Design and fabrication of micro-cube-corner array retro-reflectors

1 August 2002

Optics Communications 209 (2002) 75–83 www.elsevier.com/locate/optcom

Design and fabrication of micro-cube-corner array retro-reflectors Jinghe Yuan *, Shengjiang Chang, Sumei Li, Yanxin Zhang Key Laboratory of Opto-electronics Information Technical Science, Institute of Modern Optics, Nankai University, EMC, Tianjin 300071, China Received 19 November 2001; received in revised form 1 March 2002; accepted 22 May 2002

Abstract A novel approach of modeling and analyzing the two types of micro-cube-corner array retro-reflectors is proposed. Light tracing algorithm for determining the maximum incident angle range limited by the critical angle of total reflection, and reverse-projection (RP) algorithm for calculating the available area of an isolated unit are described. Simulation results and fabrication method of micro-cube-corner array by moving-mask technique are also given. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Retro-reflection; Reverse-projection (RP) algorithm; Moving mask technology (MMT); Continuous deep relief technology (CDRT)

1. Introduction It is well known that retro-reflectors have numerous applications, e.g. safety-devices on traffic signs [1], pseudo-conjugation devices in optical communication systems [2–7], range finders, targets in war games, etc. The most common application among these may be safety devices. Retroreflectors on bicycles and clothing enhance the visibility to the drivers, whereas those on highways greatly increase the signals return. There are many types of retro-reflectors: metalbacked cube-corner arrays, bare cube-corner ar-

*

Corresponding author. Fax: +86-22-2350-4571. E-mail address: [email protected] (J. Yuan).

rays, metal-backed micro-spheres, and bare microspheres, etc. Some detailed experimental data can be found in [1]. Because the reflectivity of total reflection is very high comparing with that of normal mirror reflection, we will only discuss retro-reflection of bare cube-corner arrays when the reflection on the three reflective planes is the total internal reflection. Retro-reflectivity and divergence are the two main parameters of retro-reflectors for safety applications. High retro-reflectivity is required for prompting the observers, and certain divergence is necessary as well because the observers may deviate from the incident direction. These two parameters are usually contrary. Micro-sphere array has larger divergence, but lower retro-reflectivity, on the other hand, micro-cube-corner array (MCCA) has

0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 6 3 0 - 9

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J. Yuan et al. / Optics Communications 209 (2002) 75–83

higher retro-reflectivity, while its fabrication is more difficult. For different applications, various approaches of modeling and analyzing retro-reflector arrays have been undertaken [8–14]. In [8], the power reflection and polarization properties have been modeled and the experimental results are supplied. In this paper, we propose a novel approach for modeling and analyzing the bare cube-corner arrays. Normally, MCCAs are fabricated by mechanical process which leads to large sizes (larger than 500 lm commonly) and little divergence. To increase the divergence, intentional flaws are made. However, the effects of the intentional flaws cannot be evaluated reliably, and therefore experimental quantification of the retro-reflectors is necessary. With the emergence of new optical micro-fabrication technology, especially the Continuous Deep Relief Technology (CDRT), it becomes possible to fabricate MCCA of smaller sizes

(about 10 lm). For the smaller size MCCA, the divergence is realized by diffraction. In this paper, the method using CDRT to fabricate MCCA is proposed.

2. Properties of MCCA As illustrated in Fig. 1, there are two types of MCCA: micro-triangle-corner array (MTCCA) where three total-reflection planes are in orthogonal-isoceles-triangles, and micro-square-corner array (MSCCA) where three total-reflection planes are in squares. In Fig. 1, O-xyz are global coordinates, and O0 -x0 y0 z0 are local coordinates. For the two types of MCCA, the maximum incident angle range limited by critical angles of total reflection are calculated in Section 2.2, and the available area of one isolated unit are calculated in Section 2.3.

Fig. 1. The structure of the two types of MCCA: (a) the structure of MTCCA; (b) the structure of MSCCA.

