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Plane kinoform lenses for axial stigmatism in finite conjugate imaging
Optics CommunlcaUons 91 ( 1992 ) 1-4 North-Holland
O P T ICS CON'I MUNICATIONS
Plane kinoform lenses for axial stigmatism in finite conjugate imaging L.N. Hazra, Y. Han and C. Delisle Centre d'Opttque, Photontque et Laser, DOpartement de Phystque, Pawllon Vachon, Unlverstte Laval. Ste-Foy, Qubbec. G1K 7P4. Canada
Received 28 January 1992, revised manuscript received 20 March 1992
We report a generalized and convenient formulation for the determination of zone spacing and blaze profile for a plane klnoform lens The method enables one to design the klnoform lens for stigmatic imaging of axml objects at any conjugate posmon Some numerical results are presented
The prospect o f the availability o f high efficiency diffractive lenses in visible and infrared wavelengths has opened up new vistas in optical system design, particularly for systems operating in m o n o c h r o m a t i c or q u a s l m o n o c h r o m a t l c light [ 1,2 ]. O v e r the years, the techniques for fabrication o f these elements have undergone steady i m p r o v e m e n t leading to significant increase in their diffraction efficiency so much so that these elements, which are also known as klnoforms [3], b i n a r y lenses [4] or phase Fresnel lenses [ 5,6 ] are gradually emerging as optical elements that can substantially ~mprove the performance o f conventional lenses when used in conjunction with the latter [7,8], and, in some apphcations, they hold the promise o f turning out Into viable and better alternatives to conventional lens systems [9-11 ]. K m o f o r m lenses have so far been used mainly for m f i m t e conjugate imaging applications and a detailed description o f the m e t h o d s for design o f such kinoforms has recently been presented [ 12 ]. The use o f klnoform lenses in optical systems, in general, calls for their operation in finite conjugate imagery. It is obvious that a plane k i n o f o r m lens designed for the imaging o f an axial object at infinity will Introduce severe spherical aberration if used for finite conjugate imaging, even at m o d e r a t e apertures [12]. Elimination o f this spherical a b e r r a t i o n calls for a specific relocation o f the full p e r i o d zones as well as
a specific change in blaze profile o f the klnoform lens for each conjugate position To the best o f our knowledge, no study on this aspect o f k l n o f o r m imaging is available in published literature. The i m p o r t a n c e o f this study can also be a p p r e c i a t e d If one recalls the need for special recording geometry in the case o f optically generated aplanatlc holographic lenses [ 13,14 ]. In this communication, we report a generalized and convenient formulation for the d e t e r m i n a t i o n o f zone spacing and blaze profile for a plane klnoform lens. The m e t h o d enables one to design the klnoform lens for stigmatic Imaging of axial objects at any conjugate position. W i t h reference to fig. l, let the klnoform A P form a stigmatic image o f the axial object point O at the point O ' . In what follows, we a d o p t the sign convention o f Born and W o l f [15 ]. The zone radii on the kinoform are defined such that the optxcal path length from O to O ' through the point P,, on the edge o f the ruth zone is [OPmO'] = [OAO']Wm2,
(1)
where the operating wavelength is 2 a n d the object distance A O = Z and the ~mage distance A O ' = Z ' . The inner a n d outer radii o f the m t h zone are A P m - 1= rm_ 1 and APm = rm respectively. F r o m eq. ( 1 ) we o b t a i n the defining expression for rm as
0030-4018/92/$05 00 © 1992 Elsevier Science Pubhshers B V All nghts reserved
1
Volume 91 n u m b e r 1,2
OPTICS COMMUNICATIONS
1 J u l y 1992
It may be noted that the expressmns ( 5 ) or (6) for the zone radii are more convenient than the ones reported earlier m zone plate t h e o ~ [ 16,17 ] The width o f the rnth zone, e,,,, IS given as e,,,=
P
P m ~ . mth Zone
rm
-
-
rm
- I
F r o m eqs. (5) or (6), the total n u m b e r o f zones m .... for given values o f T, )~, f a n d aperture radius r. . . . m a y be d e t e r m i n e d f r o m the smallest p o s m v e
A F-t
Z'
Z
O' )1 I,
T~
0
real root c~ o f the following quartlc equation
)1
m4 + 4 / ' m ~ + 4 ( T 2 + f T - r ~ , , , ) ,t )°2
Fig 1 ruth zone o f the finne conjugate kmoform
2
+,,/r,,+(Z'
=(Z'-Z)+m2.
