- Email: [email protected]
Direct image transmission through a multi-mode square optical fiber
1 December 1998
Optics Communications 157 Ž1998. 17–22
Direct image transmission through a multi-mode square optical fiber C.Y. Wu 1, A.R.D. Somervell, T.H. Barnes Department of Physics, UniÕersity of Auckland, PriÕate Bag 92019, Auckland, New Zealand Received 23 March 1998; revised 20 August 1998; accepted 25 August 1998
Abstract We explore the potential for direct image transmission through a square optical fiber. We show that when an image is sampled appropriately and its optical Fourier transform imaged on the end of a square fiber with perfectly reflecting walls, the components in the Fourier transform excite corresponding fiber modes. Specifically, even–even fiber modes carry information from one pixel only, while the odd modes carry information from neighboring pixels and give rise to cross-talk. When the odd modes are suppressed, the image can be perfectly recovered at the end of the fiber by a second optical Fourier transform. We suggest a method of suppressing the odd modes. In our system, dispersion of the mode phase velocity gives rise to different arrival times for the information in different image pixels, but has little or no effect on the output intensity distribution. We show that the square shape of the fiber is critical in forming the output image and confirm our theoretical predictions by computer simulation. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Image transmission; Multimode; Square optical-fiber
1. Introduction Direct image transmission through multimode optical fibers has attracted considerable interest over many years. The main difficulties in this type of image transmission arise from dispersion of the phase velocity of the fiber modes and angular ambiguities caused by symmetry when fibers of circular cross-section are used. Many methods have been developed to overcome these difficulties and comprehensive reviews and analysis of the subject, together with some new developments can be found in Refs. w1–4x. It is well known that an image formed in coherent light on the end of an optical fiber is expanded into a sequence of normal modes propagating along the fiber. Each mode has a characteristic amplitude distribution Žthe eigen func-
1
E-mail: [email protected]
tion. over the cross-section of the fiber. Usually, each image pixel excites several modes which propagate down the fiber at different speeds. Also, each mode carries contributions from several pixels. The image information is therefore badly corrupted after even only a short transmission distance and some way of equalising the transmission times for different modes is necessary before the image can be accurately reconstructed again. This is often achieved holographically. Our proposed system works in a different way. We arrange that each pixel in the image excites one corresponding mode in the fiber, by an optical transformation. At the output end of the fiber, a second optical transformation is applied to convert the modes back to single points of light, the reconstructed pixels. Now, eventhough the information in different pixels travels at different speeds down the fiber, we in principle avoid the mixing of information between pixels that occurs when each pixel excites several modes as in the conventional system. Al-
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 4 7 7 - 5
18
C.Y. Wu et al.r Optics Communications 157 (1998) 17–22
though the pixels arrive at the fiber output at slightly different times, the image intensity distribution is not corrupted. Consider a ‘one-dimensional’ fiber fed with a one-dimensional image. The modes here are sinusoidal functions with spatial frequencies determined by the size of the fiber and the optical wavelength. We also know that the Fourier transform of a pair of delta functions symmetrically disposed about the origin with appropriate signs, is a sinusoidal function w5x. It is therefore reasonable to expect that if an image is appropriately sampled and optically Fourier transformed before being presented to the fiber, we may be able to relate each pair of symmetrically disposed pixels to a single propagating mode, and transform that mode back to a pair of pixels again by a second optical Fourier transform at the output end of the fiber. Consideration of this simple model immediately raises two potential problems. First, each pair of pixels arrives at a different time because of modal dispersion, and second, the information in the two pixels of each symmetrical pair will be mixed together during propagation. The first of these difficulties is not too severe for ordinary lengths of fiber where the difference in mode transmission time would typically be of the order of nano- or micro-seconds, and the second problem can be overcome by transmitting only one of the symmetric pairs of pixels i.e. by restricting the input image to one half of the input field of the first Fourier transform lens. The simple one-dimensional model above can be easily extended to two dimensions with a fiber of rectangular cross-section, except that now the input image field must be restricted to a single quadrant of the input plane of the first Fourier lens. It should be noted that this system will not work for a round fiber because of the angular ambiguity, i.e. the information in input pixels in each circle centered on the optic axis of the input Fourier lens will be mixed as it passes down the fiber. In this paper we therefore limit our discussion to a transmission system consisting of a square fiber with two Fourier transform lenses. The principle of the optical Fourier transform and it’s applications are very well known, and can be found in
Refs. w1,6x. The propagation modes of a rectangular waveguide with perfectly reflecting walls and those of slab waveguides have also been thoroughly studied in the literature w7–12x. The rectangular optical waveguides are widely used in integrated optics applications w8x. Although mathematical difficulties have prevented the exact analysis of rectangular dielectric waveguides, many valuable approximations have been developed Žsee, for example Refs. w8–10x.. For simplicity, in this paper we restrict ourselves to the discussion of the ideal case of a rectangular optical fiber with perfectly reflecting walls. In Section 2 we present a short theoretical analysis which is then compared with numerical calculations, and we follow this with a brief discussion in Section 3.