J. Yuan et al. / Optics Communications 209 (2002) 75–83

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2.1. Light propagation in one isolated unit The incident beam will be refracted through the incident plane (yz plane in Fig. 1). Suppose the direction cosines of the refractive beam in coordinates O0 -x0 y0 z0 are lrl mrl krl , respectively (as in Fig. 2, lrl ¼ cosðaÞ), and the direction cosines of the reflective light beams by the three reflective planes (i.e., x0 ¼ 0, y0 ¼ 0, z0 ¼ 0) are l0 , m0 , k 0 (l0 ¼ cosðhr Þ in Fig. 2), respectively. By reflection law, we have l0 ¼ lrl

ð1Þ

and m0 ¼ mrl ;

ð2Þ

0

k ¼ krl :

ð3Þ

Thus, the final direction of the reflective beam reverses the incident beam. 2.2. Maximum incident angle range The normal mirror reflection on each reflective plane is very low compared with the total reflection. For simplicity, we only discuss the total internal reflection without reflection coating in this paper. The incident angle range in different directions is limited by the critical angle of total reflection hc . Refer to Figs. 1 and 3, the direction vector of an incident beam in coordinates O-xyz is *

Aiw ¼ ð liw

miw

T

kiw Þ ;

ð4Þ

where miw ¼ cosðbÞ, kiw ¼ cosðcÞ, and liw ¼ ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  1  m2iw  kiw . And b is the angle between in-

Fig. 3. The MCCA units structure: (a) the structure of an MTCCA unit; (b) the structure of an MSCCA unit.

cident beam and y-axis, c is the angle between incident beam and z-axis. Similarly, the direction vector of the incident beam in coordinates O0 -x0 y0 z0 is given by *

Ail ¼ ð lil

mil

kil ÞT :

Aiw and Ail are linked by pffiffi pffiffi 1 0 pffiffi2 2 6   3 2 6 pffiffi C pffiffi B pffiffi 3 6C 2 Twl ¼ B @ 3  2  6 A; pffiffi pffiffi 3 0  36 2

ð5Þ

ð6Þ

namely, *

*

Ail ¼ Twl  Aiw :

Fig. 2. Illustration of retro-reflection.

ð7Þ

The normal vector of the incident plane ABC (yz plane in coordinates O-xyz) in coordinates O0 x0 y0 z0 is pffiffi pffiffi pffiffi T * N l ¼ 33 33 33 : ð8Þ

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The direction vector of the refracted beam through incident plane ABC (yz plane) in coordinates O0 x0 y0 z0 is *

Arl ¼ ð lrl

krl ÞT :

mrl

ð9Þ

Suppose the critical angle of total reflection is hc , we can know the limits jlrl j 6 cos hc jmrl j 6 cos hc jkrl j 6 cos hc

ð10Þ

By refraction law, we have  *T * * n0 * Arl  ðArl  N l ÞN l Ail ¼ n sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

0 2 0 2 T * * * n n 2 þ ðArl  N l Þ ;  Nl 1  n n

*

ð11Þ 0

where n and n are the refractive indexes of outer medium and corner material, respectively, and *

*

Aiw ¼ T 1  Ail :

ð12Þ

When n ¼ 1:00 (air), the maximum incident angle range in different directions are displayed in Fig. 4 for 1:20 6 n0 6 2:00. For example, when n0 ¼ 1:59 (Polycarbonate), and the incident beam is perpendicular to z-axis ðc ¼ 90°Þ, the incident beam angle range can be from 58.90° to 121.10°; when the incident beam is perpendicular to y-axis ðb ¼ 90°Þ, the incident beam angle range can be

from 0° to 115.33°. Due to the pattern structure of MTCCA, the incident beam angle range can be from 0° to 180° when b ¼ 90°. 2.3. Determination of available retro-reflection area Some incident beams cannot be retro-reflected because their refractive beams cannot reach all the three reflective planes. The available area through which the incident beams can be retro-reflected is calculated as following. Incident beams can be divided into three groups according to the direction of refractive beams: 1. Two of the three direction cosines ðlrl ; mrl ; krl Þ are zero, i.e., the refractive beams are parallel to one of the corner arises. This type of beams can be retro-reflected. 2. One of the three direction cosines ðlrl ; mrl ; krl Þ is zero, i.e., the refractive beams are vertical to one of the corner arises. This type of beams can be retro-reflected as well. The above-mentioned two types of light beams are limited in fixed directions, so we would not discuss them further. 3. If ðlrl ; mrl ; krl Þ satisfy Eq. (10), only within the available area in the incident plane ABC, the incident beams can be retro-reflected. A reverseprojection (RP) algorithm is used to calculate the available area as described in the following. For a MTCCA unit as shown in Fig. 3(a) and 5, suppose the refractive beam across the corner vertex O0 intersects with the incident plane ABC at

Fig. 4. The maximum incident angle range versus the refractive index: (a) b ¼ 90°; (b) c ¼ 90°.

