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Near-field hexagonal array illumination based on fractional Talbot effect
ARTICLE IN PRESS
Optik
Optics
Optik 118 (2007) 330–334 www.elsevier.de/ijleo
Near-field hexagonal array illumination based on fractional Talbot effect Weijuan Qu, Liren Liu, De’an Liu, Ya’nan Zhi, Wei Lu Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, PR China Received 24 January 2006; accepted 3 April 2006
Abstract Hexagonal array is a basic structure widely exists in nature and adopted by optoelectronic device. A phase plate based on the fractional Talbot effect that converts a single expanded laser beam into a regular hexagonal array of uniformly illuminated apertures with virtually 100% efficiency is presented. The uniform hexagonal array illumination with a fill factor of 1/12 is demonstrated by the computer simulation. r 2006 Elsevier GmbH. All rights reserved. Keywords: Hexagonal array illumination; Fractional Talbot effect; Fill factor; Phase plate
1. Introduction The hexagonal array encircles a maximum area with the shortest boundary length of any equilateral polygon array. It is a nature-preferred economical pattern, such as a bee’s honeycomb, carbon dioxide. And it has been widely used in optical devices such as fiber couplers [1], gradient-index rods [2], photonic delay lines [3], and cellular logic image processors [4] in optical computing. It is a nonorthogonal periodic array that cannot be represented by orthogonal arrays. Recently, Peng Xi et al. [5] have studied a hexagonal array illumination based on a phase gratings with 0 and p phase difference. It is the unique property of the hexagonal array to give an array illumination in this way. Also the hexagonal array is a periodic array, so we can get the array illumination by the fractional Talbot effects as other periodic arrays [6–8]. The Talbot array illuminators (TAIs) are periodic phase diffractive elements, which can be designed and Corresponding author.
E-mail address: [email protected] (W. Qu). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.04.006
manufactured based on the theory of the fractional Talbot effect. Because it can effectively realize the high speed and concurrent optical operation, TAIs have broad applications in areas that include optical interconnection, optical computing and optoelectronic processing. Lohmann [6] and Thomas [7] were among the first to describe array illumination based on the fractional Talbot effect with experimental demonstrations at (1/4)ZT and (1/6)ZT, where ZT ¼ 2T2/l is the Talbot distance. After that, many researchers presented equations which can be used to calculate the diffraction field at the fractional Talbot distance, such as Leger and Swanson [8], Liu [9], Arrizon and Ojeda-Castaneda [10,11], Zhou et al. [12,13]. But all those equations are used to calculate the one-dimensional grating or the two-dimensional (2D) array obtained by two orthogonal gratings. We are interested in the study of the fractional Talbot effect of the array that is obtained by two nonorthogonal gratings, which is basic to the new type array illuminator such as hexagonal array illuminator. In Section 2, we present a mathematical description of the hexagonal array and prove the self-image of it. In Section 3, we analyze the fractional Talbot effect of the
ARTICLE IN PRESS W. Qu et al. / Optik 118 (2007) 330–334
hexagonal array with different fill factor in detail. Finally, we give a calculated phase distribution in the fractional Talbot plane of the hexagonal array with fill factor of 1/12, with which a hexagonal array illumination can be obtained.
