Design of plane kinoform lenses for stigmatic imaging between two prespecified points on the axis

Optics Communications 94 ( 1992 ) 203-209 North-Holland

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Design of plane kinoform lenses for stigmatic imaging between two prespecified points on the axis L.N. Hazra, Y. Hart and C. Delisle Centre d'Optique, Photonique et Laser, D~partement de Physique, Pavilion Vachon, Universit~ Laval, Ste-Foy, Quebec, G1K 7P4, Canada

Received 9 July 1992

We present a generalized approach for the determination of zone spacing and blaze profile of a plane kinoform lens that provides stigmatic imaging between any two prespecified points on the axis. A unified treatment is provided for kinoform lenses of positive and negative focal lengths. Some illustrative numerical results demonstrate the feasibility of this approach in practice.

1. Introduction Surface relief diffractive lens elements, popularly known as k i n o f o r m lenses, are rapidly finding their way into m a n y practical optical systems a n d this has given rise to a spurt o f research activities in this area [ 1-3 ]. The underlying principles o f o p e r a t i o n are usually explained in the literature for the special case o f stigmatic imaging o f an axial object at infinity, since the latter enables a c o n v e n i e n t analytical treatm e n t [ 4 - 8 ] . The general use o f a k i n o f o r m lens as a single element or a c o m p o n e n t o f a m u l t i c o m p o nent lens system, however, calls for such lenses in a finite conjugate imaging setup. In an earlier C o m m u n i c a t i o n [9], we presented results o f our p r e l i m i n a r y investigations on the dev e l o p m e n t o f techniques for the design o f plane k i n o f o r m lenses that p r o v i d e axial s t i g m a t i s m in finite conjugate imaging. It has been shown that the expression for full p e r i o d zone radii lends itself to a convenient analytical form if the focal length f and the object-to-image distance, i.e. the throw T, are used as variables, for specifying the imaging configuration, instead o f m o r e usual variables, the object and image distances, l a n d l', respectively. A seminumerical algorithm for calculation o f blaze profile was also presented. The f o r m u l a e presented in that Communication, however, presupposes the kinoform lens to be a converging one.

The relative position o f the two axial points, O and O', between which stigmatic imaging is contemplated, m a y give rise to six different imaging configurations (fig. 1). Whereas the configurations

•

0

A

O'

O'

- ~ ¢ " -. A (ii)

O)

0

O'

A (iii)

A

(v)

O'

O'

0

0

0

A (iv)

A

0

O'

(vi)

Fig. 1. Six possible imaging configurations between two axial points O and O'. AP is the kinoform lens of focal length f AO = l; AO'=I'; OO'= T. In (i), (ii) and (iv), Uis negative. In (ii), (v) and (vi), Uis positive. In (i), (iv) and (v),fis positive. In (ii), (iii) and (vi), fis negative.

0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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shown in figs. 1 (i), 1 (iv) and 1 (v) call for a converging kinoform lens AP, the configurations of figs. 1 (ii), 1 (iii) and 1 (vi) necessitate a diverging kinoform lens AP. It should be noted that, not only the blaze profile but also the location of full period zones of a diverging kinoform lens differ from those of a converging kinoform lens of the same absolute focal length, even when both are used for producing a stigmatic image of the same object point, since the object-to-image throw T is different for the two cases. The zone radii of a diverging kinoform lens are the same as those of a converging kinoform lens of the same absolute focal length only in the case of infinite conjugate imaging, where T = + ~ . The formulae for zone radii and blaze profile presented in the earlier Communication [9] pertain to the imaging geometry of fig. 1 (v). In this Communication, we present a modified approach that enables one to design a plane kinoform lens for stigmatic imaging between any two prespecified points on the axis. This approach encompasses all types of imaging geometry of fig. 1.

15 November 1992

f= - l ' l / T .

(2)

For the special case of either O or O' at infinity, we use

for l= _+oo,

T=-T- ~ ,

f=l', for l ' = _+oo,

T=_+oo, f= -l.

