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Zone plates performing generalized Hankel transforms and their metrological applications
Optics Communications 102 (1993) 391-396 North-Holland
OPTICS COMMUNICATIONS
Zone plates performing generalized Hankel transforms and their metrological applications Z. J a r o s z e w i c z l Laboratorio de Optica, Departamento de Fisica Aplicada, Facultade de Fisica, Universidade de Santiago, 15706 Santiago de Compostela, Galicia, Spain and
A. K o t o d z i e j c z y k z lnstitut fffr Technische Optik, Stuttgart Universitf~t, Pfaffenwaldring 9, W- 7000 Stuttgart 80, Germany Received 2 April 1993; revised manuscript received 10 June 1993
A modified computer-generated Fresnel zone plane with the zone distribution determined by a Fermat spiral is proposed for performing generalized Hankel transforms. The special case of these elements realizing the generalized Hankel transform of first order is of special interest from the point of view of alignment applications. The doughnut shaped focal pattern, characteristic for the whole family of proposed zone plates exhibits then the smallest diameter. Its diameter is also smaller than other solutions proposed previously for obtaining focal spots with a black dip in the centre.
F o r high-precision alignment there have been proposed recently diffractive elements with black focal spots [ 1 - 3 ] . Two reasons explain their usefulness: the d i a m e t e r o f the dark dip is usually smaller than the d i a m e t e r o f the corresponding Airy spot [ 1 ] and, since the intensity in the centre o f the focal plane is zero, the miscentering error can be detected by an infinite p r o p o r t i o n a l change o f irradiance [ 3 ]. The destructive interference in the focus o f these elements is achieved by introducing a zc phase step in the zone pattern o f the Fresnel zone plate. The phase step can be d r a w n along a circle d i v i d i n g the aperture into two equal areas, what results in a doughnut focal spot [ 1 ]; or along d i a m e t e r s d i v i d i n g the aperture into halves [ 1 ], quadrants, and in general into a phase daisy with an even n u m b e r o f sectors o f equal angle and alternately ~ phase delays [2,3]. The focal spot is then crossed by one, two or k dark spokes On leave from Institute of Applied Optics, Kamionkowska 18, 03-805 Warsaw, Poland. 2 On leave from Institute of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland.
(where 2k is the n u m b e r o f the phase daisy petals), for k > 1 forming in the centre o f the focal plane a dark hub [ 1-3 ]. The aim o f this Letter is to offer another proposal for the realization o f the focal pattern with a dark center. As a suggested solution to this problem, an element realizing the generalized Hankel transform is proposed. Its transmittance t(r, O) differs from the o r d i n a r y Fresnel zone plate by adding a linear phase term along the angular coordinate, what results in the following expression t(r, O) = P ( r ) e x p ( i n 0 ) exp( - i k r 2 / 2 j O ,
( 1)
where k=2zc/2, P ( r ) is the function in a d d i t i o n to the phase factor o f the element, n is an integer denominating the generalized Hankel transform number, f i s the focal length o f the element, and 2 is the wavelength o f the illuminating beam. By r and 0 the polar coordinates o f the element plane are meant. The zone borders o f this element are no longer designated by a set o f equiconcentric circles, like for the o r d i n a r y Fresnel zone plate, but by a set o f spirals (fig. l a ) , described as follows,
0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
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C Fig. 1. The zone pattern for: (a) on-axis spiral zone plate, ~(x, y) = -k(x2+y2)/2f+n arctan(y/x), n= 1; (b) off-axisspiral zone plate, • (x, y)=-k(x2+y2)/2f+k sin ~x+n arctan(y/x), n=l; (c) grating modulated by angular phase change, q~(x, y)=k sin ~x+n arctan(y/x), n= 1.
