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Diraction by a grating consisting of absorbing screens of dierent height. New equations
A. I. Korolkov, A. V. Shanin Decemb er 5, 2014

)>IJH=?J
The problem of diraction of a plane wave by a grating consisting of absolutely absorbing screens of dierent height is studied. It is supposed that the angle of incidence is small. The problem is considered in the parabolic approximation. Edge Green functions are introduced. The embedding formula and the spectral equation for the edge Green functions are derived. The OE-equation for the coecients of the spectral equation is constructed. The latter is solved numerically. The evolution equation describing the dependence of the edge Green functions on the geometrical parameter (the height of the screens) is proven. Using this equation, the asymptotics of the reection coecient is calculated for the principal mode.

1 Introduction
The problem of scattering of a waveguide mo de by an op en end of a waveguide with acoustically hard thin walls was studied by L. A. Weinstein [1]. The problem was solved by applying WienerHopf metho d [2]. The co ecients of scattering of the incident mo de (the mo de moving towards the op en end of the waveguide) into the reected mo des (the mo des moving back from the op en end) were obtained. This problem is of a signicant practical value. It helps to describ e oscillations in a FabryPerot resonator, which is considering as a part of a waveguide in this case. A wave pro cess in such a resonator is represented as successive reections by op en ends of the waveguide. The key result obtained by L. A. Weinstein is as follows. A high-frequency mo de (i. e. a mo de with wavelength much smaller than the width of the

1


waveguide) close to the cut-o frequency (i. e. comp osed of partial waves traveling almost p erp endicular to the axis of the waveguide) has reection co ecient close to

-1

. This result is quite surprising b ecause the op en end

of the waveguide has no reective structures. The co ecient close to The fact that the reection co ecient is close to

0

would

b e rather exp ected due to radiation of the wave energy into the op en space.

-1

explains high Q-factor

of FabryPerot resonators in the absence of fo cusing elements. The problem of high-frequency scattering close to the cut-o frequency is quite complicated since the problem cannot b e solved in terms of the geometrical theory of diraction: the eld contains multiple p enumbra comp onents. Here we mean that a primary p enumbral eld b eing scattered by an edge generates a secondary p enumbra, etc. L. A. Weinstein noticed that by using the reection principle one can reformulate the problem of diraction by a waveguide outlet as a problem of scattering on a branched surface by a p erio dic diraction grating formed by branch p oints. This formulation seems to b e more convenient due to the p ossibility to analyze this problem using the formalism of the edge Green functions. One can use a parab olic approximation in the high-frequency case. In the parab olic approximation one can can study a grating comp osed of p erfectly absorbing screens instead of the branched surface. Calculation of the Q-factor of high-frequency mo des in op en rectangular resonators leads to the problems similar to the Weinstein's one [3]. Mo des in such resonators can b e represented as families of rays propagating along closed billiard tra jectories. The losses of the mo des are due to diraction. One can asso ciate p erio dic gratings having more complex p erio ds with such problems. Unfortunately, these problems cannot b e solved using the classic WienerHopf metho d. Particularly the problem solved here can b e reduced to a matrix WienerHopf equation. Solution of this equation is unknown. A new technique of solving Weinstein's problems has b een prop osed by the authors [4, 5]. This technique do es not rely on the WienerHopf metho d. The main aim of the current work is to develop this technique further and to construct ecient numerical and asymptotical metho ds based on it. A physical problem that motivates the present study is the problem of radiation by an op en end of a plane waveguide with walls of dierent height (see Fig. 1a) ). A stationary acoustical problem is considered, the walls The incident waveguide mo de is supp osed to b e close It is necessary to nd the of the waveguide are supp osed to b e p erfectly reecting (acoustically hard) and innitely thin. to its cut-o frequency, i. e. the partial Brillouin waves propagate almost p erp endicularly to the axis of the waveguide. co ecients of scattering into the reected mo des. In the language of FabryPerot resonators this problem is related to a res-

2


a)

b)

c)

Fig. 1: Geometry of studying systems : a) Op en waveguide, b),c) Fabry-Perot resonators

onator with parallel mirrors of dierent height or with staggered mirrors(see Fig. 1b), c) ). This problem was studied in [1] using FoxLi integral equations. The integral equations were reduced to a functional equation for the reections co ecients and in the approximation of small angles some rough estimations of the mirror reection co ecient have b een made. L. A. Weinstein did not obtain any asymptotic expression for the scattering co ecients. Using the reection metho d [5] this problem can b e reduced to the problem of scattering by a p erio dic grating of p erfectly absorbing screens (see Fig. 2). In the high-frequency case the last problem can b e studied in the parab olic approximation. The reection co ecients for the initial problem corresp ond to the scattering co ecients for the grating.