+ 8T(/T-r~,a,) m (~)
/'H,nax IS g i v e n
Assigning a focal length f t o this k i n o f o r m as
l/f=l/Z'-l/Z.
/=
4T2rmax = 0 ,
(9)
b~
m ..... = I N T ( c ~ ) (3)
(10)
where I N T (c~) is the integer part o f the real n u m b e r O/.
we have
Z'Z Z'-Z
m2
-
Z'Z T "
Figs 2 - 4 d e m o n s t r a t e the variation of zone radn, rm, and zone width, ~,,,, for &fferent values o f T For each T, the k m o f o r m parameters are evaluated for a paraxlal focal length f = 100 mm. The operating wavelength is 10 lam. In order to emphasize the difference xn zone radI1 between a k m o f o r m designed
(4)
where T = Z ' - Z ~s the throw, the object to image distance. After a lengthy but strmghtforward algebraic m a m p u l a t i o n we obtain the following expression for rm from ( 2 ) and ( 3 ) ,
2 2rn2f(l+m2/2T)[l+(m2/2f)(l+mZ/2T)] rm = ( 1+ m2/ T) 2
'
(5) (l+m;t/2T)~ ,~ [ - ( l + m 2 / 2 T ) ] 2 r ' 2 = (1 + m 2 / T ) 2 zm4l+L-(1-+rn~-T-) J m222
v
(6) F o r T - , o e , the exact representaUon for the zone radii of an lnfimte conjugate k m o f o r m m a y be obtained from ( 6 ) as
T=-2
0/JS
% 8O
r2.,=2m2f+m222
(7)
This is the case o f a nonparaxml k m o f o r m ( N P K ) [12] for infinite conjugate imagery, whereas the wellknown formula for zone radii o f a paraxial kinoform ( P K ) m a y be o b t a i n e d from ( 5 ) by imposing the c o n d m o n f>> m2 and T>> m2 as
r2,~= 2 m 2 f
(8)
600
0
200
400
600
800
m
Fig 2 V a n a u o n m the difference m zone radii [ (r,,,)l ( r m ) p K ] w n h n u m b e r of zones m for different values of obJect to image throw T NPK, P K nonparaxlal and paraxlal k m o f o r m s respecUvely for T = oo -
Volume 91, number 1,2
OPTICS COMMUNICATIONS
80
60
T =E
-25
E
40 PK
E
2O
00
I
I
|
200
400
600
800
m
Fig. 3. Zone radn r,,, w~th number of zones rn for different values of obJect to image throw T NPK, PK nonparaxml and paraxml kmoforms respectivelyfor T= oo
0.12
0 10
that the width of the zones o f a kmoform or zone plate decreases quadratically wxth mcreasmg radius. But this is not necessarily true when they are designed specifically for axial stlgmatism at finite values of T. It ~s interesting to note in fig. 4 that, for smaller magmtudes of T, the zone width increases with a rise m radtus after a t t a m m g a m i n i m u m value at an lntermedmte aperture radms The practical ~mphcaUon of this effect needs further investigation. For plane kmoforms, an analytical soluUon for the blaze profile is possible for the mfinlte conjugate case [ 12 ]. In the case of finite conjugates, this seems no longer possible. This is because of the a d d m o n a l req u i r e m e n t of a n n u h n g the spherical aberration artsmg out of