2. Analysis The suggested system is sketched in Fig. 1 where L 1 and L 4 are Fourier transform lenses with focal length L f , and L 2 and L 3 are lens systems of magnification h. The optical fiber has a square cross-section with side size a and length L, and the cladding is a perfect reflector. Ei Ž x i , yi . where i s 1–6 represent the electrical field ŽTM. of the image during different stages of propagation. For simplicity, we also assume that the fiber can be regarded as being infinite in length when discussing the propagation of the light field through the fiber so that reflection from the far end of the fiber can be ignored. We will concentrate on the steady state case in this paper with the time factor expŽyi v t . suppressed. The field inside the fiber can be expressed by using the Green’s theorem w11x as E Ž x , y, z . 1
sy
H E Žx 4p S 0
0
0 , y0
.
E G Ž x , y, z < x 0 , y 0 , z 0 . E n0
d S0 z 0 s0
Ž1. where d S0 s d x 0 d y 0 , and n 0 is the outward normal of
Fig. 1. Sketch of the image transmission system consisting of a pair of Fourier lenses ŽL 1 and L 4 ., a pair of imaging lens systems ŽL 2 and L 3 ., and a perfect reflecting square fiber Žlength L..
C.Y. Wu et al.r Optics Communications 157 (1998) 17–22
the plane z 0 s 0. The Green’s function GŽ x, y, z < x 0 , y 0 , z 0 . satisfies the Helmholtz equation,
Ž=
2
q k 12
19
and similarly, ar2
E6 Ž x 6 , y6 . s hy1A
ar2
Hya r2Hyar2 E Ž x
. G Ž x , y, z < x 0 , y 0 , z 0 .
s y4pd Ž x y x 0 . d Ž y y y 0 . d Ž z y z 0 . ,
4
hk0
ž
Ž2.
=exp yi
Lf
4 , y4
.
/
Ž x 4 x 6 q y4 y6 . d x 4 d y4 .
and the boundary conditions, G s 0 on the surface, x s "ar2, y s "ar2, and z s 0. Solving the equation gives
for yar2F Ž x 4 , y4 . F ar2,y br2 F Ž x 6 , y6 . F br2. We can also combine Eqs. Ž5. – Ž7. to establish a direct relation between the input and output fields:
G Ž x , y, z < x 0 , y 0 , z 0 . `
16p s
a2
mp
ms 1 ns1
=sin
=
`
Ý Ý sin
½
mp a
a
Ž x q ar2. sin
Ž x 0 q ar2. sin
np a
np a
Ž y q ar2.
Ž y 0 q ar2.
sin Ž gmn z 0 . e ig m n z ,
z G z0
sin Ž gmn z . e ig m n z 0 ,
z - z0 ,
Ž7.
E6 Ž x 6 , y6 . s
Ž3.
(
k 1 s n1 vrc Ž ErE n 0 s yErE z 0 ., and n1 is the refractive index of the fiber material. The E-field at the front face of the fiber, E3 , can be found by noting that the lens, L 1, performs a 2-D Fourier Transform and that this is imaged onto the fiber by L 2 . This results in
Hyb r2Hybr2 E Ž x , y .
