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Real color fractional Fourier transform holograms
Optics Communications 259 (2006) 513–516 www.elsevier.com/locate/optcom
Real color fractional Fourier transform holograms Weimin Jin *, Lihong Ma, Caijie Yan Information Optics Institute, Zhejiang Normal University, Zhejiang, Jinhua 321004, PR China Received 20 May 2005; received in revised form 28 August 2005; accepted 8 September 2005
Abstract Real color fractional Fourier transform holography is proposed based on fractional Fourier transform holography. The method of fabrication, the principle of reconstruction and the design of system parameters are discussed in detail. The experiments prove real color fractional Fourier transform holograms possess the better function of anti-counterfeit than common fractional Fourier transform holograms. 2005 Elsevier B.V. All rights reserved. Keywords: Hologram; Real color; Fractional Fourier transform; Optical anti-counterfeiting
1. Introduction Lohmann [1] introduced the mathematical concept of fractional Fourier transform to information optics for the first time. He proposed a single-lens model and a doublelens model which can optically realize any fractional order Fourier transform. Soon fractional Fourier transform became a popular topic in information optics. Not only because fractional Fourier transform extended Fourier transform, but because we could understand the propagation of light, image formation and information processing from another angle of view, fractional Fourier transform had many applications [2–7]. The light field distribution in the fraction zone is related to fractional order. Using this property, fractional Fourier transform holograms recorded contain both the information of the object and the information of the system parameters [8,9]. And because the fractional order can be a new confinement condition of reconstruction and freedom degree of keep-secret, fractional Fourier transform holography is a new one of anti-counterfeit. The paper proposes a technique of making real color fractional Fourier transform holograms based on fractional Fourier
*
Corresponding author. Tel.: +86 579 2283030; fax: +86 579 2282808. E-mail address: [email protected] (W. Jin).
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.09.019
transform holography. In order to assure the precise contraposition of red, green, blue three color images and simplify the fabrication procedure, we adopt the method of making real color holograms using LCD liquid crystal screen to fabricate real color fractional Fourier transform holograms [10]. Because the technique of making the real color fractional Fourier transform holograms is more severe than common fractional Fourier transform holograms, they possess the better function of anti-counterfeit, and its reconstruction image is more beautiful and living. 2. Real color fractional Fourier transform holograms 2.1. Recording process Fractional Fourier transform holograms are the holograms which record the fractional Fourier transform distribution of object wave in the fractional Fourier transform zone. For Lohmann I, the input plane and the output plane are symmetrically placed on the both sides of the singlelens. As we all know, the point spread function of the system reveals the image formation character. So we analyze the light field distribution on the hologram plane based on the elementary fractional Fourier transform hologram. Given that O(x0, y0, z0) denotes the object point on the input plane and reference wave R is parallel beam with slant
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W. Jin et al. / Optics Communications 259 (2006) 513–516
angle h, according to Lohmann definition and paraxial condition, the light field distributions are as follows: 2 x0 þ y 20 þ x21 þ y 21 x0 x1 þ y 0 y 1 e j2p Oðx1 ; y 1 Þ ¼ A0 exp jp ; kf0 tan u kf0 sin u
X1
X2 L2
H2
RB
R2
GB RG
BB GG RR BG GR BR
ð1Þ e 1 ; y 1 Þ ¼ AR expðjkx1 sin hÞ. Rðx
ð2Þ
In the equations f0 = f sin u, u ¼ p p2, p is fractional order of fractional Fourier transform, f is the lens focal length, f0 is the standard focal length. The lens focal length f and the distance z1 of the input plane and the lens are related by the following equation: z1 ¼ f0 tan u2 ; ð3Þ f0 ¼ f sin u. Fig. 1 is an optical sketch for recording real color fractional Fourier transform holograms. It is a single-lens model (Lohmann I). A liquid crystal display (LCD) is placed on the input plane, a mobile slit S is placed between the lens and the hologram halide, and the reference wave is parallel light. After the shot color image is fed into a computer, it can be separated into red, green and blue three color images by image processing software (e.g., photoshop). The three color images are input to LCD orderly. When the red image is input, the hologram halide is exposed for the first time. Then, the green image is input, the slit S is moved, and the hologram halide is exposed for the second time. Finally the blue image is input, the slit S is moved again, and the hologram halide is exposed for the last time. In this way, a real color fractional Fourier transform hologram is recorded on the hologram halide H. 2.2. Reconstruction process Fig. 2 is an optical sketch for reconstructing real color fractional Fourier transform holograms. The hologram H is illuminated by white light along the same direction of the reference wave. Because of color blurring, the image information recorded cannot be observed when reconstruction light is directly observed. If reconstruction light is transformed by another fractional Fourier transform system with fractional order p2 matching p1 which is fractional order when recorded, that is, p1 + p2 = 2, the light field distribution of the original image is reconstructed on the output
X1
X0
L1
LCD
R
H
S Z d1
Z1
Z1
To computer
Fig. 1. Optical sketch for recording real color fractional Fourier transform holograms.
