Distortion of a simple Gaussian beam on reflection from ...jellema/MURP87.pdf · International Journal of lnfrared and Millimeter Waves, VoL 8, No. 9, 1987 DISTORTION OF A SIMPLE

International Journal of lnfrared and Millimeter Waves, VoL 8, No. 9, 1987

DISTORTION OF A SIMPLE GAUSSIAN BEAM ON REFLECTION FROM OFF-AXIS ELLIPSOIDAL

MIRRORS

J. A. Murphy

Mullard Radio Astronomy Observatory Cavendish Laboratory Cambridge CB3 0HE

England

Received May 26, 1987

Abstract:

We examine the d i s to r t ion of a simple l inear ly polarised Gaussian beam on r e f l ec t i on from an of f -axis e l l i p s o i d a l m i r r o r . E x p r e s s i o n s a r e d e r i v e d f o r t h e d i s t o r t e d r e f l e c t e d f i e l d s and f o r t h e l o s s of power i n t o h i g h e r o r d e r and c r o s s - p o l a r i s e d modes .

Keywords: Off-axis mirrors, distortion, cross-polarisation.

Gaussian beam optics,

I . I n t r o d u c t i o n

I n q u a s i - o p t i c a l s i g n a l p r o c e s s i n g s y s t e m s m i r r o r s p r o v i d e an a l t e r n a t i v e t o l e n s e s a s f o c u s s i n g e l e m e n t s f o r beams o f r a d i a t i o n [ l ] , [ 2 ] . I n l e n s s y s t e m s l o s s e s i n power i n t h e t r a n s m i t t e d beam o c c u r b e c a u s e o f a b s o r p t i o n o f t h e r a d i a t i o n i n t h e b u l k m a t e r i a l o f t h e l e n s e s and r e f l e c t i o n s f rom t h e f a c e s o f t h e l e n s . Such l o s s e s w i l l be a b s e n t i n m i r r o r s y s t e m s . A l t h o u g h l e n s e s c a n be

1165

0195-9271/87/0900-i165505.G0/0 �9 1987 Plenum Publishing Corporation

1166 Murphy

bloomed to reduce ref lect ion losses, the techniques available are awkward to implement for low refract ive index materials typ ica l ly used for making lenses, and the technique may be narrow band[3],[4].

M i r r o r s , h o w e v e r , n o r m a l l y have t o be o f f a x i s ( i f t h e beam i s n o t t o f o l d b a c k on i t s e l f ) , r e s u l t i n g i n t h e r e f l e c t e d beam s u f f e r i n g f rom c r o s s p o l a r i s a t i o n and d i s t o r t i o n e f f e c t s on t h e f i e l d p r o f i l e s [ 5 ] , [ 6 ] . D i s t o r t i o n e f f e c t s a r e a p a r t i c u l a r n u i s a n c e when d e a l i n g w i t h a n o m i n a l l y G a u s s i a n beam s y s t e m , a s one wou ld l i k e t o m a i n t a i n t h e s i m p l e G a u s s i a n p r o f i l e o f t h e beam t h r o u g h o u t t h e o p t i c a l t r a i n . I n t h i s p a p e r t h e s e e f f e c t s a r e q u a n t i f i e d f o r an e l l i p s o i d a l ( o r p a r a b o l o i d a l ) m i r r o r .

The s i t u a t i o n of i n t e r e s t i s i l l u s t r a t e d i n f i g 1; a f i a u s s i a n beam i s i n c i d e n t on a c u r v e d o f f - a x i s m i r r o r w i t h n e g l i g i b l e s p i l l - o v e r . I d e a l l ~ t h e r e f l e c t e d wave w ou l d a l s o h a v e a s i m p l e G a u s s i a n p r o f i l e and be p o l a r i s e d i n t h e same way as t h e i n c i d e n t beam. The r a d i i o f c u r v a t u r e o f t h e p h a s e f r o n t s o f t h e i n c i d e n t and " i d e a l " r e f l e c t e d beam a t t h e p o i n t o f i n t e r s e c t i o n o f t h e i r a x e s a r e R I = OC, & R z = OC 2. We would i n t u i t i v e l y e x p e c t t h e e l l i p s o i d o f r e v o l u t i o n d e f i n e d by t h e two c e n t r e s of c u r v a t u r e C, & C 2 and t h e c e n t r e o f t h e m i r r o r 0 t o be a good a p p r o x i m a t i o n t o t h e s u r f a c e r e q u i r e d t o t r a n s f o r m t h e i n c i d e n t G a u s s i a n beam i n t o t h e i d e a l r e f l e c t e d G a u s s i a n beam. I f R 2 = ~ t h e s u r f a c e w i l l be p a r a b o l o i d a l .

I n s e c t i o n 2 o f f - a x i s d i s t o r t i o n e f f e c t s on t h e f i e l d strength are covered, while in section 3 cross-polarisation effects are considered. Many of the g e o m e t r i c a l r e l a t i o n s h i p s u s e d a r e d e r i v e d i n t h e a p p e n d i x .

Reflection from Off-Axis Ellipsoidal Mirrors 1167

' I _ , d ' /

Y

of.f-axis

Fig.l. Geometry of off-axis ellipsoidal mirror.

