- Email: [email protected]
Principle and application of multiple fractional Fourier transform holography
Optics Communications 215 (2003) 53–59 www.elsevier.com/locate/optcom
Principle and application of multiple fractional Fourier transform holography Yangsu Zeng a,b,*, Yongkang Guo a, Fuhua Gao a, Jianhua Zhu a a b
Department of Physics, Sichuan University, Chengdu, 610064, PR China Department of Physics, Shaoyang University, Hunan, 422000, PR China
Received 26 June 2002; received in revised form 12 September 2002; accepted 8 November 2002
Abstract In this paper, the principle of multiple fractional Fourier transform hologram (FRTH) is presented, and its characteristics based on the particularity in recording and reconstruction are analyzed. With this method, a multiple FRTH of several objects with different fractional transform orders is fabricated on one holographic plate. It requires a matched multiple fractional Fourier transform system to reconstruct the recorded images correctly. The potential application of multiple FRTH in optical security or anti-counterfeiting system is also discussed. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Fractional Fourier Transform (FRT); Multiple holography; Optical anti-counterfeiting
1. Introduction Fractional Fourier Transform (FRT) was discussed by Namias in 1980 [1] as a mathematical tool in quantum mechanics. In 1993, Mendlovic and Ozaktas implemented the FRT optically when they studied gradient index (GRIN) fiber. In the same year, Lohmann proposed that FRT could be realized by using normal lens system. Since then, as a development and extension of Fourier transform, fractional Fourier transform has been applied in modern optics to solve the problems of light
*
Corresponding author. Tel.: +86-28-8541-2983; fax: +86-288524-2667. E-mail address: [email protected] (Y. Zeng).
propagation, imaging and information processing from a new point of view. Many new applications have been developed in recent years [2–7]. Fractional Fourier transform hologram (FRTH) is a novel hologram based on fractional Fourier transform [8,9]. It is the interference pattern of the reference wave and the fractional Fourier transform field of the object wave. That is to say, what is recorded on FRTH is not the object wave itself but the wavefront that is transformed from the object wave by a fractional Fourier transform system with a certain fractional order. So, a FRTH contains the information of both the object wave and the fractional Fourier transform order. The latter is decided by the characteristics of the fractional transform system, such as focal length and distance between object and lens. In order to reconstruct the image of
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 2 1 9 7 - 1
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Y. Zeng et al. / Optics Communications 215 (2003) 53–59
the original object, the original reference wave should be used to illuminate the FRTH, and the reconstructed wavefront should propagate through another fractional Fourier transform system whose fractional transform order must match the order of the original recording system. In this paper, based on the particularity of the FRTH in recording and reconstruction, the principles of multiple fractional Fourier transform hologram are presented. The FRTH of several objects with different fractional transform orders in different positions are recorded on one holographic plate by using a simple fractional Fourier transform system (in this paper we use Lohmann Itype optical system [10]). The images of different objects are reconstructed by different fractional Fourier transform system with relevant order. We can see that the multiple FRTH (M-FRTH) contains more anti-counterfeiting freedoms and it can be used as a new kind of optical security or anticounterfeiting system with much high anti-counterfeiting capacity.
2. The recording and reconstruction processes of M-FRTH 2.1. Recording process Fig. 1 presents the recording process of MFRTH with Lohmann I-type optical system, where L is a transform lens that is placed at zero of axis z. We suppose that several flat transparent objects g0 , g00 , g000 and g0000 are located at different positions, and they are all perpendicular to axis z. The centers of g0 is on axis z, while the centers of g00 , g000 and g0000 are
all off-axis from z. Their object functions are expressed as g0 ðx0 ; y0 ; z0 Þ, g00 ðx00 ; y00 ; z00 Þ, g000 ðx000 ; y000 ; 000 000 z000 Þ and g0000 ðx000 0 ; y0 ; z0 Þ. When the object g0 is illuminated by a monochrome light source, its fractional Fourier transform field with fractional order of P1 is obtained, expressed as g1 ðx1 ; y1 ; z1 Þ. 000 Similarly, g10 ðx01 ; y10 ; z01 Þ, g100 ðx001 ; y100 ; z001 Þ and g1000 ðx000 1 ; y1 ; 0 00 z000 Þ are fractional Fourier transform fields of g , g 1 0 0 and g0000 with respective fractional transform orders of P10 , P100 and P1000 . The fractional transform orders of P1 , P10 , P100 and P1000 are decided by the focal length and the distance between object plane and the lens. When illuminating by a reference light R and putting a holographic plate, respectively, at the planes of g1 , g10 , g100 , g1000 , the M-FRTH of several objects with different fractional transform orders are recorded on one holographic plate. 2.2. Reconstruction process Fig. 2 is the reconstruction setup of the MFRTH. Illuminating the hologram with the original reference wave R, the fractional Fourier transform fields of the object waves g0 ðx0 ; y0 ; z0 Þ, 000 000 g00 ðx00 ; y00 ; z00 Þ, g000 ðx000 ; y000 ; z000 Þ and g0000 ðx000 0 ; y0 ; z0 Þ 0 00 with fractional transform orders of P1 , P1 , P1 , P1000 will be reconstructed at the hologramÕs back surface. Because the fractional transform order changes with the distance between the transform lens and the hologram, different fractional transform orders of P2 , P20 , P200 , P2000 can be obtained when the lens L and the receiving plane are moved back and forth. If the matching conditions are satisfied, for example, P1 þ P2 ¼ 2, P10 þ P20 ¼ 2, P100 þ P200 ¼ 2, P1000 þ P2000 ¼ 2, the reconstructed images of g2 , g20 , g200 , g2000 will be obtained at different output planes.