r2/2)~f- nO/2n= m ,
(2)
lowing diffraction integral (in the scope of Fresnel approximation), c~
where m is an integer denominating the zone number. This equation can be rewritten in the form of n spiral curves known as Fermat spirals, or parabolic spirals [ 4 ],
r=a~O,
where a=x/afn/zc ,
(3)
rotated consecutively by 2nn rad. In fig. 1, together with the spiral on-axis zone plate for n = 1 (fig. la), the off-axis zone pattern is shown (fig. lb), as well as its carrier frequency alone, modulated by an angular linear change of the phase (fig. lc). The zone plate modified according to eq. ( 1 ) possesses for every sector its counterpart oppositely placed and with phase shifted by n radians. Hence, the destructive interference occurs in the focus, and the doughnut shaped focal pattern is formed. The spiral zone plate described by above equations was shown to produce phase singularities in the centre of the focal plane [ 5 ], in the sense of the definition given in ref. [ 6 ], i.e. that the phase integral around the closed path surrounding the point is equal to 2nn. It is proper to mention that the corresponding complex amplitude distribution for these singularities is described by the generalized Hankel trans-. form, according to the definition given by Papoulis [ 7 ]. Substituting the transmittance function of the spiral zone plate given by eq. ( 1 ), one can describe the complex amplitude in its focal plane by the fol392
U(p,O,J)= ~-f-iC P(r)
2n
rdrJ"
0
exp(in0)
0
X e x p [ - i ( k r p / f ) c o s ( 0 - G ) ] dO,
(4)
where C = exp [ik(f+p2/2f) ] . Using the identity 2rt
f exp(inO) e x p [ - i ( k r p / f ) cos(0-q~) ] dO 0
=2~z( - i ) " exp (inq~) Jn( krp /f) ,
(5)
we can represent the diffraction integral as the generalized Hankel transform of the nth order
U(p, ~,f) = (k/f) ( - i ) "+' exp(in0) x C t P(r)J,(krp/f) rdr.
(6)
0
Hereafter, the element performing the Hankel transform of the first order will be considered, since for this case the dip in the centre of the focal plane is the narrowest. If a circular aperture will be assumed, then the focal intensity distribution is given as the square o f the first order Hankel transform of the function circ(r/R), where R is the radius of the zone plate,
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OPTICS COMMUNICATIONS
I(p,f)=Io(2 kR!/IJ'(t)t (TYfip dt'~2 ] '
(7)
where Io = (kR2/2f) 2 and t=krp/f Taking into account the relation J'o(X) = -Jl and the identity
(x),
0
.~ (x) ] ,
(8)
where . ~ and .~ are Struve functions of zero and first order [7], the distribution of the intensity in the focal plane can be written as follows,
I (p, f) = ( Io ~2 ) ×(Jl ( kRx/f) .#o(kRp/f) -Jo( kRp/f) .~ ( kRp/f) )2 kRp/f (9) By expanding J~ function under the integral of eq. (7) into series and performing explicit integration of the polynomials the same result can be obtained as the power series, more applicable for the numerical evaluation m=oo
I(p,f)=16Io ( ~ l
[_(kRp/f)z/4]mm
1/[.) 2
kl~p
(m!)Z(Zm+ 1)
" (lO)
By the numerical evaluation of the intensity distribution one can convince, that the first maximum lies atp=O.7802f/2R, and the half-width (HW is the value ofp for which l(p, f) =ImaJ2 ) equals 0.3832f/ 2R. The further reduction of the diameter of the central dark dip can be achieved by the introduction of an apodizing function onto the aperture of the element. In the present case the simples example will be checked, i.e. the substitution of the circular aperture by an annular one. An introduction of the masking ration G = 1 -AR/R, where AR= ( R - R i ) and R i is a diameter of masking disc allows to write the respective intensity distribution in the following form
. (F(kRp/f) - GF(kGnp/f) ,2 Z(p,j)
k
p/f
.) ,
where c=re 2, F(x) = [Jl (x) .~o(x) -Jo(x) .~ (x) ] for the representation given in eq. (9) and c = 16, m=~ [ 1 _ (x)2/4] m m F ( x ) = m=,E ( - ~ - ! ~ ~
i Jo(t) dt=xJo(x) + ½reX[Jl(x) .¢fo(X)-Jo(x)
15 October 1993
(ll)
for the representation given in eq. (10). At the end, the limiting case of G approaching 1 can be immediately derived from eq. (6) giving the J~ distribution of the focal pattern (similarly as the infinitesimal annular aperture of an ordinary lens gives a Jo distribution):
l(p,f) =Io( 2AR/ R )Z[J, ( kRp/f) ]2.
