- Email: [email protected]
Irregular spot array generator with trapezoidal apertures of varying heights
1 August 1999
Optics Communications 166 Ž1999. 1–7 www.elsevier.comrlocateroptcom
Irregular spot array generator with trapezoidal apertures of varying heights Jean-Numa Gillet 1, Yunlong Sheng
)
Center for Optics, Photonics and Laser, Department of Physics, UniÕersite´ LaÕal, Quebec, Canada G1K 7P4 ´ Received 5 April 1999; received in revised form 12 May 1999; accepted 13 May 1999
Abstract Design of diffractive elements with trapezoidal apertures of varying heights is proposed, which provides closer-to-arbitrary-shape apertures for generating irregular spot array. The number of apertures is much less than that required by the previous approach with trapezoidal apertures of fixed height. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 42.40.Eq; 42.40.Jv Keywords: Diffractive optical elements; Trapezoidal-aperture topology; Irregular-spot-array generator
1. Introduction Irregular light spot array generators are recently of interest in the semiconductor industry for laser machining in the ultra-compact semiconductor-device packages, such as ball grid arrays ŽBGA. and surface-mount multi-chip modules that allows interconnections between layers. One is interested in drilling micro-holes in arbitrary locations with diameters down to 25 mm or even less. In those applications, the laser drilling stands out as clearly the best tool over the mechanical drilling. The use of diffractive spot array generator can substantially reduce cost and speed up the laser drilling process. The diffractive element, which generates an arbitrary spot pattern, is usually a kinoform designed by ) 1
Corresponding author. E-mail: [email protected] E-mail: [email protected]
the iterative Fourier transform algorithm. The irregular spot array is a two-dimensional Ž2-D. array with arbitrary spots missing, which usually cannot be separated into two 1-D vectors, so that 2-D encoding must be performed. The element for irregular spot array generation would consist of arbitrary shape apertures. Since the primitive patterns of the e-beam plotting are polygons, the trapezoidal shapes are preferred. Turunen and co-workers w1,2x introduced the stripe geometry and trapezoidal-aperture topology for 2-D phase encoding, which uses a 2-D set of trapezoids to approximate arbitrary shape apertures. In the e-beam pattern generation, the platform is moved in x-direction and the e-beam scans in y-direction. Therefore, the trapezoids in one horizontal stripe must have the same height. However, the stripes can have different heights. We propose the trapezoidal-aperture topology with varying stripe heights. The new topology adapts better than that
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 2 3 9 - 4
2
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
with fixed stripe height w2x for fitting small sharp details in arbitrary shapes. It provides more degrees of freedom to the optimization algorithm, refines the vertical structure of aperture shapes and, therefore, allows a better fitting to the arbitrary shape apertures. We use a novel algorithm w6x of iterative simulated quenching with statistical temperature rescaling for the optimization, based on the adaptive simulated annealing w7x with the ensemble approach w8x. We obtain the diffractive elements with much less trapezoidal shapes to generate irregular spot arrays than that reported in Ref. w2x to generate the same size of regular spot arrays. That can reduce significantly fabrication cost. The trapezoidal topology can be used in the design for both binary phase w2x and multilevel phase w7x elements. In this paper, we present the design of binary phase elements. Exploiting the huge number of degrees of freedom provided by the high-resolution e-beam mask generator for micro-lithography, we want to achieve a high performance of the binary-phase element. The constraint with the binaryphase element is that the spot array must be centersymmetrical, the diffraction efficiency can be lower and the reconstruction error can be higher than that of the multiple-phase element. However, the fabrication of the multilevel-phase elements is more costly, and can suffer from additional errors in mask alignment.
in a closed form by surface-integration. The Fourier transform of the aperture is Um n s s
V
yu d
4
5
=exp Ž yi2p ny . d y iD y s 2p m
exp yip n Ž yu q yd .
= exp yip m Ž b q d . =sinc m Ž d y b . q nD y yexp yip m Ž a q c . =sinc m Ž c y a . q nD y
Ž 1.
4
where the integers Ž m,n. denote diffraction orders. The above expression is valid only for m / 0. When m s 0 and n / 0, the diffracted orders U0 n cannot be computed using Eq. Ž1. with m tends to zero w3x. We re-compute the surface integration Ž1. as U0 n s s
HH
exp Ž yi2p ny . d xd y
V
Hy
yu
x 4 Ž y . y x 2 Ž y . exp Ž yi2p my . d y
d
i 2p n
exp yip n Ž yu q yd .
Ž dyc.
=exp Ž yip nD y . y Ž b y a . exp Ž ip nD y .
2. Diffraction of trapezoidal apertures
Consider a single binary-amplitude trapezoidal aperture V on an opaque screen, schematized in Fig. 1. The diffraction of the aperture may be computed
x2Ž y.
½
Hy Hx Ž y . exp Ž yi2p mx . d x
s
2.1. Binary-amplitude aperture
exp yi2p Ž mx q ny . d xd y
HH
y Ž d y b y c q a . sinc Ž nD y . 4
Ž 2.