331
Fourier series of delta function, so j(x, y) is pffiffiffi pffiffiffi 3x þ y 3x y ; jðx; yÞ ¼ comb T T X X pffiffiffi pffiffiffi 2 ¼ jTj dð 3x þ y nTÞdð 3x y mTÞ n
2. The Talbot effect of 2D amplitude-type hexagonal array As shown by Fig. 1, in the amplitude-type hexagonal array a black hexagon corresponds to luminescence and a white hexagon corresponds to nonluminescence. Assuming that the amplitude transmittance of the hexagonal array is tðx; yÞ ¼ tc ðx; yÞnjðx; yÞ,
(1)
where pffiffiffi pffiffiffi 3x þ y 3x y 2y tc ðx; yÞ ¼ rect rect rect pffiffiffi d d 3d
is the lattice-generating function, which can generate an spot array p offfiffiffiperiod T in thep two ffiffiffi directions that parallel to the line 3x þ y ¼ 0 and 3x y ¼ 0, denotes the convolution operation. The amplitude-type hexagonal array is normally illuminated by a unit-amplitude plane wave. We use the Fresnel transform formula to evaluate the amplitude distribution of the observation plane at distance z. The Fresnel transform of a two-dimensional (2D) function is defined by a convolution with a scaled quadratic phase function expðjkzÞ tc ðx; yÞnDðx; yÞ, jlz
(2)
where D(x, y) ¼ j(x, y)h(x, y), h(x, y) ¼ exp[jp(x2+ y2)/lz] is the optical transfer function in the Fresnel domain. We can write the comb function in terms of the n=1
n=0
n=-1
d
(
pffiffiffi pffiffiffi XX n m 3n þ 3m d fy Dðx; yÞ ¼F d fx T T n m " ! #) p ffiffi ffi p ffiffi ffi 2 n m 2 3n þ 3m exp jplz þ . T T 1
We define a new function as z Cðn; mÞ ¼ exp j2p 2 ðn2 þ m2 mnÞ T =2l and n2 þ m2 mn is integer. The inverse Fourier transform in Eq. (3) can be evaluated using the Fourier series expression for delta function. Hence D(x, y) becomes Dðx; yÞ ¼
T2 X X pffiffiffi Cðn; mÞ j2 3j n m ZZ pffiffi pffiffi ejpnðf y ðf x Þ= 3ÞT ejpmðf y ðf x Þ= 3ÞT 1
ej2pðf x xþf y yÞ df x df y .
n
m
pffiffiffi ð 3x y mT Þ,
m=-1
φ (x, y)
ð5Þ
Obviously, if z ¼ lT2/2l, l is integer, we can obtain the following terms: X X pffiffiffi dð 3x þ y nTÞ Dðx; yÞ ¼ jTj2 m
pffiffiffi dð 3x y mT Þ. m=1 m=0
ð4Þ
Evaluate the integration of Eq. (4), finally we can obtain the expression of D(x, y) as follows: XX pffiffiffi Dðx; yÞ ¼ jTj2 Cðn; mÞdð 3x þ y nTÞd
n
* tc (x, y)
According to convolution theorem and the property of the delta function, we can compute D(x, y) as follows:
ð3Þ
is the amplitude transmittance of the hexagonal cell, the length of the hexagonal lateral is d/2, pffiffiffi pffiffiffi 3x þ y 3x y ; jðx; yÞ ¼ comb T T
f ðx; y; zÞ ¼
m
pffiffiffi 3x þ y 1 XX exp i2pn ¼ T jTj2 n m pffiffiffi 3x y exp i2pm . T
ð6Þ
T
t(x, y)= tc (x, y)* φ (x, y)
Fig. 1. The generation of the 2D hexagonal array, the convolution operation.
denotes
It is a spot array same as the spot distribution described by function j(x, y). Therefore, the Talbot distance of the amplitude-type hexagonal array is zT ¼ T2/2l, where T is the above period of the lattice, l is the wavelength of the illuminated light.
