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The temperature dependence of the linewidth of iron group ions in MgO
Optics Communications 228 (2003) 271–278 www.elsevier.com/locate/optcom
Superresolution technology for reduction of the far-field diffraction spot size in the laser free-space communication system Jia Jia *, Changhe Zhou, Liren Liu Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800-211, Shanghai 201800, PR China Received 21 May 2003; received in revised form 29 August 2003; accepted 14 October 2003
Abstract In the free-space laser communication there is sometimes a strong need for reduction of the diffraction spot size in the far field. In this paper, instead of the usage of the larger size aperture lens in the free-space laser communication system, we introduce diffractive superresolution technology to design and fabricate a cheap pure-phase plate for realizing the smaller spot size than the usual Airy spot size, which can decrease the weight and size of the emitting lens. We have calculated 2, 3, 4, 5 circulation zones for optimizing the highest energy compression (Strehl ratio) with the constraint of the First zero ratio value G ¼ 0.8. Numerical results show that the 2- or 3-circular zone pure-phase plate can yield the highest Strehl ratio ðS 0:59Þ with the constraint of G ¼ 0.8, but the 4, 5 circular zone binary phase (0,pÞ plates are calculated to yield the result of S 0:57 with G ¼ 0.8. We have fabricated 2- and 3-circular zone binary phase plate with binary optics technology. Finally, we have established an experimental system for simulation of the freespace laser communication to verify the advantage of the superresolution phase plate. Detailed experiments are presented. Ó 2003 Elsevier B.V. All rights reserved. PACS: 42.30.Kq; 42.40.Jv; 42.79.Cj; 42.79.Sz; 42.82.Cr Keywords: Fourier optics; Computer-generated holograms; Zone plates; Optical-communication systems; Lithography
1. Introduction In the free-space laser communication application, the intensity at the far field is the Fraunhofer
*
Corresponding author. Tel.: +86215991-1214; +86216991-8800. E-mail address: [email protected] (J. Jia).
fax:
diffraction of the incident light, which is also the Fourier transform of the intensity of the incident light. The size of the diffraction spot in the receive port in the free-space laser communication system can also be calculated by the well-known Airy pattern: x ¼ 1:22ðkL=DÞ [1,2], where D is the aperture of the emitting lens and L is the distance between the emitting lens and the receiver and k is the wavelength of the laser. In the specific array
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.10.011
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J. Jia et al. / Optics Communications 228 (2003) 271–278
laser communication applications, as shown in Fig. 1, there is need for a technology that can decrease the size of the diffraction spot in the receiver port because in the free-space laser communication a smaller diffraction spot in the receive port makes the transmit data more secure. The wavelength of the laser and the distance between the emitting lens and the received are usually fixed. So a larger aperture lens is often employed to realize the small far-field spot. However, a largersized lens is very expensive, and worse, a largersized lens is usually not available due to the fabrication technology. Superresolution technology provides an attractive approach to this application. Superresolution technique is one of the methods that realize the diffraction spot smaller than the Airy spot when a superresolution pure-phase plate is employed. The first significant study of superresolution was put forward by Francia [3]. Much attention was devoted to the design of the superresolution phase plate for its applications in enhanced resolution confocal system [4], increased storage in optical disk system [5], etc. More sophisticated methods based on continuous amplitude transmittance function have also been developed [6]. Hybrid
1 Laser 1 Receiver 1 x Receiver 2 Laser 2
2 Distance L
Laser N
Receiver N
Fig. 1. Illustration of an arrayed laser communication system for the multi-channel increased transmitting capacity. Each laser should generate its Frauhofer diffraction in the far field. 1, 2 means the intensity distribution in the laser 1, 2 diffraction field. The signal channels should not disturb the nearby channel for the diffraction effect of each channel. The receivers can be put at an optimized closer distance when a superresolution phase plate is used without resorting to a larger aperture laser emitting lens.
amplitude-phase elements with two zones, as well as elements for use in optical pick-up heads have been studied [7]. Various superresolution compression parameters such as Strehl ratio (S), First zero (G), have been defined to describe the superresolution effect. The Strehl ratio is defined as the ratio between the intensity of the superresolution pattern and the intensity of the Airy pattern, both calculated at the origin. The First zero is defined as the ratio between the first zero position of the superresolution pattern and the first zero position of the airy pattern. For more details of superresolution technology and its applications, one can see [6–9]. The widely used Full-width-half-maximum (FWHM) is closely related with the first zero position (G), and the central-lobe energy percentage, the ratio of the central lobe energy at the zero order to the whole energy at all diffraction orders, is closed related with the Strehl ratio (S). As the diffraction efficiency is the most important parameter for practical applications, we will present the superresolution scheme of the pure-phase plate in this paper because of high diffraction efficiency. As to the best of our knowledge, no one has pointed out the possibility that superresolution technology can be applied in the free-space laser communication system. We introduce the superresolution technology into the free-space laser communication system to compress the far-field diffractive spot size. With this proposed technology, we typically place a filter at the exit pupil of the system and thus the smaller diffraction spot can be obtained with the same aperture lens. In this paper, we prescribe the criterion of G 6 0:8, while S should be as large as possible for practical application. If G is too small, such as G 6 0:6, the mainlobe in the diffraction spot is too small to be applied in practice. If G is too large such as G ffi 1, the too small reduction of the diffraction spot size is not interesting for practical use. In this paper, we have calculated the 2, 3, 4, 5 circular zones for optimizing the highest Strehl ratio with the constraint of the First zero value G ¼ 0:8, and have found the optimized radius and phases of the phase plate. Finally, we have set up an experimental system for simulation of the Fourier transform in the laser long-distance communication. The experimental results have verified
J. Jia et al. / Optics Communications 228 (2003) 271–278
the numerical results and confirm the effectiveness of the superresolution phase plate.