1168 Murphy

2 . Misma tch i n t h e f i e l d s t r e n g t h s .

( ! ) S i n g l e m i r r o r .

The i n c i d e n t and r e f l e c t e d f i e l d s a r e r e l a t e d by :

E r = -E i + 2 ( n . E i ) n (i)

where n i s t h e n o r m a l t o t h e s u r f a c e o f t h e m i r r o r . I n t h i s s e c t i o n we s h a l l t r e a t t h e f i e l d s as s c a l a r s and n e g l e c t c r o s s - p o l a r i s a t i o n e f f e c t s ; t h u s , we s h a l l j u s t u s e t h e c o n d i t i o n t h a t t h e a m p l i t u d e s of t h e i n c i d e n t and r e f l e c t e d f i e l d s a r e e q u a l and t h e p h a s e s d i f f e r by = on t h e s u r f a c e , v i z . E r=-E i .

C o n s i d e r t h e c a s e shown i n f i g 1 o f a G a u s s i a n beam i n c i d e n t on t h e o f f - a x i s e l l i p s o i d a l m i r r o r s u r f a c e , so t h a t t h e r e f l e c t e d beam makes an a n g l e i w i t h t h e s u r f a c e n o r m a l . The beam has a w a i s t a t A and t h e d i s t a n c e f rom A t o 0 i s d I . The ( x , y , z ) c o o r d i n a t e a x e s a r e d e f i n e d so t h a t t h e z a x i s l i e s a l o n g AO w i t h z=O a t O, and t h e x - z p l a n e i s c o i n c i d e n t w i t h t h e p l a n e o f t h e page i n f i g . 1 . S i m i l a r l y , t h e ( x ' y ' z ' ) s y s t e m i s d e f i n e d so t h a t t h e z ' a x i s l i e s a l o n g t h e BO d i r e c t i o n .

I f t h e r e were no d i s t o r t i o n s i n t r o d u c e d by t h e m i r r o r t h e n t h e r e f l e c t e d beam would be G a u s s l a n i n s h a p e w i t h a w a i s t a t z ' = - d 2. I n t h e u s u a l q u a s i - o p t i c s n o t a t i o n ( e . g . [ 7 ] ) t h e i n c i d e n t and i d e a l r e f l e c t e d f i e l d s r e s p e c t i v e l y can be w r i t t e n a s :

E i = (~Wi) -~ exp(-(r/Wi) 2 -jk((rZ/2Ri)+z+d,}+ j~i ) (2)

Eg r = (~W~)-M exp(-(r'/Wr)Z+jk{(r'Z/2Rr)+z'+dz}-j~r ) (3)

where

d, = IAOI ' dz = iBOl ' r z = x z + yz, r,Z = x,Z + y,Z

w~(z) = w~,[1 + (• ~]

W~(z') = W~211 + (x(z'+d~)/~W~,) 2]

Ri(Z) = (z+d,)[l + (-W~I/k(z+d,)) z]

R r ( Z ' ) = ( z ' + d , ) [ l + ( ~ W ~ z / x ( z ' + d , ) ) z]

r = arctan(x(z+d,)/~W~,)

~r(Z') = arctan(x(z'+dz)/~W~z )


The beam width Wi(P), and the radius of curvature of the phase front R at some other point P on the mirror surface are not the same as at 0 (see fig.l), so the "thin" lens approximation of Kogelnik and Lee[8] that we can match the two pure Gaussian amplitude profiles over the surface of the mirror is no longer true.

However, as Er=-E i at every point P we can calculate the loss in power from the fundamental Gaussian mode on reflection by writing E i in terms of the dashed coordinate system (x',y',z') and in terms of the parameters of the ideal simple Gaussian reflected beam (Woz,dz). Then E r will not be expressible as a simple Gaussian profile and the power lost from the fundamental mode will have been converted into other higher order modes. We choose to expand E r in terms of the Hermite Gaussian set of modes[8] of the form:

Emn(X',y',z';Wr(z'),Rr(Z')) =

(2m+n -l i v z , , m.n.vWr)-~.Hm(v2x /Wr).Hn(42y /Wr).

exp(-(r'/Wr)Z-jk[z'+r'Z/2Rr3+j(m+n,1)Or) (4)

where Wr, R r and O r are as given in (2) for the fundamental mode.

Proceeding with the analysis, for any point P that lies on the surface of an ellipsoid (paraboloid) it is possible to express r, Wi(z), Ri(z ) and ~i(z) in terms of an approximation in the dashed coordinate system and the parameters of the ideal reflected Gaussian i.e. r', Wr(z'), Rr(z') and o'(z'). The expressions are only true to first order in Wm/f and I/kW m and are derived in Appendix II

( i ) r 2 = (1 + x ' t a n i / f ) r ' z

( i i ) Wi(z ) = (I + x ' t a n i / f ) . W r ( z ' )

( i i i ) k ( z + r Z / R i ( z ) ) = - k ( z ' + r ' Z / R r ( Z ' ) ) (5)

(iv) ~(z) = -r + const from A(12)

1170 Murphy

Thus, transforming the amplitude terms of Ei(x,y,z) gives:

W~*exp(-(r/Wi) z) = W~*(l-( tani .x ' / f)) .

exp( - ( r ' /Wr)Z[ l - ( tan i .x ' / f ) ] )

= [1 - [1 - ( r ' /Wr)Z]( tan i .x ' / f ) ] .