We double checked Eq. Ž2. using the Green’s contour integration and obtain the same result as Eq. Ž2. Žsee Appendix A.. Finally, it is well known that the central diffraction term U00 is equal to the trapezoid area: U00 s
HH
d xd y s V s
V
Dy 2
Ž d y c q b y a.
Ž 3.
2.2. Binary-phase grating The transmittance of a single binary-phase aperture numbered by a may be expressed as Fig. 1. Binary-amplitude trapezoidal-shape aperture.
ta s Ž exp f y 1 . Aa q 1
Ž 4.
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
3
where f is the phase shift inside the aperture and Aa is the amplitude aperture function as Aa s
½
1 0
inside the aperture outside the aperture
Ž 5.
The diffraction of the ensemble of apertures for a s 1, 2, . . . in one period of the binary-phase grating is computed with the Fourier transform as Tm n s Ž exp f y 1 . Ý a
=d xd y q
1
HH
Va
exp yi2p Ž mx q ny .
1
H0 H0 exp yi2p Ž mx q ny .
s Ž exp f y 1 . Ý Uman q dm0 dn0
d xd y
Ž 6.
a
where d is the Kronecker’s delta symbol. We divide one period of the grating into R stripes of varying heights with the boundaries of the stripes indexed as r s 0, 1, 2, . . . , R, and define J q 1 transition points with j s 0, 1, 2, . . . , J on each boundary, where J is an odd integer. The points j s 0 and j s J are fixed on the vertical borders of the period. In each stripe, two transition points indexed with the same odd number j, on the upper and bottom boundaries and their next neighbor points indexed with j q 1 constitute four vertexes of a trapezoidal aperture. There are then Ž J y 1.r2 trapezoidal shape apertures in one stripe, as shown in Fig. 2. The total number of trapezoidal apertures is RŽ J y 1.r2. The complex-valued amplitude of the diffracted orders Tm n for the binary-phase grating of trapezoidal apertures, is obtained by substituting Eqs. Ž1. – Ž3. for Uman of one trapezoidal amplitude aperture a into Eq. Ž6.:
Ý Ý rs0
T00 , a real unitary term must be added, due to Kronecker’s symbols in the right-hand side of Eq. Ž7.: Ry1 Jy2
T00 s exp Ž i f . y 1
Umr nj q dm0 dn0
Ž 7.
js1 j odd
where Umr nj is the complex-valued amplitude of the diffracted order Ž m,n. of the trapezoidal amplitude aperture dug in an opaque mask; r is the number of the stripe and j s 1, 3, . . . , J y 2 is the number of the trapezoidal aperture. The Umr nj is calculated using Eq. Ž1. when m / 0, and Eq. Ž2. when m s 0 and n / 0. For the central diffraction-order amplitude
Ý Ý rs0
U00r j q 1
Ž 8.
js1 j odd
where
U00r j
is given by Eq. Ž3..
2.3. Degrees of freedom Each trapezoidal aperture has four vertexes, and all the R stripes have varying heights with a constraint that total height is given by the grating period. The number of degrees of freedom D in this configuration is then: Ds4
Ry1 Jy2
Tm n s exp Ž i f . y 1
Fig. 2. Grating period with trapezoidal apertures Rs11 and J s11.
Ž J y 1. 2
R q R y 1 s Ž 2 J y 1. R y 1
Ž 9.
Assume that the pattern to be generated is a square array of N = N spots in which there are M missing spots as < Tk l < s 0 for M missing spots at Ž k ,l . < Tu Õ < s < Tu
Ž 10 .
2
0 Õ0
< for Ž N y M y 1 .
illuminated spots at Ž u,Õ .
Ž 11 .
where Ž u 0 ,Õ 0 . is one of the illuminated diffracted order. We need to solve these N 2 y 1 non-linear
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
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equations for the positions of the Ž2 J y 1. R y 1 vertexes with as high as possible diffraction efficiency. Since the pattern generated by a binary-phase grating is center-symmetric, we need to design the grating for only half of the spot pattern with Ž N 2 y 1.r2 non-linear equations when N is odd. If all the diffracted orders Tm n would be real-valued and all the equations would be linear, then the minimum number of degrees of freedom should be equal to or larger than the number of equations, that is if J s R: JsRG
N 2
Ž 12 .
However, the diffracted orders Tm n of the trapezoidal apertures given in Eqs. Ž1., Ž2. and Ž7. are complexvalued. This is different from the regular Dammann grating w4,5x with rectangular apertures where the diffracted orders are real-valued. In our case each illuminated spot may have arbitrary phase and the number of combinations of Eq. Ž7. and of solutions can be infinitely large. Also, Eqs. Ž1., Ž2. and Ž7. are highly non-linear, the minimum number of degrees of freedom must be higher than that given by Eq. Ž12.. Our experience showed that a good starting choice for parameters R and J is R G N q 4 and J G N q 4
Ž 13 .