y=y y=0
in

2



1

y x a a a

0 =

in

Fig. 2: Geometry of the p erio dic grating The emb edding formula is derived for the problem in the current pap er. This formula expresses the scattering co ecients in terms of the directivities of the edge Green functions. Then the sp ectral equation is derived, which is the ordinary dierential equation for the directivities of the edge Green functions. An OE-equation for

3


the co ecient of this dierential equation is constructed. The OE-equation is solved numerically. This part of the pap er is close to [5]. Then the evolution equation describing the dep endence of the edge Green functions and of the co ecients of the sp ectral equation on the geometrical parameter y is derived. Asymptotic expressions for the directivities of the edge Green functions and asymptotics of the reection co ecient are built by using the evolution equation. We use the results obtained in [5] where it is p ossible.

2 Problem formulation
Consider Cartesian plane diraction equation:

(x, y ). (

Let the eld variable

u

satisfy the parab olic

1 2 + x 2ik y 2

) u = 0. x=x
m,
(1)

Perfectly absorbing screens are placed along the lines

y
m,

m Z,
(2)

xm = am, { 0 m even, ym = y m odd.

(3)

The eld on the screens is discontinuous. Perfect absorption means that the following b oundary conditions are satised on the right sides of the screens (

x = xm + 0, y < y

m ):

u(xm + 0, y ) = 0.

(4)

Such b oundary conditions can b e imp osed in the framework of the parab olic approximation since the governing equation (1) is of order one with resp ect to

x.
Short wave case is considered:

k a 1.
The total eld is represented as

(5)

u = uin + usc ,
where

u

in is the incident plane wave

} { 2 in - ik y in , uin (x, y ) = exp -ik x 2
4

(6)


and

usc

is the scattered eld. Angle

in

is small: (7)

in 1.
Boundary conditions for

usc

are as follows:

usc (xm + 0, y ) = -uin (xm , y ),

y < ym . (xm , ym )

(8)

Problem formulation should b e supplemented with conditions at the ends of the screens and with radiation condition. Near the end p oints are no sources at the end p oints. Due to the radiation condition the scattered eld should not contain comp onents coming from the area of large the total eld should b e b ounded. Boundedness of the eld guarantees that there

|y |

(this fact is taken into account when

the Fourier expansions are constructed). Geometrical p erio d of the grating along the incident wave has prop erty:

x

axis is equal to

2a

. The

uin (x + 2a, y ) = uin (x, y ),

2 = exp{-ik ain }.

Due to Flo quet theory the scattered eld should have the same prop erty:

usc (x + 2a, y ) = usc (x, y ).
So, in the upp er half-plane ( panded as series:

(9)

y > max(0, y )) {

the scattered eld can b e ex-

usc =

m=-

2 Rm exp -ik x m + ik y 2
2 in +

}
m

,

(10)

m =

2 m . ka m

(11)

The values of the square ro ot are chosen p ositive real or p ositive imaginary in consistence with radiation condition. Positive real diractions orders, while p ositive imaginary scattering co ecients corresp ond to



m corresp ond to inhomogeneous

waves existing only in the near eld. Note that

0 =

in . Our aim is to nd

Rm

.

The problem contains four geometrical parameters:

k , a, y



,

in

.

Due

to the structure of the parab olic equation the problem dep ends (up to scal k /a. The rst one ing) on two dimensionless parameters: in k a and y indicates the domain of small angles by inequality



in

k a 1.
5


Note that this condition yields also that the values (11). mismatch

1/ k a

and

R0 -1 1 - 0

for the Weinstein's problem. Note are of the same order according to

The second dimensionless parameter represents the ratio of the height y to the size of the rst Fresnel zone a/k . The size of the

rst Fresnel zone (not the wavelength!) is an imp ortant spatial scale in the

y

direction. Thus

y





k /a 1

corresp onds to small dierence of the screen heights.

3 FoxLi integral equations and matrix Wiener Hopf problem
Consider an arbitrary strip

X1 < x < X

2 without screens. It is known that

parab olic equation (1) has an explicit solution in this strip:

u(x, y ) =
-
where

u(X1 , y )g (x - X1 , y - y )dy ,

(12)

{

g (x, y ) =

k 1/2 (2 x)- 0,

1/2

exp{ik y 2 /(2x) - i /4},

x > 0, x < 0.

(13)

is the Green function of the entire plane.

Using this formula it is easy to

obtain a system of integral equations for the scattered eld lines x = x0 (= 0), y > y0 (= 0) è x = x1 (= a), y > y1 (= y ):

usc

on the half-

usc (a, y ) =
0

usc (0, y )g (a, y - y )dy -
-







0

uin (0, y )g (a, y - y )dy ,


y > y,
(14)

usc (0, y ) =
y


usc (a, y )g (a, y - y )dy -

y

uin (a, y )g (a, y - y )dy ,

y > 0.
(15)

-
Also, the Flo quet

Boundary conditions (8) are taken into account here. relation (9) is used in derivation of the second equation.