W~'exp(-(r'/Wr) ~) (6)

Similarly transforming the phase terms gives:

e x p ( - j k ( z + r 2 / a R i ) + J e i ( z ) ) =

e x p ( + d k ( z ' + r ' 2 / a R r ) - J e r ( Z ' ) + j . c o n s t )

(7)

The p h a s e t e r m s do n o t c o n t r i b u t e to any m i s m a t c h e s and t h e p h a s e f r o n t i s n o t d i s t o r t e d ( i . e . i t r e m a i n s s p h e r i c a l a t t h i s l e v e l o f a p p r o x i m a t i o n ) . I t i s i n t e r e s t i n g t o n o t e t h a t t h i s i s o n l y t r u e f o r e l l i p s o i d a l m i r r o r s d e f i n e d by C I & C z. I f a d i f f e r e n t e l l i p s o i d were u s e d b u t w i t h t h e same f o c a l l e n g t h a t 0 ( d e f i n e d by I / f = I / R ~ + I / R ~ = 1 / R I + l / R z) t h e r e wou ld be a p h a s e m i s m a t c h b e t w e e n t h e two f i e l d s on t h e s u r f a c e g i v e n by :

2 RI RI ~ R~ R~; + - (8)

f tR~

As z can be w r i t t e n t a n i . x and x = r c o s e i n p o l a r c o o r d i n a t e s , t h e s e p h a s e t e r m s a r e o f o r d e r k r 3 c o s ~ / f 2, t h e f a m i l i a r coma a b e r r a t i o n p h a s e t e r m . [ 5 ] .

On c o m b i n i n g (6) and (7) t h e r e f l e c t e d f i e l d a m p l i t u d e , E r , c an be w r i t t e n a p p r o x i m a t e l y i n t h e fo rm:

[x'l a_ x' fy '12 l (9)

The c o n s t a n t i s t o p r e s e r v e t h e t o t a l power i n t h e beam.

E r i s now i n t h e s h a p e of a d i s t o r t e d G a u s s i a n . For s m a l l v a l u e s o f r/W r E r = c o n s t . ( l - t a n i . x ' / f ( l - r 2 / W ~ ) ) a nd so E r has a maximum a t x ' = - ~ t a n i . W ~ / f and y ' = 0 .


The expression for E r given in (9) can be expanded in terms of a sum of normalised Hermite-Gaussian modes of the form Emn = hmnEg r where:

hmn = (2m+nm!n!~)-~.Hm(42X/Wr).Hn(42y/Wr).

The relevant Hermite polynomials Hm(S) are given by Ho(S)=l, HI=2s, Hz(s)=4sZ-2, H,(s)=Ss -12s.

Thus (8) becomes:

E r = const.Egr{hoo + ~u[r h~o+hlz] }

= const.(Eoo + ~u[r E3o+E1z]) (10)

where u = t a n i . W m / 2 ~ E f

As t h e c h a n g e i n p h a s e f o r t h e h i g h e r o r d e r modes o v e r t h e s u r f a c e , A#mn, w i l l be z e r o t o z e r o t h o r d e r , t h e Emn can be r e g a r d e d as r e p r e s e n t i n g h i g h e r modes of p r o p a g a t i o n . S i n c e c o n s t = l t o z e r o t h o r d e r t h e l o s s o f power ( l - n ) f rom t h e f u n d a m e n t a l w i l l be g i v e n a p p r o x i m a t e l y by t h e sum o f t h e s q u a r e s o f t h e v a r i o u s c o e f f i c i e n t s :

l-n = u z = S . t a n Z i . ( W m / f ) 2 (11)

I t m i g h t be e x p e c t e d t h a t a G a u s s i a n d i s p l a c e d from t h e - z ' a x i s wou ld g i v e a s l i g h t l y h i g h e r c o u p l i n g t o t h e d i s t o r t e d wave. However , i f a G a u s s i a n w i t h an a m p l i t u d e v a r i a t i o n o f e x p ( - C ( x - ~ ) z + ( y - 6 ) Z ] / W ~ ) i s f i t t e d to E r and t h e c o u p l i n g o p t i m i s e d a s a f u n c t i o n o f ( ~ , 6 ) t h e n t h e maximum c o u p l i n g o c c u r s when ( e , 6 ) = ( O , O ) . Thus , t h e optimum Gaussian is still ffiven by E~r and n represents the maximum coupling. Also the axis o~ propagation can be regarded as being coincident with the -z' axis. This might seem surprising since the distortion has a coma like term, [ v i z . +(tani.Wm/f)(xrZ/W~) ~ tani.(r3cos~)] in the expansion and one would expect the beam to become smeared out away from the axis of propagation. However, this term is offset by the linear term in x, [viz.-(tani.Wm/f)x], which is also an odd function of x but with opposite sign so reducing the total comatic aberration.