(12)
The positions of the first maximum and HW reach then their smallest values, respectively 0.586f/2R and
0.2912f/2R. The intensity distributions for G equal to 0, 0.5, and 0.9, calculated according to eqs. (10) and ( 11 ), are shown in figs. 2a, 2b, and 2c, respectively, where Io, defined in eq. (7), means the maximum intensity of the Airy pattern of the corresponding ordinary element. Figure 3a shows the decrease of the first maximum of the spiral zone plate with the growing masking ratio G. In turn, figs. 3b and 3c illustrate the descending values for the position of the first maximum and for the position of the HW of the spiral zone plate in the function of the growing masking ratio (lower curves), compared with respective relations for an ordinary element (upper curves), and showing clearly that the central dip becomes then narrower. The comparison of derived results with those obtained earlier for other elements producing blackfocal spots (i.e. for a zone plate with a phase jump around the circle dividing the aperture into two halves, for a zone plate with a phase jump along the diameter, and for zone plates with phase jumps dividing the aperture into quadrants) is given in table 1, for full and infinitesimal annular apertures. This comparison demonstrates distinctly the superiority of the spiral zone plate over other types of solutions presented earlier [ 1-3 ]. A zone plate with a phase jump around the circle exhibits not only significantly greater dimensions of the dip, but also its intensity decreases faster with the growing masking ratio, and in addition the further subsidiary maxima become then predominant over the first one; moreover the axial intensity dis393
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I
15 October 1993
0.20
0.020
0.3
0.15
0.015
0.2
0.10
~
0,010
0.1
ooo
0.0 0
1
2
3
4
-X 0
1
2
\)\/
oooo 3
4
0
1
2
.5
p2R/Xf
p2R/Zf
,o2R/Xf
a
b
c
4
Fig. 2. The intensity distribution across the focal spot of the spiral zone plate for: (a) circular aperture, masking ratio G= 0; (b) annular aperture, masking ratio G=0.5; (c) annular aperture, masking ratio G=0.9.
0.4
1.6
0.3
1.2
_~ o.2
\
_E
x
0,1
0.8
~
0.6
~.~
0.8
0.4
0.0 0,25
0.50
0.75
1.00
~ -
0.2
0.0 0.00
~ 0.4
0.00
0.25
~ 0.50
0.0 0.75
1.00
0.00
0.25
0.50
G
G
G
a
b
c
0.75
1.00
Fig. 3. The dependence in the function of the masking ratio G for (a) intensity of the first maximum; (b) position of the first maximum of the spiral zone plate versus the position of the first minimum of the ordinary zone plate; (c) position of HW of the spiral zone plate versus the position of the HW of the ordinary zone plate. tribution is not zero along the whole optical axis [ 1 ]. In turn, the spiral zone plate, together with the phase daisy zone plate has zero irradiance along the whole optical axis, what additionally predestines both types of elements for the axial aiming, although it seems, that for purely axial alignment the J~ Bessel beams. [9] as well as their counterparts among the logarithmic axicons are unrivaled a n d the zone plates with black spot are better suited for the sensing of the deviations of the point images in the image plate. 394
C o n t i n u i n g the comparison of the spiral zone plate with the phase daisy elements, not only the smaller diameter of the focal dip produced by the spiral zone plate should be noted (although the difference is not so distinct as in the previous case), but moreover the focal pattern is not divided by dark spokes, possessing thereby the symmetry of revolution - in consequence the signal of miscentering error will not depend on the direction of the deviation. In order to verify these results, a carrier frequency
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15 October 1993
Table 1 Positions of the first maxima and the HW for the spiral zone plate and for other types of the zone plates with black focal spot, compared with the ordinary zone plate in the cases of the full aperture (G=0) and of the infinitesimal annular aperture (g= 1 ); the unit is p2R/2f Position of~he first maximum
Position of the HW
G=0
G=I
G=0
G=I
Spiral zone plate for n= 1
0.780
0.586
0.383
0.291
Zone plate with phase a jump around the circle
1.136
0.765
0.727
0.497
Zone plate with phase a'bjump along the diameter
0.842
0.630
0.411
0.312
Zone plate with phase a'bjumps dividing the aperture into quadrants
0.873
0.682
0.554
0.439
Ordinary zone plate b
1.220
0.765
0.514
0.359
Data taken from ref. [ 1 ]. b Values for the most narrow cross sections, c Position of the first minimum. p a t t e r n with s u p e r p o s e d a n g u l a r phase change for n = 1 was d i s p l a y e d o n a C R T flat screen with 680 × 480 pixels a n d t h e n r e d u c e d p h o t o g r a p h i c a l l y in a K o d a k A H U 5460 film to a d i a m e t e r o f 6 m m (fig. l c ) . T h i s p a t t e r n was p r o g r a m m e d as a b i n a r y a m p l i t u d e display, i.e. its t r a n s m i t t a n c e was equal to
t(x,y)=O,
for for
-
1 ~ < c o s [ ~ ( x , y ) ] ~<0, 0~
,
(13)
where • (x, y ) = k sin
Fig. 4. Experimentally obtained focal spots for: (a) spiral zone plate, circular aperture; (b) ordinary zone plate, circular aperture.