S′
Z2
d2
Z2
Fig. 2. Optical sketch for reconstructing real color fractional Fourier transform holograms.
plane (x2, y2). The real image S 0 of the slit S is reconstructed on the plane which has a distance d2 from the output plane. There are nine reconstruction of real slit images. If only the design of the recording parameters is sound, the red slit real image of the red image, the green slit real image of the green image and the blue slit real image of the blue image can be overlapped in the longitudinal direction. So the real color reconstruction image can be observed where the slits are overlapped. 3. The design of the recording parameters As the illuminating wave is the reference wave in recording, that is, C(x, y) = R(x, y), p1 order fractional Fourier transform light field of the recorded object wave is reconstructed . Then, the reconstruction image of the object can be obtained after Fresnel diffraction. The centre coordinate of the reconstruction image is computed: z ¼ f sin u1 tan u1 ; x0 x¼ f sin u1 tan u1 sin h; cos u1 y0 . y¼ cos u1
ð4Þ ð5Þ ð6Þ
The magnification multiple: Transversal magnification : Longitudinal magnification :
1 . ð7Þ cos u1 h u i1 . a ¼ 1 tan2 1 2 ð8Þ
b¼
Eqs. (4)–(8) all indicate that the position and size of the reconstruction image are closely related to the order of fractional Fourier transform. So the size of the reconstruction image can be flexibly controlled by changing fractional order of the recording system. It is difficult for common Fresnel holography to do so. As shown in Fig. 2, after reconstruction wave is transformed by p2 order fractional Fourier transform which matches p1 order when recorded, that is, p1 + p2 = 2 or u1 + u2 = p, the light field distribution of original objects is reconstructed on the output plane (x2, y2) of the reconstruction system. The distance between the real image S 0 of the slit S and the output plane can be calculated by the well known lens formula,
W. Jin et al. / Optics Communications 259 (2006) 513–516
515
Fig. 3. Experiment result: (a) the original photo; (b) the reconstruction image.
f þ cos u2 ðd 1 f cos u2 Þ f. d 1 f cos u2
d2 ¼
ð9Þ
If the slit is moved a distance e when the hologram is recorded, the distance of the corresponding two real images is e0 ¼
f e. d 1 f cos u2
ð10Þ
By dispersion relation of the holograms, generally if the illuminating wave bandwidth is Dk, the dispersion of the reconstruction image in the x-direction is Dx ¼
Dk z tan h. k
ð11Þ
In the special reconstruction system of Fig. 2, Eq. (11) ought to be rewritten as Dx0 ¼ ðz2 þ d 2 Þ
Dk tan h. k
ð12Þ
4. Experiment The parameters of recording are as follows: recording wave k = 632.8 nm, lens focal length f = 25 cm, reference-object angle h = 25, p1 = 0.75, p2 = 1.25, z1 = 15.43 cm, z2 = 34.57 cm, d1 = 10 cm, d2 = 22.39 cm (computed by Eq. (9)). In the experiment, the type of LCD used is NVIEW-640 · 480 (made in America), the hologram is recorded on Tianjin I silver halide emulsion plate (made in china), and the source of light used for recording the hologram is He–Ne Laser. We select three colors wave lengths as 632.8, 546.1, 435.8 nm. The space of real images of the slits along the x-axis direction is figured out by Eq. (12): Dx0RG ¼ 3:58 cm, Dx0GB ¼ 4:56 cm. Then, by Eq. (10), the moving distance of the slits in recording can be got: eRG = 2.80 cm, eGB = 3.56 cm. Fig. 3 presents the experimental result. Fig. 3(a) is the original photo. Fig. 3(b) is the reconstruction image. The reconstruction image appears hazy. This is mainly due to that the reconstruction is with white light and noise is imported in recording and so on.