ž
=exp yi
Lf
a
2
`
`
ar2
ms1 ns1
=sin
mp a mp a
Ž x 3 q ar2. sin Ž x 4 q ar2. sin
np a np a
Ž y 3 q ar2. d x 3 d y 3 Ž y4 q ar2. e ig m n L , Ž 6 .
yi2 p Õ 1 y 3
3
ar2
ar2
4
d y3
yi2 p u 6 x 4
4
yi2 p Õ 6 y 4
Hya r2sin w np Ž y ra q 1r2. x e
=
d x4
d y4 ,
Ž9.
and
h x1 l0 L f h x6 l0 L f
,
Õ1 s
,
Õ6 s
h y1 l0 L f h y6 l0 L f
,
Ž 10 .
.
Ž 11.
Denoting the four similar integrals involved in Eq. Ž9. as Im1 , In1 , Im6 and In 6 and carrying out the integration we have a Im1 s e iŽ my1.p r2 sinc Ž mr2 y m1 . 2 m
y Ž y1 . sinc Ž mr2 q m1 . ,
Ž 12.
with similar expressions for the other three terms, if the input and output images are sampled by
Ds
ar2
Ý Ý Hya r2Hyar2 E3 Ž x 3 , y 3 .
= sin
ar2
x i s mi D,
E4 Ž x 4 , y4 . 4
Ž8.
d x3
Hya r2sin w mp Ž x ra q 1r2. x e
Ž x 1 x 3 q y1 y 3 . d x 1 d y1 ,
for ybr2F Ž x 1, y 1 . F br2, and yar2F Ž x 3 , y 3 . F ar2, where A is a complex constant multiplied by the factor, Žyie i2 k 0 L frl0 L f . w1x, and b is the size of the frame of the input image, b s h a. Then by using the Green’s function we can express the field at the output of the fiber as
yi2 p u 1 x 3
3
=
u6 s
/
ar2
Hya r2sin w np Ž y ra q 1r2. x e
1
Ž5.
s
ms1 ns1
Hya r2sin w mp Ž x ra q 1r2. x e =
u1 s
br2
hk0
br2
Fmn Ž x 1 , y 1 , x 6 , y6 .
Ž4.
1
br2
where
gmn s k 12 y Ž m2 q n2 . p 2ra2 ,
1
`
=Fmn Ž x 1 , y 1 , x 6 , y6 . d x 1 d y 1 e ig m n L ,
gmn
where
E3 Ž x 3 , y 3 . s h A
a2
`
Ý Ý Hyb r2Hybr2 E1Ž x 1 , y1 .
1
s
br2
4 A2
l0 L f b
yi s n i D s
l0 L f ha
Ž i s 1,6 . , b
s N
.
Ž 13.
In above, sincŽ x . ' sinŽ xp .rxp . We now discuss the cases of even and odd numbered modes separately. For even modes we set m s 2 m and Eq. Ž12. becomes ia Ž . ev Im1 s Ž y1 . mq1 Ž dm , m 1 y dm ,y m 1 . Ž 14. 2
C.Y. Wu et al.r Optics Communications 157 (1998) 17–22
20
Fig. 2. DFT of the fiber mode functions for m s 5: Ža. Imev Ž m1 . for an even mode and Žb. Imod Ž m1 . for an odd mode.
while for the odd modes, m s 2 m y 1, we have od Im1 s
a 2
and E6 Ž m 6 ,n 6 .
Ž y1. mq1 Ž sinc w m1 y m q 1r2x
a 2A2
qsinc Ž m1 q m y 1r2 . . .
Ž 15 .
ev od Im1 and Im1 as discrete functions of m1 , for the case m s 5 are shown in Fig. 2Ža. and 2Žb. respectively. This figure confirms our statement in the last section that each even mode relates to two symmetrically located pixels at m1 and ym1 while each odd mode may relate to several neighbuoring pixels. For Im6 , In1 and In6 we have similar expressions. By using these expressions we can write
ev od = Ž Im6 q Im6 .Ž In6ev q In6od .
Ž 16.
and E6 Ž m 6 ,n 6 . 4 A2 a2
`
`
Ý Ý
Nr2
Nr2
Ý
Ý
E1Ž m1 ,n1 .
ms 1 ns1 m 1 syNr2q1 n 1 syNr2q1
ev od = Ž Im1 q Im1 .Ž In1ev q In1od . ev od = Ž Im6 q Im6 .Ž In6ev q In6od . e ig m , n L .
Ž 17 .