ax+arctan(y/x),
a is the angle o f the carrier f r e q u e n c y ( a = 0 . 0 0 3 ) , a n d x, y are the c a r t e s i a n c o o r d i n a t e s o f the e l e m e n t plane. T h i s t r a n s m i t t a n c e , c o m b i n e d with a converging lens a n d i l l u m i n a t e d by a p l a n e wave ( 2 = 6 3 2 8 n m ) has p r o d u c e d in its focal p l a n e ( f = 9 5 0 n m ) the desired d o u g h n u t focal pattern. In fig. 4a there is s h o w n the i n t e n s i t y p a t t e r n o f the exp e r i m e n t a l l y o b t a i n e d e l e m e n t . In t u r n , in fig. 4b the Airy spot o f a c o r r e s p o n d i n g o r d i n a r y e l e m e n t with the s a m e a p e r t u r e a n d focal length is s h o w n for c o m parison. T h e focus o f the spiral zone plate was t h e n t a k e n with a C C D c a m e r a , m e a s u r e d along one o f the d i r e c t i o n s with the help o f a D T 2 8 5 1 D a t a T r a n s l a t i o n i m a g e processing b o a r d a n d c o m p a r e d with the theoretical curve. T h e result, p r e s e n t e d in fig. 5 shows that in spite of the m o d e s t m e a n s used for the e x p e r i m e n t , the c o i n c i d e n c e is q u i t e good.
1.00 '0~C0.10"75 l 50 0.25
0.00
0.0 0.5 1.0 1.5 2.0 p2R/Xf
Fig. 5. Experimentally measured intensity of the spiral zone plate focal spot presented in fig. 4a versus theoretical prediction presented in fig. 2a. 395
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OPTICS COMMUNICATIONS
Much more difficult to achieve is the same minim u m intensity for all directions. In conclusion, it was shown, that by introduction of an angular phase change, elements capable of realizing generalized Hankel transforms are possible to produce, in the form of spiral zone plates. Moreover, the parameters corresponding to the element realizing the generalized Hankel transform of the first order predominate over other solutions proposed for obtaining a focal pattern with a dip in the c e n t e r thus, it can be useful for alignment purposes. This work was performed during the stay of Z. Jaroszewicz at the University Santiago de Compostela. The fellowship for his stay from the Spanish Ministry of Education and Science is gratefully acknowledged. At the same time A. Kolodziejczyk was incorporated at the University of Stuttgart as a fellow of a Alexander von H u m b o l d t Fellowship. The authors gratefully acknowledge apt and helpful remarks of the Referee.
396
15 October 1993
References [ 1]S. Bar~i Vifias, Z. Jaroszewicz, A. Ko|odziejczyk and M. Sypek, Appl. Optics 31 (1992) 192. [2 ] J. Ojeda-Castafieda,P. Andr6sand M. Martinez-Corral,Appl. Optics 31 (1992) 4600. [ 3 ] J. Ojeda-Castafiedaand G. Ramirez, Optics Lett. 18 ( 1983 ) 87. [4 ] E. Niczyporowicz,Planar curves (Polish ScientificPublishers, Warsaw, 1991)p. 437 (in Polish). [5] N.R. Heckenberg, R. McDuff, C.P. Smith and A.G. White, Optics Lett. 17 (1992) 221. [ 6 ] A.G. White, C.P. Smith, N.R. Heckenberg, H. RubinszteinDunlop, R. McDuff, C.O. Weiss and Chr. Tamm, J. Mod. Optics 38 (1991) 2531. [7]A. Papoulis, Systems and transforms with applications in optics (McGraw-Hill,New York, 1968)p. 163. [8]J. Spanier and K.B. Oldham, An atlas of functions (Hemisphere PublishingCorporation, Washington, 1987) p. 518. [9 ] A. Vasara, J. Tutunen and A.T. Friberg,J. Opt. Soc. Am. A 6 (1989) 1748.