obhque l n o d e n c e on the plane interface. A modificaUon of a semi-numerical procedure for the design of a s p h e n c surfaces [ 15 ] is used for the purpose. With reference to fig. 5, let Q R S O ' be an arbitrary ray path through the ruth blazed zone of the kmoform producing stigmaUc image O' of an object point O at a distance Z from the vertex A which is consldered as origin of the coordinate system. The image distance AO' = Z ' and the refractive index of the optical material of the kmoform is/t. Let the angle of i n o d e n c e and refraction at the point R of the ray QRS be I and I' respectively. We have
E
v
slnl'-
sm I --fl
cos/'=
[f1222+(f12--1)r2] 1/2
E ~o
0.0E
_ 25
j 5 ,
1July 1992
r /Ix~ZS+ r 2 ' / l ~ r
(11)
,
2
(12)
0.04
Q 0.02 0
i 200
-400 l 400
NPK
i 600
p
PK, 400 800
m
Fig 4 Zone width ~,, with number of zones m for different values of object to image throw T NPK, PK nonparaxml and paraxlal klnoforms respectivelyfor T= specially for a given T a n d those of the paraxlal kmoform for the same value o f f fig. 2 shows the difference m zone radu [ (rm)r-- (rm)pK] wtth m. For smaller magnitudes of T, a direct plot of (rm) T versus m, for different values of T, xs sufficient to demonstrate the difference (fig. 3). It is usually thought
I'm_ 1
I
~
- "- \
A ~ I - I
O' Z'
O
,I Z
,I
Fig. 5 Geometry and notaUon for the design of the blaze profile for a kmoform for fimte conjugates
Volume 91, number 1,2
OPTICS COMMUNICATIONS
w h e r e r = A R . T h e c o n d i t i o n for axial s t l g m a t l s m o f the k l n o f o r m i m p l i e s that, for any r c o r r e s p o n d m g to the m t h zone, 1.e., for r,, i<-%r<,Nrm,the optical path length [ O R S O ' ] is g t v e n by [ O R S O ' ] = [OABO'
] + (m-
1 )/t
=(Z'-Z)+m2.
(13)
Let RS = t,,. N o t e that the m a x t m u m thickness o f the k m o f o r m ts A B = d = 2 / ( ~ t - 1 ). E x p r e s s i o n ( 13 ) can be used to o b t a i n a q u a d r a t i c relatton d e f i n i n g tm as
At2n + Bt,,, + C = 0 ,
T h i s w o r k was s u p p o r t e d by the F o r m a t i o n de C h e r c h e u r s et A i d e ~i la R e c h e r c h e o f the Q u d b e c G o v e r n m e n t a n d the N a t u r a l Sciences a n d Engineering Research Council of Canada
References 2) ,
(15)
B=Z{u[(Z'-Z)+m2+x/~+r
-(Z'
Finally, it should be m e n t i o n e d that all kanoforms in p r a c t i c e are m a d e on a substrate o f finite thickness. A m o d i f i c a t i o n o f the a b o v e t r e a t m e n t enables one to a c c o u n t for this thickness in o b t a i n i n g exact p a r a m e t e r s o f the k l n o f o r m .
(14)
where A=(l-ct
l July 1992
2]
cosl'+rslnI')},
C=2(Z'-Z)(Z-m2-~r - 2 m 2 ~
-m2~. 2 .