We start with such a high number of R and J. During the optimization process some stripes and trapezoids can be eliminated or fused in the optimized results.
3. Optimization We use a novel iterative simulated quenching with statistical temperature re-scaling algorithm w6x for the optimization in the design. The method is based on the adaptive simulated annealing w7x and the ensemble approach w8x. A heuristic demonstration in the adaptive simulated annealing w7x shows that a global minimum statistically can be achieved with a given density of probability in the D-dimensional configuration space and an exponential cooling schedule as T Ž k . s T0 exp Ž yck Q r D .
Ž 14 .
where k is time step, c is a constant, and the quenching factor Q s 1. However, because of the huge number of degrees of freedom D in the hologram design, this annealing schedule is impossible to implement in practice with Q s 1. We propose a cooling schedule Ž14. with Q s D 4 1 that leads to a quenching instead of annealing w6x. At the end of a quenching, the system is out of equilibrium and is frozen. Then, we consider the actual frozen state as one of the probable states of a Boltzmann equilibrium system at a temperature Tr with the mean energy ² E :r and variance of energy sr 2 , so that the heat capacity w6,8x can be defined as
Cs
dE s dT
Ee y ² E :r Te y Tr
s
sr 2 Tr2
Ž 15 .
where ² E :r and sr 2 are calculated with the ensemble average using the energies E recorded in a number of last Metropolis steps, under hypothesis that the random walk in the configuration space is ergodic. Then, we re-start a new quenching with the schedule Ž14. at the re-scaling temperature T0 s Tr . After iterating a small number of simulated quenching with temperature re-scaling, we obtained the final energy, much lower than that achieved in a conventional simulated annealing with a fast exponential cooling schedule. Locations of the trapezoid corners and heights of stripes were initially set randomly. A first stripe was chosen randomly. Its top boundary was moved randomly in vertical direction that implies the alteration of top boundary for all the trapezoids in this stripe. Then a second move of the stripe top boundary was performed. After the two vertical moves, the trapezoids in that stripe were altered one-by-one in a random order. For each trapezoid the four vertex abscissas were moved simultaneously five times. After each move, the cost function was evaluated. The move was accepted with a probability according to the Gibbs–Boltzmann distribution as that in the conventional simulated annealing process. The process is repeated for a new stripe chosen randomly among the remaining non-visited stripes. After all stripes within one grating period were altered, the temperature is decrease using the exponential cooling schedule mentioned in Eq. Ž14.. All the above
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
5
process is then repeated for the new temperature. The incremental moves of the trapezoid corners decrease with the temperature w7x. The number of iterations for a single simulated quenching loops was of five hundreds. The number of iterations of the quenching loop was between ten for 9 = 9 irregular spot array and twenty for 17 = 17 irregular spot array. A fusion process was performed after the optimization. The trapezoidal apertures with both top and bottom widths below a threshold length l, with l s 1r256 of the grating period, and the stripes of heights below l were eliminated. The neighbor trapezoidal apertures in the same stripe with both top and bottom interstices below l were fused together. The elimination of the structures of the size smaller than l, did not alter the grating performance seriously. 4. Design results The design of binary-phase elements for generating a 9 = 9 spot array with 3 = 3 central orders missing, a 15 = 15 array and a 17 = 17 array with 7 = 7 central orders missing is given in Table 1, where the data for the diffraction efficiency, reconstruction error and zero-order suppression are from numerical simulation. One designed grating period is shown in Fig. 3a; its diffraction pattern is shown in Fig. 3b. The grating period contained initially 21 stripes of 21 trapezoids. The total number of trapezoidal apertures was initially 840. The final total number of trapezoidal apertures remaining is only of 115. That is a
Table 1 Designs with varying-height trapezoidal-aperture binary-phase elements
Fig. 3. Ža. Designed hologram plotted as a 256=256-pixel array; Žb. 17=17-spot array with 7=7 orders missing obtained from the hologram shown in Ža..
Irregular arrays with varying-height apertures Dimensions
SJ
R
h Ž%.
´ Ž%.
< T00 < 2
9=9 Ž3=3 missing. 15=15 Ž7=7 missing. 17=17 Ž7=7 missing.
38 98 115
13 18 20
72.7 70.1 71.1
2.4 2.2 3.2
2.0=10 -5 3.1=10 -5 5.0=10 -6
S J s total number of trapezoids; Rs number of stripes; h s diffraction efficiency; ´ s reconstruction error; < T00 < 2 s zero-order intensity.