Fig. 1. The optical setup for the recording of M-FRTH.
Y. Zeng et al. / Optics Communications 215 (2003) 53–59
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Fig. 2. Reconstruction of the M-FRTH.
3. The principles of recording and reconstruction of M-FRTH
ð2Þ
1
3.1. Recording principle To simplify the discussions, we only analyze the double exposure fractional Fourier transform hologram (D-FRTH) to explain the recording and reconstruction principles of M-FRTH. The FRTH records the distribution of the object wave in fractional Fourier domain by using holographic interference method while the D-FRTH records the FRTH of two different objects with different fractional transform orders on one holographic plate, the recording scheme is shown in Fig. 3, where g0 ðx0 Þ and g00 ðx0 Þ are two object functions, g1 ðx1 Þ and g10 ðx1 Þ are fractional Fourier transform functions of g0 ðx0 Þ and g00 ðx0 Þ with fractional transform orders of P1 and P10 , respectively (for Eq. (1), refer [3]). g1 ðx1 Þ ¼ F P1 ½g0 ðx0 Þ Z 1 BP1 ðx0 ; x1 Þg0 ðx0 Þ dx0 ; ¼
0 g10 ðx01 Þ ¼ F P1 g00 ðx0 Þ Z 1 BP10 ðx0 ; x01 Þg00 ðx0 Þ dx0 ; ¼
ð1Þ
where BP1 and BP10 are the kernel functions of the corresponding FRT. For Lohmann I-type optical system [10], the focal length f and the distance z between input plane and the lens should satisfy the following conditions: f ¼ f1 = sinðP p=2Þ; z ¼ f1 tanðP p=4Þ;
ð3Þ
where f1 ¼ f sinðP p=2Þ is called standard focal length, P is the fractional transform order, different z corresponds to different P. With (1)–(3), we can get Z g1 ðx1 Þ ¼ g0 ðx0 Þ exp ip x20 þ x21 =½kf sinðP1 p=2ÞtgðP1 p=2Þ
exp i2px0 x1 = kf sin2 ðP1 p=2Þ dx0 ;
1
Fig. 3. Recording of the D-FRTH.
ð4Þ
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Y. Zeng et al. / Optics Communications 215 (2003) 53–59
g10 ðx01 Þ
¼
Z
g00 ðx0 Þ exp ip x20 þ x02 1
=½kf sinðP10 p=2ÞtgðP10 p=2Þ
exp i2px0 x01 = kf sin2 ðP10 p=2Þ dx0 ; ð5Þ where k is the recording wavelength. Supposing R is the recording reference wave, t1 and t2 are exposure time. Under the linear recording condition, the amplitude transmission function sH of the hologram is proportional to the exposure time 2
2
sH / t1 jg1 ðx1 Þ þ Rj þ t2 jg10 ðx01 Þ þ Rj 2 2 ¼ t1 jg1 j þ jRj þ t1 g1 R þ t1 g1 R 2 2 þ t2 jg10 j þ jRj þ t2 g10 R þ t2 g10 R :
0
ð6Þ
The fractional Fourier transform fields of the object waves g0 ðx0 Þ and g00 ðx0 Þ with fractional transform orders of P1 and P10 will be reconstructed by illuminating the hologram with the original reference wave R. According to expression (6), the reconstructed wave is h i i ¼ t1 jg1 ðx1 Þj2 þ jRj2 R þ t1 g1 ðx1 Þ R2 þ t1 g1 ðx1 ÞjRj2 h 2 i þ t2 g10 x01 þ jRj2 R þ t2 g10 ðx01 Þ R2 ð7Þ
In Eq. (7), the first and the fourth parts are the zero-order wave and the halo wave; the second and the fifth parts denote conjugate images; the third and the sixth parts denote the original images. When the original image components are only considered, we can get 2
2
2
2
t1 jRj g1 ðx1 Þ ¼ t1 jRj F P1 ½g0 ðx0 Þ; 0
t2 jRj g10 ðx01 Þ ¼ t2 jRj F P1 ½g00 ðx0 Þ:
ð8Þ
The complex amplitude is A ¼ t1 jRj2 g1 ðxÞ þ t2 jRj2 g10 ðx01 Þ; and the intensity is: I / AA .