(3)

2.2. Calculation of zone radii The zone radii of a converging kinoform lens are defined such that the optical path length from O to O' through the point Pm o n the outer edge of the mth zone is (fig. 2 ( i ) ) [OPmO'] = [OAO']

+m2,

(4)

where the operating wavelength is 2. For the case of a diverging kinoform lens (fig. 2 (ii)), the zone radii are defined by the relation

(5)

[OPmO'] = [ O A O ' ] - m 3 . ,

From eqs. (4) and (5), we can have a common relation for the zone radii of a plane kinoform lens as

2. Design procedure In what follows, we adopt the sign convention of Born and Wolf [ 10]. More specifically, the convergence angle of the incident ray U =/_. POA is positive in figs. 1 (ii), 1 (v) and 1 (vi), and it is negative in figs. 1 (i), 1 (iii) and 1 (iv). The axial object and image points, O and O', respectively, are specified by given values for l = AO and l' = AO'. Light is assumed to travel from left to right and the directed distances l and l' have their signs accordingly as positive when directed from left to right and negative when directed from right to left.

where s=+l

, for f > 0 , for f < 0 .

(7)

Let/_ PmOA = Um a n d / - P m O ' A = U ' . From eq. (6), we obtain the following expression for rm

2.1. Determination of the value o f f

rmT~~

For a given set of values for l and l', the object to image distance, the throw T, is given by

T=(I'-I).

(1)

The required value of focal length of the kinoform lens is determined from the relation 204

(6)

[OPmO'] = [OAO'] +ms2,

A

I---l'

t--I O)

O'

0

"~-T~ >i

A

~--r O)

0

O'

)1

Fig. 2. mth zone of a plane kinoform lens of (i) positive focal length, (ii) negative focal length.

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r,, + r_.___~_m _ ( l ' - l ) + m s 2 . sin Um sin U "

(8)

We have rm

sin Um= +- - x f T ~ r 2 , sin U " = +_

(9)

( l0)

rm x/(l')2+r~ "

The plus and minus signs in eqs. (9) and (10) are determined according to the sign of the corresponding angles. From eqs. ( 8 ) - ( 10 ), by substitution, we get

-T-w/~+r~ + _ ~

=(l'-l)+ms2.

(ll)

Squaring b o t h sides we get

-T-~

N/~2-I- r 2 m2s2,~2

=-r2+

2

-l'l+ms2(l'-l).

(12)

Further squaring leads to the following expression for rm in terms of l and l' rZm={ - 2 m s 2 ( l ' - l ) l ' l + m 2 s 2 2 2 [ ( l ' - l ) 2 - l ' l ]

2.3. Evaluation of zone profile With reference to fig. 3, let QRSO' be an arbitrary path through the mth blazed zone of the kinoform lens producing stigmatic image O' of an axial object point O at a distance l from the vertex A. The vertex A is considered as origin of a rectangular coordinate system with the Z axis along the optical axis AO and Y-axis along AP on a meridional section of the rotationally symmetric kinoform lens. The image distance A O ' = l' and the refractive index of the optical material of the kinoform is/~. Let the angles of incidence and refraction at the point R of the ray QRS be I and I ' respectively. The convergence angle of the incident ray QR with the optical axis is U and the same for the outgoing ray SO' is U'. By applying the law of refraction at the point R, we get /~ s i n / ' = s i n I .

(17)

U= - I , (13)

Using eqs. (1) and (2), we obtain from eq. ( 1 3 ) r 2 m

values of zone radii are to be determined from the expressions (14) or ( 15 ), where the suitable value of s is to be substituted as per eq. (7) in accordance with the sign of focal length of the kinoform lens.

In accordance with our notation

+ m3s32 3( l'--l) + m4s4,~.4/4} ×[(l'-l)2+(ms2)2+2ms2(l'-l)] -' .

15 November 1992

(18)

so that (19)

sin I ' = - ( 1/p) sin U

{2ms2f(1 + ms2/2T) × [ 1 + (ms2/2jO ( 1 + ms2/ZT) ] } Y X (1 + m s 2 / T ) -2 .