6


Equations (14) and (15) form the system of FoxLi integral equations for the considered problem. Apparently, this system can b e solved by iterations. Such a solution is not however helpful for nding the co ecients

Rm

due to

slow convergence of the iterative series. Moreover, the iterative series do es not enable one to establish asymptotic prop erties of the scattering co ecients. That is why we prop ose another metho d for solving this problem (the metho d of emb edding formula, sp ectral equation and OE-equation). A matrix WienerHopf equation can b e constructed from system (14), (15). Intro duce Fourier transforms for the unknown functions:

^ U0 ( ) =
0

usc (0, y )e-iy dy ,

(16)

^ U1 ( ) =
y


usc (a, y )e

-i (y -y )

dy .

(17)

Also, intro duce auxiliary funcitons

0 ^ W0 ( ) =
- y


usc (-0, y )e-iy dy ,

(18)

^ W1 ( ) =
-

usc (a - 0, y )e-

i (y -y )

dy .

(19)

Rewrite the system (14), (15) as follows:

(

^ W0 ^ W1

)

(

+K

^ U0 ^ U1

) = r,
(20)

where

(

K( ) =

2 1 - exp{ik ain - i y - ia 2 /(2k )} - exp{i y - ia 2 /(2k )} 1

) ,
(21)

r( ) =
Unknowns

-i + k

(
in

exp{i y - ia /(2k )} 2 exp{ik ain - i y - ia 2 /(2k )}
2



) .

(22)

^^ U0 , U1

in (20) should b e regular and decreasing functions in the

lower half-plane of the complex variable



.

Functions

^ ^ W0 , W

1 should b e

regular and decrease in the upp er half-plane. WienerHopf problem derived here is a matrix one, and the authors are not aware of its solution.

7


4 Edge Green functions
The main idea of emb edding is to reduce the initial problem to two (in the current case) simpler auxiliary problems. Intro duce the edge Green functions

v

0 and

v

1 as solutions of the following inhomogeneous parab olic equations:

(

1 2 + x 2ik y 2 )

) v0 = (x - 0, y ) ,
(23)

and

(

1 2 + x 2ik y 2

v1 = (x - a - 0, y - y )

(24)

on the plane with p erfectly absorbing screens describ ed earlier. One can see that the edge Green functions are generated by p oint sources. The sources are shown in Fig. 3. Arguments

x-0

è

x-a-0

mean that the sources are

a)

b)

Fig. 3: Problems for the edge Green functions: case a) corresp onds to case b) corresp onds to placed to the right of

v

0,

v

1.

(0, 0) and (a, y ). This lay-out guarantees that function v0 (x, y ) is equal to g (x, y ) in the area 0 < x < a and function v1 (x, y ) is equal to g (x - a, y - y ) in the area a < x < 2a. By using geometrical p erio dicity intro duce edge Green functions vn for any n Z: v
m+2n

(x, y ) = vm (x - 2an, y ), Vm

m = 0, 1. v
m as co ecients

Dene the directivities

of the edge Green functions

of the leading terms of the asymptotic expansions:

( vm (x, y ) = g (x - xm , y - ym )Vm

y - ym x - xm

)

( + o (x - xm )

-1/2

)

,

m = 0 , 1.
(25)

8


The parab olic equation and formula (12) enables one to construct a formal representation for

vm (x, y ).

Obviously, for

vm (x, y ) = 0
Then

x < xm .

(26)

vm (x, y ) = g (x - xm , y - ym ) for xm < x < xm+1 , vm (x, y ) = vm (xn - 0, y )g (x - xn , y - y )dy for xn < x < x
yn
Thus, the expression for directivities

(27)

n+1

.

(28)

vm (x, y )

in the strip

xn < x < x

n+1 contains

n-m

nested integrals. However, such a solution is not useful for calculation of the

Vm ().

Intro duce the values that play an imp ortant role b elow, namely the edge values of the edge Green functions:

zm,n =

xxn -0

lim vm (xn , yn )

m < n.

(29)

The limiting pro cedure is necessary b ecause solution of the parab olic equation obtained by (28) is generally discontinuous at the p oint

(xn , yn ).

The left

limit is taken due to the continuity of the eld on the left side of the screen. Prove an imp ortant formula that will b e used in derivation of the sp ectral and evolution equation. Consider a more general problem. Let the ends of the absorbing screens have arbitrary co ordinates conditions (2), (3)). problem. Let values

(xm , ym )

(i. e. we discard

By analogy, intro duce edge Green functions for this

Obviously, edge Green functions are still expressed by formulae

(26), (27), (28). Consider a family of such problems indexed by parameter



.

y

m in this family b e smo oth enough functions of



. We denote

the edge Green functions of this family by the edge Green functions by

vm (, x, y )

and the edge values of

zm,n ()

(by analogy with (29)).