2. Review of superresolution theory In a traditional sense, the Airy resolution is defined by the distance from the first zero position of the main peak intensity spot. Superresolution technology means that we can realize a smaller diffraction spot than an Airy diffraction spot. A pure-phase plate that is put before (or after ) the diffraction-limited lens is involved to realize the superresolution effect, as is shown in Fig. 2. According to the superresolution diffractive theory [6] the normalized field measured at the observation plane is given by: Z 1 uðm; lÞ ¼ P ðqÞ expðjlq2 =2ÞJ0 ðmqÞqdq; ð1Þ
uð0; lÞ ¼
Z
273
1
P ðqÞ expðjlq2 =2Þdq;
where m ¼ 0 in the axial plane. So in the focal plane, the radial amplitude distribution is the Hankel transform of the pupil function, while the axis amplitude distribution is the Fourier transform of the pupil function. In the superresolution field, there are many methods to design the pure-phase plate. Due to ease of fabrication, we choose the circular binary phase-only plate in this paper. The structure of the 3 zone circular binary phase-only plate is shown in Fig. 3. If we put the circular phase-only plate into the optical system, the diffractive field can be described as wðnÞ ¼
N X
expði/j Þ½a2j 2J1 ðaj nÞ=aj n
j¼1
a2j1 2J1 ðaj1 nÞ=aj1 n;
0
ð4Þ
where aj and /j represent the radius and phase of the jth zone, respectively. If the binary phases are chosen to be only 0 and /, we can rewrite Eq. (4) as
m ¼ kr sinðaÞ; 2
l ¼ 4kzðsinðaÞ=2Þ ;
wðnÞ ¼ 2J1 ðnÞ=n ½1 expði/0 Þð1Þ k ¼ 2p=k;
ð3Þ
0
sinðaÞ ¼ NA;
where l, m are the axial and radial coordinates, P ðqÞ is the pupil function (which is also called the diffractive phase plate). From Eq. (1), we can get Z 1 uðm; 0Þ ¼ P ðqÞJ0 ðmtÞdt; ð2Þ 0
where l ¼ 0 in the focal plane
Superresolution pure-phase plate
Collimated laser beam (λ)
N 1 X
ð1Þj a2j 2J1 ðaj nÞ=aj n:
N þ1
ð5Þ
j¼1
In numerical simulations, we will apply Eq. (5) to calculate the 2, 3, 4, 5 circular zones of the phase plate. According to the compression criterion, we will find the optimized parameters: the radius and phase value of each zone.
focal plane
D CCD camera
Fig. 2. The superresolution experimental system for simulation of the far-field diffraction in the long-distance free-space laser communication system.
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J. Jia et al. / Optics Communications 228 (2003) 271–278
0
Φ 0 1.0 a
b
Fig. 3. The normalized structure of the 3 zone binary purephase plate, a, b are the radii of the phase-transition circles between the phase 0 and U.
3. Optimization algorithms and numerical results We have used the global searching algorithms in our computer simulations. The global searching algorithm is to find the optimized result in the whole searching areas with changing each radius at a minimum searching step. The advantage of this algorithm is that it can find the optimized result, but the searching time is quite long. Numerical results of the Strehl ratio (S) and First zero ðS ¼ 0:8Þ and the central-lobe energy percentage with the radius (a) and the phase are given in Table 1. From Table 1, we know that the largest Strehl ratio S ¼ 0:5911, which is the Strehl ratio of the 2 zone phase plate. From numerical results of the First zero and Strehl ratio with two radius a and b of the 3 zone phase plate, we find the best optimized result under the constraint
of G ¼ 0:8 as: a ¼ 0:09; b ¼ 0:36; U ¼ 0:9p; G ¼ 0:7979; SMAX ¼ 0:5835. The central-lobe energy percentage that is closely related with the Strehl ratio is calculated and given in Table 1. The maximum central-lobe energy percentage is 39% in Table 1 while it is 84% for the Airy diffraction distribution. The far-field intensity distribution of this optimized numerical result is shown in Fig. 4. From Fig. 4, we can see that the superresolution technology can yield a smaller diffractive size in the far field than that in the Airy spot. We have also studied the effect of the multiple phase modulation for the 3 zone plate with the same optimized two radius a ¼ 0:09 and b ¼ 0:36. The inner zone phase is always assumed to be zero. The optimized phases are as follows: /1 ¼ 0:00p; /2 ¼ 0:06p; /3 ¼ 0:86p, and the Strehl ratio Smax ¼ 0:5905. Note that this optimized Strehl ratio S ¼ 0:59 with the multiple phases is a little higher than the optimized S ¼ 0:5835 with the binary phases. This result illustrates that the multiple-phase modulation can indeed improve the performance but at a much higher cost for fabrication than the binary phase modulation. The numerical simulation results of the 4 and 5 zone phase plates give the maximum S 0:57 with G ¼ 0:8. We realized from our numerical result that the S could not be increased with the increase of the zone numbers. Numerical comparison of the 2- and 3-zones plate in Table 1 is given in Fig. 4. From Fig. 4, we can see that FWHM (Superresolved) is smaller than the FWHM (Airy). If the maximum intensity of the superresolved pattern is normalized from 0.59 to 1, the FWHM keeps unchanged because of the fixed first zero position.