5. Conclusion Real color fractional Fourier transform rainbow holograms not only record the information of the object, but also record the information of the system parameters, such as fractional order p1, lens focal length f, the distance between the object and the lens z1, the distance between the slit and the hologram d1, the moving distance between the slits eRG and eGB. The color and the position and the size of the reconstruction images of real color fractional Fourier transform rainbow holograms are related to fractional order p2 and illuminating waves. If the real color image can be reconstructed in the special fractional Fourier transform system, the hologram is true; otherwise, the hologram is fake. If only the parameters of the recording system are kept secret, real color fractional Fourier transform rainbow holography can be a new anti-counterfeit technique. Real color fractional Fourier transform holograms possess the better function of anti-counterfeit than common fractional Fourier transform holograms as a result of the increase of the recording and reconstruction system parameters. Simultaneously, a real color fractional Fourier transform hologram can be synthesized into a double-hologram with a common rainbow hologram. The integral hologram can be fabricated into an embossed hologram. The technique could be extensively applied to the fabrication of currency, certificates and brands. Acknowledgements This work was supported by Zhejiang Provincial Natural Science Foundation of China (No. M603212) and Zhejiang Provincial Science and Technology Program of China (No. 2004C31089). References [1] [2] [3] [4]
A.W. Lohmann, J. Opt. Soc. Am. A 10 (10) (1993) 2181. G. Dorsch, A.W. Lohmann, Appl. Opt. 34 (20) (1995) 4111. G. Dorsch, A.W. Lohmann, Appl. Opt. 34 (32) (1995) 7599. L.M. Bernardo, O.D.D. Soares, J. Opt. Soc. Am. A 11 (10) (1994) 2622.
516
W. Jin et al. / Optics Communications 259 (2006) 513–516
[5] H.M. Ozaktas, D. Mendlovic, Opt. Lett. 19 (21) (1994) 1678. [6] S.W. Xie, F. Gao, Y.S. Zeng, F.H. Gao, Y.K. Guo, Chinese J. Laser 30 (5) (2003) 431 (in Chinese). [7] N.K. Nishchal, J. Joseph, K. Singh, Opt. Commun. 235 (2004) 253.
[8] Y.K. Guo, Q.Z. Huang, J.L. Du, Acta Opt. Sin. 19 (6) (1999) 821 (in Chinese). [9] Y.S. Zeng, Y.K. Guo, F.H. Gao, J.H. Zhu, Opt. Commun. 215 (2003) 53. [10] W.M. Jin, Y. Li, Laser J. 25 (4) (2004) 69 (in Chinese).
Real color fractional Fourier transform holograms Weimin Jin *, Lihong Ma, Caijie Yan Information Optics Institute, Zhejiang Normal University, Zhejiang, Jinhua 321004, PR China Received 20 May 2005; received in revised form 28 August 2005; accepted 8 September 2005
Abstract Real color fractional Fourier transform holography is proposed based on fractional Fourier transform holography. The method of fabrication, the principle of reconstruction and the design of system parameters are discussed in detail. The experiments prove real color fractional Fourier transform holograms possess the better function of anti-counterfeit than common fractional Fourier transform holograms. 2005 Elsevier B.V. All rights reserved. Keywords: Hologram; Real color; Fractional Fourier transform; Optical anti-counterfeiting
1. Introduction Lohmann [1] introduced the mathematical concept of fractional Fourier transform to information optics for the first time. He proposed a single-lens model and a doublelens model which can optically realize any fractional order Fourier transform. Soon fractional Fourier transform became a popular topic in information optics. Not only because fractional Fourier transform extended Fourier transform, but because we could understand the propagation of light, image formation and information processing from another angle of view, fractional Fourier transform had many applications [2–7]. The light field distribution in the fraction zone is related to fractional order. Using this property, fractional Fourier transform holograms recorded contain both the information of the object and the information of the system parameters [8,9]. And because the fractional order can be a new confinement condition of reconstruction and freedom degree of keep-secret, fractional Fourier transform holography is a new one of anti-counterfeit. The paper proposes a technique of making real color fractional Fourier transform holograms based on fractional Fourier
*
Corresponding author. Tel.: +86 579 2283030; fax: +86 579 2282808. E-mail address: [email protected] (W. Jin).