Òheorem 1 The fol lowing formula is valid
vm (, x, y ) = ( ym yn ) ym vm (, x, y ) - . zm,n ()vn (, x, y ) - y =m+1 x
an integer parameter

(30)

n

Proof
to

Intro duce for each

n(x),

such that

x

x xn(x) + 1 n(x) - m.

n(x)

<

. The theorem can b e proven easily by induction with resp ect Formulae (27), (28) should b e used.

9


Formula (27) works for

n(x) = m

, and this case is easy to verify (there is

no summation in the right side of (30)). This is the base of induction. Make the inductive step. Supp ose that the statement (30) is valid for

l - 1.

Then for

n(x) - m = l

n(x) - m =

we have

vm =
y

vm (x
m+l

m+l

- 0, y )g (x - xm+l , y - y )dy =

-

ym+l vm (xm+l - 0, ym+l )g (x - xm+l , y - ym+l ) vm (xm+l - 0, y ) + g (x - xm+l , y - y )dy .
y
m+l

The derivative in the last integral can b e transformed using (30) (here we use the induction conjecture). Taking into account that


y
and also
m+l

vn (xm+l - 0, y )g (x - x

m+l

, y - y )dy = vn (x, y ),

vm (, xm+l - 0, y ) g (x - xm+l , y - y )dy = ym+l ) ( vm (, x, y ) ym -zm,m+l () vm+l (, x, y ) + , y
we obtain (30) for



n(x) - m = l

.

5 Embedding formula
An emb edding formula links the scattering co ecients of the initial problem with the directivities of the edge Green functions. The formula is given by the following theorem:

Òheorem 2 For m corresponding to real positive m the fol lowing identity
is valid:



Rm =

1 n=0

2 2 Vn (in )Vn (m ) exp {ik xn (m - in )/2 - ik yn (m - in )} . 2ik am (in + m )

(31)

10


Sketch of the proof

The pro of of this theorem is very close to the pro of

of corresp onding statement from [5]. Here we list only the main steps of the derivation of the emb edding formula. Apply dierential op erator

H
to the total eld

+ ik y y

in

(32)

u(x, y ).

It was shown in [5] that the eld

w(x, y ) = H [u](x, y ) um (xm , ym )
at the

satises the parab olic equation, the b oundary conditions on the screens, and the radiation condition, but it has sources with amplitudes vertices (xm

,y

m ). Thus, using uniqueness of the solution one can represent

w(x, y )

as a sup erp osition of the edge Green functions

v

n:

H [u](x, y ) =
The values of the eld

n=-

u(xn - 0, yn )vn (x, y ).

(33)

u(xm - 0, ym )

at the end p oints of the screens can b e

expressed through the edge Green functions with the help of the recipro city theorem for the parab olic equation. The expression is as follows:

{ } 2 u(xm - 0, ym ) = exp -ik xm in /2 + ik ym in Vm (in ).
elds we obtain (31).

(34)

Substituting this expression into (33) and considering the directivities of the (31) takes the simplest form for the case of mirror reection, i. e. for

R0

:

R0 =

(V0 (in ))2 + (V1 (m ))2 . 2 4ik ain

(35)

Thus, in order to solve the original problem we need to nd the directivities

V0 , V

1.

6 Spectral equation
Òheorem 3 A rowvector comprised of directivities (V0 (), V1 ()) satises
the ordinary dierential equation

d (V0 , V1 ) = (V0 , V1 )(y )C(, y )-1 (y ), d
with the initial condition

(36)

V0 (+) = 1,
11

V1 (+) = 1,

(37)


where

( ) 1 0 (y ) = , 0 exp{-ik y } ( ) C0,0 C0,1 C(, y ) = , C1,0 C1,1


(38)

(39)

C0,0 (, y ) = C1,0 (, y ) = C0,1 (, y ) = C1,1 (, y ) =




(x2m - x0 )z0

,2m

{ } exp ik (x2m - x0 )2 /2 , } - x0 )2 /2 , } - x1 )2 /2 , } - x1 )2 /2 .

(40)

m=1

(x2m+1 - x0 )z (x2m+2 - x1 )z (x2m+1 - x1 )z

0,2m+1

{ exp ik (x2m { exp ik (x2m { exp ik (x2m

+1

(41)

m=0 m=0 m=1
(43) (42)

1,2m+2

+2

1,2m+1

+1

Proof
m

Derivation of the sp ectral equation is very close to the derivation of

the corresp onding equation from [5]. Here we use (30) for this. Fix value of (2) but dierent values as 0 or 1. Intro duce a family of problems having the same values ym :

xn

as in

yn = yn + (xn - xm ).
Due to the fact that op erator

Km = (x - xm )

- ik y y
if a solution for

(44)

is the symmetry op erator of the parab olic equation in the classical sense, it is p ossible to construct solutions for all



=0

is known:

vn (, x, y ) = vn (x, y - (x - xm )) exp{ik (y - ym ) + ik 2 (x - xm )/2}.
Besides, it follows from the last equality that

Km [vm ] = -

vm . =0
.