reduction by a factor more than 7. The diffraction efficiency h for the 17 = 17-spot pattern reaches 71.1%. This value is close to its upper bound of diffraction efficiency calculated in Ref. w9x as Ž0.848. 2 s 71.9% for a 17 = 17 regular-and-separable binary phase Ž0, p . 2-D spot array. The reconstruction error ´ inside the 17 = 17-spot pattern is
6
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
3.2%, which could be acceptable for the application. The zero-order intensity was only of 5.0 = 10y6 of the incident power for the 17 = 17 array. We discovered that the number of stripes and total number of trapezoids in one period is about 3–4 times less than that reported in Ref. w2x, which used the fixed-height trapezoidal-aperture topology. This is probably because in the new topology the trapezoids with variable heights fit more efficiently the arbitrary shapes with sharp narrow details. In addition, Ref. w2x used a two steps optimization: first a simulated annealing until pattern non-uniformity and diffraction efficiency reach some predefined target values, then a downhill algorithm to improve pattern uniformity. The trapezoid-corner mobility is reduced in this optimization. Only few merging of trapezoids would occur. The design in Ref. w2x is for regular spot arrays. Our design is for large irregular arrays, where the extinction of a large number of diffracted orders imposes heavy constraints on the optimization. Furthermore, the extinction of zero order implies that the amount of area of zero-phase is equal to that of p-phase as shown by Eqs. Ž3., Ž8. and Ž10., which must be achieved in the optimization.
Appendix A The Green’s theorem for two differentiable functions P Ž x, y . and QŽ x, y . is formulated as
E QŽ x , y.
HH
Ex
V
s
y
E
E P Ž x, y. Ey
d xd y
Pd x q Qd y
Ž A1.
C
If QŽ x, y . is independent of coordinate x, we obtain:
HH
y
E P Ž x, y. Ey
V
d xd y s EC Pd x
Ž A2.
Let y
E P Ž x, y.
s exp yi2p Ž mx q ny .
Ey
Ž A3.
in Eq. ŽA2. then P Ž x, y. s
1
exp yi2p Ž mx q ny . q g Ž x .
i2p n
Ž A4. where g Ž x . is independent of coordinate y EC g Ž x . d x ' 0
5. Conclusion We have designed binary-phase elements with trapezoidal apertures of varying heights for generating irregular spot arrays. The new design requires 3–4 times less trapezoids and stripes than that needed by the previous fixed-height trapezoid topology, because the trapezoids with varying heights fit well the arbitrary shape apertures. That can reduce significantly fabrication cost. We have shown the optimization results for odd-number spot arrays with central orders missing and with high diffraction efficiencies and small reconstruction errors.
and we can write the Fourier transform Um n in Eq. Ž1. as a contour integration along horizontal differential d x: Um n s
1 i2p n
½H
b
a
exp yi2p Ž mx q nyd . d x
d
Hb exp yi2p Ž mx q ny Ž x . .
q
2
dx
c
Hd exp yi2p Ž mx q ny .
q
u
dx
a
Hc exp yi2p Ž mx q ny Ž x . .
q
4
i s Acknowledgements
Ž A5.
2p n
y Ž d y b . exp
dx
5
yip n Ž yu q yd .
=exp yip m Ž b q d . sinc m Ž d y b . We acknowledge Mr. Peter Kung and Claude Beaulieu from QPS Technology for their assistance in e-beam mask coding.
qnD y q Ž c y a . exp yip n Ž yu q yd . =exp yip m Ž a q c . sinc m Ž c y a .
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
qnD y q Ž d y c . exp Ž yi2p nyu .
References
=exp yip m Ž c q d . sinc m Ž d y c .
w1x J. Turunen, A. Vasara, J. Westerholm, Opt. Commun. 74 Ž1989. 245–252. w2x A. Vasara, M.R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J.M. Miller, T. Jaakkola, S. Kuisma, Appl. Opt. 31 Ž1992. 3320–3336. w3x P. Blair, H. Lupken, M.R. Taghizadeh, F. Wyrowski, Appl. ¨ Opt. 36 Ž1997. 4713–4721. w4x H. Dammann, K. Gortler, Opt. Commun. 3 Ž1971. 312–315. ¨ w5x H. Dammann, E. Klotz, Opt. Acta 24 Ž1971. 505–515. w6x J.-N. Gillet, Y. Sheng, ICO XVIII, San Francisco, CA, August, 1999, in press. w7x L. Ingber, J. Math. Comput. Modelling 18 Ž1993. 29–57. w8x G. Ruppeiner, J.M. Pedersen, P. Salamon, J. Phys. I 1 Ž1991. 455–470. w9x U. Krackhardt, J.N. Mait, N. Streibl, Appl. Opt. 31 Ž1992. 27–37.
y Ž b y a . exp Ž yi2p nyd . exp yip m = Ž a q b . sinc m Ž b y a .
4
Ž A6.
where y 2 Ž x ., y4 Ž x ., yu and yd are defined in Fig. 1. When integers m / 0 and n / 0, Eq. ŽA6. is identical to Eq. Ž1.. Now, if we substitute m by 0 into Eq. ŽA6., we find back above Eq. Ž2. for U0 n : U0 n s
7
i
exp yip n Ž yu q yd . Ž d y c . 2p n =exp Ž yip nD y . y Ž b y a . exp Ž ip nD y . y Ž d y b y c q a . sinc Ž nD y . 4
Ž A7.