As an e x a m p l e c o n s i d e r t h e c a s e o f a p a r a b o l o i d a l m i r r o r where t h e a n g l e o f i n c i d e n c e i i s 45 ~ and Wm,=f/4 ( s e e f i g . 2 ) .

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Horn

IIE~y'=o, f= 0)I 2

| {arbJl-rary un

ifs)

i~

i 0

-1 0

1 X)W

r +---

2

Fig.2.

Reflection

of Gaussian

beam from an off-axis

paraboloidal

mirror

(antenna)

where

W =f/4.

m

Reflection from Off-Axis Eliipsoidal Mirrors 1173

Thus , a t t h e m i r r o r (where Rr=~) :

E r ~ { l - ~ ( x ' / W r ) + ~ ( x ' r ' 2 / W r 3 ) } e x p ( - r ' 2 / W ~ ) .

I n f i g . 2 IEr l z i s p l o t t e d as a f u n c t i o n of x ' f o r y '=O and z ' = O . The r e f l e c t e d wave i s c l e a r l y d i s t o r t e d and t h e maximum of t h e d i s t o r t e d wave l i e s a t x ' = - ~ t a n i . W ~ / f =-~W m .

I f t h i s m i r r o r i s u s e d i n a r e f e c t o r a n t e n n a s y s t e m f e d by a c o r r u g a t e d h o r n (wh ich p r o d u c e s an a p p r o x i m a t e l y G a u s s i a n beam) w i t h n e g l i g i b l e s p i l l - o v e r a t t h e a n t e n n a , t h e n i n t h e f a r f i e l d t h e h i g h e r o r d e r modes a r e o u t o f p h a s e w i t h t h e f u n d a m e n t a l . T h i s i s b e c a u s e f o r Eoo, ~ o o = ~ r ( ~ ) - ~ r ( O ) = ~ w h i l e f o r E d, ~ I z = ~ 3 o = 4 ~ O o o = 2 ~ . Thus , i n t h e f a r f i e l d E f f i s g i v e n by :

E f f ~ Eoo - ~ juE43E3o+E1z3 .

However, t h e a m p l i t u d e of E f f i s now g i v e n by:

JEf f [ = ( E f f * E f f ) ~

(hoo + ~uZ[43 hao+hlz3Z)Egr

= Eoo

In other words the distortions in the field amplitude is now only of second order and the far field beam shape will appear more Gaussian since the distortion field E d is now in quadrature with the fundamental mode.

(ii) Multiple m i r r o r s .

I f a number o f o f f - a x i s e l l i p s o i d a l m i r r o r s a r e u s e d i n t h e o p t i c a l t r a i n i t i s n e c e s s a r y t o compute how t h e p h a s e s o f t h e c o m p o n e n t beam modes of t h e " d i s t o r t i o n " f i e l d E d = ~ [43 E3o+E1z] v a r y r e l a t i v e to t h e f u n d a m e n t a l as t h e beam p r o p a g a t e s b e t w e e n t h e m i r r o r s of t h e s y s t e m . T h i s r e l a t i v e p h a s e as a f u n c t i o n of z ' f o r b o t h E~o and E1z i s g i v e n by :

~ r = 3 ~ r )

= 3 [ t a n ( X ( d z + z ' ) / ~ W ~ z ) - t a n ( X d z / ~ W ~ z ) 3 . (12)

1174 Murphy

Thus, at a second mirror a distance d~ from the waist of the beam ref lec ted from the f i r s t mirror the incident f i e l d , Ei2, wi l l be of the form:

Ei2 = Eoo + u , e x p ( - j 3 A ~ l , z ) . E d

where ~ t , 2 = A ~ o ( d 2 , d l )

(13)

I f t h i s m i r r o r i s a l s o of t h e c o r r e c t e l l i p s o i d a l s h a p e and t h e e r r o r s i n t r o d u c e d on t h e f u n d a m e n t a l mode a r e a g a i n o f f i r s t o r d e r ( and so s e c o n d o r d e r ( n e g l i g i b l e } on Ed) t h e r e f l e c t e d f i e l d f rom t h e s e c o n d m i r r o r w i l l be of t h e g e n e r a l fo rm:

Er2 = Eoo + (u z • ulexp(-j~r d . (14)

The sign of the phase term depends on the or ienta t ion of the coordinate system defined by the re f l ec t ion at the second mirror r e l a t i v e to f i r s t . This wi l l be i l l u s t r a t e d l a te r in an example.

For a system of n mirrors the to t a l loss in power wi l l be of the general form:

1 - n = I [ u n • e x p ( - j ~ O n _ l . n ) [ U n _ l

e x p ( - j ~ O n _ z , n _ l ) E . , .

• e x p ( - j a o l , z ) u l ] . . . ] ] ! z (15)

By choosing mirrors careful ly i t is therefore possible to eliminate these d i s to r t ions . For example, in the case of two mirrors this implies that ~r = n~ so that both terms in the coef f ic ien t of E d are real .