(14)

2

f l + m s 2 / 2 T ~ 2 2--2 if l + m s 2 / 2 T ' ~ _ ) m X +L(-l~i)2mszy,

O ~ Pm

Uf•X• s

Alternatively, we have

rm = ~ - i ~

Y

,r

(15)

& Bo

Bm-1 r

1.r1

~

xxx\

since s2= 1. For T ~ +oe, eq. (15) reduces to

r 2 =m222+2ms2f.

o, o, z

(16)

Since, by definition, s = - 1 for f < 0, eq. (16) shows that, in the case of infinite conjugate imaging, the values of zone radii of a converging or a diverging kinoform lens of the same absolute focal length are same. In the general case, with a finite value of T, the

,

o

o,-z

'%7 i-- I (i)

'-I (ii)

Fig. 3. First and mth blazed zones o f a plane kinoform lens of (i) positive focal length, and (ii) negative focal length.

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and cos I ' = ( 1//t)x//.t2- sin 2 U

(20)

15 November 1992

Squaring both sides we obtain the following quadratic relation defining tm as

At~ + Btm + C= O,

By geometry, we have

(25)

where r + tm sin I ' sin U ' = + x/(l,_tm cos I') 2+ (r+ tm sin I ' ) 2,

A = ( 1 - / t 2) , (21)

(26)

B=2{/~(T+ps2+r/sin U) -(1/lt)[(T+l)x/It2-sin2U+rsinU]},

where tm=RS and r = A R . The plus and minus signs in the expression for sin U' correspond to the sign of the angle U'. The condition for axial stigmatism of the kinoform lens implies that, for any ray passing through the mth zone, i.e. for rm_, <~r~rm, the optical path length [ORSO'] is given by

The coordinates of the point S, on the blaze profile, are

[ORSO'] = (1'- l) +ps2,

[tm COS I', (r+tm sin I ' ) ]

(22)

where for f > 0,

(28)

(29)

,

,

p=m-1, s=-I.

(23)

The difference in the value of p for f > 0 and f < 0 should be noted. It is so, because, in the case of a lens with f > 0, the optical path length of any ray from O to O' through a specific blazed zone of the kinoform lens is equal to the optical path length of the ray from O to O' passing through the outer edge of that particular zone. On the contrary, in the case of a lens with f < 0, the optical path length of any ray from O to O' through a specific blazed zone of the kinoform lens is equal to the optical path length of the ray from O to O' passing through the inner edge of that zone. For example, in fig. 3(i), for the ruth zone, [ORSO'] = [OPmO'] and for the first zone [ O G H O ' ] = [OP~O']; whereas, in fig. 3(ii), for the mth zone [ORSO'] = [OPm_~O'] and for the first zone [ O G H O ' ] = [OAO']. Also, note that the maximum thickness of the kinoform is A B = d = 2 / (/~- 1). Expanding eq. (22) and using eq. (21),

+-x/(l'--lm COSI') + (r+tm sin 1,)2 = [ (l'-l) +ps2+r/sin U] -lit,,. 206

- P22 2 •

=-[~x/'It2-sin2 U ' ( r - t~msin U)] p~m

s=+l forf<0,

C= 2 T ( l - ps2- si--~-U)- 2pS2 s i ~

(27)

For a set of values of r, where rm_ ~~
IZx/+rr -f 2 ,

where the plus or minus sign is determined by the sign of the angle U. According to the sign convention followed herein, for all positive values of r, U is negative when 1<0 and U is positive when l>0. Obviously the above relations cannot be directly used when either the object point O or the image point O' is at infinity. By rearranging we can circumvent indeterminacy in the expressions. For the case l ' ~ + ~ , we get

tm=#(ps2--l+

r__f__ .2

s i n U ) ( ~* - N /

/1~2

F2 .~--I - -l 2--t-r2J

" (31)

As l-, + oo, the limiting form for B and C can be obtained from eqs. (27) and (28) as Bt~ ±oo = 2 [ ( / t - 1 )l'+Itps2] and

(24)

(30)

Cl~ ±~ = r 2- 2ps21'-p22 2 .

(32)

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OPTICS C O M M U N I C A T I O N S

Thus, for I-~ + ~ , tm is to b e d e t e r m i n e d f r o m the quadratic equation

15 November 1992

3.5 3

( 1 - / 1 2 ) t 2 + 2 [ ( / 1 - 1 )l'+/1ps,~]tm 2.5

+ (r2-2ps21'-p222) =0.