In the special cases when both m and n are even and both are odd we have respectively E6 Ž m 6 ,n 6 . a 2A2 s 4
`
`
Ý Ý
Nr2
Nr2
Ý
Ý
4
`
`
Ý Ý
E1Ž m1 ,n1 .
ms 1 ns1 m 1 syNr2q1 n 1 syNr2q1
= Ž dm, m 1 y dm ,y m 1 .Ž dm , m 6 y dm ,y m 6 . = Ž dn, n 1 y dn ,y n 1 .Ž dn , n 6 y dn ,y n 6 . e ig 2 m ,2 n L
Ž 18 .
Nr2
Nr2
Ý
Ý
E1Ž m1 ,n1 .
ms 1 ns1 m 1 syNr2q1 n 1 syNr2q1
= sinc Ž m1 y m q 12 . q sinc Ž m1 q m y 12 . = sinc Ž m 6 y m q 12 . q sinc Ž m 6 q m y 12 . = sinc Ž n1 y n q 12 . q sinc Ž n1 q n y 12 . = sinc Ž n 6 y n q 12 . q sinc Ž n 6 q n y 12 . =e ig 2 my 1 ,2 ny 1 L .
ev od q Im1 Fmn Ž x 1 , y 1 , x 6 , y6 . s Ž Im1 .Ž In1ev q In1od .
s
s
Ž 19 .
Expanding the right-hand side terms in Eq. Ž18. and carrying out the summation we can see that carried through by even–even modes, m s 2 m1, n s 2 n1, each input pixel E1Ž m1,n1 . will be seen on the output screen as four images, E6 Ž m1,n1 ., E6 Žym1, n1 ., E6 Ž m1,y n1 . and E6Žym1,y n1 . meanwhile each output image E6Ž m 6 ,n 6 . will be the superposition of the images of four symmetrical input pixels, E1Ž m 6 ,n 6 ., E1Žym 6 ,n 6 ., E1Ž m 6 ,y n 6 . and E1Žym 6 ,y n 6 . with the same phase velocity. This ambiguity can be resolved by restricting the input image inside one quadrant as shown in Fig. 3Ža.. Through this treatment, we will have four unambiguous images in the whole output screen as shown in Fig. 3Žb. which is numerically calculated according to Eqs. Ž5. – Ž7. for a sampling matrix of N = N, N s 64. From Eq. Ž18. it is also expected that if the effects of the odd numbered modes are suppressed it should be possible to recover the input image perfectly without crosstalk, which is shown in Fig. 3Žc. when only the even–even modes are accounted for. Finally, as a comparison we show in Fig. 3Žd. the output image directly transmitted through the fiber without using the Fourier lenses, L 1 and L 4 . Now we show that the crosstalk Žthe effect of odd numbered modes as shown in Fig. 3Žb.. can be removed
C.Y. Wu et al.r Optics Communications 157 (1998) 17–22
21
Fig. 3. Numerical calculation of the amplitude of the images: Ža. Imput image; Žb. output image with both even and odd modes included; Žc. output image when even modes only are included and Žd. out put image without using the Fourier lenses, L 1 and L 4 .
theoretically by properly combining the four images in the ev output screen. In fact, if we expand the product Ž Im6 q od .Ž ev od . Im6 In6 q In6 in Eq. Ž17. referring to the expressions of ev od Im1 and Im1 in the form ev od q Im6 Ž Im6 .Ž In6ev q In6od . ev ev ev od od ev od od s Im6 In6 q Im6 In6 q Im6 In6 q Im6 In6
s Ž a 2r4 .
yŽ d
m, m 6 y d m ,y m 6
This combination can be achieved by changing the phase of E6Žym 6 ,n 6 . and E6Ž m 6 ,y n 6 . by p , and adding the images interferometrically. This has also been confirmed numerically by adding up the four terms in the left-hand side of Eq. Ž21. calculated from Eq. Ž7.. This leads to an output image exactly the same as that shown in Fig. 3Žc. except for an increase of the intensity by a factor of 4.
.Ž dn , n y dn ,y n . 6
6
q i Ž dm, m 6 y dm ,y m 6 . Ž sinc Ž n 6 y n q 1r2 . qsinc Ž n 6 q n y 1r2 . . q i Ž sinc Ž m 6 y m q 1r2 . qsinc Ž m 6 q m y 1r2 . . Ž dn , n 6 y dn ,y n 6 .