Figs. 3(a) and (b) show two configurations where th is has been achieved. For s implici ty 2i=90 ~ and Wm/f=~ in both cases. In the f i r s t case, f ig .3 (a ) :

~r = [tan(X(dz+d~)/~W~2)-tan(xdz/~W~z)] = ~/3

so that A~ r = ~ and the f i e ld p ro f i l e incident on the second mirror is e s sen t i a l ly an inverted image of the p ro f i l e of the re f lec ted beam from the f i r s t mirror. In th is case i f the second mirror is orientated so that the re f lec ted beam propagates as shown in f ig . 3(a) then the re f lec ted beam wil l again have a true Gaussian shape.

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G~,o. Beam

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Fig.3. Amplitude distortion for system consisting of two off-axis

ellipsoidal mirrors, Case (a

): A~12 = ~/3. Case (b):A~12 = O.

Cases (c)

& (d):A~12= ~/4.

=

1176 Murphy

In t h e s e c o n d c a s e , ( b ) , kW~2~(d~+d2) so t h a t t h e p h a s e s l i p p a g e , ~ e r ( d z , d ~ ) , b e t w e e n t h e m i r r o r s i s e s s e n t i a l l y z e r o . C l e a r l y , t h e f i e l d p r o f i l e o f t h e i n c i d e n t beam on t h e s e c o n d m i r r o r i s d i s t o r t e d i n t h e same s e n s e a s t h e r e f l e c t e d beam f rom t h e f i r s t m i r r o r . T h e r e f o r e , i f t h e s e c o n d m i r r o r i s o r i e n t a t e d a s shown t h e beam f rom t h e s y s t e m w i l l h a v e a t r u e G a u s s i a n s h a p e o n c e m o r e .

I n t h e more g e n e r a l c a s e f i g s . 3 ( c ) and (d) w h e r e t h e p h a s e s l i p p a g e i s ~/4, f o r e x a m p l e , b u t ( s t i l l } u~=u 2 t h e d i s t o r t e d beam i s g i v e n by:

E r = Eoo + 1/8r d

Eoo + 1/8r d

f o r c a s e ( c )

f o r c a s e (d) (16)

The l o s s i n power i s g i v e n by :

l - n = ( ( 1 - 1 / 4 2 ) 2 + 1 / 2 ) / 1 2 8 = ( 2 - r = 0 . 0 0 7 f o r ( c )

and

l - n = ( ( I + I / r z + I / 2 ) / 1 2 8 = ( 2 + 4 2 } / 1 2 8 = 0 . 0 3 f o r (d)

Thus , c l e a r l y , when a number o f m i r r o r s a r e u s e d a c a r e f u l d e s i g n o f t h e c o n f i g u r a t i o n c a n r e d u c e t h e o v e r a l l d i s t o r t i o n t o n e g l i g i b l e l e v e l s .


3. C r o s s p o l a r i s a t i o n e f f e c t s .

I n t h i s s e c t i o n t h e v e c t o r n a t u r e o f t h e E f i e l d i s i n c l u d e d i n t h e a n a l y s i s . As was g i v e n i n ( I ) t h e law of r e f l e c t i o n a t t h e p o i n t P on t h e m i r r o r i s g i v e n by: E r ( P ) = - E i ( P ) + 2 { n ( P ) . E ( P ) } n ( P ) where n ( P ) i s t h e n o r m a l t o t h e m i r r o r a t P. I t i s c o n v e n i e n t t o u s e an a p p r o x i m a t i o n f o r n ( P ) i n t h e ( x , y , z ) c o o r d i n a t e s y s t e m d e r i v e d i n t h e a p p e n d i x . The r e l a t i o n s h i p s a r e t r u e t o f i r s t o r d e r i n Wm/f.

n x = s i n i - cosi (x/2f)

ny = - y /2 fcos i

n z = -cos i - s in i (x/2f)

(17)

where f = RiRz/(Rt+Rz) (the "focal length" of the mirror.)

The i n c i d e n t beam t o z e r o t h o r d e r c a n be t r e a t e d a s t h e v e c t o r sum o f a p a i r o f x and y p o l a r i s e d f i e l d s . T h e r e i s a s m a l l E z componen t b e c a u s e o f t h e V.E = 0 c o n d i t i o n . For E=Exi+Eyj+Ezk t h i s c o n d i t i o n g i v e s :

Ez=[(2/ikWZ)-(I/R)][XEx+YEy],

assuming aEz/OZ=-ikE z (the paraxial approximation).

F i r s t t r e a t i n g the case where the incident r ad ia t ion is l i n e a r l y polar ised in the y d i r e c t i o n (E x = 0), Er, the r e f l e c t e d f i e l d , as given by (1) and (17) can to f i r s t order be wr i t ten :

Erx = - t a n i . ( y / f ) . E i y - s i n 2 i . E i z

Ery = _ E i y

Erz = c o s 2 i . E i z + ( y / f ) . E i y

Then transforming (Erx,Ery,Erz) c o o r d i n a t e s y s t e m , one g e t s :

(18)

to the (x',y',z')

Erx , = c o s 2 i . E r x + s i n 2 i . E r z = t a n i . y / f . E r y

E r z , = - s i n 2 i . E r x + c o s 2 i . E r z = E i z - ( y / f ) . E r y

(19)

1178 Murphy

Then, as YEiy=y'Ery and Eiz = [2/ikW z - i/Rt].YEiy. these reduce to:

Erx, = [ t an i / f ] .Y 'Ery '

t - - . I E Erz = [- 2/ikwZ I/Rz] Y ry (2o)

The f i r s t of these terms Erx , represents cross-polarisat ion losses from the fundamental incident mode Eiy into the f i r s t order mode Y'Ery' in the x' direct ion. As already discussed the second term does not represent any loss, but rather the condition that there exists a small Erz, term so that V.E = 0 for the reflected wave.