Optics Communications 166 Ž1999. 1–7 www.elsevier.comrlocateroptcom
Irregular spot array generator with trapezoidal apertures of varying heights Jean-Numa Gillet 1, Yunlong Sheng
)
Center for Optics, Photonics and Laser, Department of Physics, UniÕersite´ LaÕal, Quebec, Canada G1K 7P4 ´ Received 5 April 1999; received in revised form 12 May 1999; accepted 13 May 1999
Abstract Design of diffractive elements with trapezoidal apertures of varying heights is proposed, which provides closer-to-arbitrary-shape apertures for generating irregular spot array. The number of apertures is much less than that required by the previous approach with trapezoidal apertures of fixed height. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 42.40.Eq; 42.40.Jv Keywords: Diffractive optical elements; Trapezoidal-aperture topology; Irregular-spot-array generator
1. Introduction Irregular light spot array generators are recently of interest in the semiconductor industry for laser machining in the ultra-compact semiconductor-device packages, such as ball grid arrays ŽBGA. and surface-mount multi-chip modules that allows interconnections between layers. One is interested in drilling micro-holes in arbitrary locations with diameters down to 25 mm or even less. In those applications, the laser drilling stands out as clearly the best tool over the mechanical drilling. The use of diffractive spot array generator can substantially reduce cost and speed up the laser drilling process. The diffractive element, which generates an arbitrary spot pattern, is usually a kinoform designed by ) 1
Corresponding author. E-mail: [email protected] E-mail: [email protected]
the iterative Fourier transform algorithm. The irregular spot array is a two-dimensional Ž2-D. array with arbitrary spots missing, which usually cannot be separated into two 1-D vectors, so that 2-D encoding must be performed. The element for irregular spot array generation would consist of arbitrary shape apertures. Since the primitive patterns of the e-beam plotting are polygons, the trapezoidal shapes are preferred. Turunen and co-workers w1,2x introduced the stripe geometry and trapezoidal-aperture topology for 2-D phase encoding, which uses a 2-D set of trapezoids to approximate arbitrary shape apertures. In the e-beam pattern generation, the platform is moved in x-direction and the e-beam scans in y-direction. Therefore, the trapezoids in one horizontal stripe must have the same height. However, the stripes can have different heights. We propose the trapezoidal-aperture topology with varying stripe heights. The new topology adapts better than that
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 2 3 9 - 4
2
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
with fixed stripe height w2x for fitting small sharp details in arbitrary shapes. It provides more degrees of freedom to the optimization algorithm, refines the vertical structure of aperture shapes and, therefore, allows a better fitting to the arbitrary shape apertures. We use a novel algorithm w6x of iterative simulated quenching with statistical temperature rescaling for the optimization, based on the adaptive simulated annealing w7x with the ensemble approach w8x. We obtain the diffractive elements with much less trapezoidal shapes to generate irregular spot arrays than that reported in Ref. w2x to generate the same size of regular spot arrays. That can reduce significantly fabrication cost. The trapezoidal topology can be used in the design for both binary phase w2x and multilevel phase w7x elements. In this paper, we present the design of binary phase elements. Exploiting the huge number of degrees of freedom provided by the high-resolution e-beam mask generator for micro-lithography, we want to achieve a high performance of the binary-phase element. The constraint with the binaryphase element is that the spot array must be centersymmetrical, the diffraction efficiency can be lower and the reconstruction error can be higher than that of the multiple-phase element. However, the fabrication of the multilevel-phase elements is more costly, and can suffer from additional errors in mask alignment.
in a closed form by surface-integration. The Fourier transform of the aperture is Um n s s
V
yu d
4
5
=exp Ž yi2p ny . d y iD y s 2p m
exp yip n Ž yu q yd .
= exp yip m Ž b q d . =sinc m Ž d y b . q nD y yexp yip m Ž a q c . =sinc m Ž c y a . q nD y
Ž 1.
4
where the integers Ž m,n. denote diffraction orders. The above expression is valid only for m / 0. When m s 0 and n / 0, the diffracted orders U0 n cannot be computed using Eq. Ž1. with m tends to zero w3x. We re-compute the surface integration Ž1. as U0 n s s
HH
exp Ž yi2p ny . d xd y
V
Hy
yu
x 4 Ž y . y x 2 Ž y . exp Ž yi2p my . d y
d
i 2p n
exp yip n Ž yu q yd .
Ž dyc.
=exp Ž yip nD y . y Ž b y a . exp Ž ip nD y .
2. Diffraction of trapezoidal apertures
Consider a single binary-amplitude trapezoidal aperture V on an opaque screen, schematized in Fig. 1. The diffraction of the aperture may be computed
x2Ž y.
½
Hy Hx Ž y . exp Ž yi2p mx . d x
s
2.1. Binary-amplitude aperture
exp yi2p Ž mx q ny . d xd y
HH
y Ž d y b y c q a . sinc Ž nD y . 4
Ž 2.