Table 1 Numerical result of the multi-zones superresolution phase plate with the prescribed First zero (G ¼ 0:8) Zone no.
Search step
2 3 4
0.01 0.03 0.08
5
0.10
Optimized numerical result Normalized radius
U ðpÞ
Strehl ratio
Central lobe energy (%)
a ¼ 0.34 a ¼ 0.09, b ¼ 0.36 a ¼ 0.16, b ¼ 0.24, c ¼ 0.40 a ¼ 0.1, b ¼ 0.2, c ¼ 0.4, d ¼ 0.9
1.0 0.9 0.9
0.5911 0.5835 0.5645
38.83 36.06 35.06
0.9
0.5587
13.16
See Fig. 3 for the meaning of the normalized radius a, b.
J. Jia et al. / Optics Communications 228 (2003) 271–278
275
superresolved pattern(3-zones) superresolved pattern(2-zones) Airy pattern
0.9
Normalized Intensity
0.8 0.7 0.6
1/2FWHM (Airy)
0.5
1/2FWHM (Superresolved)
0.4 0.3 0.2 0.1 0 0
0.5
1.0
1.5
2.0
2.5
3.0
Relative radial distance
Fig. 4. Theoretical far-field intensity distribution of the Airy (solid line) and the superresolution diffraction (dashed line) patterns with the 2- and 3-zones plates in Table 1. The superresolution pattern generated by the 2- and 3-zones plates are quite close to each other. The receiver 2 can be at the first zero position of the superresolution pattern that is 0.8 times closer to receiver 1 than the Airy pattern. FWHM is the Full-width-half-maximum of the diffraction distribution. We can see that the FWHM (Superresolved) is smaller than the FWHM (Airy).
Thus, the FWHM of the superresolved distribution is indeed reduced compared with the FWHM of the Airy distribution. From Fig. 4, we can also see that there is little difference between 2- and 3zones plates. Fabrication of the three zone phase plate can demonstrate the capability that we can also fabricate the 2-zone phase plate. So we choose the 3-zone binary phase plate for the experiment, which is given in the next section.
4. Fabrication and experimental results VLSI techniques have been adopted in our experiment to obtain the phase plate. The mask is made with electro-beam writing method. We have fabricated two three-zone mask with the apertures of 2 and 20 mm that are designed from Table 1 as follows: a ¼ 91:3 lm, b ¼ 360 lm, r3 ¼ 1 mm; and a ¼ 913 lm, b ¼ 3600 lm, r3 ¼ 10 mm, respectively. The phase plate of the aperture size of 2 mm can generate a larger central diffraction spot that is more easily captured by the CCD camera than that with the aperture size of 20 mm. Larger-sized
phase plates can also be fabricated in principle. Then a thin layer of photoresist is spun onto a glass substrate. We transfer the mask pattern into photoresist through photolithographic technology. In our experiment, the photoresist is Shipley S1818. The contact copy error is within 0.5 lm. We use the wet chemical etching (WCE) method [10,11] to transfer the photoresist pattern into the glass substrate. The glass substrate has a refractive index of n ¼ 1:52. The expected etching depth corresponding to 0.9p in Table 1 is 548 nm. We use the Taylor Hobson step height standand to measure the surface relief profile of the phase plate. The surface relief profile of the superresolution phase plate is shown in Fig. 5. The measure depth is 518 nm, the phase error is smaller than 5%. We have set up an experimental system, as shown in Fig. 2, to simulate the far-field diffraction free-space laser communication. The He–Ne laser with a wavelength of 633 nm is used as the light source. A diffraction-limited lens with a focal length f ¼ 550 mm is used to yield the Airy diffraction pattern in its focal pattern that is captured
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J. Jia et al. / Optics Communications 228 (2003) 271–278
Fig. 5. The surface relief profile of the fabricated superresolution phase plate of the 3-zones plate in Table 1.
by a CCD camera and shown in Fig. 6(a). When the fabricated phase plate of Fig. 5 is inserted and aligned with the aperture of the lens, a superresolution diffraction pattern is generated, which is shown in Fig. 6(b). Comparison of the Airy pattern and the superresolution pattern is done offline in the computer and shown in Fig. 7. From Fig. 7, we can see that the experimental results are in good agreement with the theoretical results, and that the fabricated superresolution phase plate do generate a smaller central spot ðG 0:8Þ than the Airy spot. This result demonstrated the possibility that a superresolution phase plate can be incorporated in free-space laser communication system for reduction of the far-field diffraction spot size.