The f i r s t order normalised Hermite-Gaussian mode has

the form (2y'/Wm)Eoo(42x'/Wm,42y'/Wm), where Eoo is the

normalised zeroth order mode. Thus, the coefficient of the normalised Eol mode is ~tani(Wm/f ) and the fract ional loss in power is given by:

1-n = ~. tanZi . (Wm/f) 2 (21)

Similarly for the case where Ey = 0 (radiation polarised in the x direction) the reflected f ie ld components can he written as:

Erx = - cos2i.Eix - sin2i.Eiz - sin2i (x/f).Eix

Ery = - tani (y/f) Eix

Erz = + cos2i.Eiz - sin2i.Eix + cos2i (x/f).Eix

(22) Then, on transforming to the dashed coordinate system, one gets:

Erx, = _ Eix

E ' = - [tani/f] YEix ry

grz, = [2/ikW z + I/Rz].XEix

(23)

Thus, here also the loss in power is : l -n = ~tanZi(Wm/f) z

As an example, we again take the case of an off-axis e l l ipso id where 2i = 90 ~ and Wm/f = ~ ( i . e . F2). The reflected f ie ld l ines for the x & y polarised cases are

" 2

r

Incident field polarised in y direction

-2

-2

Incident field polarised in x direction

xVw

r

o

o

o

Fig.4.

Reflected

fields

for case of linearly

polarised

incident

fields.

=

1180 Murphy

shown in fig.4. The loss in power from the fundamental mode polarised in the x or the y direction is thus 1.5%.

If the optical system consists of a number of mirrors the s i tua t ion can be analysed in the same way as for scalar losses in section 2(b). A phase slippage will appear between the cross polarised f ie ld and the fundamental as the beam propagates af ter ref lec t ion from the f i r s t mirror. He re , however, the re la t ive phase slippage term is given by:

a c r ( d z , z ) = [tan(x(d~+z')/,W~,)-tan(xdz/=W~z)].

Thus for the case of a two mirror system only in the case where tan(k(dz+d])/=W~z)) = 0 or = (as shown in figs.5(a) and (b)) can the cross polar isat ion losses be eliminated

Conclusions

We have obtained expressions for the coefficients of the higher order modes generated when a simple Gaussian beam is incident on an off-axis e l l ipso ida l mirror, expressed in terms of the beam radius at the mirror, W m, the focal length of the mirror, f, and the angle by which the axis of the beam is deflected, 2i.

The scalar losses due to the amplitude (and phase) mismatch can be written as: aP = M.(Wm/f)2.tanZi.

For any incident polar isat ion the cross polar isat ion losses can be written as: aP = ~.(Wm/f)Z.tanZi

In order to be able to estimate the effect for a system consist ing of a number of off-axis mirrors i t is necessary to calculate the phase slippages of the higher order modes between the mirrors. The loss in power is then given by:

ZiP = I Z~ [~.(Wm/f)j .tan(ij).exp(-JZ~.a~k~k+,)]l z,

where ~Qk~k+* is the fundamental mode phase slippage between mirror k and mirror k+l, with ~=3, 8=I/242 for scalar losses and ~=I, ~=~ for cross-polarisation losses.


I I Case (a)

/

Fig.5. Eliminating off-axis effects in a

two mirror system.

Case (a): WO2<<Wm , i.e a~12=

Case (b): W02 = Wm, i.e A~I2 = 0

1182 Murphy

A p p e n d i x I

A_n a p p r o x i m a t i o n f o r t h e n o r m a l t o t h e m i r r o r s u r f a c e i n t e r m s o f t h e c o o r d i n a t e s y s t e m d e f i n e d bY t h e i n c o m i n g beam.

In this section we derive the expressions for the normal to the mirror surface in terms of the local coordinate system determined by the direction of the incident wave.

In a g l o b a l C a r t e s i a n c o o r d i n a t e s y s t e m (X,Y,Z) an e l l i p s o i d o f r e v o l u t i o n i s d e s c r i b e d i n t h e u s u a l n o t a t i o n by :

(X2+YZ)/bZ + ZZ/a z = 1. A(1)

The g e o m e t r y i s i l l u s t r a t e d i n f i g . 1 . L e t O d e n o t e t h e p o i n t on t h e e l l i p s o i d l o c a t e d a d i s t a n c e R, f rom t h e n e a r f o c u s C 1 and R z f rom t h e f a r f o c u s C z. 2 i i s t h e a n g I e d e f i n e d by CIO and CzO ; a & b a r e r e l a t e d t o R1, a z and 2 i by 2a=Rt+Rz and b Z = R t R z c o s Z i . We w i s h t o t r a n s f o r m i n t o a l o c a l c o o r d i n a t e s y s t e m ( x , y , z ) whose o r i g i n i s a t O so t h a t t h e x , y p l a n e l i e s p e r p e n d i c u l a r t o t h e d i r e c t i o n C,O. The i n c l i n a t i o n o f t h e ClO d i r e c t i o n t o t h e Z a x i s i s o t and Cz0 i s �9 z . F o r c o n v e n i e n c e we d e n o t e c o s * t and s i n o I as c & s . Then t h e c o o r d i n a t e s y s t e m s a r e r e l a t e d by :

