Waveguide lens

NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH

Nuclear Instruments and Methods in Physics Research A 331 (1993) 627-629 North-Holland

Section A

Waveguide

lens

S.A. Lutsenko and N.A. Vinokurov Budker Institute of Nuclear Physics, Lavrentyev Prospekt 11, Novosibirsk 90, Russian Federation

A focusing device, which is a specially shaped waveguide, is considered. Such a device would be useful for focusing laser beams, especially intensive, and transferring radiation into a small gap device, like an undulator. Results of a numerical calculation on the focusing properties and some applications are presented and discussed.

1. Introduction

If

the

initial

dependence

is

symmetrical,

i.e.

E p ( - x , z ) = E p ( x , z), the synchronization length is Let us first consider the two-dimensional problem concerning the propagation of the monochromatic radiation between two parallel conducting planes. Let the electric field E be directed along the y axis; the equation of planes x = +_g/2 and z is the direction of propagation. So we deal with T E waves. T h e n for the mode

(1+2x/g)

Ey = sin ~rp ~

e i(kzz-ckt)

(1)

the dispersion law is ar2p 2 k z = ~'k 2 -

'rr2p2/g

2 = k -

2kg-----7,

(2)

where k = o)/c, p is the m o d e number, and we assume kg/('rrp) >> 1, i.e. waves go from left to right paraxially. If there is any field d e p e n d e n c e on x at z = 0 we may easily find E(x, z). In particular for the distance L the initial dependence will be reproduced: E(x, L) = E(x, 0) e i~3 (/3 is some constant phase). To obtain it we must satisfy the synchronism condition of each m o d e with the fundamental m o d e ( p = 1):

(k~ - k l ) L = 2-rrqp,

2. Waisted waveguides The evolution of a paraxial wave can be described by a parabolic equation: 0u ~2u 2 i k - - + - - = 0. (5) ()Z

(3)

where qv is an integer number, or ,.rr 2

2kg 2 (p2 _ 1 ) L = 2"rrqp.

sufficiently shorter: g2 L~ ) = - - n . (6) A Thus we can say that a waveguide with length L 1 provides the optical image of an object at its entrance, 1 and a waveguide with length 2L 1 provides the inverted image. This effect makes it possible to use the waveguide as an element of linear optics. For example, the waveguide with a varying gap, in particular any waisted waveguide, may be used as an optical system with a magnification. As a waveguide of resonant length transfers a Gaussian b e a m to an (almost) Gaussian, it may be included easily into the optical resonator with the end mirror. In this paper we describe both the m e t h o d of calculation and some applications of such a waveguide.

(4)

.(x, z)lx=_+~)= 0.

(5')

By the following transformation: S=Z~

Condition (4) is satisfied by

X

= F(z----5'

g2

Ln = 8 T n ,

0X 2

In the case of T E mode, when u(x, z) = Ey, the corresponding boundary condition on the waveguide surface is

(5) u

w(~:, s) [ ik~ 2 ~ exp[ ~ - - F ' F )

;

where n is an integer. Elsevier Science Publishers B.V.

XI. OPTICS

S.A. Lutsenko, N.A. ~nokurov / Waveguidelens

628

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, i

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[ i

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i

The condition of synchronism for symmetric modes on every interval is

.

i

w2[(2n+1) 2-1]

R

8k

c

ds

joq F~(s) = 2~N~,

7/'2[(2/7, q- 1) 2 - 1] fl+l I as 8k ll F2(s ) ggl

1

g2

1

~ll

ql

hl

q

2~M~, (8)

where q, ql, Mn, and N~ are integers.

b

2.2. Hyperbolic waveguide

Fig. 1. (a) A waveguide with cuffs; (b) An arbitrary waisted wavegu!de.