(16)
2) (17)
T h e c o o r d i n a t e s o f the p o i n t S, on the blaze profile, are [ t,,, cos I', ( r - tm sin I' ) ]. F o r a set o f v a l u e s o f r m (r,, i, r,,,) eqs. ( 1 1 ) - ( 1 7 ) are used to determ i n e the c o r r e s p o n d m g v a l u e s o f tm w h i c h are t h e n a p p l t e d to e v a l u a t e the c o o r d i n a t e s o f the p o i n t s on the blaze profile. T h e blaze profile can be determ i n e d to the d e s i r e d level o f a c c u r a c y w i t h suitable chotce o f the n u m b e r o f samples r in (r,n_ 1, rm ). N o t e that the l o w e r edge o f the blaze profile, for the m t h zone, P,,,_ 1 B . . . . 1, IS not parallel to the ax~s and tt lS u n i q u e l y k n o w n after the c o o r d i n a t e s o f the p o i n t Bin_ 1, are d e t e r m i n e d
[ 1] G J Swanson, Binary optics technology, Techmcal report 914 (Lincoln Laboratory, MIT, USA, 1991 ) [2] R E Fisher, OE Reports, 67 (1989) 1 [ 3 ] J A Jordan, Jr., P M Hlrsch, L.B Lesem and D L VanRooy, Appl OpUcs 9 (1970) 1883 [ 4 ] A D Kathman and S K Pltalo, Proc Soc Photo-Opt Instrum Eng 1354 (1990) 297 [5]K Mlyamoto, J Opt Soc Am 51 (1961) 17 [6] G G Sllussarev, Sov Phys Doklady 2 (1957) 161 [7 ] G J Swanson, Binary optics technology, Techmcal report 854 (Lincoln Laboratory, MIT,USA,1990) p 27 [ 8 ] D Fakhs and G M Morns, Phot Spectra 25 ( 1991 ) 131 [9] S T Bobrov and G T Grelsukh, Avtometnya 6 (1985) 3 [ 10] D A Buralh and G M Morns, Appl Optics 28 (1989) 3950 [ 11 ] D A Buralh and G M Morns, Appl Optics 30 ( 1991 ) 2151 [ 12 ] D A Buralll and G M Morns, Appl Optics 28 ( 1989 ) 976 [ 13] W T Welford, Optics Comm 9 (1973) 268 [ 14] W T Welford, Optics Comm 15 (1975) 46 [ 15 ] M Born and E Wolf, Principles of optics (Pergamon Press, Oxford, 1980)pp 191, 197 [16]J Klrz, J OptlcsSoc Am 64 (1974) 301 [ 17 ] V G Schmahl and D Rudolph, Optlk 29 (1969) 577
O P T ICS CON'I MUNICATIONS
Plane kinoform lenses for axial stigmatism in finite conjugate imaging L.N. Hazra, Y. Han and C. Delisle Centre d'Opttque, Photontque et Laser, DOpartement de Phystque, Pawllon Vachon, Unlverstte Laval. Ste-Foy, Qubbec. G1K 7P4. Canada
Received 28 January 1992, revised manuscript received 20 March 1992
We report a generalized and convenient formulation for the determination of zone spacing and blaze profile for a plane klnoform lens The method enables one to design the klnoform lens for stigmatic imaging of axml objects at any conjugate posmon Some numerical results are presented
The prospect o f the availability o f high efficiency diffractive lenses in visible and infrared wavelengths has opened up new vistas in optical system design, particularly for systems operating in m o n o c h r o m a t i c or q u a s l m o n o c h r o m a t l c light [ 1,2 ]. O v e r the years, the techniques for fabrication o f these elements have undergone steady i m p r o v e m e n t leading to significant increase in their diffraction efficiency so much so that these elements, which are also known as klnoforms [3], b i n a r y lenses [4] or phase Fresnel lenses [ 5,6 ] are gradually emerging as optical elements that can substantially ~mprove the performance o f conventional lenses when used in