Apply formula (30) to the left-hand side of the last equality and put It yields

Km [vm ](x, y ) =



(xn - xm )zm,n vn (x, y ).
12

n=m+1


Consider the directivities of the elds in the right and in the left. result, obtain (36).

As the

In [5] it was shown that the sp ectral equation can b e formulated in a more general way. A similar statement is valid in our case.

Òheorem 4 The fol lowing ordinary dierential equation is valid:

where

( ? V0 ? V1

,0 ,0

? V0, ? V1,

)

1 1

( ? V = ?0, V1,

0 0

) ? V0,1 (y )C( 2 + 2p)-1 (y ), ? V1,1

(45)

? V0,0 (, p) = ? V1,0 (, p) =

l=0

~ V2l,0 () exp {ik (x2l - x0 )p} , () exp {ik (x - x0 )p} ,

(46)

l=0

~ V2

l+1,0

2l+1

(47)

? V0,1 (, p) = ? V1,1 (, p) = ~ Vm,n () =
y m

l=1

~ V2l,1 () exp {ik (x2l - x1 )p} , () exp {ik (x - x1 )p} ,

(48)

l=0

~ V2

l+1,1

2l+1

(49)

- exp{ik (xm - xn )2 /2}
-

vn (xm - 0, y ) exp{-ik (y - yn )}dy , ~ Vm,m () 1, ~ Vm,n () 0, m < n,

m > n,
(50)

and p has an arbitrary complex value with a nonnegative imaginary part.
Values

? V0,0 (, p)

are directly connected with the directivities:

? ? V0 () = V0,0 (, 0) + V1,0 (, 0),

? ? V1 () = V0,1 (, 0) + V1,1 (, 0).

(51)

These relations are close to (28) from [5]. Then,

Vm () = 1-

n=m+1

y } n 2 vm (xn -0, y ) exp{ik (ym -y )}dy . exp ik (xn - xm ) 2

{

-

(52)

13


The values intro duced here can b e interpreted as follows. Formula (52) expresses the directivities of the edge Green functions through the integrals of the eld on the surface of the screens. The rst term that do es not contain an integral (the unity) represents a contribution from the p oint source. Contributions of dierent screens can b e treated separately. These contributions are closely related to the directivities on the branched surface intro duced in [5]. Parameter

p

has no physical meaning, it is the variable of the dis-

crete Fourier transform with resp ect to the numb er of the screen. Sp ectral equation (36) follows from (45) when p = 0. Thus when matrix C( , y ) is constructed it will b e p ossible to determine

V0 , V

1 as a solution of the sp ectral equation (36) with the initial condition

(37). Formulae (40), (41), (42), (43) cannot b e used for calculation of the coecient

C()

due to slow convergence of the series there.

7 OEequation
Intro duce a notation for a solution of a matrix ordinary dierential equation. Consider an equation of the rst order:

d X( ) = X( )K( ). d
Solve it along contour

(53)

h

with start p oint

1

and end p oint



2 . Also set the

following initial condition:

X(1 ) = I.
By denition

OEh [K( ) d ] X(2 ).

(54)

Òheorem 5 Function C(, y ) satises the fol lowing equation:
OE [(y ) C(
for any p (Im[p] 0),



2 + 2p) -1 (y )d] = T(p)

(55)

(

T(p) =

) 1 - exp{ik ap} , - exp{ik ap} 1

(56)

contour is the real axis passed in the negative direction. in the left-hand side is an independent variable, while p is a parameter.
The pro of of this theorem is similar to the pro of of the corresp onding statement in [5].

14


8 Evolution equation of the rst kind
Intro duce matrix

( D D(, y ) = D
m=1

0,0 1,0

D D

)
0,1 1,1

,

(57)

D0,0 (, y ) = D1,0 (, y ) =


z0

,2m

{ } exp ik (x2m - x0 )2 /2 , { exp ik (x2 { exp ik (x2 { exp ik (x2 } - x0 )2 /2 , } - x1 )2 /2 , } - x1 )2 /2 .