OPTICS COMMUNICATIONS
Zone plates performing generalized Hankel transforms and their metrological applications Z. J a r o s z e w i c z l Laboratorio de Optica, Departamento de Fisica Aplicada, Facultade de Fisica, Universidade de Santiago, 15706 Santiago de Compostela, Galicia, Spain and
A. K o t o d z i e j c z y k z lnstitut fffr Technische Optik, Stuttgart Universitf~t, Pfaffenwaldring 9, W- 7000 Stuttgart 80, Germany Received 2 April 1993; revised manuscript received 10 June 1993
A modified computer-generated Fresnel zone plane with the zone distribution determined by a Fermat spiral is proposed for performing generalized Hankel transforms. The special case of these elements realizing the generalized Hankel transform of first order is of special interest from the point of view of alignment applications. The doughnut shaped focal pattern, characteristic for the whole family of proposed zone plates exhibits then the smallest diameter. Its diameter is also smaller than other solutions proposed previously for obtaining focal spots with a black dip in the centre.
F o r high-precision alignment there have been proposed recently diffractive elements with black focal spots [ 1 - 3 ] . Two reasons explain their usefulness: the d i a m e t e r o f the dark dip is usually smaller than the d i a m e t e r o f the corresponding Airy spot [ 1 ] and, since the intensity in the centre o f the focal plane is zero, the miscentering error can be detected by an infinite p r o p o r t i o n a l change o f irradiance [ 3 ]. The destructive interference in the focus o f these elements is achieved by introducing a zc phase step in the zone pattern o f the Fresnel zone plate. The phase step can be d r a w n along a circle d i v i d i n g the aperture into two equal areas, what results in a doughnut focal spot [ 1 ]; or along d i a m e t e r s d i v i d i n g the aperture into halves [ 1 ], quadrants, and in general into a phase daisy with an even n u m b e r o f sectors o f equal angle and alternately ~ phase delays [2,3]. The focal spot is then crossed by one, two or k dark spokes On leave from Institute of Applied Optics, Kamionkowska 18, 03-805 Warsaw, Poland. 2 On leave from Institute of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland.
(where 2k is the n u m b e r o f the phase daisy petals), for k > 1 forming in the centre o f the focal plane a dark hub [ 1-3 ]. The aim o f this Letter is to offer another proposal for the realization o f the focal pattern with a dark center. As a suggested solution to this problem, an element realizing the generalized Hankel transform is proposed. Its transmittance t(r, O) differs from the o r d i n a r y Fresnel zone plate by adding a linear phase term along the angular coordinate, what results in the following expression t(r, O) = P ( r ) e x p ( i n 0 ) exp( - i k r 2 / 2 j O ,
( 1)
where k=2zc/2, P ( r ) is the function in a d d i t i o n to the phase factor o f the element, n is an integer denominating the generalized Hankel transform number, f i s the focal length o f the element, and 2 is the wavelength o f the illuminating beam. By r and 0 the polar coordinates o f the element plane are meant. The zone borders o f this element are no longer designated by a set o f equiconcentric circles, like for the o r d i n a r y Fresnel zone plate, but by a set o f spirals (fig. l a ) , described as follows,
0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
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C Fig. 1. The zone pattern for: (a) on-axis spiral zone plate, ~(x, y) = -k(x2+y2)/2f+n arctan(y/x), n= 1; (b) off-axisspiral zone plate, • (x, y)=-k(x2+y2)/2f+k sin ~x+n arctan(y/x), n=l; (c) grating modulated by angular phase change, q~(x, y)=k sin ~x+n arctan(y/x), n= 1.