(34) 2 E

T h e roots o f this e q u a t i o n are

E 1.5 >-

--q+_qx/1 -- ( 1 --IZ 2 ) (r2--2ps21'--p222)/q 2 t,.=

(1_/12)

1

, (35)

0.5 0

where

q= (/1- 1 )1'+/1ps2.

-0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

(36)

A t r = 0 , tm=d for f > 0 a n d tm=O for f < 0 . T h i s b o u n d a r y c o n d i t i o n i m p l i e s t h a t t h e p o s i t i v e sign o f the s q u a r e r o o t is to be r e t a i n e d a n d so tm is g i v e n by

Z (.am) o) 3.5

- q + q ~ / 1 - ( 1 _ / 1 2 ) (r2_2ps21,_p222)/q2 tin=

''"

3

(1_/12)

(37)

2.5

By further algebraic m a n i p u l a t i o n s , we can recast the

2

I ....

I I '' rl'*-"l'

"

'1 ' ' " 1 ' ' " 1 "

/ ~

~

E 1.5

0.5 0 O

I'~ I

A

i I

I

r

O'

O

O'

~'1

l~---i

~'1

t-- T -~

A

I ~r-I

T (i)

(ii)

- 100 m m , / ' = - (200/3) mm, T= (400/3) mm. Table 1 Zone radii (rm)a and (rm)2 in m m for selected values o f m .

1

10 20 50 100 200 500 1000

,~l .... ~.... I .... ~.... I .... I .... I,, -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Z (.am)

Fig. 4. Plane kinoform lens AP with object distance 1= - 2 0 0 mm. In (i), f = 1 0 0 m m , l ' = 2 0 0 m m , T = 4 0 0 m m . In (ii), f =

m

-0.5

(r~)l

(rm)2

1.414 4.472 6.325 10.003 14.151 20.025 31.721 45.000

1.414 4.476 6.335 10.041 14.258 20.329 32.945 48.568

0i) Fig. 5. Blaze profiles on the first five zones of the kinoform lenses of fig. 4(i) and 4(ii) in (i) and (ii), respectively.

a b o v e r e l a t i o n ( 3 7 ) for tm in the case of/--* + ~ , i n t o a f o r m s i m i l a r to the s t a n d a r d optical design sag formula

ps2 tm-- ( u - - l )

cr z + l + ~ / 1 - - ( K + l ) c Z r 2'

(38)

where c-

1

( 1 - #)l' +ps2

(39)

and K= _/12.

(40) 207

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15 November 1992

Table 2 (Z, Y) coordinates of points on the blaze profile for the I 0th, the 50th and the 100th zones of a plane kinoform lens of focal length f= - 100 mm; l= 100 mm; 1'= oo; 2 = 10 ~tm;/1= 4; width of the mth zone ~ = rm - r m _ ~; r = r m _ ~+ a~,~; r, Z and Yare in mm. a

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

m= 10

m=50

r

Z

Y

r

Z

Y

r

Z

Y

4.24360 4.26656 4.28953 4.31249 4.33546 4.35842 4.38139 4.40436 4.42732 4.45029 4.47325

0.00000 0.00032 0.00065 0.00098 0.00131 0.00165 0.00198 0.00231 0.00265 0.00299 0.00333

4.24360 4.26656 4.28952 4.31248 4.33544 4.35841 4.38137 4.40433 4.42729 4.45025 4.47322

9.91161 9.92170 9.93179 9.94188 9.95197 9.96205 9.97214 9.98223 9.99232 10.00240 10.01249

0.00000 0.00033 0.00066 0.00100 0.00133 0.00166 0.00200 0.00233 0.00266 0.00300 0.00333

9.91161 9.92169 9.93177 9.94185 9.95193 9.96201 9.97209 9.98217 9.99225 10.00233 10.01241

14.10603 14.11317 14.12031 14.12746 12.13460 14.14174 14.14888 14.15602 14.16316 14.17031 14.17745

0.00000 0.00033 0.00066 0.00100 0.00133 0.00166 0.00200 0.00233 0.00266 0.00300 0.00333

14.10603 14.11316 14.12029 14.12742 14.13455 14.14168 14.14881 14.15594 14.16307 14.17020 14.17733

The expression (38) is a generalized version of the formula for blaze profile of kinoform lenses when l ~ oo [ 8]. In the form presented here, it is valid for kinoform lenses of both positive a n d negative focal lengths.