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.09.019
transform holography. In order to assure the precise contraposition of red, green, blue three color images and simplify the fabrication procedure, we adopt the method of making real color holograms using LCD liquid crystal screen to fabricate real color fractional Fourier transform holograms [10]. Because the technique of making the real color fractional Fourier transform holograms is more severe than common fractional Fourier transform holograms, they possess the better function of anti-counterfeit, and its reconstruction image is more beautiful and living. 2. Real color fractional Fourier transform holograms 2.1. Recording process Fractional Fourier transform holograms are the holograms which record the fractional Fourier transform distribution of object wave in the fractional Fourier transform zone. For Lohmann I, the input plane and the output plane are symmetrically placed on the both sides of the singlelens. As we all know, the point spread function of the system reveals the image formation character. So we analyze the light field distribution on the hologram plane based on the elementary fractional Fourier transform hologram. Given that O(x0, y0, z0) denotes the object point on the input plane and reference wave R is parallel beam with slant
514
W. Jin et al. / Optics Communications 259 (2006) 513–516
angle h, according to Lohmann definition and paraxial condition, the light field distributions are as follows: 2 x0 þ y 20 þ x21 þ y 21 x0 x1 þ y 0 y 1 e j2p Oðx1 ; y 1 Þ ¼ A0 exp jp ; kf0 tan u kf0 sin u
X1
X2 L2
H2
RB
R2
GB RG
BB GG RR BG GR BR
ð1Þ e 1 ; y 1 Þ ¼ AR expðjkx1 sin hÞ. Rðx
ð2Þ
In the equations f0 = f sin u, u ¼ p p2, p is fractional order of fractional Fourier transform, f is the lens focal length, f0 is the standard focal length. The lens focal length f and the distance z1 of the input plane and the lens are related by the following equation: z1 ¼ f0 tan u2 ; ð3Þ f0 ¼ f sin u. Fig. 1 is an optical sketch for recording real color fractional Fourier transform holograms. It is a single-lens model (Lohmann I). A liquid crystal display (LCD) is placed on the input plane, a mobile slit S is placed between the lens and the hologram halide, and the reference wave is parallel light. After the shot color image is fed into a computer, it can be separated into red, green and blue three color images by image processing software (e.g., photoshop). The three color images are input to LCD orderly. When the red image is input, the hologram halide is exposed for the first time. Then, the green image is input, the slit S is moved, and the hologram halide is exposed for the second time. Finally the blue image is input, the slit S is moved again, and the hologram halide is exposed for the last time. In this way, a real color fractional Fourier transform hologram is recorded on the hologram halide H. 2.2. Reconstruction process Fig. 2 is an optical sketch for reconstructing real color fractional Fourier transform holograms. The hologram H is illuminated by white light along the same direction of the reference wave. Because of color blurring, the image information recorded cannot be observed when reconstruction light is directly observed. If reconstruction light is transformed by another fractional Fourier transform system with fractional order p2 matching p1 which is fractional order when recorded, that is, p1 + p2 = 2, the light field distribution of the original image is reconstructed on the output
X1
X0
L1
LCD
R
H
S Z d1
Z1
Z1
To computer
Fig. 1. Optical sketch for recording real color fractional Fourier transform holograms.
S′
Z2
d2
Z2
Fig. 2. Optical sketch for reconstructing real color fractional Fourier transform holograms.
plane (x2, y2). The real image S 0 of the slit S is reconstructed on the plane which has a distance d2 from the output plane. There are nine reconstruction of real slit images. If only the design of the recording parameters is sound, the red slit real image of the red image, the green slit real image of the green image and the blue slit real image of the blue image can be overlapped in the longitudinal direction. So the real color reconstruction image can be observed where the slits are overlapped. 3. The design of the recording parameters As the illuminating wave is the reference wave in recording, that is, C(x, y) = R(x, y), p1 order fractional Fourier transform light field of the recorded object wave is reconstructed . Then, the reconstruction image of the object can be obtained after Fresnel diffraction. The centre coordinate of the reconstruction image is computed: z ¼ f sin u1 tan u1 ; x0 x¼ f sin u1 tan u1 sin h; cos u1 y0 . y¼ cos u1
ð4Þ ð5Þ ð6Þ
The magnification multiple: Transversal magnification : Longitudinal magnification :
1 . ð7Þ cos u1 h u i1 . a ¼ 1 tan2 1 2 ð8Þ
b¼
Eqs. (4)–(8) all indicate that the position and size of the reconstruction image are closely related to the order of fractional Fourier transform. So the size of the reconstruction image can be flexibly controlled by changing fractional order of the recording system. It is difficult for common Fresnel holography to do so. As shown in Fig. 2, after reconstruction wave is transformed by p2 order fractional Fourier transform which matches p1 order when recorded, that is, p1 + p2 = 2 or u1 + u2 = p, the light field distribution of original objects is reconstructed on the output plane (x2, y2) of the reconstruction system. The distance between the real image S 0 of the slit S and the output plane can be calculated by the well known lens formula,
W. Jin et al. / Optics Communications 259 (2006) 513–516
515
Fig. 3. Experiment result: (a) the original photo; (b) the reconstruction image.