(58)

m=0

z0

,2m+1

m+1

(59)

D0,1 (, y ) =


m=0

z1

,2m+2

m+2

(60)

D1,1 (, y ) =
Obviously,

m=1

z1

,2m+1

m+1

(61)

C=

1 D . ik

(62)

Òheorem 6 Rowvector comprised of directivities (V0 (), V1 ()) satises the
fol lowing dierential equation with respect to y :

d ~ (V0 , V1 ) = (V0 , V1 )(y )D(, y )-1 (y ), dy
where

(63)

( ~ D(, y ) =

0 -D1

,0

) D0,1 . 0 y

as

(64)

Proof

To prove the theorem use the formula (30). Take



. This yields

vm (x, y ) vm (x, y ) =- (n - m )zm,n vn (x, y ) - m , y y n=m+1
where

n = 1

for o dd

n

and

n = 0

for even

n

. Particularly,

v0 (x, y ) =- z y l=0 v1 (x, y ) = z y l=0

0,2l+1

v

2l+1

(x, y ), v1 (x, y ) . y

(65)

1,2l+2

v

2l+2

(x, y ) -

(66)

15


Consider the directivities in the previous equations:

V0 = -V1 exp{-ik y } z y l=0 V1 = V0 exp{ik y } z y l=0

0,2l+1

{ exp ik (x2 {

1,2l+2

exp ik (x2

l+2

2 l+1 - x0 ) 2 } 2 - x1 ) . 2

} ,
(67)

(68)

The last pair of equations can b e rewritten in terms of (63). Evolution equation can b e written in more general form for values (as for the sp ectral equation).

? Vm,n (, p)

Òheorem 7 The fol lowing ordinary dierential equation is valid :
y


( )( ? ? ? V0,0 V0,1 V0 ?1,0 V1,1 = V1 ? ? V

,0 ,0

) ? V0,1 -1 ~ 2 ?1,1 (y )D( + 2p) (y ). V

(69)

Proof

To prove this statement return to (65)(66). Multiply the left-hand

side and the right-hand side by

- exp{ik (xm - xn )2 /2 - ik (y - yn )},
and integrate them along the line

n = 0, 1
m . This yields

x = xm - 0, - < y < y
+1

~ z Vm,0 () = - y l=0 ~ z1 Vm,1 () = y l=0
Intro duce the values

0,2l+1

exp{ik (x2l

~ - x0 )2 /2 - ik y }Vm,2l+1 ,(70) .
(71)

,2l+2

exp{ik (x2l

+2

~ - x1 )2 /2 + ik y }Vm,

2l+2

al,n () = z

n,l

exp{ik (xl - xn )2 /2}.

(72)

Equations (70)(71) can b e rewritten in the following way:

~ a Vm,0 () = - y l=0 ~ Vm,1 () = a y l=0

2l+1,0 Vm,2l+1

~

() exp{-ik y },

(73)

2l+2,1 Vm,2l+2

~

() exp{ik y }. a

(74)

It is easy to notice that functions o dicity conditions

~ Vm,n

and co ecients

l,n satisfy the p eri-

~ ~ Vm,n () = Vm+2,n+2 (),
16

a

m,n

() = am

+2,n+2

().


Moreover, the right-hand sides of the equations are of a convolution nature. This fact enables one to apply the discrete Fourier transform to equations (73) and (74). arbitrary as follows Let us multiply (73) and (74) by

p

with

Im[p] 0

and make a summation

exp{ik (xm - xn )p} for over all m. The result is

y
where



( )( ? ? ? V0,0 V0,1 V = ?0 ? ? V1,0 V1,1 V1

,0 ,0

? V0, ? V1,

)
1 1

) 0 a0,1 ? (y ) -1 (y ), -a1,0 0 ?


(

(75)

a1,0 (, p) = ?

l=0

a

2l+1,0

() exp {ik (x () exp {ik (x

2l+1

- x0 )p} , - x1 )p} .

(76)

a0,1 (, p) = ?

l=0

a

2l+2,1

2l+2

(77)

Besides, by comparing (59)(60) with (76)(77) we get the identity

(

) 0 a0,1 ? ~ (, p) = D( 2 + 2p). -a1,0 0 ?

(78)

Equation (63) also follows from this result.

9 Evolution equation of the second kind
Equation (79) will b e called the evolution equation of the second kind.

Òheorem 8 Matrix D(, y ) satises the fol lowing equation:

where

(

D y

)

[ ] [ D ] ~ ~ = - k y D, + ,D ,
2

(79)

( ) 00 = . 01 ? (Vm,n )

(80)

Proof
matrix

To prove the theorem consider equations (45) and (69) written for with

p = 0 and ( ? V0,0 ? y V1,0

use the obvious prop erty

? V0 ? V1

)
,1 ,1

= y

) ( ? ? V0,0 V0,1 . ? ? V1,0 V1,1

(81)

17


After a simple algebra obtain matrix equation

1 ik y



) ( [ ] D -1 ( ~ -1 ) 1 D ~ = + D , D -1 . ik
-1
on the left and by

(82)

Multiply the equation by taken into account that



on the right. Also, it is

-
Thereby obtain (79).