ARTICLE IN PRESS 332
W. Qu et al. / Optik 118 (2007) 330–334
3. Simulation of the fractional Talbot effect In order to getting the phase plate to implement the hexagonal array illumination, we must find a fractional Talbot plane with uniform image. In another word, there is equal amplitude but unequal phase distribution of the wavefield in the plane. The length of the cell’s lateral and the period of the whole array must obey the superposition theorem [14]. A double-wedge unit is taken out of the hexagonal array, as shown by Fig. 2. Comparison of this three hexagonal array unit tell us that only when the relation d:T ¼ 2:3k between d and T is satisfied, a tightly arranged hexagonal array can be obtained, where k is a nonzero positive integer, d is two times of the length of the hexagonal cell’s lateral, T is the period of the hexagonal array in the direction that parallel to the line pffiffiffi 3x þ y ¼ 0. The analytical mathematical operation of the fractional Talbot effects of the hexagonal array is very difficult. Use the mathematical description of the hexagonal array given by Section 2, we simulate the Talbot and the fractional Talbot effect of two amplitude-type hexagonal arrays with different fill factor (defined as the ratio of the area of luminescence and the whole area of the unit). The Talbot image and fractional Talbot images of the amplitude-type hexagonal array with a fill factor of 3/16 is shown by Fig. 3. Fig. 3(a) is the amplitude-type hexagonal array. Fig. 3(b) is the Talbot image at distance zT ¼ T2/2l. There are edge effects due to the limited computation boundary. Fig. 3(c) and (d) is the fractional Talbot images that no overlap occurs. As analysis above, we cannot obtain the fractional Talbot image with equal intensity distribution. We investigate the amplitude-type hexagonal array with a fill factor of 1/12 shown by Fig. 4(a). We can obtain only one fractional Talbot image with equal intensity and unequal phase distribution shown by Fig. 4(f) at distance z ¼ T2/12lU
(b)
(a)
(c)
(d)
(e)
Fig. 3. The amplitude-type hexagonal array with fill factor 1/4(a), simulation of the Talbot image at distance zT ¼ T2/ 2l(b), and the fractional Talbot images at distance z ¼ T2/ 3l(c), z ¼ T2/4l(d), z ¼ T2/6l(e), l ¼ 633 nm.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 4. The amplitude-type hexagonal array with fill factor 1/6(a), simulation of the Talbot image at distance zT ¼ T2/ 2l(b), and the fractional Talbot images at distance z ¼ T2/ 3l(c), z ¼ T2/4l(d), z ¼ T2/6l(e), z ¼ T2/12l(f), l ¼ 633 nm.
4. Implementation of the hexagonal array illumination d/2
T
(a)
(b)
(c)
Fig. 2. Double-wedge unit of the hexagonal array with d/T ¼ 2/3(a), d/T ¼ 1/2 (b), d/T ¼ 1/3(c).
The phase distribution in the fractional Talbot plane at distance z ¼ T2/12l is shown by Fig. 5(a). In order to giving a clearer observation, we take a unit same as the double-wedge unit of the hexagonal array out of the whole phase distribution and mesh it. There are obvious bulges and sags at a certain regular. It likes a binaryoptics phase plate with two phase levels. Using a unitamplitude plane wave normally illuminates the phase
ARTICLE IN PRESS W. Qu et al. / Optik 118 (2007) 330–334
333
efficiency of this effective two-dimensional array illuminator is up to 96%.
5. Conclusions (a)
(b)
Fig. 5. The phase distribution in the plane at distance z ¼ T2/ 12l(a), a unit of three-dimensional phase distribution (b), l ¼ 633 nm.
15
the amplitude type hexagonal array the hexagonal array illumination
Intensity
10
The mathematic description of the amplitude-type hexagonal array in this paper is available for computer simulation. According to the fractional Talbot effect simulation results, the implementation of the hexagonal array illumination based on the fractional Talbot effect only fits for the hexagonal array with fill factor of 1/(3k), where k is a nonzero positive integer. Phase plate can be obtained from the calculation of the fractional Talbot image of the amplitude-type hexagonal array. The theoretical efficiency of the obtained array illuminator is up to 96%.
5
Acknowledgements 0 -0.04
(b)
-0.02
0.00
0.02
0.04
The size of the array (mm)
15
the amplitude type hexagonal array the hexagonal array illumination
The work is supported by the National Natural Science Foundation of China (Grant No. 60177016) and the Science and Technique Minister of China (granted 2002CCA03500).
Intensity
(a) 10
References 5
0 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04
(c)
The size of the array (mm)
Fig. 6. The hexagonal array illumination with fill factor 1/6 (a) obtained by the phase distribution shown by Fig. 5(a), the curve of the transverse intensity shown by the white doted line (b), the curve of the vertical intensity shown by the white doted line (c), l ¼ 633 nm.