r2/2)~f- nO/2n= m ,
(2)
lowing diffraction integral (in the scope of Fresnel approximation), c~
where m is an integer denominating the zone number. This equation can be rewritten in the form of n spiral curves known as Fermat spirals, or parabolic spirals [ 4 ],
r=a~O,
where a=x/afn/zc ,
(3)
rotated consecutively by 2nn rad. In fig. 1, together with the spiral on-axis zone plate for n = 1 (fig. la), the off-axis zone pattern is shown (fig. lb), as well as its carrier frequency alone, modulated by an angular linear change of the phase (fig. lc). The zone plate modified according to eq. ( 1 ) possesses for every sector its counterpart oppositely placed and with phase shifted by n radians. Hence, the destructive interference occurs in the focus, and the doughnut shaped focal pattern is formed. The spiral zone plate described by above equations was shown to produce phase singularities in the centre of the focal plane [ 5 ], in the sense of the definition given in ref. [ 6 ], i.e. that the phase integral around the closed path surrounding the point is equal to 2nn. It is proper to mention that the corresponding complex amplitude distribution for these singularities is described by the generalized Hankel trans-. form, according to the definition given by Papoulis [ 7 ]. Substituting the transmittance function of the spiral zone plate given by eq. ( 1 ), one can describe the complex amplitude in its focal plane by the fol392
U(p,O,J)= ~-f-iC P(r)
2n
rdrJ"
0
exp(in0)
0
X e x p [ - i ( k r p / f ) c o s ( 0 - G ) ] dO,
(4)
where C = exp [ik(f+p2/2f) ] . Using the identity 2rt
f exp(inO) e x p [ - i ( k r p / f ) cos(0-q~) ] dO 0
=2~z( - i ) " exp (inq~) Jn( krp /f) ,
(5)
we can represent the diffraction integral as the generalized Hankel transform of the nth order
U(p, ~,f) = (k/f) ( - i ) "+' exp(in0) x C t P(r)J,(krp/f) rdr.
(6)
0
Hereafter, the element performing the Hankel transform of the first order will be considered, since for this case the dip in the centre of the focal plane is the narrowest. If a circular aperture will be assumed, then the focal intensity distribution is given as the square o f the first order Hankel transform of the function circ(r/R), where R is the radius of the zone plate,
Volume 102, number 5,6
OPTICS COMMUNICATIONS
I(p,f)=Io(2 kR!/IJ'(t)t (TYfip dt'~2 ] '
(7)
where Io = (kR2/2f) 2 and t=krp/f Taking into account the relation J'o(X) = -Jl and the identity
(x),
0
.~ (x) ] ,
(8)
where . ~ and .~ are Struve functions of zero and first order [7], the distribution of the intensity in the focal plane can be written as follows,
I (p, f) = ( Io ~2 ) ×(Jl ( kRx/f) .#o(kRp/f) -Jo( kRp/f) .~ ( kRp/f) )2 kRp/f (9) By expanding J~ function under the integral of eq. (7) into series and performing explicit integration of the polynomials the same result can be obtained as the power series, more applicable for the numerical evaluation m=oo
I(p,f)=16Io ( ~ l
[_(kRp/f)z/4]mm
1/[.) 2
kl~p
(m!)Z(Zm+ 1)
" (lO)
By the numerical evaluation of the intensity distribution one can convince, that the first maximum lies atp=O.7802f/2R, and the half-width (HW is the value ofp for which l(p, f) =ImaJ2 ) equals 0.3832f/ 2R. The further reduction of the diameter of the central dark dip can be achieved by the introduction of an apodizing function onto the aperture of the element. In the present case the simples example will be checked, i.e. the substitution of the circular aperture by an annular one. An introduction of the masking ration G = 1 -AR/R, where AR= ( R - R i ) and R i is a diameter of masking disc allows to write the respective intensity distribution in the following form
. (F(kRp/f) - GF(kGnp/f) ,2 Z(p,j)
k
p/f
.) ,
where c=re 2, F(x) = [Jl (x) .~o(x) -Jo(x) .~ (x) ] for the representation given in eq. (9) and c = 16, m=~ [ 1 _ (x)2/4] m m F ( x ) = m=,E ( - ~ - ! ~ ~
i Jo(t) dt=xJo(x) + ½reX[Jl(x) .¢fo(X)-Jo(x)
15 October 1993
(ll)
for the representation given in eq. (10). At the end, the limiting case of G approaching 1 can be immediately derived from eq. (6) giving the J~ distribution of the focal pattern (similarly as the infinitesimal annular aperture of an ordinary lens gives a Jo distribution):
l(p,f) =Io( 2AR/ R )Z[J, ( kRp/f) ]2.