X = X O + CX + sz

y = y

Z = Z o - sx + cz

A(Z)

W i t h t h e s e e x p r e s s i o n s i n s e r t e d i n e q u a t i o n A ( 1 ) , we g e t :

x Z [ ( c Z / b Z ) + ( s Z / a Z ) ] + yZ/bZ + z Z [ ( s Z / b Z ) + ( c Z / a Z ) ]

+ 2 x z E ( 1 / b Z ) - ( 1 / a Z ) ] c s + 2 x [ ( X o c / b Z ) - ( Z o s / a Z ) ] A(3)

+ 2 z E ( X o s / b Z ) + ( Z o c / a Z ) ] = 0

U s i n g : b z = a Z ( 1 - e Z ) , Xo = sR t , Zo= c R l + a e and R , s e = f s i n 2 i , w h e r e f = R t R 2 / ( R t + R z ) ( t h e " f o c a l l e n g t h " o f t h e m i r r o r ) :

Reflectionfrom Off-Axis[]lipsoidalMirrors 1183

x 2 ( l _ e Z s Z ) + y z + z Z ( l _ e 2 c Z ) + 2 x z ( e Z c s )

- 4 x f s i n i . c o s i + 4 z f c o s Z i = 0 A ( 4 )

F o r s m a l l v a l u e s o f x , y , z r e l a t i v e t o f t h e l a s t t w o t e r m s d o m i n a t e . T h u s : ~ x / A z = c o t i a n d we c a n t a k e z = t a n i x

An a p p r o x i m a t e v a l u e f o r n ( p ) f o r t h e m i r r o r s u r f a c e c a n now b e c a l c u l a t e d , n i s d e f i n e d b y : n = v o / [ V o [ , w h e r e f o r t h e e l l i p s o i d s u r f a c e r i s g i v e n b y :

r = Ax z + By z + Cz z +Dxz +Ex + Fz = 0 .

( A , B e t c a s g i v e n b y e q u a t i o n A ( 3 ) a b o v e } .

n x = s i n i - c o s i . x / 2 f

ny = - y/2fcosi

n z = - cosi - sini.x/2f A(5)

Appendix II

Transforming from the inciden t beam coordinate

system(x~y,z) to the reflected beam coordinate system (x',y',z').

I n t h i s s e c t i o n we d e r i v e t h e e x p r e s s i o n f o r t h e t r a n s f o r m a t i o n b e t w e e n t h e p a r a m e t e r s u s e d t o d e s c r i b e t h e i n c i d e n t b e a m a n d t h e p a r a m e t e r s u s e d t o d e s c r i b e t h e r e f l e c t e d b e a m s .

The d i s t a n c e o f r o m t h e o r i g i n 0 t o some p o i n t o n t h e s u r f a c e P s a t i s f i e s t h e t w o r e l a t i o n s h i p s :

p2 = xZ+yZ+zZ = x'Z+Y'z+z'Z A ( 6 )

2 a = [pZ+R~+2p.Rt] ~ + [pZ+R~+2p. Rz] ~

= [ x Z + Y Z + ( z + R t ) Z ] ~ + [ x ' Z + y ' Z + ( z ' + R z ) Z ] ~ A ( 7 )

F o r d i s t a n c e s p s m a l l c o m p a r e d t o R 1 o r R 2 o n e f i n d s t h a t s i n c e R I + R z = 2 a t o t h i r d o r d e r i n x / f , y / f , z / f :

Z + (xZ+yZ)/2(Rl+Z) + Z'+ (x'Z+y'Z)/2(R+z') = 0 A(8)

I ~ 4 Mushy

(i.e. terms up to order (xZ+yz)Z/R ~)

We can derive a useful approximate relationship between r and r' from this. From A(6) we get that:

r 2 - r ' 2 = z ' 2 - z 2 = ( 2 z ' - ( z ' + z ) ) ( z ' + z )

and a s ( z ' + z ) ( z ' and z ' + z = - ( r Z / 2 R , + r ' Z / 2 R z )

r z _ r , Z = - 2 z ' . ( r ' Z / 2 R z + r Z / 2 R , )

= -2z'.(r'Z/2f + (rZ-r'Z)/2R1)

= -2z' (r'Z/Zf - 2z.(r'Z/2fRi +...))