2
¼ t1 jRj F P1 þP2 ½g0 ðx0 Þ; n o 0 0 2 g20 ðx02 Þ ¼ F P2 t2 jRj F P1 ½g00 ðx0 Þ
ð10Þ
0
¼ t2 jRj2 F P1 þP2 ½g00 ðx0 Þ:
3.2. Reconstruction
þ t2 g10 ðx01 ÞjRj2 :
In order to obtain the images of objects g0 ðx0 Þ and g00 ðx0 Þ, the fractional Fourier transform operations with fractional orders of P2 and P20 should be performed. And their fractional orders of P2 and P20 must match the recording systemÕs fractional transform orders of P1 and P10 . So the amplitude distributions of the two original image components on the output planes x2 and x02 can be expressed as n o 2 g2 ðx2 Þ ¼ F P2 t1 jRj F P1 ½g0 ðx0 Þ
ð9Þ
Based on the periodicity of the FRT, when P1 þ P2 ¼ 4 or P10 þ P20 ¼ 4, we can observe the isometric and erect image of the original object on the output plane. Specially, when P1 þ P2 ¼ 2;
g2 ðx2 Þ / F P2 F P1 ½g0 ðx0 Þ ¼ F P2 þP1 ½g0 ðx0 Þ ¼ g0 ðx0 Þ;
P10 þ P20 ¼ 2;
g20 ðx02 Þ / F P2 F P1 ½g00 ðx0 Þ
0
0
0
0
¼ F P2 þP1 ½g00 ðx0 Þ ¼ g00 ðx0 Þ: ð11Þ The reconstructed wavefront is the corresponding object wavefront by coordinate inversion. At two different output planes, we can observe the isometric and inverted images of the original objects, respectively. So utilizing the periodicity of FRT and the dependency of reconstructed image position on the fractional transform order of the reconstruction system, we can observe the respective reconstructed images of different objects in different positions while other information can be considered as background noise. From the above discussions, we can find that the M-FRTH records not only the information of several different objects, but also the information of the corresponding systems, such as the focal length f and the distances z1 ; z2 ; z3 ; . . . between the objects and the lens. The reconstructed images of M-FRTH usually are located in different positions, while the reconstructed images of the common multiple exposure Fourier transform hologram
Y. Zeng et al. / Optics Communications 215 (2003) 53–59
can be observed only in one position and will superimpose and interfere with each other severely. The distances between the several reconstructed images of M-FRTH can be changed in a large spatial range by properly choosing the distances z1 ; z01 ; z001 ; . . . between the object and the lens. If the objects g0 ðx0 Þ; g00 ðx0 Þ; g000 ðx0 Þ; . . . are designed in different positions on the object plane, the influence of zero-order noise to a certain reconstructed image can be ignored and the interference among the reconstructed images can also be reduced greatly.
4. Experiment In the experiment, we fabricate a triple fractional Fourier transform hologram. The objects used in the experiment are transparent films that contain picture and characters, and the focal length f of the lens used for the recording and reconstruction of M-FRTH is always 300 mm. The recording process can be divided into three steps:
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(1) Illuminate the transparent object g0 ðx0 Þ (picture (a) in Fig. 4) with collimated He–Ne laser beam. The angle between the reference wave R and the object wave is about 30°. The fractional Fourier transform order is P1 ¼ 1:5 and the exposure time is t1 ¼ 10 s; (2) keep the reference wave R and the illuminating wave unchanged, replace g0 ðx0 Þ with a transparent object g00 ðx0 Þ (letter ‘‘E’’ in Fig. 4(b)). The fractional Fourier transform order is P10 ¼ 1:25. Expose the same holographic plate again, and the exposure time is t2 ¼ 13 s; (3) still keep the reference wave R and the illuminating wave unchanged, replace g00 ðx0 Þ with another offaxis transparent object g000 ðx0 Þ (letter ‘‘B’’ in Fig. 4(c)). The fractional Fourier transform order is P100 ¼ 1:00. Expose the same holographic plate the third time, and the exposure time is t3 ¼ 16 s. After development and fixation, a M-FRTH is obtained. By illuminating the M-FRTH with the original reference wave and decoding with the order-matched FRT systems, the reconstructed images are observed. Pictures (d–f) in Fig. 4 show the corresponding images captured with CCD camera. In
Fig. 4. The objects used for the recording of M-FRTH and the relevant reconstructed images, (a–c) are the objects, (d–f) are the reconstructed images. (d) Fractional orders of the recording and reconstruction are P1 ¼ 1:5 and P2 ¼ 0:5; the distance between reconstruction image and the lens is Z2 ¼ 87:9 mm; the off-axis distance is X2 ¼ 0 mm; center coordinates of the image is g2 ( 0, 0, 87.9). (e) Fractional orders of the recording and reconstruction are P10 ¼ 1:25 and P20 ¼ 0:75; the distance between reconstruction image and the lens is Z20 ¼ 185:2 mm; the off-axis distance is X20 ¼ 90 mm; center coordinates of the image is g20 (90, 0, 185.2). (f) Fractional orders of the recording and reconstruction are P100 ¼ 1:00 and P200 ¼ 1:00; the distance between reconstruction image and the lens is Z200 ¼ 300 mm; the off-axis distance is Y200 ¼ 110 mm; center coordinates of the image is g200 (0, 110, 300).