f þ cos u2 ðd 1 f cos u2 Þ f. d 1 f cos u2
d2 ¼
ð9Þ
If the slit is moved a distance e when the hologram is recorded, the distance of the corresponding two real images is e0 ¼
f e. d 1 f cos u2
ð10Þ
By dispersion relation of the holograms, generally if the illuminating wave bandwidth is Dk, the dispersion of the reconstruction image in the x-direction is Dx ¼
Dk z tan h. k
ð11Þ
In the special reconstruction system of Fig. 2, Eq. (11) ought to be rewritten as Dx0 ¼ ðz2 þ d 2 Þ
Dk tan h. k
ð12Þ
4. Experiment The parameters of recording are as follows: recording wave k = 632.8 nm, lens focal length f = 25 cm, reference-object angle h = 25, p1 = 0.75, p2 = 1.25, z1 = 15.43 cm, z2 = 34.57 cm, d1 = 10 cm, d2 = 22.39 cm (computed by Eq. (9)). In the experiment, the type of LCD used is NVIEW-640 · 480 (made in America), the hologram is recorded on Tianjin I silver halide emulsion plate (made in china), and the source of light used for recording the hologram is He–Ne Laser. We select three colors wave lengths as 632.8, 546.1, 435.8 nm. The space of real images of the slits along the x-axis direction is figured out by Eq. (12): Dx0RG ¼ 3:58 cm, Dx0GB ¼ 4:56 cm. Then, by Eq. (10), the moving distance of the slits in recording can be got: eRG = 2.80 cm, eGB = 3.56 cm. Fig. 3 presents the experimental result. Fig. 3(a) is the original photo. Fig. 3(b) is the reconstruction image. The reconstruction image appears hazy. This is mainly due to that the reconstruction is with white light and noise is imported in recording and so on.
5. Conclusion Real color fractional Fourier transform rainbow holograms not only record the information of the object, but also record the information of the system parameters, such as fractional order p1, lens focal length f, the distance between the object and the lens z1, the distance between the slit and the hologram d1, the moving distance between the slits eRG and eGB. The color and the position and the size of the reconstruction images of real color fractional Fourier transform rainbow holograms are related to fractional order p2 and illuminating waves. If the real color image can be reconstructed in the special fractional Fourier transform system, the hologram is true; otherwise, the hologram is fake. If only the parameters of the recording system are kept secret, real color fractional Fourier transform rainbow holography can be a new anti-counterfeit technique. Real color fractional Fourier transform holograms possess the better function of anti-counterfeit than common fractional Fourier transform holograms as a result of the increase of the recording and reconstruction system parameters. Simultaneously, a real color fractional Fourier transform hologram can be synthesized into a double-hologram with a common rainbow hologram. The integral hologram can be fabricated into an embossed hologram. The technique could be extensively applied to the fabrication of currency, certificates and brands. Acknowledgements This work was supported by Zhejiang Provincial Natural Science Foundation of China (No. M603212) and Zhejiang Provincial Science and Technology Program of China (No. 2004C31089). References [1] [2] [3] [4]
A.W. Lohmann, J. Opt. Soc. Am. A 10 (10) (1993) 2181. G. Dorsch, A.W. Lohmann, Appl. Opt. 34 (20) (1995) 4111. G. Dorsch, A.W. Lohmann, Appl. Opt. 34 (32) (1995) 7599. L.M. Bernardo, O.D.D. Soares, J. Opt. Soc. Am. A 11 (10) (1994) 2622.
516
W. Jin et al. / Optics Communications 259 (2006) 513–516
[5] H.M. Ozaktas, D. Mendlovic, Opt. Lett. 19 (21) (1994) 1678. [6] S.W. Xie, F. Gao, Y.S. Zeng, F.H. Gao, Y.K. Guo, Chinese J. Laser 30 (5) (2003) 431 (in Chinese). [7] N.K. Nishchal, J. Joseph, K. Singh, Opt. Commun. 235 (2004) 253.
[8] Y.K. Guo, Q.Z. Huang, J.L. Du, Acta Opt. Sin. 19 (6) (1999) 821 (in Chinese). [9] Y.S. Zeng, Y.K. Guo, F.H. Gao, J.H. Zhu, Opt. Commun. 215 (2003) 53. [10] W.M. Jin, Y. Li, Laser J. 25 (4) (2004) 69 (in Chinese).