' +A (x)e where A+(x) are new unknown functions, slowly varying in comparison with the rapidly varying factors e J The equations for A +(x} are obtained by averaging over the period of rapid motion; the coefficients p(x ), o (x) are related to the slow motion and are regarded as constants in the averaging procedure. The final form of the equations is d A+ kv+(x ) = j A (x )exp( 2jlkx ), kv (x) A+ (x )exp(2jhx ), 0163-1829/94/50(6)/3631(5)/$06. 00 50 3631 1994 The American Physical Society

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Page 1: Bragg diffraction of waves in one-dimensional doubly periodic media

PHYSICAL REVIE% B VOLUME 50, NUMBER 6 1 AUGUST 1994-II

Bragg difFraction of waves in one-dimensional doubly periodic media

F. G. BassInstitute ofRadiophysics and Electronics, Ukraine Academy ofSciences, 310085, Proskura, 12, Kharkov, Ukraine

G. Ya. Slepyan, S. T. Zavtrak, and A. V. GurevichInstitute ofNuclear Problems, Belarus State University, Bobruiskaya, 11, 220050, Minsk, Belarus

(Received 28 January 1994)

The Bragg diffraction of waves in one-dimensional doubly periodic media is analyzed by means ofKogelnik's coupled-waves technique. The spectrum problem and the problem of reflection from a half-

space and from a layer are considered. It is shown that a devil' s-staircase type of spectrum causes

characteristic peaks and valleys in the frequency dependence of the reflection coeScient.

I. INTRODUCTION

The energy spectrum of periodic media is investigatedin detail in connection with different problems of solid-state physics, mechanics, radiophysics, optics, etc. ' Al-though the physical nature of periodic media and ofperiodic configurations can vary greatly, the main quali-tative feature of the spectrum —its band structure—remains in all cases. This band structure determines thequalitative features of wave reflection and transmissioncharacteristics.

Of considerable interest is the wave process describedby the equation

d A +k [I+f(x)]A =0,

The analysis of this question is the main subject of thiswork. At the same time we also consider the spectralproblem for the case when both periods are approximate-ly equal. Our approach for solving this problem isdifferent from that used in Refs. 6 and 9-11. It permitssolving the spectral and boundary problems in a unified

way. This approach is based on Kogelnik's theory ofcoupled waves. The function f(x } for simplicity is tak-en to be

f(x)= tttcpo2sqx+ls, cos2q, x,where pp „q, and q, are given real numbers. The case ofa general form for the doubly periodic function f(x ) canbe analyzed in a similar way, and does not possess anyqualitatively difFerent features.

where f(x ) is a given function representing the superpo-sition of two functions with diff'erent incommensurate

periods.Equation (1) describes different physical processes:

wave propagation in quasicrystals, light diffraction incrystals from two ultrasonic waves with nearly equal fre-quencies, monoatomic film on a crystal substrate with adifferent period, mercury chains, etc. Artificially creat-ed multilayered doubly periodic structures and crystal su-perlattices are promising as highly efficient x-rayreflectors and high-Q x-ray resonators. '

Two types of problems can be studied on the basis ofEq. (1):spectral problems (k is the eigenvalue sought) and"reflection-transmission" problems (in this case, k isgiven and Eq. (1) must be solved simultaneously with cor-responding boundary conditions).

The spectral problems for (1) are considered in Refs. 6and 9—11. The effects of Anderson localization andfeatures of a devil*s-staircase type of spectrum are re-vealed. However, in these works very specific models andtechniques are used: In Ref. 9, f (x) is defined by theKronig-Penny approximation, while in Refs. 6, 10, and11 the tight-binding model with periodic level modula-tion is used (this is valid when the periods differ greatly).But the main thing is that from these results it is notclear how the features revealed in the spectrum appear inthe reflection and transmission coefficients.

II. THE SPECTRUM

Let us assume that qi=q+c, where ~c~ &&q, andrepresent f(x}in the following form:

f(x ) =is(x )cos2qx —o (x )sin2qx,

where p(x ) =pp+picos2cx and o'(x ) =lsisin2cx.Then, for ~f(x)~ &&1, Eq. (1) with the coefficient (2)

can be solved by Kogelnik's method of coupled waves.In accordance with Ref. 2, solution (1) can be representedas

A(x)=A+(x)e' +A (x)e

where A+(x) are new unknown functions, slowly varyingin comparison with the rapidly varying factors e J

The equations for A +(x}are obtained by averaging overthe period of rapid motion; the coefficients p(x ),o (x) arerelated to the slow motion and are regarded as constantsin the averaging procedure.