plate, we can obtain the hexagonal array illumination with a fill factor of 1/12 as shown by Fig. 6(a) at distance z ¼ T2/12l behind it. The curve of the transverse and vertical intensity indicated by the white doted line of the illumination is shown by Fig. 6(b) and (c), respectively. It is obvious that the hexagonal array illumination is located at the same place as the original amplitude-type array. But the intensity is more than 12 times of that of the original array. In the paraxial approximation, the technique is 100% efficient for an infinite array. For a finite array, , there is a slight distortion occurring in the apertures close to the edge and a small amount of power is deposited outside the array. But the theoretical
[1] D.B. Mortimore, J.W. Arkwright, Monolithic wavelength-flattened 1 7 single-mode fused fiber couplers: theory, fabrication, and analysis, Appl. Opt. 30 (1991) 650–659. [2] S. Nemoto, J. Kida, Retroreflector using gradient-index rods, Appl. Opt. 30 (1991) 815–822. [3] N. Madamopoulos, N.A. Riza, Demonstration of an alldigital 7-bit 33-channel photonic delay line for phasedarray radars, Appl. Opt. 39 (2000) 4168–4181. [4] L. Liu, X. Liu, B. Cui, Optical programmable cellular logic array for image processing, Appl. Opt. 30 (1991) 943–946. [5] P. Xi, C. Zhou, E. Dai, L. Liu, Generation of near-field hexagonal array illumination with a phase grating, Opt. Lett. 27 (2002) 228–230. [6] A.W. Lohmann, An array illuminator based on the Talbot effect, Optik 79 (1988) 41–45. [7] A.W. Lohmann, J.A. Thomas, Making an array illuminator based on the Talbot effect, Appl. Opt. 29 (1990) 4337–4340. [8] J.R. Leger, G.J. Swanson, Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes, Opt. Lett. 15 (1990) 288–290. [9] L. Liu, Lau cavity and phase locking of laser arrays, Opt. Lett. 14 (1989) 1312–1314. [10] V. Arrizon, J. Ojeda-Castaneda, Multilevel phase gratings for array illuminators, Appl. Opt. 33 (1994) 5925–5931.
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[11] V. Arrizon, J. Ojeda-Castaneda, Fresnel diffraction of substructured gratings: matrix description, Opt. Lett. 20 (1995) 118–120. [12] C. Zhou, L. Liu, Simple equations for the calculation of a multilevel phase grating for Talbot array illumination, Opt. Commun. 115 (1995) 40–44.
[13] C. Zhou, S. Stankovic, T. Tschudi, Analytic phase-factor equations for Talbot array illuminations, Appl. Opt. 38 (1999) 284–290. [14] J.T. Winthrop, C.R. Worthington, Theory of Fresnel image; I: plane periodic objects in monochromatic light, J. Opt. Soc. Am. A 55 (1965) 373–381.
Optik
Optics
Optik 118 (2007) 330–334 www.elsevier.de/ijleo
Near-field hexagonal array illumination based on fractional Talbot effect Weijuan Qu, Liren Liu, De’an Liu, Ya’nan Zhi, Wei Lu Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, PR China Received 24 January 2006; accepted 3 April 2006
Abstract Hexagonal array is a basic structure widely exists in nature and adopted by optoelectronic device. A phase plate based on the fractional Talbot effect that converts a single expanded laser beam into a regular hexagonal array of uniformly illuminated apertures with virtually 100% efficiency is presented. The uniform hexagonal array illumination with a fill factor of 1/12 is demonstrated by the computer simulation. r 2006 Elsevier GmbH. All rights reserved. Keywords: Hexagonal array illumination; Fractional Talbot effect; Fill factor; Phase plate
1. Introduction The hexagonal array encircles a maximum area with the shortest boundary length of any equilateral polygon array. It is a nature-preferred economical pattern, such as a bee’s honeycomb, carbon dioxide. And it has been widely used in optical devices such as fiber couplers [1], gradient-index rods [2], photonic delay lines [3], and cellular logic image processors [4] in optical computing. It is a nonorthogonal periodic array that cannot be represented by orthogonal arrays. Recently, Peng Xi et al. [5] have studied a hexagonal array illumination based on a phase gratings with 0 and p phase difference. It is the unique property of the hexagonal array to give an array illumination in this way. Also the hexagonal array is a periodic array, so we can get the array illumination by the fractional Talbot effects as other periodic arrays [6–8]. The Talbot array illuminators (TAIs) are periodic phase diffractive elements, which can be designed and Corresponding author.