Fig. 6. The experimental image of the diffraction spot of Airy pattern (a) and the superresolution pattern (b) with the 3-zones plate of Fig. 5. We can see that the central spot of (b) is smaller and especially the first ring of (b) is a little wider and brighter than that of the well-known Airy pattern (a).
5. Conclusion Reduction of the far-field diffraction size in the free-space laser communication system is always interesting. The wavelength, the distance, and the aperture size of the emitting lens are usually considered to this end. The wavelength that is related with the available laser and the distance between the transmitter and the receiver are usually fixed, so a larger sized lens has to be fabricated at a much higher cost and sometimes unavailable due to the limit of the fabrication technology. The superresolution technology can realize the smaller central diffraction spot size in the far field than the usual Airy spot size. So the superresolution technology is very interesting in the free-space laser communication system. A superresolution phase plate is much cheaper than the larger aperture lens, which is the essential motivation of this work. We have calculated the 2, 3, 4, 5 circular zones for optimizing the highest energy compression (Strehl ratio) with the constraint of the First zero value G ¼ 0:8. Numerical results show that the 2circular zone pure-phase plate can yield the highest Strehl ratio (S ¼ 0.59) with the constraint of G ¼ 0:8. The 3 zone phase plate with 0.9p phase modulation yields the largest Strehl ratio S ¼ 0:58 with the constraint of G ¼ 0:8. The 3-circular zone pure-phase plates with the respective phases of U1 U2 U3 have also been calculated. At the same
J. Jia et al. / Optics Communications 228 (2003) 271–278 1 0.9
277
experiment theory
Airy pattern
0.8 0.7 0.6
Superresolution pattern
0.5 0.4 0.3 0.2 0.1 0 0
50
100
150
200
250
300
350
Fig. 7. Experimental comparison of the superresolution diffraction distribution with the Airy diffraction with the superresolution phase plate of Fig. 6, which are represented by the dotted line (experiments) and the solid line (theory), respectively. Experimental results verify that a smaller central spot (G0.8) than the Airy spot has been achieved.
time the 4, 5 circular zone binary phase (0,pÞ plates have been calculated to yield the result of S ¼ 0:57 with G ¼ 0:8. We realized from our numerical results that it might be not necessary to calculate more zone phase plate for a higher Strehl ratio. The efficiency of the superresolution phase plate may be enhanced by using a multi-phase modulation, which is the future work. It should be noted that using a stronger or a weaker laser intensity should yield the same normalized Airy diffraction pattern in the far field, the first zero position will not be changed. Air turbulence, absorption and scattering are not considered in this paper. The preassumed condition using a superresolution phase plate is that the emitting lens should be diffraction-limited. If the emitting lens is not diffraction limited, it is not necessary to use the superresolution phase plate proposed in this paper. In summary, we have studied the superresolution technology for generating the small diffraction spot size in the far field. We have presented numerical simulation results and we have also given experimental results as a simulation of reduction of the far-field diffraction size in the laser freespace communication system. We have set up an experimental system to detect such a smaller diffraction spot size in the far field that is generated
by using the superresolution plate above. Two phase plate with an aperture size of 2 and 20 mm has been fabricated with the microlithingraphic technology. We have obtained the experimental image of the smaller diffraction spot using the superresolution phase plate in this experimental system. More larger-sized superresolution phase plate can be fabricated in principle. But the smaller diffraction spot resulting from the superresolution technology always causes the energy loss in the mainlobe. And the lossing energy is transferred into the sidelobes. Careful avoidance of the intensity in the sidelobes would not cause the serious communication problem. So this technology can be applied when the laser energy is not the main concern in the communication. Experimental result has verified the possibility that by using a superresolution phase plate matching the emitting laser lens, a smaller central diffraction spot can be obtained in the far-field than Airy pattern.
Acknowledgements The authors acknowledge financial support from the National Outstanding Youth Foundation of China (60125512, 60177016).
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J. Jia et al. / Optics Communications 228 (2003) 271–278
References [1] M. Born, E. Wolf, Principles of Optics, seventh ed., University Press, Cambridge, UK, 1999. [2] J.W. Goodman, Introduction to the Fourier Optics, second ed., McGraw-Hill, New York, 1996. [3] G.T. di Francia, Nuovo Cimento Suppl. 9 (1952) 426. [4] D.J. Brady, Opt. Lett. 24 (1999) 811. [5] M. Bertero, J. Opt. So. Am. A 15 (1998) 2275.
[6] T.R.M. Sales, G.M. Morris, J. Opt. Soc. Am. A. 14 (1997) 1637. [7] T.R.M. Sales, G.M. Morris, Opt. Lett. 22 (1997) 582. [8] I.J. Cox, C.J.R. Sheppard, T. Wilson, J. Opt. Soc. Am. 72 (1982) 1287. [9] H. Ando, Jpn. J. Appl. Phys. 31 (1992) 557. [10] C. Zhou, L. Liu, Appl. Opt. 34 (1995) 5961. [11] P. Xi, C. Zhou, E. Dai, L. Liu, Opt. Lett. 27 (2002) 228.