(12)
The positions of the first maximum and HW reach then their smallest values, respectively 0.586f/2R and
0.2912f/2R. The intensity distributions for G equal to 0, 0.5, and 0.9, calculated according to eqs. (10) and ( 11 ), are shown in figs. 2a, 2b, and 2c, respectively, where Io, defined in eq. (7), means the maximum intensity of the Airy pattern of the corresponding ordinary element. Figure 3a shows the decrease of the first maximum of the spiral zone plate with the growing masking ratio G. In turn, figs. 3b and 3c illustrate the descending values for the position of the first maximum and for the position of the HW of the spiral zone plate in the function of the growing masking ratio (lower curves), compared with respective relations for an ordinary element (upper curves), and showing clearly that the central dip becomes then narrower. The comparison of derived results with those obtained earlier for other elements producing blackfocal spots (i.e. for a zone plate with a phase jump around the circle dividing the aperture into two halves, for a zone plate with a phase jump along the diameter, and for zone plates with phase jumps dividing the aperture into quadrants) is given in table 1, for full and infinitesimal annular apertures. This comparison demonstrates distinctly the superiority of the spiral zone plate over other types of solutions presented earlier [ 1-3 ]. A zone plate with a phase jump around the circle exhibits not only significantly greater dimensions of the dip, but also its intensity decreases faster with the growing masking ratio, and in addition the further subsidiary maxima become then predominant over the first one; moreover the axial intensity dis393
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OPTICS COMMUNICATIONS
I
15 October 1993
0.20
0.020
0.3
0.15
0.015
0.2
0.10
~
0,010
0.1
ooo
0.0 0
1
2
3
4
-X 0
1
2
\)\/
oooo 3
4
0
1
2
.5
p2R/Xf
p2R/Zf
,o2R/Xf
a
b
c
4
Fig. 2. The intensity distribution across the focal spot of the spiral zone plate for: (a) circular aperture, masking ratio G= 0; (b) annular aperture, masking ratio G=0.5; (c) annular aperture, masking ratio G=0.9.
0.4
1.6
0.3
1.2
_~ o.2
\
_E
x
0,1
0.8
~
0.6
~.~
0.8
0.4
0.0 0,25
0.50
0.75
1.00
~ -
0.2
0.0 0.00
~ 0.4
0.00
0.25
~ 0.50
0.0 0.75
1.00
0.00
0.25
0.50
G
G
G
a
b
c
0.75
1.00
Fig. 3. The dependence in the function of the masking ratio G for (a) intensity of the first maximum; (b) position of the first maximum of the spiral zone plate versus the position of the first minimum of the ordinary zone plate; (c) position of HW of the spiral zone plate versus the position of the HW of the ordinary zone plate. tribution is not zero along the whole optical axis [ 1 ]. In turn, the spiral zone plate, together with the phase daisy zone plate has zero irradiance along the whole optical axis, what additionally predestines both types of elements for the axial aiming, although it seems, that for purely axial alignment the J~ Bessel beams. [9] as well as their counterparts among the logarithmic axicons are unrivaled a n d the zone plates with black spot are better suited for the sensing of the deviations of the point images in the image plate. 394
C o n t i n u i n g the comparison of the spiral zone plate with the phase daisy elements, not only the smaller diameter of the focal dip produced by the spiral zone plate should be noted (although the difference is not so distinct as in the previous case), but moreover the focal pattern is not divided by dark spokes, possessing thereby the symmetry of revolution - in consequence the signal of miscentering error will not depend on the direction of the deviation. In order to verify these results, a carrier frequency
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OPTICS COMMUNICATIONS
15 October 1993
Table 1 Positions of the first maxima and the HW for the spiral zone plate and for other types of the zone plates with black focal spot, compared with the ordinary zone plate in the cases of the full aperture (G=0) and of the infinitesimal annular aperture (g= 1 ); the unit is p2R/2f Position of~he first maximum
Position of the HW
G=0
G=I
G=0
G=I
Spiral zone plate for n= 1
0.780
0.586
0.383
0.291
Zone plate with phase a jump around the circle
1.136
0.765
0.727
0.497
Zone plate with phase a'bjump along the diameter
0.842
0.630
0.411
0.312
Zone plate with phase a'bjumps dividing the aperture into quadrants
0.873
0.682
0.554
0.439
Ordinary zone plate b
1.220
0.765
0.514
0.359
Data taken from ref. [ 1 ]. b Values for the most narrow cross sections, c Position of the first minimum. p a t t e r n with s u p e r p o s e d a n g u l a r phase change for n = 1 was d i s p l a y e d o n a C R T flat screen with 680 × 480 pixels a n d t h e n r e d u c e d p h o t o g r a p h i c a l l y in a K o d a k A H U 5460 film to a d i a m e t e r o f 6 m m (fig. l c ) . T h i s p a t t e r n was p r o g r a m m e d as a b i n a r y a m p l i t u d e display, i.e. its t r a n s m i t t a n c e was equal to
t(x,y)=O,
for for
-
1 ~ < c o s [ ~ ( x , y ) ] ~<0, 0~
,
(13)
where • (x, y ) = k sin
Fig. 4. Experimentally obtained focal spots for: (a) spiral zone plate, circular aperture; (b) ordinary zone plate, circular aperture.