1

= -ik , y

-1 = ik . y y = 0

(83)

Evolution equation of the second kind can b e integrated at result is

. The

D(, 0) = y D0,1 (, 0)
m=0

(

(D0,1 ) 0

2

0 -(D0,1 )2

) .
(84)

The value of

can b e found with the help of (60):

D0,1 (, 0) =
The values

z0

,2m+1

{ exp ik (x2

m+1

} )2 /2 ,

(85)

z

0,n at

y = 0

are represented by the following nested integrals: (86)

z

0,n

=

. . . g (a, y1 ) g (a, y2 - y1 ) . . . g (a, yn - yn-1 ) g (x - xn , y - yn ) dy1 . . . dyn .
0 0
Integrals of the class of (86) are calculated explicitly in [6]:

D0,1 (, 0) =
It follows from (84) that

k 2 ai n=1

(

einka n3/2

2

/2

einka - (2n)3/2

2

) .
(87)

~ D (, 0) = 0. y
Therefore matrix

(88)

~ D

y (( )2 ) ~ , y ) = D(, 0) + O ~ D( y k /a ,
ob eys the following asymptotics for



k /a 1:
(89)

where

( ~ D(, 0) =

) 0 D0,1 (, 0) . -D0,1 (, 0) 0
18

(90)


10 Asymptotic estimation of the coecient R0
Intro duce the following notation:

=y



k /a.
:

(91) Search the

Consider again the evolution equation of the rst kind (63). solution as an expansion in a small parameter

Vm (, y ) = V
where

(0)

() + V

(1) m

(2) () + 2 Vm () + ...,

m = 0, 1,


(92) ), which

V

(0)

is a solution of the classical Weinstein problem (y

=0

can b e calculated explicitly and which has the following form

[4]:

1 erfc( 0 V = exp - 2 n=1
where

{



-in/2k a) n

} ,
(93)

erfc(z)

is the complementary error function:

2 erfc(z) =
we get:


z



e

-

2

d .
. In the rst approximation (94)

Substitute (89) and (92) into (63) and expand in

Vm = V 0 - (-1)m D0,1 (, 0)V 0 () + ...
term having rst order in

One can notice (by substituting (94) into emb edding formula (31)) that the



do es not contribute to the scattering co ecient

R0

. This result is quite obvious from the geometrical p oint of view. In order to compute the quadratic term (with resp ect to



) it is necessary

to use relation (88). Due to (63),

( ) ~ 2 D ~ ~ (V0 , V1 ) = (V0 , V1 ) D2 + + ik [D, ] -1 . (y )2 y
We are interested in the value of the second derivative at

(95)

y = 0

, therefore

=I

. Also

~ D = [D, ].
Using (95) and (88), we get the following expression in the second approximation

) a 2 ( (D0,1 (, 0))2 - ik D0,1 (, 0) V 0 ()+... Vm = V -(-1) D0,1 (, 0)V () - k2
0 m 0
(96)

19


Eventually by substituting last relation into the emb edding formula we obtain the following expression for the reection co ecient

R0

:

(V 0 ())2 (1 + iaD0,1 (, 0) 2 ) R0 = + o( 2 ). 2 2ik a
For small angles

(97)

formula (97) takes the following form: ( ) ( ) 1 ka ka 3 R0 = -1 - (1 - i) - (1 + i)2-5/2 (2 2 - 1) 2 + o( 2 ). 2 2
(98) It follows from (98) that a high-frequency mo de close to the cut-o in a waveguide with staggered walls has is reected from the op en end of the waveguide with the co ecient close to -1 provided that y k /a 1. This 2 follows from the fact that the term containing is prop ortional to . This fact is similar to the Weinstein's result.

The authors supp ose that the co ecient R0 ( , y ) tends to -1 for ar bitrary y as 0. However, this statement cannot b e proved by the
technique develop ed here.

11 Numerical results
Here we determine the co ecient equation (55). Intro duce new variables

C

(

) + 2p
2
by direct solving of the OE-

= k a2 + ,
OE-equation changes as follows:

= 2k ap.

[ OE
where

( )

] B( , )d = T ( ), 2 k a( - ) ( - ka )

Im[ ] 0, ( ) ,

(99)

- B( , ) = C ( ) ka ( ) 1 - exp{i /2} T ( ) = , - exp{i /2} 1 ( ) C ( ) = C /(k a) .
-1
20

(100)


Im[t] g'(b) b
Fig. 4:

Re[t]
Contour ( ).

and

shown in Fig. 4. It is necessary to determine C ( ) on the p ositive half of the real axis in or der to calculate reection co ecients Rm . However, matrix C ( ) transforms as follows:



is going along contour





C ( + 2 n) = n C ( )-1 , n ( ) 1 0 n = , 0 exp{-i n}
1 and

(101)

where

(102)

therefore it is sucient to solve the OE-equation only on the segment OE-equation is solved along contours

(0, 2 )

.