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Fig. 4(d), the transform order of the reconstruction system is P2 ¼ 0:5, and the distance between reconstruction image and the lens is Z2 ¼ 87:9 mm. The off-axis distance is X2 ¼ 0 mm; center coordinates of the image is g2 (0, 0, 87.9). Here, only the image of the picture can be seen clearly on the output plane, while the other two images are blurred background noises and cannot be recognized. In Fig. 4(e), the transform order of the reconstruction system is P20 ¼ 0:75, and the distance between reconstruction image and the lens is Z20 ¼ 185:2 mm. The off-axis distance is X20 ¼ 90 mm, center coordinates of the image is g20 (90, 0, 185.2). In this case, only the image of ‘‘E’’ is clear on the output plane, while the other two images are blurred background noises and cannot be recognized. In Fig. 4(f), the transform order of the reconstruction system is P200 ¼ 1:00, and the distance between reconstruction image and the lens is Z200 ¼ 300 mm. The off-axis distance is Y200 ¼ 110 mm, center coordinates of the image is g200 ( 0, 110, 300). Similarly, only the image of ‘‘B’’ is clear on the output plane, and the other two images are blurred noises and cannot be recognized. It can be seen that, if the FRTH of three objects are recorded with different fractional transform orders of P1 , P10 , P100 on the same holographic plate, the positions of the reconstruction planes with matched fractional transform orders of P2 , P20 , P200 are different.
5. Anti-counterfeiting characteristics of M-FRTH The M-FRTH records not only the information of different objects, but also the information of the corresponding recording systems, such as the focal length f of the lens and the distances of z1 ; z2 ; z3 ; . . . between the objects and the lens. Thus we can encode the scale of each object during the recording process of the M-FRTH. During the reconstruction process, in order to decode the multiple object patterns, each transform order of fractional Fourier transform system should match the transform order of the corresponding recording system. If we can observe the clear images of all recorded objects in the output planes of respective fractional Fourier transform systems, the
hologram is original and true. Otherwise it is forged. Due to the particularity of recording and reconstruction processes, the M-FRTH has a much high reliability and anti-counterfeiting ability when compared with common multiple exposure hologram or single exposure FRTH, and can be widely used in the fabrication of the trademarks, identity cards and the cash.
6. Conclusion Based on the theory of fractional Fourier transform, a recording method of M-FRTH is developed by moving the object planes parallelly. Compared with conventional recording method of rotating the holographic plate, our method can precisely design the positions of reconstructed images in three-dimensional space, and tune the reconstructed images in large spatial range. Moreover, due to the particularity of M-FRTH, it requires several matched fractional Fourier transform systems to reconstruct the recorded images correctly, which increases the anti-counterfeiting ability of M-FRTH. M-FRTH is possessed of quite a few advantages over conventional multiple hologram, and can be considered as a new way to fabricate multiple holograms.
Acknowledgements This research is supported by National Natural Science Foundation of China and Key State Lab of Microfabrication Optical Technology, Institute of Optical-electrical Technology, the Chinese Academy of Sciences.
References [1] V. Namias, J. Inst. Math. Appl. 25 (3) (1980) 241. [2] D. Mendlovic, H.M. Ozaktas, J. Opt. Soc. Am. (A) 10 (9) (1993) 1875. [3] A.W. Lohmann, J. Opt. Soc. Am. (A) 10 (10) (1993) 2181. [4] H.M. Ozaktas, D. Mendlovic, J. Opt. Soc. Am. A 12 (4) (1995) 743.