We double checked Eq. Ž2. using the Green’s contour integration and obtain the same result as Eq. Ž2. Žsee Appendix A.. Finally, it is well known that the central diffraction term U00 is equal to the trapezoid area: U00 s
HH
d xd y s V s
V
Dy 2
Ž d y c q b y a.
Ž 3.
2.2. Binary-phase grating The transmittance of a single binary-phase aperture numbered by a may be expressed as Fig. 1. Binary-amplitude trapezoidal-shape aperture.
ta s Ž exp f y 1 . Aa q 1
Ž 4.
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
3
where f is the phase shift inside the aperture and Aa is the amplitude aperture function as Aa s
½
1 0
inside the aperture outside the aperture
Ž 5.
The diffraction of the ensemble of apertures for a s 1, 2, . . . in one period of the binary-phase grating is computed with the Fourier transform as Tm n s Ž exp f y 1 . Ý a
=d xd y q
1
HH
Va
exp yi2p Ž mx q ny .
1
H0 H0 exp yi2p Ž mx q ny .
s Ž exp f y 1 . Ý Uman q dm0 dn0
d xd y
Ž 6.
a
where d is the Kronecker’s delta symbol. We divide one period of the grating into R stripes of varying heights with the boundaries of the stripes indexed as r s 0, 1, 2, . . . , R, and define J q 1 transition points with j s 0, 1, 2, . . . , J on each boundary, where J is an odd integer. The points j s 0 and j s J are fixed on the vertical borders of the period. In each stripe, two transition points indexed with the same odd number j, on the upper and bottom boundaries and their next neighbor points indexed with j q 1 constitute four vertexes of a trapezoidal aperture. There are then Ž J y 1.r2 trapezoidal shape apertures in one stripe, as shown in Fig. 2. The total number of trapezoidal apertures is RŽ J y 1.r2. The complex-valued amplitude of the diffracted orders Tm n for the binary-phase grating of trapezoidal apertures, is obtained by substituting Eqs. Ž1. – Ž3. for Uman of one trapezoidal amplitude aperture a into Eq. Ž6.:
Ý Ý rs0
T00 , a real unitary term must be added, due to Kronecker’s symbols in the right-hand side of Eq. Ž7.: Ry1 Jy2
T00 s exp Ž i f . y 1
Umr nj q dm0 dn0
Ž 7.
js1 j odd
where Umr nj is the complex-valued amplitude of the diffracted order Ž m,n. of the trapezoidal amplitude aperture dug in an opaque mask; r is the number of the stripe and j s 1, 3, . . . , J y 2 is the number of the trapezoidal aperture. The Umr nj is calculated using Eq. Ž1. when m / 0, and Eq. Ž2. when m s 0 and n / 0. For the central diffraction-order amplitude
Ý Ý rs0
U00r j q 1
Ž 8.
js1 j odd
where
U00r j
is given by Eq. Ž3..
2.3. Degrees of freedom Each trapezoidal aperture has four vertexes, and all the R stripes have varying heights with a constraint that total height is given by the grating period. The number of degrees of freedom D in this configuration is then: Ds4
Ry1 Jy2
Tm n s exp Ž i f . y 1
Fig. 2. Grating period with trapezoidal apertures Rs11 and J s11.
Ž J y 1. 2
R q R y 1 s Ž 2 J y 1. R y 1
Ž 9.
Assume that the pattern to be generated is a square array of N = N spots in which there are M missing spots as < Tk l < s 0 for M missing spots at Ž k ,l . < Tu Õ < s < Tu
Ž 10 .
2
0 Õ0
< for Ž N y M y 1 .
illuminated spots at Ž u,Õ .
Ž 11 .
where Ž u 0 ,Õ 0 . is one of the illuminated diffracted order. We need to solve these N 2 y 1 non-linear
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
4
equations for the positions of the Ž2 J y 1. R y 1 vertexes with as high as possible diffraction efficiency. Since the pattern generated by a binary-phase grating is center-symmetric, we need to design the grating for only half of the spot pattern with Ž N 2 y 1.r2 non-linear equations when N is odd. If all the diffracted orders Tm n would be real-valued and all the equations would be linear, then the minimum number of degrees of freedom should be equal to or larger than the number of equations, that is if J s R: JsRG
N 2
Ž 12 .
However, the diffracted orders Tm n of the trapezoidal apertures given in Eqs. Ž1., Ž2. and Ž7. are complexvalued. This is different from the regular Dammann grating w4,5x with rectangular apertures where the diffracted orders are real-valued. In our case each illuminated spot may have arbitrary phase and the number of combinations of Eq. Ž7. and of solutions can be infinitely large. Also, Eqs. Ž1., Ž2. and Ž7. are highly non-linear, the minimum number of degrees of freedom must be higher than that given by Eq. Ž12.. Our experience showed that a good starting choice for parameters R and J is R G N q 4 and J G N q 4
Ž 13 .
We start with such a high number of R and J. During the optimization process some stripes and trapezoids can be eliminated or fused in the optimized results.