The final form of the equations is

d A+ kv+(x )=j A (x )exp( —2jlkx ),kv (x)

A+ (x )exp(2jhx ),

0163-1829/94/50(6)/3631(5)/$06. 00 50 3631 1994 The American Physical Society

Page 2: Bragg diffraction of waves in one-dimensional doubly periodic media

3632 BASS, SLEPYAN, ZAVTRAK, AND GUREVICH 50

where

v+(x ) =go+ p, exp(+2jex ), 5=k —q .

It is clear that since v+=v, system (4) satisfies theenergy-conservation law

It is conventional to transform (4) by eliminating oneof the unknown functions (for definiteness, A ). Then,for A+, we obtain the equation

k =[6 —(k@0/4) ]'

As a result, we obtain the equations

kp+=j a (x )exp( —2jb,x ),4

kp= —j a+ (x )exp(2jbx ),

where 5=k —e,

(8)

d A+

dx

1 d dA+[v+(x )exp( —2jbx ) ]exp(2jhx)

v~ x dx dx2

kv+v A+ =0 . (6)

PiP+=@ok

Pi

ppk

2kPo Roe

4 2

2kpo

d 0 Q2X

Po

4—p, b,e exp(2jex )

k2pipocos(2sx) a =0 . (7)

As a result of the transformations carried out, the ini-tial problem is reduced to (7). It is an equation of type(1), but with periodic coefficients, whose oscillationperiod is determined by s. Equation (7) describes wave

processes in a certain conventional periodic medium witha complex refractive index, whose real and imaginaryparts are oscillatory. In this case the latter acquires posi-tive and negative values. However, it should be notedthat neither this periodic medium nor the function a (x )

sought has a direct physical significance [a(x } is directlyconnected not with the wave field A(x} but with theslowly varying amplitude of its partial wave]. It issignificant that Eq. (7) is related to the class of Hill-typesystems, and can therefore can be investigated by con-ventional methods.

The further analysis of Eq. (7) depends on whether theparameters correspond either to the aHowed or the for-bidden bands. Let us consider the allowed band charac-terized by the condition l b l

- k@0/4. In principle,diferent approaches are possible here, but the simplest isa secondary application of Kogelnik's method of coupledwaves with an averaging over the period m/c. . In analogywith what was done above, let us assume

a{x}=a+{x)ej"+a (x)e

where a + (x ) are new unknown functions and

Let us introduce a new unknown function a(x) by therelation A+ =ae"' ', where

II(x)=,' f"g(q)dZ,

2jy, ,e exp(2jex )Q(x )= —2jh+

pa+ p, exp(2jsx }

Let us restrict our consideration to the case of small

}ui, and in the equation for a(x ) drop the terms 0(pi).Then according to (6), the equation will have the form

'2

System (8) does not satisfy the law of energy conserva-tion in the form (5). This fact is due to the above-mentioned indirect connection of the function's a + (x )

value with the initial physical field.%hen the condition

u+p-4

is satisfied, system (8} describes the regime of wave non-transmission. When the opposite inequality is satisfied, itdescribes the regime of transmission. Thus, we are led tothe first step of a devil' s-staircase type of spectrum: Theallowed bands under the influence of the double periodi-city acquire an additional small-scale band structure. In-side these bands the new forbidden microbands appear,whose frequency width is determined by the equality

u lpcpil

32lko —ql'

where ko is the wave number

k =q+(e2+q2p2/16)i

(10)

corresponding to exact synchronism and which is defined

by the equation k =s. A description of the higher stepsof the devil's staircase requires taking into account higherapproximations in the averaging method, and this is aproblem for another discussion.

In the case of the forbidden band, lb, l ~k@0/4, theanalysis based on the method of coupled waves is not ap-plicable. A quantitative analysis of the spectral proper-ties can be carried out if we consider 5 to be small. Thisassumption is satisfied in the central part of the forbiddenband. Then in Eq. (7) we can drop the terms proportion-al to 6 and reduce it to the Mathieu equation.

Using the results of the analysis of the Mathieu equa-tion given in Ref. 12, we can conc1ude that due to thedouble periodicity, narrow-frequency bands of wave

propagation are possible inside the forbidden band.Their physical nature is analogous to that of the Tammsurface states in semi-infinite crystals. '

Page 3: Bragg diffraction of waves in one-dimensional doubly periodic media

50 &RAGG DD"E'RACTION OF WAVES IN ONE-DIMENSIONAL. . . 3633

III. THE SRAGG REFLECTION COEI FlCIENTFROM A HALF-SPACE AND FROM A LAYER

OF A DOUBLY PERIODIC MEDIUM

In this section we shall calculate on the basis of theabove-mentioned approach the refiection of a plane wavefrom the layer 0&x &L characterized by the functionf (x) given by Eq. (2). Outside the layer (for x &0 andx )L), we assume f(x )=0. Let us represent the solutionin the following form:

A(x)=e'~+Re ', x &0

A( x)=A+( x)e J +A (x)e J, 0&x &L

A(x)=Tej, x)L

where

[exp( 2—yL )—p R „/p+ ]r~~ — exp(2jsL )

[1—p R „exp( 2—yL )/p+ ]

1+p r~exp(2j eL )/p+

[1—exp( 2y—L ) ][1—p R „exp( 2y—L )/p+ )

JkP+

4(jE—y)where

(17)

(18)

(19)

where R and T are the required reflection and transmis-sion coefficients and A+(x) are the solutions of system(4)

The boundary conditions for (4), as in Ref. 14, have theform

A+(0) =1,A (L)=0 .