Superresolution technology for reduction of the far-field diffraction spot size in the laser free-space communication system Jia Jia *, Changhe Zhou, Liren Liu Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800-211, Shanghai 201800, PR China Received 21 May 2003; received in revised form 29 August 2003; accepted 14 October 2003
Abstract In the free-space laser communication there is sometimes a strong need for reduction of the diffraction spot size in the far field. In this paper, instead of the usage of the larger size aperture lens in the free-space laser communication system, we introduce diffractive superresolution technology to design and fabricate a cheap pure-phase plate for realizing the smaller spot size than the usual Airy spot size, which can decrease the weight and size of the emitting lens. We have calculated 2, 3, 4, 5 circulation zones for optimizing the highest energy compression (Strehl ratio) with the constraint of the First zero ratio value G ¼ 0.8. Numerical results show that the 2- or 3-circular zone pure-phase plate can yield the highest Strehl ratio ðS 0:59Þ with the constraint of G ¼ 0.8, but the 4, 5 circular zone binary phase (0,pÞ plates are calculated to yield the result of S 0:57 with G ¼ 0.8. We have fabricated 2- and 3-circular zone binary phase plate with binary optics technology. Finally, we have established an experimental system for simulation of the freespace laser communication to verify the advantage of the superresolution phase plate. Detailed experiments are presented. Ó 2003 Elsevier B.V. All rights reserved. PACS: 42.30.Kq; 42.40.Jv; 42.79.Cj; 42.79.Sz; 42.82.Cr Keywords: Fourier optics; Computer-generated holograms; Zone plates; Optical-communication systems; Lithography
1. Introduction In the free-space laser communication application, the intensity at the far field is the Fraunhofer
*
Corresponding author. Tel.: +86215991-1214; +86216991-8800. E-mail address: [email protected] (J. Jia).
fax:
diffraction of the incident light, which is also the Fourier transform of the intensity of the incident light. The size of the diffraction spot in the receive port in the free-space laser communication system can also be calculated by the well-known Airy pattern: x ¼ 1:22ðkL=DÞ [1,2], where D is the aperture of the emitting lens and L is the distance between the emitting lens and the receiver and k is the wavelength of the laser. In the specific array
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.10.011
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J. Jia et al. / Optics Communications 228 (2003) 271–278
laser communication applications, as shown in Fig. 1, there is need for a technology that can decrease the size of the diffraction spot in the receiver port because in the free-space laser communication a smaller diffraction spot in the receive port makes the transmit data more secure. The wavelength of the laser and the distance between the emitting lens and the received are usually fixed. So a larger aperture lens is often employed to realize the small far-field spot. However, a largersized lens is very expensive, and worse, a largersized lens is usually not available due to the fabrication technology. Superresolution technology provides an attractive approach to this application. Superresolution technique is one of the methods that realize the diffraction spot smaller than the Airy spot when a superresolution pure-phase plate is employed. The first significant study of superresolution was put forward by Francia [3]. Much attention was devoted to the design of the superresolution phase plate for its applications in enhanced resolution confocal system [4], increased storage in optical disk system [5], etc. More sophisticated methods based on continuous amplitude transmittance function have also been developed [6]. Hybrid
1 Laser 1 Receiver 1 x Receiver 2 Laser 2
2 Distance L
Laser N
Receiver N
Fig. 1. Illustration of an arrayed laser communication system for the multi-channel increased transmitting capacity. Each laser should generate its Frauhofer diffraction in the far field. 1, 2 means the intensity distribution in the laser 1, 2 diffraction field. The signal channels should not disturb the nearby channel for the diffraction effect of each channel. The receivers can be put at an optimized closer distance when a superresolution phase plate is used without resorting to a larger aperture laser emitting lens.
amplitude-phase elements with two zones, as well as elements for use in optical pick-up heads have been studied [7]. Various superresolution compression parameters such as Strehl ratio (S), First zero (G), have been defined to describe the superresolution effect. The Strehl ratio is defined as the ratio between the intensity of the superresolution pattern and the intensity of the Airy pattern, both calculated at the origin. The First zero is defined as the ratio between the first zero position of the superresolution pattern and the first zero position of the airy pattern. For more details of superresolution technology and its applications, one can see [6–9]. The widely used Full-width-half-maximum (FWHM) is closely related with the first zero position (G), and the central-lobe energy percentage, the ratio of the central lobe energy at the zero order to the whole energy at all diffraction orders, is closed related with the Strehl ratio (S). As the diffraction efficiency is the most important parameter for practical applications, we will present the superresolution scheme of the pure-phase plate in this paper because of high diffraction efficiency. As to the best of our knowledge, no one has pointed out the possibility that superresolution technology can be applied in the free-space laser communication system. We introduce the superresolution technology into the free-space laser communication system to compress the far-field diffractive spot size. With this proposed technology, we typically place a filter at the exit pupil of the system and thus the smaller diffraction spot can be obtained with the same aperture lens. In this paper, we prescribe the criterion of G 6 0:8, while S should be as large as possible for practical application. If G is too small, such as G 6 0:6, the mainlobe in the diffraction spot is too small to be applied in practice. If G is too large such as G ffi 1, the too small reduction of the diffraction spot size is not interesting for practical use. In this paper, we have calculated the 2, 3, 4, 5 circular zones for optimizing the highest Strehl ratio with the constraint of the First zero value G ¼ 0:8, and have found the optimized radius and phases of the phase plate. Finally, we have set up an experimental system for simulation of the Fourier transform in the laser long-distance communication. The experimental results have verified
J. Jia et al. / Optics Communications 228 (2003) 271–278
the numerical results and confirm the effectiveness of the superresolution phase plate.