3. Optimization We use a novel iterative simulated quenching with statistical temperature re-scaling algorithm w6x for the optimization in the design. The method is based on the adaptive simulated annealing w7x and the ensemble approach w8x. A heuristic demonstration in the adaptive simulated annealing w7x shows that a global minimum statistically can be achieved with a given density of probability in the D-dimensional configuration space and an exponential cooling schedule as T Ž k . s T0 exp Ž yck Q r D .
Ž 14 .
where k is time step, c is a constant, and the quenching factor Q s 1. However, because of the huge number of degrees of freedom D in the hologram design, this annealing schedule is impossible to implement in practice with Q s 1. We propose a cooling schedule Ž14. with Q s D 4 1 that leads to a quenching instead of annealing w6x. At the end of a quenching, the system is out of equilibrium and is frozen. Then, we consider the actual frozen state as one of the probable states of a Boltzmann equilibrium system at a temperature Tr with the mean energy ² E :r and variance of energy sr 2 , so that the heat capacity w6,8x can be defined as
Cs
dE s dT
Ee y ² E :r Te y Tr
s
sr 2 Tr2
Ž 15 .
where ² E :r and sr 2 are calculated with the ensemble average using the energies E recorded in a number of last Metropolis steps, under hypothesis that the random walk in the configuration space is ergodic. Then, we re-start a new quenching with the schedule Ž14. at the re-scaling temperature T0 s Tr . After iterating a small number of simulated quenching with temperature re-scaling, we obtained the final energy, much lower than that achieved in a conventional simulated annealing with a fast exponential cooling schedule. Locations of the trapezoid corners and heights of stripes were initially set randomly. A first stripe was chosen randomly. Its top boundary was moved randomly in vertical direction that implies the alteration of top boundary for all the trapezoids in this stripe. Then a second move of the stripe top boundary was performed. After the two vertical moves, the trapezoids in that stripe were altered one-by-one in a random order. For each trapezoid the four vertex abscissas were moved simultaneously five times. After each move, the cost function was evaluated. The move was accepted with a probability according to the Gibbs–Boltzmann distribution as that in the conventional simulated annealing process. The process is repeated for a new stripe chosen randomly among the remaining non-visited stripes. After all stripes within one grating period were altered, the temperature is decrease using the exponential cooling schedule mentioned in Eq. Ž14.. All the above
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
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process is then repeated for the new temperature. The incremental moves of the trapezoid corners decrease with the temperature w7x. The number of iterations for a single simulated quenching loops was of five hundreds. The number of iterations of the quenching loop was between ten for 9 = 9 irregular spot array and twenty for 17 = 17 irregular spot array. A fusion process was performed after the optimization. The trapezoidal apertures with both top and bottom widths below a threshold length l, with l s 1r256 of the grating period, and the stripes of heights below l were eliminated. The neighbor trapezoidal apertures in the same stripe with both top and bottom interstices below l were fused together. The elimination of the structures of the size smaller than l, did not alter the grating performance seriously. 4. Design results The design of binary-phase elements for generating a 9 = 9 spot array with 3 = 3 central orders missing, a 15 = 15 array and a 17 = 17 array with 7 = 7 central orders missing is given in Table 1, where the data for the diffraction efficiency, reconstruction error and zero-order suppression are from numerical simulation. One designed grating period is shown in Fig. 3a; its diffraction pattern is shown in Fig. 3b. The grating period contained initially 21 stripes of 21 trapezoids. The total number of trapezoidal apertures was initially 840. The final total number of trapezoidal apertures remaining is only of 115. That is a
Table 1 Designs with varying-height trapezoidal-aperture binary-phase elements
Fig. 3. Ža. Designed hologram plotted as a 256=256-pixel array; Žb. 17=17-spot array with 7=7 orders missing obtained from the hologram shown in Ža..
Irregular arrays with varying-height apertures Dimensions
SJ
R
h Ž%.
´ Ž%.
< T00 < 2
9=9 Ž3=3 missing. 15=15 Ž7=7 missing. 17=17 Ž7=7 missing.
38 98 115
13 18 20
72.7 70.1 71.1
2.4 2.2 3.2
2.0=10 -5 3.1=10 -5 5.0=10 -6
S J s total number of trapezoids; Rs number of stripes; h s diffraction efficiency; ´ s reconstruction error; < T00 < 2 s zero-order intensity.