Ž sinc Ž m6 y m q 1r2. q sinc Ž m6 q m y 1r2. . = Ž sinc Ž n 6 y n q 1r2 . q sinc Ž n 6 q n y 1r2 . . 4 ,
q
Ž 20. then it is easy to see that 1 4 w E6 Ž m 6 ,n 6 . y E6 Ž ym 6 ,n 6 . y E6 Ž m 6 ,y n 6 . qE6 Ž ym 6 ,y n 6 . x a 2A2 s 4
`
`
Ý Ý
Nr2
Nr2
Ý
Ý
E1Ž m1 ,n1 .
ms 1 ns1 m 1 syNr2q1 n 1 syNr2q1
= Ž dm, m 1 y dm ,y m 1 . i Ž dm , m 6 y dm ,y m 6 . = Ž dn, n 1 y dn ,y n 1 .Ž dn , n 6 y dn ,y n 6 . e ig 2 m ,2 n L s a2A2 E1Ž m 6 ,n 6 . e ig 2 m 6 ,2 n 6 L .
Ž 21.
3. Discussion Ž1. When the finite length of the fiber is taken into account, the Green’s function, Eq. Ž3., will be slightly altered and the factor, e ig m n L, in Eq. Ž8. will be replaced correspondingly by wŽ1 q R .rŽ1 q Re i2g m n L .xe ig m n L. Since the normal reflection coefficient R Žamplitude. is about 0.2 from the glassrair interface the terms in square brackets have been treated as unity in this analysis. Moreover, because its influence is restricted to single pixels, it will not have significant effect on the overall picture. Ž2. Ordinary round section fibers do not have the image transmission characteristics described above. This is mainly because of the angular ambiguity or ‘azimuthal uniformity’ indicated in Ref. w3x as mentioned before. Besides, the spatial distribution of the zero points of the mode functions is also different for different modes. This causes additional crosstalk when a fixed sampling rate is used.
22
C.Y. Wu et al.r Optics Communications 157 (1998) 17–22
Ž3. This approach can, in principle, be used for an imperfect reflecting square fiber as long as it supports sufficient guided wave modes to carry the pixels.
from the New Zealand Institute for Industrial Research and Development. References
4. Conclusion The possibility of direct image transmission through a perfect reflecting square fiber is demonstrated theoretically. The key point of the new approach is the introduction of a pair of Fourier lenses and the use of a square fiber.
Acknowledgements We wish to thank Professor A.C. Kibblewhite and Professor G.L. Austin, both of the Physics Department, University of Auckland, for their help and support during this work. We are also indebted to Dr. S.M. Tan of the Physics Department, University of Auckland, and Dr. Tomohiro Shirai, Optical Engineering Division, Mechanical Engineering Laboratory, Japan, for interesting discussions and to Dr. D.D. Wu for help with preparation of the manuscript. Finally, we acknowledge the financial support
w1x A. Yariv, Optical Electronics in Modern Communications, 5th ed., Oxford University Press, 1997. w2x A. Yariv, J. Opt. Soc. Am. 66 Ž1976. 306. w3x A.A. Friesem, U. Levy, Y. Silberberg, Proc. IEEE 71 Ž1983. 208. w4x J.-Y. Son, V.I. Bobrinev, H.-W. Jeon, Y.-H. Cho, Y.-S. Eom, Appl. Optics 35 Ž1996. 273. w5x R.N. Bracewell, The Fourier Transform and its Applications, 2nd ed., McGraw-Hill, 1986. w6x F.T.S. Yu and Suganda Jutamulia, Optical Signal Processing, Computing, and Neural Networks, Wiley, New York, 1992. w7x R. Ulrich, W. Prettl, Appl. Phys. 1 Ž1973. 55. w8x T. Tamir ŽEd.., Guided-wave Optoelectronics, 2nd ed., Springer, Berlin, 1990. w9x M.J. Adams, An Introduction to Optical Waveguides, Wiley, New York, 1981. w10x R.E. Collin, Field Theory of Guided Waves, 2nd ed., IEEE Press, 1991. w11x P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. w12x L.B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, Prentice-Hall, Englewood Cliffs, NJ, 1973.