2. Review of superresolution theory In a traditional sense, the Airy resolution is defined by the distance from the first zero position of the main peak intensity spot. Superresolution technology means that we can realize a smaller diffraction spot than an Airy diffraction spot. A pure-phase plate that is put before (or after ) the diffraction-limited lens is involved to realize the superresolution effect, as is shown in Fig. 2. According to the superresolution diffractive theory [6] the normalized field measured at the observation plane is given by: Z 1 uðm; lÞ ¼ P ðqÞ expðjlq2 =2ÞJ0 ðmqÞqdq; ð1Þ
uð0; lÞ ¼
Z
273
1
P ðqÞ expðjlq2 =2Þdq;
where m ¼ 0 in the axial plane. So in the focal plane, the radial amplitude distribution is the Hankel transform of the pupil function, while the axis amplitude distribution is the Fourier transform of the pupil function. In the superresolution field, there are many methods to design the pure-phase plate. Due to ease of fabrication, we choose the circular binary phase-only plate in this paper. The structure of the 3 zone circular binary phase-only plate is shown in Fig. 3. If we put the circular phase-only plate into the optical system, the diffractive field can be described as wðnÞ ¼
N X
expði/j Þ½a2j 2J1 ðaj nÞ=aj n
j¼1
a2j1 2J1 ðaj1 nÞ=aj1 n;
0
ð4Þ
where aj and /j represent the radius and phase of the jth zone, respectively. If the binary phases are chosen to be only 0 and /, we can rewrite Eq. (4) as
m ¼ kr sinðaÞ; 2
l ¼ 4kzðsinðaÞ=2Þ ;
wðnÞ ¼ 2J1 ðnÞ=n ½1 expði/0 Þð1Þ k ¼ 2p=k;
ð3Þ
0
sinðaÞ ¼ NA;
where l, m are the axial and radial coordinates, P ðqÞ is the pupil function (which is also called the diffractive phase plate). From Eq. (1), we can get Z 1 uðm; 0Þ ¼ P ðqÞJ0 ðmtÞdt; ð2Þ 0
where l ¼ 0 in the focal plane
Superresolution pure-phase plate
Collimated laser beam (λ)
N 1 X
ð1Þj a2j 2J1 ðaj nÞ=aj n:
N þ1
ð5Þ
j¼1
In numerical simulations, we will apply Eq. (5) to calculate the 2, 3, 4, 5 circular zones of the phase plate. According to the compression criterion, we will find the optimized parameters: the radius and phase value of each zone.
focal plane
D CCD camera
Fig. 2. The superresolution experimental system for simulation of the far-field diffraction in the long-distance free-space laser communication system.
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J. Jia et al. / Optics Communications 228 (2003) 271–278
0
Φ 0 1.0 a
b
Fig. 3. The normalized structure of the 3 zone binary purephase plate, a, b are the radii of the phase-transition circles between the phase 0 and U.
3. Optimization algorithms and numerical results We have used the global searching algorithms in our computer simulations. The global searching algorithm is to find the optimized result in the whole searching areas with changing each radius at a minimum searching step. The advantage of this algorithm is that it can find the optimized result, but the searching time is quite long. Numerical results of the Strehl ratio (S) and First zero ðS ¼ 0:8Þ and the central-lobe energy percentage with the radius (a) and the phase are given in Table 1. From Table 1, we know that the largest Strehl ratio S ¼ 0:5911, which is the Strehl ratio of the 2 zone phase plate. From numerical results of the First zero and Strehl ratio with two radius a and b of the 3 zone phase plate, we find the best optimized result under the constraint
of G ¼ 0:8 as: a ¼ 0:09; b ¼ 0:36; U ¼ 0:9p; G ¼ 0:7979; SMAX ¼ 0:5835. The central-lobe energy percentage that is closely related with the Strehl ratio is calculated and given in Table 1. The maximum central-lobe energy percentage is 39% in Table 1 while it is 84% for the Airy diffraction distribution. The far-field intensity distribution of this optimized numerical result is shown in Fig. 4. From Fig. 4, we can see that the superresolution technology can yield a smaller diffractive size in the far field than that in the Airy spot. We have also studied the effect of the multiple phase modulation for the 3 zone plate with the same optimized two radius a ¼ 0:09 and b ¼ 0:36. The inner zone phase is always assumed to be zero. The optimized phases are as follows: /1 ¼ 0:00p; /2 ¼ 0:06p; /3 ¼ 0:86p, and the Strehl ratio Smax ¼ 0:5905. Note that this optimized Strehl ratio S ¼ 0:59 with the multiple phases is a little higher than the optimized S ¼ 0:5835 with the binary phases. This result illustrates that the multiple-phase modulation can indeed improve the performance but at a much higher cost for fabrication than the binary phase modulation. The numerical simulation results of the 4 and 5 zone phase plates give the maximum S 0:57 with G ¼ 0:8. We realized from our numerical result that the S could not be increased with the increase of the zone numbers. Numerical comparison of the 2- and 3-zones plate in Table 1 is given in Fig. 4. From Fig. 4, we can see that FWHM (Superresolved) is smaller than the FWHM (Airy). If the maximum intensity of the superresolved pattern is normalized from 0.59 to 1, the FWHM keeps unchanged because of the fixed first zero position.