E-mail address: [email protected] (W. Qu). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.04.006
manufactured based on the theory of the fractional Talbot effect. Because it can effectively realize the high speed and concurrent optical operation, TAIs have broad applications in areas that include optical interconnection, optical computing and optoelectronic processing. Lohmann [6] and Thomas [7] were among the first to describe array illumination based on the fractional Talbot effect with experimental demonstrations at (1/4)ZT and (1/6)ZT, where ZT ¼ 2T2/l is the Talbot distance. After that, many researchers presented equations which can be used to calculate the diffraction field at the fractional Talbot distance, such as Leger and Swanson [8], Liu [9], Arrizon and Ojeda-Castaneda [10,11], Zhou et al. [12,13]. But all those equations are used to calculate the one-dimensional grating or the two-dimensional (2D) array obtained by two orthogonal gratings. We are interested in the study of the fractional Talbot effect of the array that is obtained by two nonorthogonal gratings, which is basic to the new type array illuminator such as hexagonal array illuminator. In Section 2, we present a mathematical description of the hexagonal array and prove the self-image of it. In Section 3, we analyze the fractional Talbot effect of the
ARTICLE IN PRESS W. Qu et al. / Optik 118 (2007) 330–334
hexagonal array with different fill factor in detail. Finally, we give a calculated phase distribution in the fractional Talbot plane of the hexagonal array with fill factor of 1/12, with which a hexagonal array illumination can be obtained.
331
Fourier series of delta function, so j(x, y) is pffiffiffi pffiffiffi 3x þ y 3x y ; jðx; yÞ ¼ comb T T X X pffiffiffi pffiffiffi 2 ¼ jTj dð 3x þ y nTÞdð 3x y mTÞ n
2. The Talbot effect of 2D amplitude-type hexagonal array As shown by Fig. 1, in the amplitude-type hexagonal array a black hexagon corresponds to luminescence and a white hexagon corresponds to nonluminescence. Assuming that the amplitude transmittance of the hexagonal array is tðx; yÞ ¼ tc ðx; yÞnjðx; yÞ,
(1)
where pffiffiffi pffiffiffi 3x þ y 3x y 2y tc ðx; yÞ ¼ rect rect rect pffiffiffi d d 3d
is the lattice-generating function, which can generate an spot array p offfiffiffiperiod T in thep two ffiffiffi directions that parallel to the line 3x þ y ¼ 0 and 3x y ¼ 0, denotes the convolution operation. The amplitude-type hexagonal array is normally illuminated by a unit-amplitude plane wave. We use the Fresnel transform formula to evaluate the amplitude distribution of the observation plane at distance z. The Fresnel transform of a two-dimensional (2D) function is defined by a convolution with a scaled quadratic phase function expðjkzÞ tc ðx; yÞnDðx; yÞ, jlz
(2)
where D(x, y) ¼ j(x, y)h(x, y), h(x, y) ¼ exp[jp(x2+ y2)/lz] is the optical transfer function in the Fresnel domain. We can write the comb function in terms of the n=1
n=0
n=-1
d
(
pffiffiffi pffiffiffi XX n m 3n þ 3m d fy Dðx; yÞ ¼F d fx T T n m " ! #) p ffiffi ffi p ffiffi ffi 2 n m 2 3n þ 3m exp jplz þ . T T 1
We define a new function as z Cðn; mÞ ¼ exp j2p 2 ðn2 þ m2 mnÞ T =2l and n2 þ m2 mn is integer. The inverse Fourier transform in Eq. (3) can be evaluated using the Fourier series expression for delta function. Hence D(x, y) becomes Dðx; yÞ ¼
T2 X X pffiffiffi Cðn; mÞ j2 3j n m ZZ pffiffi pffiffi ejpnðf y ðf x Þ= 3ÞT ejpmðf y ðf x Þ= 3ÞT 1
ej2pðf x xþf y yÞ df x df y .