Optics Communications 157 Ž1998. 17–22
Direct image transmission through a multi-mode square optical fiber C.Y. Wu 1, A.R.D. Somervell, T.H. Barnes Department of Physics, UniÕersity of Auckland, PriÕate Bag 92019, Auckland, New Zealand Received 23 March 1998; revised 20 August 1998; accepted 25 August 1998
Abstract We explore the potential for direct image transmission through a square optical fiber. We show that when an image is sampled appropriately and its optical Fourier transform imaged on the end of a square fiber with perfectly reflecting walls, the components in the Fourier transform excite corresponding fiber modes. Specifically, even–even fiber modes carry information from one pixel only, while the odd modes carry information from neighboring pixels and give rise to cross-talk. When the odd modes are suppressed, the image can be perfectly recovered at the end of the fiber by a second optical Fourier transform. We suggest a method of suppressing the odd modes. In our system, dispersion of the mode phase velocity gives rise to different arrival times for the information in different image pixels, but has little or no effect on the output intensity distribution. We show that the square shape of the fiber is critical in forming the output image and confirm our theoretical predictions by computer simulation. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Image transmission; Multimode; Square optical-fiber
1. Introduction Direct image transmission through multimode optical fibers has attracted considerable interest over many years. The main difficulties in this type of image transmission arise from dispersion of the phase velocity of the fiber modes and angular ambiguities caused by symmetry when fibers of circular cross-section are used. Many methods have been developed to overcome these difficulties and comprehensive reviews and analysis of the subject, together with some new developments can be found in Refs. w1–4x. It is well known that an image formed in coherent light on the end of an optical fiber is expanded into a sequence of normal modes propagating along the fiber. Each mode has a characteristic amplitude distribution Žthe eigen func-
1
E-mail: [email protected]
tion. over the cross-section of the fiber. Usually, each image pixel excites several modes which propagate down the fiber at different speeds. Also, each mode carries contributions from several pixels. The image information is therefore badly corrupted after even only a short transmission distance and some way of equalising the transmission times for different modes is necessary before the image can be accurately reconstructed again. This is often achieved holographically. Our proposed system works in a different way. We arrange that each pixel in the image excites one corresponding mode in the fiber, by an optical transformation. At the output end of the fiber, a second optical transformation is applied to convert the modes back to single points of light, the reconstructed pixels. Now, eventhough the information in different pixels travels at different speeds down the fiber, we in principle avoid the mixing of information between pixels that occurs when each pixel excites several modes as in the conventional system. Al-
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 4 7 7 - 5
18
C.Y. Wu et al.r Optics Communications 157 (1998) 17–22
though the pixels arrive at the fiber output at slightly different times, the image intensity distribution is not corrupted. Consider a ‘one-dimensional’ fiber fed with a one-dimensional image. The modes here are sinusoidal functions with spatial frequencies determined by the size of the fiber and the optical wavelength. We also know that the Fourier transform of a pair of delta functions symmetrically disposed about the origin with appropriate signs, is a sinusoidal function w5x. It is therefore reasonable to expect that if an image is appropriately sampled and optically Fourier transformed before being presented to the fiber, we may be able to relate each pair of symmetrically disposed pixels to a single propagating mode, and transform that mode back to a pair of pixels again by a second optical Fourier transform at the output end of the fiber. Consideration of this simple model immediately raises two potential problems. First, each pair of pixels arrives at a different time because of modal dispersion, and second, the information in the two pixels of each symmetrical pair will be mixed together during propagation. The first of these difficulties is not too severe for ordinary lengths of fiber where the difference in mode transmission time would typically be of the order of nano- or micro-seconds, and the second problem can be overcome by transmitting only one of the symmetric pairs of pixels i.e. by restricting the input image to one half of the input field of the first Fourier transform lens. The simple one-dimensional model above can be easily extended to two dimensions with a fiber of rectangular cross-section, except that now the input image field must be restricted to a single quadrant of the input plane of the first Fourier lens. It should be noted that this system will not work for a round fiber because of the angular ambiguity, i.e. the information in input pixels in each circle centered on the optic axis of the input Fourier lens will be mixed as it passes down the fiber. In this paper we therefore limit our discussion to a transmission system consisting of a square fiber with two Fourier transform lenses. The principle of the optical Fourier transform and it’s applications are very well known, and can be found in
Refs. w1,6x. The propagation modes of a rectangular waveguide with perfectly reflecting walls and those of slab waveguides have also been thoroughly studied in the literature w7–12x. The rectangular optical waveguides are widely used in integrated optics applications w8x. Although mathematical difficulties have prevented the exact analysis of rectangular dielectric waveguides, many valuable approximations have been developed Žsee, for example Refs. w8–10x.. For simplicity, in this paper we restrict ourselves to the discussion of the ideal case of a rectangular optical fiber with perfectly reflecting walls. In Section 2 we present a short theoretical analysis which is then compared with numerical calculations, and we follow this with a brief discussion in Section 3.