Y. Zeng et al. / Optics Communications 215 (2003) 53–59 [5] D. Mendlovic, Y. Bitran, R.G. Dorsch, Appl. Opt. 34 (32) (1995) 7541. [6] G. Unnikrishnan, Kehar Singh, Opt. Eng. 39 (11) (2000) 2853. [7] Shutian Liu, Quanlin Mi, Banghe Zhu, Opt. Lett. 26 (16) (2001) 1242.
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[8] Guo Yongkang, Huang Qizhong, Du Jinglei, Opt. Acta 6 (19) (1999) 821 (in Chinese). [9] Zeng Yangsu, Xie Shiwei, Zhang Yixiao, Gao Fuhua, Guo Yongkang, Chin. J. Lasers 6 (28) (2001) (in Chinese). [10] G. Dorsch, A.W. Lohmann, Appl. Opt. 34 (20) (1995) 4111.
Principle and application of multiple fractional Fourier transform holography Yangsu Zeng a,b,*, Yongkang Guo a, Fuhua Gao a, Jianhua Zhu a a b
Department of Physics, Sichuan University, Chengdu, 610064, PR China Department of Physics, Shaoyang University, Hunan, 422000, PR China
Received 26 June 2002; received in revised form 12 September 2002; accepted 8 November 2002
Abstract In this paper, the principle of multiple fractional Fourier transform hologram (FRTH) is presented, and its characteristics based on the particularity in recording and reconstruction are analyzed. With this method, a multiple FRTH of several objects with different fractional transform orders is fabricated on one holographic plate. It requires a matched multiple fractional Fourier transform system to reconstruct the recorded images correctly. The potential application of multiple FRTH in optical security or anti-counterfeiting system is also discussed. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Fractional Fourier Transform (FRT); Multiple holography; Optical anti-counterfeiting
1. Introduction Fractional Fourier Transform (FRT) was discussed by Namias in 1980 [1] as a mathematical tool in quantum mechanics. In 1993, Mendlovic and Ozaktas implemented the FRT optically when they studied gradient index (GRIN) fiber. In the same year, Lohmann proposed that FRT could be realized by using normal lens system. Since then, as a development and extension of Fourier transform, fractional Fourier transform has been applied in modern optics to solve the problems of light
*
Corresponding author. Tel.: +86-28-8541-2983; fax: +86-288524-2667. E-mail address: [email protected] (Y. Zeng).
propagation, imaging and information processing from a new point of view. Many new applications have been developed in recent years [2–7]. Fractional Fourier transform hologram (FRTH) is a novel hologram based on fractional Fourier transform [8,9]. It is the interference pattern of the reference wave and the fractional Fourier transform field of the object wave. That is to say, what is recorded on FRTH is not the object wave itself but the wavefront that is transformed from the object wave by a fractional Fourier transform system with a certain fractional order. So, a FRTH contains the information of both the object wave and the fractional Fourier transform order. The latter is decided by the characteristics of the fractional transform system, such as focal length and distance between object and lens. In order to reconstruct the image of
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 2 1 9 7 - 1
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Y. Zeng et al. / Optics Communications 215 (2003) 53–59
the original object, the original reference wave should be used to illuminate the FRTH, and the reconstructed wavefront should propagate through another fractional Fourier transform system whose fractional transform order must match the order of the original recording system. In this paper, based on the particularity of the FRTH in recording and reconstruction, the principles of multiple fractional Fourier transform hologram are presented. The FRTH of several objects with different fractional transform orders in different positions are recorded on one holographic plate by using a simple fractional Fourier transform system (in this paper we use Lohmann Itype optical system [10]). The images of different objects are reconstructed by different fractional Fourier transform system with relevant order. We can see that the multiple FRTH (M-FRTH) contains more anti-counterfeiting freedoms and it can be used as a new kind of optical security or anticounterfeiting system with much high anti-counterfeiting capacity.