3. Numerical results For a given set of i n p u t parameters l, 1', 2 a n d #, the approach presented in the earlier section can be followed to determine the zone radii a n d blaze profile of a kinoform lens yielding stigmatic imaging between any two prespecified axial points. In fig. 4 ( i ) , we have an imaging geometry where l = - 2 0 0 m m and l ' = 200 m m so that f = 100 m m and T = 400 m m . Let the zone radii of the kinoform lens be specified by ( r m ) t . For the imaging geometry shown in fig. 4 ( i i ) , l = - 2 0 0 m m , and l ' = - ( 2 0 0 / 3) m m so that f = - 1 0 0 m m a n d T = ( 4 0 0 / 3 ) mm. The zone radii of this kinoform lens are specified by (rm)2. The values of (rm) 1 a n d (rm)2 for some selected values of m are given in table 1 for an operating wavelength 2 = 10 ~tm. The blaze profiles on the first five zones of the two kinoform lenses corresponding to figs. 4 ( i ) and 4 ( i i ) are shown in figs. 5 (i) and 5 (ii), respectively for a kinoform material of refractive i n d e x / ~ = 4.0: As an alternative example, table 2 gives the values 208

m= 100

of (Z, Y) coordinates on the required blaze profile for the 10th, the 50th a n d the 100th zones of a kinoform lens providing stigmatic image at I ' = oo of the object point at l = I00 mm. The values o f / t a n d 2 are the same as those of the last example. For each zone, (Z, Y) coordinates corresponding to eleven equispaced values of r in (rm_ 1, rm) are given. It may be noted that the values of Z remain practically unchanged with higher values of m. The corresponding values of Y are, however, different. In fact, for large values of m, the blaze profile over a single zone is almost linear. It is the slope of this blaze profile that changes from one zone to the other. This feature can also be appreciated from a closer look at the blaze profiles presented in figs. 5 (i) and 5 (ii).

4. Concluding remarks We have presented a convenient method for the determination of constructional parameters of a plane kinoform ]ens that yields stigmatic imaging between any two prespecified points on the axis. Our approach deals with the requirements of positive and negative kinoform lenses in such a m a n n e r that the parameters of the required kinoform lens can be directly obtained from the input specifications for l, I', /~ a n d 2 with the help of a single computer algorithm based on the treatment presented above. In practice, these lenses are to be made on sub-

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strates of finite thickness. The width, shape a n d material of the substrate will influence imaging properties of the k i n o f o r m lens substantially. It is also possible to take account of these effects in determ i n i n g exact parameters of the kinoform by modifying the approach presented here. We i n t e n d to deal with this aspect of the problem in a forthcoming Communication.

Acknowledgements This work was supported by the F o r m a t i o n de Chercheurs et Aide ~ la Recherche of the Qu6bec G o v e r n m e n t a n d the Natural Sciences a n d Engineering Research Council of Canada.

15 November 1992

References [ 1] W.D. Veldkampand T.J. McHugh, ScientificAmerican266 (1992) 92. [2] D. Faklis and G.M. Morris, Phot. Spectra 25 ( 1991 ) 205; 131. [3] See, for example, papers in: 1992 Technical Digest (Opt. Soc. Am) Series Vol. 9 on Diffractive Optics: Design, Fabrication and Applications. [4] G.G. Sliussarev,Sov. Phys. Doklady 2 (1957) 161. [5] A.I. Tudorovskii, Opt. Spectrosc. 6 (1959) 126. [6] K. Miyamoto, J. Opt. Soc. Am. 51 ( 1961 ) 17. [ 7 ] J.A. Jordan Jr., P.M. Hirsch, L.B. Lesem and D.L. VanRooy, Appl. Optics 9 (1970) 1883. [8 ] D.A. Buralli, G.M. Morris and J.R. Rogers, Appl. Optics 28 (1989) 451. [9 ] L.N. Hazra, Y. Han and C. Delisle,OpticsComm. 91 (1992) 1. [10] M. Born and E. Wolf, Principles of optics (Pergamon, Oxford, 1980) p. 191.

209