2. Analysis The suggested system is sketched in Fig. 1 where L 1 and L 4 are Fourier transform lenses with focal length L f , and L 2 and L 3 are lens systems of magnification h. The optical fiber has a square cross-section with side size a and length L, and the cladding is a perfect reflector. Ei Ž x i , yi . where i s 1–6 represent the electrical field ŽTM. of the image during different stages of propagation. For simplicity, we also assume that the fiber can be regarded as being infinite in length when discussing the propagation of the light field through the fiber so that reflection from the far end of the fiber can be ignored. We will concentrate on the steady state case in this paper with the time factor expŽyi v t . suppressed. The field inside the fiber can be expressed by using the Green’s theorem w11x as E Ž x , y, z . 1
sy
H E Žx 4p S 0
0
0 , y0
.
E G Ž x , y, z < x 0 , y 0 , z 0 . E n0
d S0 z 0 s0
Ž1. where d S0 s d x 0 d y 0 , and n 0 is the outward normal of
Fig. 1. Sketch of the image transmission system consisting of a pair of Fourier lenses ŽL 1 and L 4 ., a pair of imaging lens systems ŽL 2 and L 3 ., and a perfect reflecting square fiber Žlength L..
C.Y. Wu et al.r Optics Communications 157 (1998) 17–22
the plane z 0 s 0. The Green’s function GŽ x, y, z < x 0 , y 0 , z 0 . satisfies the Helmholtz equation,
Ž=
2
q k 12
19
and similarly, ar2
E6 Ž x 6 , y6 . s hy1A
ar2
Hya r2Hyar2 E Ž x
. G Ž x , y, z < x 0 , y 0 , z 0 .
s y4pd Ž x y x 0 . d Ž y y y 0 . d Ž z y z 0 . ,
4
hk0
ž
Ž2.
=exp yi
Lf
4 , y4
.
/
Ž x 4 x 6 q y4 y6 . d x 4 d y4 .
and the boundary conditions, G s 0 on the surface, x s "ar2, y s "ar2, and z s 0. Solving the equation gives
for yar2F Ž x 4 , y4 . F ar2,y br2 F Ž x 6 , y6 . F br2. We can also combine Eqs. Ž5. – Ž7. to establish a direct relation between the input and output fields:
G Ž x , y, z < x 0 , y 0 , z 0 . `
16p s
a2
mp
ms 1 ns1
=sin
=
`
Ý Ý sin
½
mp a
a
Ž x q ar2. sin
Ž x 0 q ar2. sin
np a
np a
Ž y q ar2.
Ž y 0 q ar2.
sin Ž gmn z 0 . e ig m n z ,
z G z0
sin Ž gmn z . e ig m n z 0 ,
z - z0 ,
Ž7.
E6 Ž x 6 , y6 . s
Ž3.
(
k 1 s n1 vrc Ž ErE n 0 s yErE z 0 ., and n1 is the refractive index of the fiber material. The E-field at the front face of the fiber, E3 , can be found by noting that the lens, L 1, performs a 2-D Fourier Transform and that this is imaged onto the fiber by L 2 . This results in
Hyb r2Hybr2 E Ž x , y .
ž
=exp yi
Lf
a
2
`
`
ar2
ms1 ns1
=sin
mp a mp a
Ž x 3 q ar2. sin Ž x 4 q ar2. sin
np a np a
Ž y 3 q ar2. d x 3 d y 3 Ž y4 q ar2. e ig m n L , Ž 6 .
yi2 p Õ 1 y 3
3
ar2
ar2
4
d y3
yi2 p u 6 x 4
4
yi2 p Õ 6 y 4
Hya r2sin w np Ž y ra q 1r2. x e
=
d x4
d y4 ,
Ž9.
and
h x1 l0 L f h x6 l0 L f
,
Õ1 s
,
Õ6 s
h y1 l0 L f h y6 l0 L f
,
Ž 10 .