2. The recording and reconstruction processes of M-FRTH 2.1. Recording process Fig. 1 presents the recording process of MFRTH with Lohmann I-type optical system, where L is a transform lens that is placed at zero of axis z. We suppose that several flat transparent objects g0 , g00 , g000 and g0000 are located at different positions, and they are all perpendicular to axis z. The centers of g0 is on axis z, while the centers of g00 , g000 and g0000 are
all off-axis from z. Their object functions are expressed as g0 ðx0 ; y0 ; z0 Þ, g00 ðx00 ; y00 ; z00 Þ, g000 ðx000 ; y000 ; 000 000 z000 Þ and g0000 ðx000 0 ; y0 ; z0 Þ. When the object g0 is illuminated by a monochrome light source, its fractional Fourier transform field with fractional order of P1 is obtained, expressed as g1 ðx1 ; y1 ; z1 Þ. 000 Similarly, g10 ðx01 ; y10 ; z01 Þ, g100 ðx001 ; y100 ; z001 Þ and g1000 ðx000 1 ; y1 ; 0 00 z000 Þ are fractional Fourier transform fields of g , g 1 0 0 and g0000 with respective fractional transform orders of P10 , P100 and P1000 . The fractional transform orders of P1 , P10 , P100 and P1000 are decided by the focal length and the distance between object plane and the lens. When illuminating by a reference light R and putting a holographic plate, respectively, at the planes of g1 , g10 , g100 , g1000 , the M-FRTH of several objects with different fractional transform orders are recorded on one holographic plate. 2.2. Reconstruction process Fig. 2 is the reconstruction setup of the MFRTH. Illuminating the hologram with the original reference wave R, the fractional Fourier transform fields of the object waves g0 ðx0 ; y0 ; z0 Þ, 000 000 g00 ðx00 ; y00 ; z00 Þ, g000 ðx000 ; y000 ; z000 Þ and g0000 ðx000 0 ; y0 ; z0 Þ 0 00 with fractional transform orders of P1 , P1 , P1 , P1000 will be reconstructed at the hologramÕs back surface. Because the fractional transform order changes with the distance between the transform lens and the hologram, different fractional transform orders of P2 , P20 , P200 , P2000 can be obtained when the lens L and the receiving plane are moved back and forth. If the matching conditions are satisfied, for example, P1 þ P2 ¼ 2, P10 þ P20 ¼ 2, P100 þ P200 ¼ 2, P1000 þ P2000 ¼ 2, the reconstructed images of g2 , g20 , g200 , g2000 will be obtained at different output planes.
Fig. 1. The optical setup for the recording of M-FRTH.
Y. Zeng et al. / Optics Communications 215 (2003) 53–59
55
Fig. 2. Reconstruction of the M-FRTH.
3. The principles of recording and reconstruction of M-FRTH
ð2Þ
1
3.1. Recording principle To simplify the discussions, we only analyze the double exposure fractional Fourier transform hologram (D-FRTH) to explain the recording and reconstruction principles of M-FRTH. The FRTH records the distribution of the object wave in fractional Fourier domain by using holographic interference method while the D-FRTH records the FRTH of two different objects with different fractional transform orders on one holographic plate, the recording scheme is shown in Fig. 3, where g0 ðx0 Þ and g00 ðx0 Þ are two object functions, g1 ðx1 Þ and g10 ðx1 Þ are fractional Fourier transform functions of g0 ðx0 Þ and g00 ðx0 Þ with fractional transform orders of P1 and P10 , respectively (for Eq. (1), refer [3]). g1 ðx1 Þ ¼ F P1 ½g0 ðx0 Þ Z 1 BP1 ðx0 ; x1 Þg0 ðx0 Þ dx0 ; ¼
0 g10 ðx01 Þ ¼ F P1 g00 ðx0 Þ Z 1 BP10 ðx0 ; x01 Þg00 ðx0 Þ dx0 ; ¼
ð1Þ
where BP1 and BP10 are the kernel functions of the corresponding FRT. For Lohmann I-type optical system [10], the focal length f and the distance z between input plane and the lens should satisfy the following conditions: f ¼ f1 = sinðP p=2Þ; z ¼ f1 tanðP p=4Þ;
ð3Þ
where f1 ¼ f sinðP p=2Þ is called standard focal length, P is the fractional transform order, different z corresponds to different P. With (1)–(3), we can get Z g1 ðx1 Þ ¼ g0 ðx0 Þ exp ip x20 þ x21 =½kf sinðP1 p=2ÞtgðP1 p=2Þ
exp i2px0 x1 = kf sin2 ðP1 p=2Þ dx0 ;
1
Fig. 3. Recording of the D-FRTH.
ð4Þ
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Y. Zeng et al. / Optics Communications 215 (2003) 53–59
g10 ðx01 Þ
¼
Z
g00 ðx0 Þ exp ip x20 þ x02 1
=½kf sinðP10 p=2ÞtgðP10 p=2Þ
exp i2px0 x01 = kf sin2 ðP10 p=2Þ dx0 ; ð5Þ where k is the recording wavelength. Supposing R is the recording reference wave, t1 and t2 are exposure time. Under the linear recording condition, the amplitude transmission function sH of the hologram is proportional to the exposure time 2
2
sH / t1 jg1 ðx1 Þ þ Rj þ t2 jg10 ðx01 Þ þ Rj 2 2 ¼ t1 jg1 j þ jRj þ t1 g1 R þ t1 g1 R 2 2 þ t2 jg10 j þ jRj þ t2 g10 R þ t2 g10 R :
0
ð6Þ
The fractional Fourier transform fields of the object waves g0 ðx0 Þ and g00 ðx0 Þ with fractional transform orders of P1 and P10 will be reconstructed by illuminating the hologram with the original reference wave R. According to expression (6), the reconstructed wave is h i i ¼ t1 jg1 ðx1 Þj2 þ jRj2 R þ t1 g1 ðx1 Þ R2 þ t1 g1 ðx1 ÞjRj2 h 2 i þ t2 g10 x01 þ jRj2 R þ t2 g10 ðx01 Þ R2 ð7Þ
In Eq. (7), the first and the fourth parts are the zero-order wave and the halo wave; the second and the fifth parts denote conjugate images; the third and the sixth parts denote the original images. When the original image components are only considered, we can get 2
2
2
2
t1 jRj g1 ðx1 Þ ¼ t1 jRj F P1 ½g0 ðx0 Þ; 0
t2 jRj g10 ðx01 Þ ¼ t2 jRj F P1 ½g00 ðx0 Þ:
ð8Þ
The complex amplitude is A ¼ t1 jRj2 g1 ðxÞ þ t2 jRj2 g10 ðx01 Þ; and the intensity is: I / AA .