conjunction with the latter [7,8], and, in some apphcations, they hold the promise o f turning out Into viable and better alternatives to conventional lens systems [9-11 ]. K m o f o r m lenses have so far been used mainly for m f i m t e conjugate imaging applications and a detailed description o f the m e t h o d s for design o f such kinoforms has recently been presented [ 12 ]. The use o f klnoform lenses in optical systems, in general, calls for their operation in finite conjugate imagery. It is obvious that a plane k i n o f o r m lens designed for the imaging o f an axial object at infinity will Introduce severe spherical aberration if used for finite conjugate imaging, even at m o d e r a t e apertures [12]. Elimination o f this spherical a b e r r a t i o n calls for a specific relocation o f the full p e r i o d zones as well as
a specific change in blaze profile o f the klnoform lens for each conjugate position To the best o f our knowledge, no study on this aspect o f k l n o f o r m imaging is available in published literature. The i m p o r t a n c e o f this study can also be a p p r e c i a t e d If one recalls the need for special recording geometry in the case o f optically generated aplanatlc holographic lenses [ 13,14 ]. In this communication, we report a generalized and convenient formulation for the d e t e r m i n a t i o n o f zone spacing and blaze profile for a plane klnoform lens. The m e t h o d enables one to design the klnoform lens for stigmatic Imaging of axial objects at any conjugate position. W i t h reference to fig. l, let the klnoform A P form a stigmatic image o f the axial object point O at the point O ' . In what follows, we a d o p t the sign convention o f Born and W o l f [15 ]. The zone radii on the kinoform are defined such that the optxcal path length from O to O ' through the point P,, on the edge o f the ruth zone is [OPmO'] = [OAO']Wm2,
(1)
where the operating wavelength is 2 a n d the object distance A O = Z and the ~mage distance A O ' = Z ' . The inner a n d outer radii o f the m t h zone are A P m - 1= rm_ 1 and APm = rm respectively. F r o m eq. ( 1 ) we o b t a i n the defining expression for rm as
0030-4018/92/$05 00 © 1992 Elsevier Science Pubhshers B V All nghts reserved
1
Volume 91 n u m b e r 1,2
OPTICS COMMUNICATIONS
1 J u l y 1992
It may be noted that the expressmns ( 5 ) or (6) for the zone radii are more convenient than the ones reported earlier m zone plate t h e o ~ [ 16,17 ] The width o f the rnth zone, e,,,, IS given as e,,,=
P
P m ~ . mth Zone
rm
-
-
rm
- I
F r o m eqs. (5) or (6), the total n u m b e r o f zones m .... for given values o f T, )~, f a n d aperture radius r. . . . m a y be d e t e r m i n e d f r o m the smallest p o s m v e
A F-t
Z'
Z
O' )1 I,
T~
0
real root c~ o f the following quartlc equation
)1
m4 + 4 / ' m ~ + 4 ( T 2 + f T - r ~ , , , ) ,t )°2
Fig 1 ruth zone o f the finne conjugate kmoform
2
+,,/r,,+(Z'
=(Z'-Z)+m2.
+ 8T(/T-r~,a,) m (~)
/'H,nax IS g i v e n
Assigning a focal length f t o this k i n o f o r m as
l/f=l/Z'-l/Z.
/=
4T2rmax = 0 ,
(9)
b~
m ..... = I N T ( c ~ ) (3)
(10)
where I N T (c~) is the integer part o f the real n u m b e r O/.