B u t r ' Z / f ) z r ' Z / f R , , t h e r e f o r e : r 2 - r ' 2

On r e a r r a n g i n g we g e t :

r z = [I - z'/f].r 'z A(9)

= [i + x'tani/f].r 'z as

= z r ' Z / f .

z' = -x'.tani

Appendix III

The relationships between W, R, & �9 for the incident and

reflected wave~

(i) The beam radius W

Assuming that the off-axis angle 2i is not greater than 90 ~ and that the beam width at the mirror is much

less than R i or R r, we can make approximations:

w~(p) : w~i[i + (• 2] = Wg,[l + (• z + 2(kdl/=Wg~)Z(z/d,)]

= W~ [I + 2(kd,/~Wo,Wm)Z(z/dl)]

where W~ = Wg1[l+(xd~/=W~l) 2] Then, since (• and (•

W~ = W~[l + 2z/R,]

w~ = w~[1 + 2z'/R~]

Reflection from Off-A~s Ellipsoidal Minors

Thus: Wi(z) = [I - z'/f].Wr(Z') = [I + tani.x'/f]Wr(Z') A(10)

(ii) Phase front radius of curvature R)

by:

1185

The i n c i d e n t phase f r o n t r a d i u s o f c u r v a t u r e i s g i v e n

Ri : (dx+z)[l + (=Wgi/X(d1+z)) 2]

: R t + z - z(kRI/=W~)Z(I + z/dr) -i

as ~wZ,/xd1=xR1/#Wg

Thus, I/R i - I/(R,+z) = z(X/=Wg) z

The equation of the ellipsoid to second order gives:

z + rZ/2(R1+z) + z' + r'z/Z(Rz+z ') = 0

Thus : k(z+z'+rZ/2Ri+r 'z/2Rr)

= k(rZ/2)[I/Ri-i/(R1.z)] - k(r'2/Z)[i/Rr-i/(R,+z')]

= 2(kWm)-1(zrZ+z'r 'z)lw~

= 2(z+z' )r' Z/kW~

= 0

as (z+z') is zero of first order, and I/kW m is small.

k(z+z'+r2/ZRi+r'Z/2Rr) = 0 to second order. A(ll)

(iii) Phase s l i p p a g e

To f i n d t he r e l a t i o n s h i p between e ( z ) and ~ ( z ' ) we note (see section 3.2.1) that: r and as arctan(A+A)=arctan(A) + A/(I+A zJ

e(z) = arctan(Xd~/~W~1) + kz/(~W~i{l+(Xdl/~W~i)z})

= arctan(Xdl/~W~1) + xz/~W~

as W~ = W~i{l+(Xd~/~Wgi) 2}

1186 Murphy

Similarly, for the dashed coordinate system:

~'(z') = arctan(-xdz/~Wgz ) - xz'/~W~

(the negative sign arises because the reflected wave is travelling in the -z' direction).

* ( z ) = | + x t z + z ' ) / = W ~ + c o n s t .

w h e r e c o n s t . = a r c t a n ( X d l / = W ~ o ) + a r c t a n ( x d z / ~ W ~ o )

= ~ ' ( z ' ) - r ' Z / k f W ~ + c o n s t .

= o ' ( z ' ) + c o n s t , t o s e c o n d o r d e r . A ( 1 2 )

Acknowledgement

The a u t h o r w o u l d l i k e t o e x p r e s s h i s s i n c e r e t h a n k s t o Dr . R a c h a e l Padman f o r v a l u a b l e d i s c u s s i o n s and f o r h e l p f u l comments i n p r e p a r i n g t h i s m a n u s c r i p t .

R e f e r e n c e s :

[I] R.W. Barker, D.W. Bly, M.J. Cox, R.E. Hills, J.A. Nurphy, R. Padman, A.P.G.R. Russel, D.M.A. Wilson, S.W. Withington and D.R. Vizard, "Millimetre and submillimetre receivers for UKIRT.', SPIE, Voi.598(1985), 203.

[2 ] N.R. E r i c k s o n , "A v e r y low n o i s e s i n g l e s i d e - b a n d r e c e i v e r f o r 200 -260GHz" , IEEE T r a n s . ~rrT-33, 1179, ( 1 9 8 5 ) .

[3 ] R. Padman, " R e f l e c t i o n and c r o s s - p o l a r i s a t i o n p r o p e r t i e s o f g r o o v e d d i e l e c t r i c p a n e l s " , I g g g T r a n s Antennas Propagat., A P - 2 6 , 7 4 1 ( 1 9 7 9 ) .

[4] H. J a s i k ( e d ) , Antenna Engineering Handbook, 1St ed. , M c G r a w - H i l l (New York 1 9 6 1 ) , C h a p t e r 14, "Lens t y p e r a d i a t o r s " , by S . B . Cohn.


[5] T .S . Chu and R . H . T u r r i n , " D e p o l a r i s a t i o n p r o p e r t i e s o f off-set reflector antennas," IEEE Trans. Antennas Propagat., AP-21, 339, (1973)

[6] T .S . Chu and R.W. Eng land , "An e x p e r i m e n t a l b r a o d b a n d imag ing f e e d " , Bell System Technical Journal, v o l . 6 2 , 1233-1250.

[7] P. Goldsmith, " Quasi-optical techniques at millimeter and submillieter wavelengths", in "infrared and Millimeter Waves", vol.6, 277 (Academic Press).

[8] H. K o g e l n i k and T. L i , " L a s e r Beams and r e s o n a t o r s " , Proc IEEE., V01.54, 1312, (1966)

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