When

F"F 3 = const. ¢* F(s) = 1]"~2S2q- g2/4 is put in eq. (2):

the waveguide becomes planar (~ = +_ 1), but with a little complication of the source equation 0w 1 2ik-- + - -

Os

~2w

F2(s) ~,2

kZ~2F"Fw = 0,

(6)

Ow Os

2ik--+

(6')

The last system can be solved analytically in a few cases, when variables can be parted.

]

2 ~2w[=0;

the variables can be parted:

kozg/2 •(_+1)

(9)

~" - keeg/2~2~ = - ( 2 p + 1 ) E , =0,

(9')

ag/2 2iS"-

2.1. The planar waveguide with cuffs (fig. la)

ko~g

w =_ (~)S(s), 1

w( + 1, s) = 0.

ko~g/2 [ 1 02W / ~ [ k~-g/2 ~)~2

F----5-(2p+ 1 ) S = 0 .

(9")

The eigenmodes can be written as

For any interval I, II, III, eq. (6) with boundary condition (6') gives

--i'rrZn

-s ds

× D (k Cva77se), (7,

where the constants A i are determined by the conditions in the planes s = s2, s = s3:

w(i)(e, s) exp( ike2F'F] 2

I1,~

=w(i 1)(" s) exp( ik~2zF'F) s~' or ik~:2

w(i)(~, si ) =w(i 1)(~, si ) e x p ( - - - 2 - a F . ,F.), where i = 2, 3 (fig. 1) and

A F' = F'(s i + O) - F'(s i - 0).

00)

where Dp is a parabolic cylinder function, which is a solution of the equation

d2D~ ~dt

+[2p+l-tZ]Dp=0,

and with a definite parity Dp(-t)= +Dp(t), and Pn = pn(k~/~7/2 ) are the roots of Dp(~) =0. (7) The phase advance for the nth even mode on the whole waveguide length is determined by the expression

S.A. Lutsenko, N.A. Vinokurov / Waveguidelens

I

629

1

Fig. 3. Scheme of the unloading mirror method.

The result of this optimization by numerical methods is shown in fig. 2a. Since the m i n i m u m of integral (8) corresponds approximately to the modes synchronism, the given result depends weakly on the divergence radius p of the initial bunch. Note, that a similar picture can be reached for any waisted waveguide with insufficient quantitative divergences. If we choose, for example, the waveguide to be shaped by two cylindrical surfaces

b

g .

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F(s)=R-

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Fig. 2. Images of a best fit (a) and best focusing (b) waveguides.

from which one can see that it is impossible to synchronize all modes. For practical use, however, it is enough to solve the following problem. Let we have a wave bunch [2] with a Gaussian energy distribution in the transverse direction:

lj2 2

1

R~2-s 2 + g,

we would get a similar picture. For some practical use a similar problem about the best focusing of an initial Gaussian bunch can be interesting. If we would define the best focusing as a maximum of intensity [ u ( 0 , / / 2 ) t 2 on the main optical plane x = 0 of the outcoming bunch, we get a typical transverse distribution of this bunch as shown in fig. 2b. The part of the energy inside the petal is usually about 5%. In order to demonstrate a practical application we will consider here the method of optical cavity mirror unloading by such a focusing waveguide. It is schematically shown in fig. 3. W h e n the waveguide is absent, the maximum intensity on the right mirror is sufficiently larger (figs. 4a and 4b).

(x2)

2 ~ c r exp - 2-----2 . It is required to find such waveguide parameters (a, g), g = 2or, for arriving at a best fit image of the initial bunch. We can, for example, minimize the following integral:

fF_F[lUf(X)12--lui(x)]2] dx = f_11[ I w f ( ~ ) l 2 - Iwi(~)121 dE,

ui(x ) = u(x, - l / 2 ) ,

u f ( x ) = u(x, l/2),

wi(~ ) = w(~, - l / 2 ) ,

w f ( ( ) = w((, l/2).

-I ............... J

/ (11)

Fig. 4. Energy distribution on a mirror of the optical cavity with (a) and without (b) a focusing waveguide. XI. OPTICS