we have
Z'Z Z'-Z
m2
-
Z'Z T "
Figs 2 - 4 d e m o n s t r a t e the variation of zone radn, rm, and zone width, ~,,,, for &fferent values o f T For each T, the k m o f o r m parameters are evaluated for a paraxlal focal length f = 100 mm. The operating wavelength is 10 lam. In order to emphasize the difference xn zone radI1 between a k m o f o r m designed
(4)
where T = Z ' - Z ~s the throw, the object to image distance. After a lengthy but strmghtforward algebraic m a m p u l a t i o n we obtain the following expression for rm from ( 2 ) and ( 3 ) ,
2 2rn2f(l+m2/2T)[l+(m2/2f)(l+mZ/2T)] rm = ( 1+ m2/ T) 2
'
(5) (l+m;t/2T)~ ,~ [ - ( l + m 2 / 2 T ) ] 2 r ' 2 = (1 + m 2 / T ) 2 zm4l+L-(1-+rn~-T-) J m222
v
(6) F o r T - , o e , the exact representaUon for the zone radii of an lnfimte conjugate k m o f o r m m a y be obtained from ( 6 ) as
T=-2
0/JS
% 8O
r2.,=2m2f+m222
(7)
This is the case o f a nonparaxml k m o f o r m ( N P K ) [12] for infinite conjugate imagery, whereas the wellknown formula for zone radii o f a paraxial kinoform ( P K ) m a y be o b t a i n e d from ( 5 ) by imposing the c o n d m o n f>> m2 and T>> m2 as
r2,~= 2 m 2 f
(8)
600
0
200
400
600
800
m
Fig 2 V a n a u o n m the difference m zone radii [ (r,,,)l ( r m ) p K ] w n h n u m b e r of zones m for different values of obJect to image throw T NPK, P K nonparaxlal and paraxlal k m o f o r m s respecUvely for T = oo -
Volume 91, number 1,2
OPTICS COMMUNICATIONS
80
60
T =E
-25
E
40 PK
E
2O
00
I
I
|
200
400
600
800
m
Fig. 3. Zone radn r,,, w~th number of zones rn for different values of obJect to image throw T NPK, PK nonparaxml and paraxml kmoforms respectivelyfor T= oo
0.12
0 10
that the width of the zones o f a kmoform or zone plate decreases quadratically wxth mcreasmg radius. But this is not necessarily true when they are designed specifically for axial stlgmatism at finite values of T. It ~s interesting to note in fig. 4 that, for smaller magmtudes of T, the zone width increases with a rise m radtus after a t t a m m g a m i n i m u m value at an lntermedmte aperture radms The practical ~mphcaUon of this effect needs further investigation. For plane kmoforms, an analytical soluUon for the blaze profile is possible for the mfinlte conjugate case [ 12 ]. In the case of finite conjugates, this seems no longer possible. This is because of the a d d m o n a l req u i r e m e n t of a n n u h n g the spherical aberration artsmg out of obhque l n o d e n c e on the plane interface. A modificaUon of a semi-numerical procedure for the design of a s p h e n c surfaces [ 15 ] is used for the purpose. With reference to fig. 5, let Q R S O ' be an arbitrary ray path through the ruth blazed zone of the kmoform producing stigmaUc image O' of an object point O at a distance Z from the vertex A which is consldered as origin of the coordinate system. The image distance AO' = Z ' and the refractive index of the optical material of the kmoform is/t. Let the angle of i n o d e n c e and refraction at the point R of the ray QRS be I and I' respectively. We have
E
v
slnl'-
sm I --fl
cos/'=
[f1222+(f12--1)r2] 1/2
E ~o
0.0E
_ 25
j 5 ,
1July 1992
r /Ix~ZS+ r 2 ' / l ~ r
(11)
,
2
(12)
0.04
Q 0.02 0
i 200
-400 l 400
NPK
i 600
p
PK, 400 800
m
Fig 4 Zone width ~,, with number of zones m for different values of object to image throw T NPK, PK nonparaxml and paraxlal klnoforms respectivelyfor T= specially for a given T a n d those of the paraxlal kmoform for the same value o f f fig. 2 shows the difference m zone radu [ (rm)r-- (rm)pK] wtth m. For smaller magnitudes of T, a direct plot of (rm) T versus m, for different values of T, xs sufficient to demonstrate the difference (fig. 3). It is usually thought
I'm_ 1
I
~
- "- \
A ~ I - I
O' Z'
O
,I Z
,I
Fig. 5 Geometry and notaUon for the design of the blaze profile for a kmoform for fimte conjugates
Volume 91, number 1,2
OPTICS COMMUNICATIONS
w h e r e r = A R . T h e c o n d i t i o n for axial s t l g m a t l s m o f the k l n o f o r m i m p l i e s that, for any r c o r r e s p o n d m g to the m t h zone, 1.e., for r,, i<-%r<,Nrm,the optical path length [ O R S O ' ] is g t v e n by [ O R S O ' ] = [OABO'
] + (m-
1 )/t
=(Z'-Z)+m2.