2
¼ t1 jRj F P1 þP2 ½g0 ðx0 Þ; n o 0 0 2 g20 ðx02 Þ ¼ F P2 t2 jRj F P1 ½g00 ðx0 Þ
ð10Þ
0
¼ t2 jRj2 F P1 þP2 ½g00 ðx0 Þ:
3.2. Reconstruction
þ t2 g10 ðx01 ÞjRj2 :
In order to obtain the images of objects g0 ðx0 Þ and g00 ðx0 Þ, the fractional Fourier transform operations with fractional orders of P2 and P20 should be performed. And their fractional orders of P2 and P20 must match the recording systemÕs fractional transform orders of P1 and P10 . So the amplitude distributions of the two original image components on the output planes x2 and x02 can be expressed as n o 2 g2 ðx2 Þ ¼ F P2 t1 jRj F P1 ½g0 ðx0 Þ
ð9Þ
Based on the periodicity of the FRT, when P1 þ P2 ¼ 4 or P10 þ P20 ¼ 4, we can observe the isometric and erect image of the original object on the output plane. Specially, when P1 þ P2 ¼ 2;
g2 ðx2 Þ / F P2 F P1 ½g0 ðx0 Þ ¼ F P2 þP1 ½g0 ðx0 Þ ¼ g0 ðx0 Þ;
P10 þ P20 ¼ 2;
g20 ðx02 Þ / F P2 F P1 ½g00 ðx0 Þ
0
0
0
0
¼ F P2 þP1 ½g00 ðx0 Þ ¼ g00 ðx0 Þ: ð11Þ The reconstructed wavefront is the corresponding object wavefront by coordinate inversion. At two different output planes, we can observe the isometric and inverted images of the original objects, respectively. So utilizing the periodicity of FRT and the dependency of reconstructed image position on the fractional transform order of the reconstruction system, we can observe the respective reconstructed images of different objects in different positions while other information can be considered as background noise. From the above discussions, we can find that the M-FRTH records not only the information of several different objects, but also the information of the corresponding systems, such as the focal length f and the distances z1 ; z2 ; z3 ; . . . between the objects and the lens. The reconstructed images of M-FRTH usually are located in different positions, while the reconstructed images of the common multiple exposure Fourier transform hologram
Y. Zeng et al. / Optics Communications 215 (2003) 53–59
can be observed only in one position and will superimpose and interfere with each other severely. The distances between the several reconstructed images of M-FRTH can be changed in a large spatial range by properly choosing the distances z1 ; z01 ; z001 ; . . . between the object and the lens. If the objects g0 ðx0 Þ; g00 ðx0 Þ; g000 ðx0 Þ; . . . are designed in different positions on the object plane, the influence of zero-order noise to a certain reconstructed image can be ignored and the interference among the reconstructed images can also be reduced greatly.