2

shown in Fig. 5. We solve the

I m[ ]

1 2

0

N
0

2
Fig. 5:

Re[ ]

Êîíòóðû 1 è 2 .

OE-equation using a gradient pro cedure. Both contours start at the p oint



0 having large imaginary part. Therefore we set

C (0 ) = 0.

21


Contour

1

is divided into small segments by no des

N = , C

where



is a small p ositive value. (We cho ose a nonzero

OE-equation has singularities at the p oints step by step at the p oints OE-equation (99) with parameter at the no des follows

n = 0, 1, 2, ..., N . Let because the = n, n Z.) Then we nd
n,



1 , 2 , 3 , ...N = n .

. To do this we have to solve the We supp ose that C ( ) is known

1 , 2 , 3 , ...

n-1 . The co ecient at the no de



n is found by the

gradient pro cedure. The starting value for gradient pro cedure is chosen as

C ( )(0 ) = C ( )( [

n-1

),

(103)

i. e. we cho ose the value of the previous p oint as the starting approximation. The residual is calculated:

OE



] B( , n ) - T(n ) 2 k a( - n )

(104)

Using the gradient pro cedure we make the residual small enough. Then we move to the next step and so on. The same pro cedure is rep eated on contour known on the segment along contours determined.

2

. Co ecient

C ( ) V0 , V1

b ecomes b ecome

1,

2,

(0, 2 ). Then the 1,2 + 2 , 1,2 + 4 Rm
.

sp ectral equation (36) is solved etc. Directivities

Finally by applying the emb edding formula (31) we nd the

scattering co ecients

To verify the validity of the numerical results we use direct calculation based on formulae similar to (28). Numerical results are shown in Fig. 6 and Fig. 7. Fig. 6 displays the dep endence of the scattering co ecient 2 in for

R0

on

p = ak

y/

(a/k ) = 1/4.

Fig. 7 displays the same dep endence for

y / (a/k ) = 1/3.

Solid lines corresp ond to calculations p erformed using the sp ectral equation metho d. The results of the direct computations are plotted by dotted lines. One can see that the agreement is reasonable. The emb edding formula is not checked, since it is assumed to b e a well-established result in the diraction theory (e. g. see [7]).

12 Conclusion
The metho d of sp ectral equation and emb edding formula [5] is applied in the present pap er to a problem of diraction by two-height grating. Some

22


??

t|R0 |

??

??

??

???

???

???

t/
? ? ?? ? ?? ? ??

Fig. 6:

Dependence p|R0 (p)| on p for y / (a/k ) = 1/4. The solid line corresponds to spectral equation method and the dotted line corresponds to direct computations
imp ortant results are obtained using this technique. Namely, an asymptotic formula (98) for the scattering co ecient

R0

obtained using the evolution

equation is the main result of the pap er. This formula allows one to calculate Q-factor of FabryPerot resonators with staggered mirrors or mirrors of dierent height under the assumptions of small displacement (y k /a 1). The work is supp orted by RF Government grant 11.G34.31.0066, Scientic scho ol grant 2631.2012.2, RFBR grant 12-02-00114.

References
[1] L. A. Weinstein, Op en resonators ans op en waveguides. Golem Press, Boulder, Colo.,1969.

23


??

t|R0 |
??

??

??

???

???

???

t/
? ? ?? ? ?? ? ??

Fig. 7:

Dependence p|R0 (p)| on p for y / (a/k ) = 1/3. The solid line corresponds to spectral equation method and the dotted line corresponds to direct computations
[2] L. A. Weinstein, The theory of Diraction and the Factorization Metho d. Golem Press, Boulder, Colo.,1969.

[3] E. D. Shabalina, N. V. Shirgina, and A. V. Shanin, High Frequency Mo des in a Two Dimensional Rectangular Ro om with Windows, Acoustical Physics, 56(2010), pp. 525 - 536. [4] A. V. Shanin, Weinstein's Difraction Problem: Emb edding Formula and Sp ectral Equation in Parab olic Approximation, SIAM J. Appl. Math. (2009), pp. 1201-1218. [5] A. V. Shanin, Diraction of a high-frequency grazing wave by a grating with a complicated p erio d, Zap. nauch. sem. POMI RAN, 409(2012), p. 212-239.

24


[6] J. Bo ersma, On certain multiple integrals o ccuring in a waveguide scattering problem, SIAM J. Math. Anal., 9(1978), pp. 377-393. [7] C. M. Linton, P. McIver, Handb o ok of Mathematical Techniques for Wave/Structure Interactions, Chapman & Hall/CRC, 2001.

25