(13)
Let RS = t,,. N o t e that the m a x t m u m thickness o f the k m o f o r m ts A B = d = 2 / ( ~ t - 1 ). E x p r e s s i o n ( 13 ) can be used to o b t a i n a q u a d r a t i c relatton d e f i n i n g tm as
At2n + Bt,,, + C = 0 ,
T h i s w o r k was s u p p o r t e d by the F o r m a t i o n de C h e r c h e u r s et A i d e ~i la R e c h e r c h e o f the Q u d b e c G o v e r n m e n t a n d the N a t u r a l Sciences a n d Engineering Research Council of Canada
References 2) ,
(15)
B=Z{u[(Z'-Z)+m2+x/~+r
-(Z'
Finally, it should be m e n t i o n e d that all kanoforms in p r a c t i c e are m a d e on a substrate o f finite thickness. A m o d i f i c a t i o n o f the a b o v e t r e a t m e n t enables one to a c c o u n t for this thickness in o b t a i n i n g exact p a r a m e t e r s o f the k l n o f o r m .
(14)
where A=(l-ct
l July 1992
2]
cosl'+rslnI')},
C=2(Z'-Z)(Z-m2-~r - 2 m 2 ~
-m2~. 2 .
(16)
2) (17)
T h e c o o r d i n a t e s o f the p o i n t S, on the blaze profile, are [ t,,, cos I', ( r - tm sin I' ) ]. F o r a set o f v a l u e s o f r m (r,, i, r,,,) eqs. ( 1 1 ) - ( 1 7 ) are used to determ i n e the c o r r e s p o n d m g v a l u e s o f tm w h i c h are t h e n a p p l t e d to e v a l u a t e the c o o r d i n a t e s o f the p o i n t s on the blaze profile. T h e blaze profile can be determ i n e d to the d e s i r e d level o f a c c u r a c y w i t h suitable chotce o f the n u m b e r o f samples r in (r,n_ 1, rm ). N o t e that the l o w e r edge o f the blaze profile, for the m t h zone, P,,,_ 1 B . . . . 1, IS not parallel to the ax~s and tt lS u n i q u e l y k n o w n after the c o o r d i n a t e s o f the p o i n t Bin_ 1, are d e t e r m i n e d
[ 1] G J Swanson, Binary optics technology, Techmcal report 914 (Lincoln Laboratory, MIT, USA, 1991 ) [2] R E Fisher, OE Reports, 67 (1989) 1 [ 3 ] J A Jordan, Jr., P M Hlrsch, L.B Lesem and D L VanRooy, Appl OpUcs 9 (1970) 1883 [ 4 ] A D Kathman and S K Pltalo, Proc Soc Photo-Opt Instrum Eng 1354 (1990) 297 [5]K Mlyamoto, J Opt Soc Am 51 (1961) 17 [6] G G Sllussarev, Sov Phys Doklady 2 (1957) 161 [7 ] G J Swanson, Binary optics technology, Techmcal report 854 (Lincoln Laboratory, MIT,USA,1990) p 27 [ 8 ] D Fakhs and G M Morns, Phot Spectra 25 ( 1991 ) 131 [9] S T Bobrov and G T Grelsukh, Avtometnya 6 (1985) 3 [ 10] D A Buralh and G M Morns, Appl Optics 28 (1989) 3950 [ 11 ] D A Buralh and G M Morns, Appl Optics 30 ( 1991 ) 2151 [ 12 ] D A Buralll and G M Morns, Appl Optics 28 ( 1989 ) 976 [ 13] W T Welford, Optics Comm 9 (1973) 268 [ 14] W T Welford, Optics Comm 15 (1975) 46 [ 15 ] M Born and E Wolf, Principles of optics (Pergamon Press, Oxford, 1980)pp 191, 197 [16]J Klrz, J OptlcsSoc Am 64 (1974) 301 [ 17 ] V G Schmahl and D Rudolph, Optlk 29 (1969) 577