reduction by a factor more than 7. The diffraction efficiency h for the 17 = 17-spot pattern reaches 71.1%. This value is close to its upper bound of diffraction efficiency calculated in Ref. w9x as Ž0.848. 2 s 71.9% for a 17 = 17 regular-and-separable binary phase Ž0, p . 2-D spot array. The reconstruction error ´ inside the 17 = 17-spot pattern is
6
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
3.2%, which could be acceptable for the application. The zero-order intensity was only of 5.0 = 10y6 of the incident power for the 17 = 17 array. We discovered that the number of stripes and total number of trapezoids in one period is about 3–4 times less than that reported in Ref. w2x, which used the fixed-height trapezoidal-aperture topology. This is probably because in the new topology the trapezoids with variable heights fit more efficiently the arbitrary shapes with sharp narrow details. In addition, Ref. w2x used a two steps optimization: first a simulated annealing until pattern non-uniformity and diffraction efficiency reach some predefined target values, then a downhill algorithm to improve pattern uniformity. The trapezoid-corner mobility is reduced in this optimization. Only few merging of trapezoids would occur. The design in Ref. w2x is for regular spot arrays. Our design is for large irregular arrays, where the extinction of a large number of diffracted orders imposes heavy constraints on the optimization. Furthermore, the extinction of zero order implies that the amount of area of zero-phase is equal to that of p-phase as shown by Eqs. Ž3., Ž8. and Ž10., which must be achieved in the optimization.
Appendix A The Green’s theorem for two differentiable functions P Ž x, y . and QŽ x, y . is formulated as
E QŽ x , y.
HH
Ex
V
s
y
E
E P Ž x, y. Ey
d xd y
Pd x q Qd y
Ž A1.
C
If QŽ x, y . is independent of coordinate x, we obtain:
HH
y
E P Ž x, y. Ey
V
d xd y s EC Pd x
Ž A2.
Let y
E P Ž x, y.
s exp yi2p Ž mx q ny .
Ey
Ž A3.
in Eq. ŽA2. then P Ž x, y. s
1
exp yi2p Ž mx q ny . q g Ž x .
i2p n
Ž A4. where g Ž x . is independent of coordinate y EC g Ž x . d x ' 0
5. Conclusion We have designed binary-phase elements with trapezoidal apertures of varying heights for generating irregular spot arrays. The new design requires 3–4 times less trapezoids and stripes than that needed by the previous fixed-height trapezoid topology, because the trapezoids with varying heights fit well the arbitrary shape apertures. That can reduce significantly fabrication cost. We have shown the optimization results for odd-number spot arrays with central orders missing and with high diffraction efficiencies and small reconstruction errors.
and we can write the Fourier transform Um n in Eq. Ž1. as a contour integration along horizontal differential d x: Um n s
1 i2p n
½H
b
a
exp yi2p Ž mx q nyd . d x
d
Hb exp yi2p Ž mx q ny Ž x . .
q
2
dx
c
Hd exp yi2p Ž mx q ny .
q
u
dx
a
Hc exp yi2p Ž mx q ny Ž x . .
q
4
i s Acknowledgements
Ž A5.
2p n
y Ž d y b . exp
dx
5
yip n Ž yu q yd .
=exp yip m Ž b q d . sinc m Ž d y b . We acknowledge Mr. Peter Kung and Claude Beaulieu from QPS Technology for their assistance in e-beam mask coding.
qnD y q Ž c y a . exp yip n Ž yu q yd . =exp yip m Ž a q c . sinc m Ž c y a .
J.-N. Gillet, Y. Sheng r Optics Communications 166 (1999) 1–7
qnD y q Ž d y c . exp Ž yi2p nyu .
References
=exp yip m Ž c q d . sinc m Ž d y c .
w1x J. Turunen, A. Vasara, J. Westerholm, Opt. Commun. 74 Ž1989. 245–252. w2x A. Vasara, M.R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J.M. Miller, T. Jaakkola, S. Kuisma, Appl. Opt. 31 Ž1992. 3320–3336. w3x P. Blair, H. Lupken, M.R. Taghizadeh, F. Wyrowski, Appl. ¨ Opt. 36 Ž1997. 4713–4721. w4x H. Dammann, K. Gortler, Opt. Commun. 3 Ž1971. 312–315. ¨ w5x H. Dammann, E. Klotz, Opt. Acta 24 Ž1971. 505–515. w6x J.-N. Gillet, Y. Sheng, ICO XVIII, San Francisco, CA, August, 1999, in press. w7x L. Ingber, J. Math. Comput. Modelling 18 Ž1993. 29–57. w8x G. Ruppeiner, J.M. Pedersen, P. Salamon, J. Phys. I 1 Ž1991. 455–470. w9x U. Krackhardt, J.N. Mait, N. Streibl, Appl. Opt. 31 Ž1992. 27–37.
y Ž b y a . exp Ž yi2p nyd . exp yip m = Ž a q b . sinc m Ž b y a .
4
Ž A6.
where y 2 Ž x ., y4 Ž x ., yu and yd are defined in Fig. 1. When integers m / 0 and n / 0, Eq. ŽA6. is identical to Eq. Ž1.. Now, if we substitute m by 0 into Eq. ŽA6., we find back above Eq. Ž2. for U0 n : U0 n s
7
i
exp yip n Ž yu q yd . Ž d y c . 2p n =exp Ž yip nD y . y Ž b y a . exp Ž ip nD y . y Ž d y b y c q a . sinc Ž nD y . 4
Ž A7.