Table 1 Numerical result of the multi-zones superresolution phase plate with the prescribed First zero (G ¼ 0:8) Zone no.
Search step
2 3 4
0.01 0.03 0.08
5
0.10
Optimized numerical result Normalized radius
U ðpÞ
Strehl ratio
Central lobe energy (%)
a ¼ 0.34 a ¼ 0.09, b ¼ 0.36 a ¼ 0.16, b ¼ 0.24, c ¼ 0.40 a ¼ 0.1, b ¼ 0.2, c ¼ 0.4, d ¼ 0.9
1.0 0.9 0.9
0.5911 0.5835 0.5645
38.83 36.06 35.06
0.9
0.5587
13.16
See Fig. 3 for the meaning of the normalized radius a, b.
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superresolved pattern(3-zones) superresolved pattern(2-zones) Airy pattern
0.9
Normalized Intensity
0.8 0.7 0.6
1/2FWHM (Airy)
0.5
1/2FWHM (Superresolved)
0.4 0.3 0.2 0.1 0 0
0.5
1.0
1.5
2.0
2.5
3.0
Relative radial distance
Fig. 4. Theoretical far-field intensity distribution of the Airy (solid line) and the superresolution diffraction (dashed line) patterns with the 2- and 3-zones plates in Table 1. The superresolution pattern generated by the 2- and 3-zones plates are quite close to each other. The receiver 2 can be at the first zero position of the superresolution pattern that is 0.8 times closer to receiver 1 than the Airy pattern. FWHM is the Full-width-half-maximum of the diffraction distribution. We can see that the FWHM (Superresolved) is smaller than the FWHM (Airy).
Thus, the FWHM of the superresolved distribution is indeed reduced compared with the FWHM of the Airy distribution. From Fig. 4, we can also see that there is little difference between 2- and 3zones plates. Fabrication of the three zone phase plate can demonstrate the capability that we can also fabricate the 2-zone phase plate. So we choose the 3-zone binary phase plate for the experiment, which is given in the next section.
4. Fabrication and experimental results VLSI techniques have been adopted in our experiment to obtain the phase plate. The mask is made with electro-beam writing method. We have fabricated two three-zone mask with the apertures of 2 and 20 mm that are designed from Table 1 as follows: a ¼ 91:3 lm, b ¼ 360 lm, r3 ¼ 1 mm; and a ¼ 913 lm, b ¼ 3600 lm, r3 ¼ 10 mm, respectively. The phase plate of the aperture size of 2 mm can generate a larger central diffraction spot that is more easily captured by the CCD camera than that with the aperture size of 20 mm. Larger-sized
phase plates can also be fabricated in principle. Then a thin layer of photoresist is spun onto a glass substrate. We transfer the mask pattern into photoresist through photolithographic technology. In our experiment, the photoresist is Shipley S1818. The contact copy error is within 0.5 lm. We use the wet chemical etching (WCE) method [10,11] to transfer the photoresist pattern into the glass substrate. The glass substrate has a refractive index of n ¼ 1:52. The expected etching depth corresponding to 0.9p in Table 1 is 548 nm. We use the Taylor Hobson step height standand to measure the surface relief profile of the phase plate. The surface relief profile of the superresolution phase plate is shown in Fig. 5. The measure depth is 518 nm, the phase error is smaller than 5%. We have set up an experimental system, as shown in Fig. 2, to simulate the far-field diffraction free-space laser communication. The He–Ne laser with a wavelength of 633 nm is used as the light source. A diffraction-limited lens with a focal length f ¼ 550 mm is used to yield the Airy diffraction pattern in its focal pattern that is captured
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Fig. 5. The surface relief profile of the fabricated superresolution phase plate of the 3-zones plate in Table 1.
by a CCD camera and shown in Fig. 6(a). When the fabricated phase plate of Fig. 5 is inserted and aligned with the aperture of the lens, a superresolution diffraction pattern is generated, which is shown in Fig. 6(b). Comparison of the Airy pattern and the superresolution pattern is done offline in the computer and shown in Fig. 7. From Fig. 7, we can see that the experimental results are in good agreement with the theoretical results, and that the fabricated superresolution phase plate do generate a smaller central spot ðG 0:8Þ than the Airy spot. This result demonstrated the possibility that a superresolution phase plate can be incorporated in free-space laser communication system for reduction of the far-field diffraction spot size.