4. Experiment In the experiment, we fabricate a triple fractional Fourier transform hologram. The objects used in the experiment are transparent films that contain picture and characters, and the focal length f of the lens used for the recording and reconstruction of M-FRTH is always 300 mm. The recording process can be divided into three steps:
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(1) Illuminate the transparent object g0 ðx0 Þ (picture (a) in Fig. 4) with collimated He–Ne laser beam. The angle between the reference wave R and the object wave is about 30°. The fractional Fourier transform order is P1 ¼ 1:5 and the exposure time is t1 ¼ 10 s; (2) keep the reference wave R and the illuminating wave unchanged, replace g0 ðx0 Þ with a transparent object g00 ðx0 Þ (letter ‘‘E’’ in Fig. 4(b)). The fractional Fourier transform order is P10 ¼ 1:25. Expose the same holographic plate again, and the exposure time is t2 ¼ 13 s; (3) still keep the reference wave R and the illuminating wave unchanged, replace g00 ðx0 Þ with another offaxis transparent object g000 ðx0 Þ (letter ‘‘B’’ in Fig. 4(c)). The fractional Fourier transform order is P100 ¼ 1:00. Expose the same holographic plate the third time, and the exposure time is t3 ¼ 16 s. After development and fixation, a M-FRTH is obtained. By illuminating the M-FRTH with the original reference wave and decoding with the order-matched FRT systems, the reconstructed images are observed. Pictures (d–f) in Fig. 4 show the corresponding images captured with CCD camera. In
Fig. 4. The objects used for the recording of M-FRTH and the relevant reconstructed images, (a–c) are the objects, (d–f) are the reconstructed images. (d) Fractional orders of the recording and reconstruction are P1 ¼ 1:5 and P2 ¼ 0:5; the distance between reconstruction image and the lens is Z2 ¼ 87:9 mm; the off-axis distance is X2 ¼ 0 mm; center coordinates of the image is g2 ( 0, 0, 87.9). (e) Fractional orders of the recording and reconstruction are P10 ¼ 1:25 and P20 ¼ 0:75; the distance between reconstruction image and the lens is Z20 ¼ 185:2 mm; the off-axis distance is X20 ¼ 90 mm; center coordinates of the image is g20 (90, 0, 185.2). (f) Fractional orders of the recording and reconstruction are P100 ¼ 1:00 and P200 ¼ 1:00; the distance between reconstruction image and the lens is Z200 ¼ 300 mm; the off-axis distance is Y200 ¼ 110 mm; center coordinates of the image is g200 (0, 110, 300).
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Y. Zeng et al. / Optics Communications 215 (2003) 53–59
Fig. 4(d), the transform order of the reconstruction system is P2 ¼ 0:5, and the distance between reconstruction image and the lens is Z2 ¼ 87:9 mm. The off-axis distance is X2 ¼ 0 mm; center coordinates of the image is g2 (0, 0, 87.9). Here, only the image of the picture can be seen clearly on the output plane, while the other two images are blurred background noises and cannot be recognized. In Fig. 4(e), the transform order of the reconstruction system is P20 ¼ 0:75, and the distance between reconstruction image and the lens is Z20 ¼ 185:2 mm. The off-axis distance is X20 ¼ 90 mm, center coordinates of the image is g20 (90, 0, 185.2). In this case, only the image of ‘‘E’’ is clear on the output plane, while the other two images are blurred background noises and cannot be recognized. In Fig. 4(f), the transform order of the reconstruction system is P200 ¼ 1:00, and the distance between reconstruction image and the lens is Z200 ¼ 300 mm. The off-axis distance is Y200 ¼ 110 mm, center coordinates of the image is g200 ( 0, 110, 300). Similarly, only the image of ‘‘B’’ is clear on the output plane, and the other two images are blurred noises and cannot be recognized. It can be seen that, if the FRTH of three objects are recorded with different fractional transform orders of P1 , P10 , P100 on the same holographic plate, the positions of the reconstruction planes with matched fractional transform orders of P2 , P20 , P200 are different.
5. Anti-counterfeiting characteristics of M-FRTH The M-FRTH records not only the information of different objects, but also the information of the corresponding recording systems, such as the focal length f of the lens and the distances of z1 ; z2 ; z3 ; . . . between the objects and the lens. Thus we can encode the scale of each object during the recording process of the M-FRTH. During the reconstruction process, in order to decode the multiple object patterns, each transform order of fractional Fourier transform system should match the transform order of the corresponding recording system. If we can observe the clear images of all recorded objects in the output planes of respective fractional Fourier transform systems, the
hologram is original and true. Otherwise it is forged. Due to the particularity of recording and reconstruction processes, the M-FRTH has a much high reliability and anti-counterfeiting ability when compared with common multiple exposure hologram or single exposure FRTH, and can be widely used in the fabrication of the trademarks, identity cards and the cash.
6. Conclusion Based on the theory of fractional Fourier transform, a recording method of M-FRTH is developed by moving the object planes parallelly. Compared with conventional recording method of rotating the holographic plate, our method can precisely design the positions of reconstructed images in three-dimensional space, and tune the reconstructed images in large spatial range. Moreover, due to the particularity of M-FRTH, it requires several matched fractional Fourier transform systems to reconstruct the recorded images correctly, which increases the anti-counterfeiting ability of M-FRTH. M-FRTH is possessed of quite a few advantages over conventional multiple hologram, and can be considered as a new way to fabricate multiple holograms.
Acknowledgements This research is supported by National Natural Science Foundation of China and Key State Lab of Microfabrication Optical Technology, Institute of Optical-electrical Technology, the Chinese Academy of Sciences.
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