J. Yuan et al. / Optics Communications 209 (2002) 75–83

Fig. 5. The RP Algorithm, For a MTCCA unit, A0 , B0 , C 0 are the symmetric points of A, B, C with respect to Q0 , respectively.

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point Q0 , and A0 , B0 and C 0 are the symmetrical points of A, B and C with respect to Q0 , thus, the overlapping area of the triangle A0 B0 C 0 and the triangle ABC is the available area. This RP algorithm can be demonstrated as following. In Fig. 3(a), A1 , B1 and C1 are the virtual images of A, B, C with respect to the three reflective planes, respectively. So the beam which intersects with the incident plane ABC at point A, after reflected by the three reflective planes, will be along the line that reverses the direction of the incident beam and is through the point A1 . A0 is the projection point of A1 on plane ABC. B0 and C 0 can be obtained similarly (in Fig. 5).

Fig. 6. (a)The available area at different incident angles for a MTCCA unit; (b) the available area at different incident angles for a MSCCA unit.

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If the incident beam is perpendicular to the incident plane ABCðliw ¼ 1; b ¼ 90°; c ¼ 90°Þ, according to the RP algorithm, we can reasonably consider that the available area is the overlapping area of two equilateral triangles which have inverse directions and have the same center point, i.e., the available area is an equilateral hexagon. If the incident angle is not zero, the available area is still an overlapping area of two equilateral triangles. It may be a parallelogram or a hexagon, as shown in Fig. 6(a). We also obtained the available area of a MSCCA unit by using the RP algorithm, as in Fig. 6(b). The ratios of the available area to the whole incident area at different incident angles are shown

in Fig. 7. Fig. 7(a) is a plot of the ratios versus c when b ¼ 90° for MTCCA, Fig. 7(b) is the ratios versus b when c ¼ 90° for MTCCA, Fig. 7(c) is the ratios versus c when b ¼ 90° for MSCCA, and Fig. 7(d) is the ratios versus beta when c ¼ 90° for MSCCA. From Fig. 7, we can conclude that: 1. When the incident angle is zero ðliw ¼ 1, b ¼ 90°; c ¼ 90°Þ, the ratio is the largest and equal to 66.7% for a MTCCA unit and 100% for a MSCCA unit. 2. As the incident angle increases, the ratios decrease more slowly for MTCCA unit than for MSCCA unit especially near the vertical incident direction.

Fig. 7. The ratios of the available area to the whole incident area at different incident angles: (a) for MTCCA, b ¼ 90, the ratios versus c; (b) for MTCCA, c ¼ 90, the ratios versus b; (c) for MSCCA, b ¼ 90, the ratios versus c; (d) for MSCCA, c ¼ 90, the ratios versus b.

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3. Fabrication method of MCCA Traditionally, cube-corner array was fabricated by using mechanical process, i.e., using the limited optical process equipments to cut, grind, polish or coat on the optical parts surfaces. However, it is difficult to miniaturize the optical parts and to make an array apparatus in this approach. To increase the divergence, intentional flaws have to be made. With the development of the micro-electronic technology and the optical micro-fabrication technology, especially the Continuous Deep Relief Technology (CDRT), it is possible to manufacture MCCAs. The first step of using CDRT to fabricate MCCA is to make an appropriate mask, the second step is to produce relief figures on resist, and the third step is to transfer the relief figures onto the substrate by the Reactive Ion Etching Technology. There are many advantages in using CDRT to fabricate MCCA, such as simple process, miniaturization, low cost, etc. In this section, we present the fabrication of MTCCA by using Gray-level Mask Technology (GMT) and the fabrication of MSCCA by using Moving Mask Technology (MMT).

Fig. 8. (a) The gray-level mask of MTCCA; (b) MTCCA made by GMT (magnified 300), with poor linearity and homogeneity.

3.1. Fabrication of MTCCA by using GMT GMT includes, to design a gray-level mask, and then to obtain the wanted relief figure after exposure on resist materials and development. The advantage of GMT is that its mask can be designed and made easily. However, it is difficult to control the linearity and homogeneity of relief figure. Fig. 8(a) and (b) are a gray-level mask of MTCCA and its relief figure (magnified 300).

and homogeneity in GMT, but it is only used in the fabrication of continuous optical elements with symmetry. For using MMT to make continuous exposure distribution the design of binary mask is the key. For instance, Fig. 9 is a moving-mask of one-

3.2. Fabrication of MSCCA by using MMT MMT, employing a projection exposure system, downsizes and projects the binary mask pattern onto the resist materials. During the exposure, the mask is translated in a linear or rotary motion, to obtain a continuous exposure distribution with linear symmetry or rotary symmetry on the resist. MMT overcomes the problem of poor linearity

Fig. 9. A one-dimensional MCCA moving-mask.