(12)

where gp=b, —p, s/(pp+p|} and the prime signifies thederivative with respect to x. The required reflectioncoefficient R is expressed through the function a(x) bythe equality

R=a(0) . (14)

The secondary averaging procedure is correct if thecondition sL 1 is satisfied. As a result of this pro-cedure, we proceed to Eqs. (8), and the boundary condi-tions to them follow from (13). They can be representedin the form

To use Eq. (7), it is necessary to express A~ in (12}through the function a(x ). Then we obtain the boundaryconditions to (7) in the following forms:

kv (0)a'(0)+jqpa(0)= —j(13)

a(L)=0,

Of particular interest is the case of a semi-infinitestructure (L ~ 0o ) for which we have

(1+R„)R =ap(1+r, ) 1+r)R „ (20)

1+R1+RpR

(21)

where R „is given by Eq. (19).Let us analyze in detail the result (20). In this case let

us consider that the value R „ is the coefficient of Braggreflection for conventional periodic media described byEq. (7}and characterized by the spatial frequency 2s.

In the range of parameters corresponding to the in-equality (9), ~R „~=(p+ /p )'~ (in this case the inequali-

ty ~R „~ 1 is possible, which does not contradict physi-cal meaning, since R „does not correspond directly tothe real physical field); on departing from the regiondefined by this inequality, ~R „~ decreases quickly and be-comes small. In this case, the second periodicity scaleexpresses itself in a trivial fashion: under its in8uence theedges of the allowed and forbidden bands are slightlyshifted [formally it implies that the value of r, differsfrom (b, —k )/(b, +k )].

The most interesting effects arise in the forbidden mi-crobands, where ~R „~-1. For the analysis of this case itis convenient to present Eq. (20) in the form R =RpN,where

a+(0)+ ] ra(0)=(1+r~ )ap

a (L)= a+(L )exp(2jkL —),where

gp—k(1 —p /4)

gp+k(1 —p+ /4)

(15)and Rp is the reflection coefficient for singly periodicmedia which corresponds to p,|=0. The coefficient N in-side the forbidden microband is complex valued and be-comes real only at the edges of the microband. At one ofthese edges, R „=(1+2 Ys/k ) '~; at the other,

R „=—(1+2Y./k)'",

R =ap(1+r&) (1+7)(1+r,r)

(16)

kv (0)ao=-[8np «Z+ —u —)1—

By solving system (8) with the boundary conditions(15), as was done in Ref. 1, and using the equality (14), wereceive an expression for 8 in the allowed band:

where

4hkpo

Correspondingly, at one of these edges, ~N ~) 1 (in this

case, )N( &2); at the other of these edges, )N) becomessmall, of the order of e,. Thus, oscillations on the curve of~R (5) ~

arise; its qualitative nature is shown in Figs. 1-4.These oscillations decrease directly at the edge of the for-

Page 4: Bragg diffraction of waves in one-dimensional doubly periodic media

3634 BASS, SLEPYAN, ZAVTRAK, AND GUREVICH 50

1.00— 'I .00—

0.80

0.60— 0 60

IRI

0.40—

0.20—

' (II)

I

I

t

IRI

0.40

0.20

1.00 1.10 1.20 1.300.00

1 1.20 1.301.10I I I I I I I I I l I I I I I I I I I l I I I I I I I I I

FIG. 1. The forbidden microband structure in the curve of!R ( Y)! for different slow modulation amplitudes p,{8/q =0.001 pp=0. 01 ): {I) p) =0.001 (II) p] =0.00025 and(III) I,——0.

FIG. 3. The forbidden microband structure in the curve of!R ( Y)! for different slow modulation amplitudes Iu,

(c,/q=0. 001, pp=0. 01): (I) p&= —0.001, (II) p&= —0.0005,and (III) p& =0.

bidden and allowed bands, where Ra~ 1, and nothing ofinterest appears far from Bragg resonance due to the rap-id decrease of R. The most interesting features arepresent in the vicinities of edges of the allowed band,wher e R p 0.3-0.7. Similar oscillations should alsooccur for a layer of finite thickness, but in this case theywill be superimposed on the oscillatory curve! R (b, )!.