n
m
pffiffiffi ð 3x y mT Þ,
m=-1
φ (x, y)
ð5Þ
Obviously, if z ¼ lT2/2l, l is integer, we can obtain the following terms: X X pffiffiffi dð 3x þ y nTÞ Dðx; yÞ ¼ jTj2 m
pffiffiffi dð 3x y mT Þ. m=1 m=0
ð4Þ
Evaluate the integration of Eq. (4), finally we can obtain the expression of D(x, y) as follows: XX pffiffiffi Dðx; yÞ ¼ jTj2 Cðn; mÞdð 3x þ y nTÞd
n
* tc (x, y)
According to convolution theorem and the property of the delta function, we can compute D(x, y) as follows:
ð3Þ
is the amplitude transmittance of the hexagonal cell, the length of the hexagonal lateral is d/2, pffiffiffi pffiffiffi 3x þ y 3x y ; jðx; yÞ ¼ comb T T
f ðx; y; zÞ ¼
m
pffiffiffi 3x þ y 1 XX exp i2pn ¼ T jTj2 n m pffiffiffi 3x y exp i2pm . T
ð6Þ
T
t(x, y)= tc (x, y)* φ (x, y)
Fig. 1. The generation of the 2D hexagonal array, the convolution operation.
denotes
It is a spot array same as the spot distribution described by function j(x, y). Therefore, the Talbot distance of the amplitude-type hexagonal array is zT ¼ T2/2l, where T is the above period of the lattice, l is the wavelength of the illuminated light.
ARTICLE IN PRESS 332
W. Qu et al. / Optik 118 (2007) 330–334
3. Simulation of the fractional Talbot effect In order to getting the phase plate to implement the hexagonal array illumination, we must find a fractional Talbot plane with uniform image. In another word, there is equal amplitude but unequal phase distribution of the wavefield in the plane. The length of the cell’s lateral and the period of the whole array must obey the superposition theorem [14]. A double-wedge unit is taken out of the hexagonal array, as shown by Fig. 2. Comparison of this three hexagonal array unit tell us that only when the relation d:T ¼ 2:3k between d and T is satisfied, a tightly arranged hexagonal array can be obtained, where k is a nonzero positive integer, d is two times of the length of the hexagonal cell’s lateral, T is the period of the hexagonal array in the direction that parallel to the line pffiffiffi 3x þ y ¼ 0. The analytical mathematical operation of the fractional Talbot effects of the hexagonal array is very difficult. Use the mathematical description of the hexagonal array given by Section 2, we simulate the Talbot and the fractional Talbot effect of two amplitude-type hexagonal arrays with different fill factor (defined as the ratio of the area of luminescence and the whole area of the unit). The Talbot image and fractional Talbot images of the amplitude-type hexagonal array with a fill factor of 3/16 is shown by Fig. 3. Fig. 3(a) is the amplitude-type hexagonal array. Fig. 3(b) is the Talbot image at distance zT ¼ T2/2l. There are edge effects due to the limited computation boundary. Fig. 3(c) and (d) is the fractional Talbot images that no overlap occurs. As analysis above, we cannot obtain the fractional Talbot image with equal intensity distribution. We investigate the amplitude-type hexagonal array with a fill factor of 1/12 shown by Fig. 4(a). We can obtain only one fractional Talbot image with equal intensity and unequal phase distribution shown by Fig. 4(f) at distance z ¼ T2/12lU
(b)
(a)
(c)
(d)
(e)
Fig. 3. The amplitude-type hexagonal array with fill factor 1/4(a), simulation of the Talbot image at distance zT ¼ T2/ 2l(b), and the fractional Talbot images at distance z ¼ T2/ 3l(c), z ¼ T2/4l(d), z ¼ T2/6l(e), l ¼ 633 nm.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 4. The amplitude-type hexagonal array with fill factor 1/6(a), simulation of the Talbot image at distance zT ¼ T2/ 2l(b), and the fractional Talbot images at distance z ¼ T2/ 3l(c), z ¼ T2/4l(d), z ¼ T2/6l(e), z ¼ T2/12l(f), l ¼ 633 nm.
4. Implementation of the hexagonal array illumination d/2
T
(a)
(b)
(c)
Fig. 2. Double-wedge unit of the hexagonal array with d/T ¼ 2/3(a), d/T ¼ 1/2 (b), d/T ¼ 1/3(c).