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dimensional MCCA, and Fig. 10(a) shows the method to achieve the one-dimensional MCCA by using this mask. Suppose the exposure intensity is I and exposure time is t, make the mask to move a period in the time t along the direction v, the exposure on the lines 1, 2 and 3 will be 0, I t and 0, respectively. After development, we can attain the relief (Fig. 10(b)), which is a one-dimensional MCCA. By using a proper mask, with different moving processes, we can fabricate a desirable MSCCA, as shown in Fig. 11 (magnified 300).

Fig. 10. Illustration of a one-dimensional MCCA fabrication by using MMT.

4. Discussion and conclusion 1. MCCA retro-reflectors have large incident angle range. For instance, when n0 ¼ 1:59 (Polycarbonate in air), for the incident beam perpendicular to z-axis ðc ¼ 90°Þ, the incident angle range can be from 58.90° to 121.10°; for the incident beam perpendicular to y-axis ðb ¼ 90°Þ, the incident angle range can be from 0° to 115.33°. By noticing the arrangement of MTCCA, it is obvious that the incident angle range can be from 0° to 180°. 2. The maximum ratio of the available area to the whole incident area is equal to 66.7% for MTCCA unit, and 100% for MSCCA unit. As the incident angle increases, the ratio decreases more slowly for MTCCA than for MSCCA, especially near the vertical incident direction. 3. The limit on the incident angle, in the direction of y-axis and negative z-axis ðc > 90°Þ, is mainly due to critical angle of total reflection, and in the direction of positive z-axis ðc < 90°Þ, is mainly due to the shape of the cube-corner. Because the shape of cube-corner cannot be changed, n0 should be increased in order to increase the incident angle range. 4. Although the mask can be easily made, when using GMT to make MTCCA, it is difficult to control the linearity and homogeneity of relief figures in practice. By using MMT to make MSCCA, however, we can obtain a relief figure with high degree of linearity and homogeneity in the fabrication of symmetry relief figures.

Acknowledgement This project is supported by Tianjin United Scientific Research Center on Opto-electronics.

References

Fig. 11. An MSCCA fabricated by MMT (magnified 300) with perfect linearity and homogeneity.

[1] G.H. Seward, P.S. Cort, Opt. Eng. 38 (1) (1999) 164. [2] G.J. Olson, H.W. Mocker, N.A. Demma, J.B. Ross, Appl. Opt. 34 (12) (1995) 2033. [3] H.H. Barrett, S.F. Tacobs, Opt. Lett. 4 (6) (1979) 190. [4] P. Mathieu, P.A. Belanger, Appl. Opt. 19 (14) (1980) 2262. [5] T.R. O’Meara, Opt. Eng. 21 (2) (1982) 271.

J. Yuan et al. / Optics Communications 209 (2002) 75–83 [6] Guo-Sheng Zhou, Lee W. Casperson, Appl. Opt. 20 (9) (1981) 1621. [7] V.K. Orlov, Ya.Z. Virnik, S.P. Vorotilin, V.B. Gerasimov, Yu.A. Kalinin, A.Ya. Sagalovich, Sov. J. Quantum Electron. 8 (6) (1978) 799. [8] D.C. O’Brien, G.E. Faulkner, D.J. Edwards, Appl. Opt. 38 (19) (1999) 4137. [9] Takuso Sato, Yoshihide Nagura, Osamu Lkeda, Takeshi Hatsuzawa, Appl. Opt. 21 (10) (1982) 1778.

83

[10] Shaomin Wang, E. Bernabeu, J. Alda, Appl. Opt. 32 (22) (1993) 4279. [11] Xichun Hong, Weigang Huang, Shaomin Wang, Acta Phys. Sin. 31 (12) (1982) 1655 (in Chinese). [12] Shaomin Wang, J. Hangzhou Univ. 10 (4) (1983) 476 (in Chinese). [13] Shaomin Wang, J. Hangzhou Univ. 11 (1) (1984) 79 (in Chinese). [14] S.F. Jacobs, Opt. Eng. 21 (2) (1982) 281.