It should be noted that even under the conditions ofsmall IMII and pl, the value of the amplitude reSectioncoefficient depends significantly on the phases of theslowly modulating functions p,(x ) and cr(x) at the valuesof x corresponding to the boundaries of the doublyperiodic media. This is clear from the different qualita-tive natures of the curves of !R (b )! corresponding todifFerent signs of (p&yII ) (Figs. 1 and 2 and Figs. 3 and 4).

q pp~

(I I I)(IV)

1.00—

0.800.80

IRI

0.600.60

0.400.40

0.200 20

I (I)l

I

0.001.00

I I I I I I I I l I I I I I I I I I l I I I I I I I I I

1.20 0.001 1.3000 1.20

I I I I I I I I I l I I I I I I I I I l I I I I I I I I I

FIG. 2. The forbidden microband structure in the curve of!R(Y)! for different slow modulation periods a (pl=0. 001,pp=0. 01): (I) c/q =0.001, (II) c/q =0.0008, (rrr) c/q =0.0005,and (IV) c/q=0. 00025. Curve (V) corresponds to the case

pi =0.

FIG. 4. The forbidden microband structure in the curve of!R(Y)! for different slow modulation periods a (pl = —0.001,pp=0. 01 ): (I) E/q =0.001 (II) 8/q =0.0008 and (III)c./q =0.00025. Curve (IV) corresponds to the case p& =0.

Page 5: Bragg diffraction of waves in one-dimensional doubly periodic media

50 BRAGG DIi j.'+ACTION OF %AVES IN ONE-DIMENSIONAL. . . 3635

1V. DISCUSSION

The analysis carried out here shows that for doublyperiodic structures with nearly equal periods, the devia-tion from the exact periodicity causes the characterizedpeaks on the curves of ~R (5)

~in the vicinities of edges of

the allowed bands. Inside the forbidden bands, sharpdips appear, which are caused by the influence of narrowallowed microbands (see the end of Sec. II). The first stepof the "devil's staircase" manifests itself. A descriptionof the fine structure of these peaks and dips requires tak-ing into account higher steps of the devil's staircase andhigher approximations of the averaging method, respec-tively.

l.et us consider some possible extensions. The exten-sion of the present analysis to the case when f(x ) in (1) isthe sum of two arbitrary periodic functions is quite obvi-ous. The analysis is more complicated for two- or three-dimensional diffracting systems which are described bypartial-differential equations instead of (I) (Maxwell sequations, Schrodinger s equation, etc.). In this case thedouble periodicity can be caused by the inhomogeneity ofthe medium (the variable coefficient in the equation), bythe form of the boundary, or by a combination of both ofthese factors. As an example of the latter case, one canindicate a periodically corrugated radiowaveguide, filledby a periodic medium with a different period. Finally, letus note the case when double periodicity is caused by theinteraction of several periodic boundaries with differentperiods (for example, a plane dielectric waveguide onboth surfaces of which corrugations with differentperiods are present).

It is natural to investigate all of the above-mentionedproblems by the use of the coupled-wave method, but in-stead of plane waves, one should use the eigenmodes ofthe corresponding regular waveguides. In this case theanalysis will be similar to the one described in this paper;the physical conclusions will be similar also.

A certain diSculty appears when the structure con-sidered is formed by the double periodicity of an irregularboundary. This difficulty is connected with the fact thatthe required fields and coupled waves used for its repre-sentation can satisfy nonidentical boundary conditions.To overcome this difBculty, it is necessary to use a specialnonorthogonal coordinate transformation. ' ' As a re-sult, the boundaries become regular and inside them is an"efFective" inhomogeneous and anisotropic medium,whose parameters are fully determined by the boundaryequations. Thus, all the problems of the above-mentioned class can be solved in an unfiled way.

Results similar to those described in this paper will beobtained for parametric light scattering (parametricfiuorescence) in periodic media with a Pockels nonlineari-ty. ' In the approximation of the given (very strong)pump wave, the equations for the amplitudes of the idlerand signal waves have the form (4). In this case theperiodicity of v+(x) is caused, for example, by the periodicity of the nonlinear permittivity or by the slow spatialperiodic modulation of the pump wave amplitude. Theforbidden microbands are interpreted as new areas ofpump wave decay instability. So, the parametric scatter-ing spectrum will have the structure of a "devil's stair-case."

In conclusion, let us note two more classes of problemsconnected with doubly periodic media. The radiationfrom charged particles transversing such media is in-teresting. Obviously, the peaks and dips revealed in thispaper will manifest themselves in the characteristics, ofthe radiation spectrum. Finally, nonlinear effects in dou-bly periodic media are very important. The devil' s-staircase type of spectrum is formed as a result of thecomplex interaction of numerous Q-switched modes withnearly equal frequencies. So, it may be expected that theinclusion of nonlinearity causes the resonant bridges anda dynamic chaotic intensity significantly lower than inother wave systems.

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