The phase distribution in the fractional Talbot plane at distance z ¼ T2/12l is shown by Fig. 5(a). In order to giving a clearer observation, we take a unit same as the double-wedge unit of the hexagonal array out of the whole phase distribution and mesh it. There are obvious bulges and sags at a certain regular. It likes a binaryoptics phase plate with two phase levels. Using a unitamplitude plane wave normally illuminates the phase
ARTICLE IN PRESS W. Qu et al. / Optik 118 (2007) 330–334
333
efficiency of this effective two-dimensional array illuminator is up to 96%.
5. Conclusions (a)
(b)
Fig. 5. The phase distribution in the plane at distance z ¼ T2/ 12l(a), a unit of three-dimensional phase distribution (b), l ¼ 633 nm.
15
the amplitude type hexagonal array the hexagonal array illumination
Intensity
10
The mathematic description of the amplitude-type hexagonal array in this paper is available for computer simulation. According to the fractional Talbot effect simulation results, the implementation of the hexagonal array illumination based on the fractional Talbot effect only fits for the hexagonal array with fill factor of 1/(3k), where k is a nonzero positive integer. Phase plate can be obtained from the calculation of the fractional Talbot image of the amplitude-type hexagonal array. The theoretical efficiency of the obtained array illuminator is up to 96%.
5
Acknowledgements 0 -0.04
(b)
-0.02
0.00
0.02
0.04
The size of the array (mm)
15
the amplitude type hexagonal array the hexagonal array illumination
The work is supported by the National Natural Science Foundation of China (Grant No. 60177016) and the Science and Technique Minister of China (granted 2002CCA03500).
Intensity
(a) 10
References 5
0 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04
(c)
The size of the array (mm)
Fig. 6. The hexagonal array illumination with fill factor 1/6 (a) obtained by the phase distribution shown by Fig. 5(a), the curve of the transverse intensity shown by the white doted line (b), the curve of the vertical intensity shown by the white doted line (c), l ¼ 633 nm.
plate, we can obtain the hexagonal array illumination with a fill factor of 1/12 as shown by Fig. 6(a) at distance z ¼ T2/12l behind it. The curve of the transverse and vertical intensity indicated by the white doted line of the illumination is shown by Fig. 6(b) and (c), respectively. It is obvious that the hexagonal array illumination is located at the same place as the original amplitude-type array. But the intensity is more than 12 times of that of the original array. In the paraxial approximation, the technique is 100% efficient for an infinite array. For a finite array, , there is a slight distortion occurring in the apertures close to the edge and a small amount of power is deposited outside the array. But the theoretical
[1] D.B. Mortimore, J.W. Arkwright, Monolithic wavelength-flattened 1 7 single-mode fused fiber couplers: theory, fabrication, and analysis, Appl. Opt. 30 (1991) 650–659. [2] S. Nemoto, J. Kida, Retroreflector using gradient-index rods, Appl. Opt. 30 (1991) 815–822. [3] N. Madamopoulos, N.A. Riza, Demonstration of an alldigital 7-bit 33-channel photonic delay line for phasedarray radars, Appl. Opt. 39 (2000) 4168–4181. [4] L. Liu, X. Liu, B. Cui, Optical programmable cellular logic array for image processing, Appl. Opt. 30 (1991) 943–946. [5] P. Xi, C. Zhou, E. Dai, L. Liu, Generation of near-field hexagonal array illumination with a phase grating, Opt. Lett. 27 (2002) 228–230. [6] A.W. Lohmann, An array illuminator based on the Talbot effect, Optik 79 (1988) 41–45. [7] A.W. Lohmann, J.A. Thomas, Making an array illuminator based on the Talbot effect, Appl. Opt. 29 (1990) 4337–4340. [8] J.R. Leger, G.J. Swanson, Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes, Opt. Lett. 15 (1990) 288–290. [9] L. Liu, Lau cavity and phase locking of laser arrays, Opt. Lett. 14 (1989) 1312–1314. [10] V. Arrizon, J. Ojeda-Castaneda, Multilevel phase gratings for array illuminators, Appl. Opt. 33 (1994) 5925–5931.
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