Fig. 6. The experimental image of the diffraction spot of Airy pattern (a) and the superresolution pattern (b) with the 3-zones plate of Fig. 5. We can see that the central spot of (b) is smaller and especially the first ring of (b) is a little wider and brighter than that of the well-known Airy pattern (a).
5. Conclusion Reduction of the far-field diffraction size in the free-space laser communication system is always interesting. The wavelength, the distance, and the aperture size of the emitting lens are usually considered to this end. The wavelength that is related with the available laser and the distance between the transmitter and the receiver are usually fixed, so a larger sized lens has to be fabricated at a much higher cost and sometimes unavailable due to the limit of the fabrication technology. The superresolution technology can realize the smaller central diffraction spot size in the far field than the usual Airy spot size. So the superresolution technology is very interesting in the free-space laser communication system. A superresolution phase plate is much cheaper than the larger aperture lens, which is the essential motivation of this work. We have calculated the 2, 3, 4, 5 circular zones for optimizing the highest energy compression (Strehl ratio) with the constraint of the First zero value G ¼ 0:8. Numerical results show that the 2circular zone pure-phase plate can yield the highest Strehl ratio (S ¼ 0.59) with the constraint of G ¼ 0:8. The 3 zone phase plate with 0.9p phase modulation yields the largest Strehl ratio S ¼ 0:58 with the constraint of G ¼ 0:8. The 3-circular zone pure-phase plates with the respective phases of U1 U2 U3 have also been calculated. At the same
J. Jia et al. / Optics Communications 228 (2003) 271–278 1 0.9
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experiment theory
Airy pattern
0.8 0.7 0.6
Superresolution pattern
0.5 0.4 0.3 0.2 0.1 0 0
50
100
150
200
250
300
350
Fig. 7. Experimental comparison of the superresolution diffraction distribution with the Airy diffraction with the superresolution phase plate of Fig. 6, which are represented by the dotted line (experiments) and the solid line (theory), respectively. Experimental results verify that a smaller central spot (G0.8) than the Airy spot has been achieved.
time the 4, 5 circular zone binary phase (0,pÞ plates have been calculated to yield the result of S ¼ 0:57 with G ¼ 0:8. We realized from our numerical results that it might be not necessary to calculate more zone phase plate for a higher Strehl ratio. The efficiency of the superresolution phase plate may be enhanced by using a multi-phase modulation, which is the future work. It should be noted that using a stronger or a weaker laser intensity should yield the same normalized Airy diffraction pattern in the far field, the first zero position will not be changed. Air turbulence, absorption and scattering are not considered in this paper. The preassumed condition using a superresolution phase plate is that the emitting lens should be diffraction-limited. If the emitting lens is not diffraction limited, it is not necessary to use the superresolution phase plate proposed in this paper. In summary, we have studied the superresolution technology for generating the small diffraction spot size in the far field. We have presented numerical simulation results and we have also given experimental results as a simulation of reduction of the far-field diffraction size in the laser freespace communication system. We have set up an experimental system to detect such a smaller diffraction spot size in the far field that is generated
by using the superresolution plate above. Two phase plate with an aperture size of 2 and 20 mm has been fabricated with the microlithingraphic technology. We have obtained the experimental image of the smaller diffraction spot using the superresolution phase plate in this experimental system. More larger-sized superresolution phase plate can be fabricated in principle. But the smaller diffraction spot resulting from the superresolution technology always causes the energy loss in the mainlobe. And the lossing energy is transferred into the sidelobes. Careful avoidance of the intensity in the sidelobes would not cause the serious communication problem. So this technology can be applied when the laser energy is not the main concern in the communication. Experimental result has verified the possibility that by using a superresolution phase plate matching the emitting laser lens, a smaller central diffraction spot can be obtained in the far-field than Airy pattern.
Acknowledgements The authors acknowledge financial support from the National Outstanding Youth Foundation of China (60125512, 60177016).
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References [1] M. Born, E. Wolf, Principles of Optics, seventh ed., University Press, Cambridge, UK, 1999. [2] J.W. Goodman, Introduction to the Fourier Optics, second ed., McGraw-Hill, New York, 1996. [3] G.T. di Francia, Nuovo Cimento Suppl. 9 (1952) 426. [4] D.J. Brady, Opt. Lett. 24 (1999) 811. [5] M. Bertero, J. Opt. So. Am. A 15 (1998) 2275.
[6] T.R.M. Sales, G.M. Morris, J. Opt. Soc. Am. A. 14 (1997) 1637. [7] T.R.M. Sales, G.M. Morris, Opt. Lett. 22 (1997) 582. [8] I.J. Cox, C.J.R. Sheppard, T. Wilson, J. Opt. Soc. Am. 72 (1982) 1287. [9] H. Ando, Jpn. J. Appl. Phys. 31 (1992) 557. [10] C. Zhou, L. Liu, Appl. Opt. 34 (1995) 5961. [11] P. Xi, C. Zhou, E. Dai, L. Liu, Opt. Lett. 27 (2002) 228.