ax+arctan(y/x),
a is the angle o f the carrier f r e q u e n c y ( a = 0 . 0 0 3 ) , a n d x, y are the c a r t e s i a n c o o r d i n a t e s o f the e l e m e n t plane. T h i s t r a n s m i t t a n c e , c o m b i n e d with a converging lens a n d i l l u m i n a t e d by a p l a n e wave ( 2 = 6 3 2 8 n m ) has p r o d u c e d in its focal p l a n e ( f = 9 5 0 n m ) the desired d o u g h n u t focal pattern. In fig. 4a there is s h o w n the i n t e n s i t y p a t t e r n o f the exp e r i m e n t a l l y o b t a i n e d e l e m e n t . In t u r n , in fig. 4b the Airy spot o f a c o r r e s p o n d i n g o r d i n a r y e l e m e n t with the s a m e a p e r t u r e a n d focal length is s h o w n for c o m parison. T h e focus o f the spiral zone plate was t h e n t a k e n with a C C D c a m e r a , m e a s u r e d along one o f the d i r e c t i o n s with the help o f a D T 2 8 5 1 D a t a T r a n s l a t i o n i m a g e processing b o a r d a n d c o m p a r e d with the theoretical curve. T h e result, p r e s e n t e d in fig. 5 shows that in spite of the m o d e s t m e a n s used for the e x p e r i m e n t , the c o i n c i d e n c e is q u i t e good.
1.00 '0~C0.10"75 l 50 0.25
0.00
0.0 0.5 1.0 1.5 2.0 p2R/Xf
Fig. 5. Experimentally measured intensity of the spiral zone plate focal spot presented in fig. 4a versus theoretical prediction presented in fig. 2a. 395
Volume 102, number 5,6
OPTICS COMMUNICATIONS
Much more difficult to achieve is the same minim u m intensity for all directions. In conclusion, it was shown, that by introduction of an angular phase change, elements capable of realizing generalized Hankel transforms are possible to produce, in the form of spiral zone plates. Moreover, the parameters corresponding to the element realizing the generalized Hankel transform of the first order predominate over other solutions proposed for obtaining a focal pattern with a dip in the c e n t e r thus, it can be useful for alignment purposes. This work was performed during the stay of Z. Jaroszewicz at the University Santiago de Compostela. The fellowship for his stay from the Spanish Ministry of Education and Science is gratefully acknowledged. At the same time A. Kolodziejczyk was incorporated at the University of Stuttgart as a fellow of a Alexander von H u m b o l d t Fellowship. The authors gratefully acknowledge apt and helpful remarks of the Referee.
396
15 October 1993
References [ 1]S. Bar~i Vifias, Z. Jaroszewicz, A. Ko|odziejczyk and M. Sypek, Appl. Optics 31 (1992) 192. [2 ] J. Ojeda-Castafieda,P. Andr6sand M. Martinez-Corral,Appl. Optics 31 (1992) 4600. [ 3 ] J. Ojeda-Castafiedaand G. Ramirez, Optics Lett. 18 ( 1983 ) 87. [4 ] E. Niczyporowicz,Planar curves (Polish ScientificPublishers, Warsaw, 1991)p. 437 (in Polish). [5] N.R. Heckenberg, R. McDuff, C.P. Smith and A.G. White, Optics Lett. 17 (1992) 221. [ 6 ] A.G. White, C.P. Smith, N.R. Heckenberg, H. RubinszteinDunlop, R. McDuff, C.O. Weiss and Chr. Tamm, J. Mod. Optics 38 (1991) 2531. [7]A. Papoulis, Systems and transforms with applications in optics (McGraw-Hill,New York, 1968)p. 163. [8]J. Spanier and K.B. Oldham, An atlas of functions (Hemisphere PublishingCorporation, Washington, 1987) p. 518. [9 ] A. Vasara, J. Tutunen and A.T. Friberg,J. Opt. Soc. Am. A 6 (1989) 1748.