.
Ž 11.
Denoting the four similar integrals involved in Eq. Ž9. as Im1 , In1 , Im6 and In 6 and carrying out the integration we have a Im1 s e iŽ my1.p r2 sinc Ž mr2 y m1 . 2 m
y Ž y1 . sinc Ž mr2 q m1 . ,
Ž 12.
with similar expressions for the other three terms, if the input and output images are sampled by
Ds
ar2
Ý Ý Hya r2Hyar2 E3 Ž x 3 , y 3 .
= sin
ar2
x i s mi D,
E4 Ž x 4 , y4 . 4
Ž8.
d x3
Hya r2sin w mp Ž x ra q 1r2. x e
Ž x 1 x 3 q y1 y 3 . d x 1 d y1 ,
for ybr2F Ž x 1, y 1 . F br2, and yar2F Ž x 3 , y 3 . F ar2, where A is a complex constant multiplied by the factor, Žyie i2 k 0 L frl0 L f . w1x, and b is the size of the frame of the input image, b s h a. Then by using the Green’s function we can express the field at the output of the fiber as
yi2 p u 1 x 3
3
=
u6 s
/
ar2
Hya r2sin w np Ž y ra q 1r2. x e
1
Ž5.
s
ms1 ns1
Hya r2sin w mp Ž x ra q 1r2. x e =
u1 s
br2
hk0
br2
Fmn Ž x 1 , y 1 , x 6 , y6 .
Ž4.
1
br2
where
gmn s k 12 y Ž m2 q n2 . p 2ra2 ,
1
`
=Fmn Ž x 1 , y 1 , x 6 , y6 . d x 1 d y 1 e ig m n L ,
gmn
where
E3 Ž x 3 , y 3 . s h A
a2
`
Ý Ý Hyb r2Hybr2 E1Ž x 1 , y1 .
1
s
br2
4 A2
l0 L f b
yi s n i D s
l0 L f ha
Ž i s 1,6 . , b
s N
.
Ž 13.
In above, sincŽ x . ' sinŽ xp .rxp . We now discuss the cases of even and odd numbered modes separately. For even modes we set m s 2 m and Eq. Ž12. becomes ia Ž . ev Im1 s Ž y1 . mq1 Ž dm , m 1 y dm ,y m 1 . Ž 14. 2
C.Y. Wu et al.r Optics Communications 157 (1998) 17–22
20
Fig. 2. DFT of the fiber mode functions for m s 5: Ža. Imev Ž m1 . for an even mode and Žb. Imod Ž m1 . for an odd mode.
while for the odd modes, m s 2 m y 1, we have od Im1 s
a 2
and E6 Ž m 6 ,n 6 .
Ž y1. mq1 Ž sinc w m1 y m q 1r2x
a 2A2
qsinc Ž m1 q m y 1r2 . . .
Ž 15 .
ev od Im1 and Im1 as discrete functions of m1 , for the case m s 5 are shown in Fig. 2Ža. and 2Žb. respectively. This figure confirms our statement in the last section that each even mode relates to two symmetrically located pixels at m1 and ym1 while each odd mode may relate to several neighbuoring pixels. For Im6 , In1 and In6 we have similar expressions. By using these expressions we can write
ev od = Ž Im6 q Im6 .Ž In6ev q In6od .
Ž 16.
and E6 Ž m 6 ,n 6 . 4 A2 a2
`
`
Ý Ý
Nr2
Nr2
Ý
Ý
E1Ž m1 ,n1 .
ms 1 ns1 m 1 syNr2q1 n 1 syNr2q1
ev od = Ž Im1 q Im1 .Ž In1ev q In1od . ev od = Ž Im6 q Im6 .Ž In6ev q In6od . e ig m , n L .
Ž 17 .
In the special cases when both m and n are even and both are odd we have respectively E6 Ž m 6 ,n 6 . a 2A2 s 4
`
`
Ý Ý
Nr2
Nr2
Ý
Ý
4
`
`
Ý Ý
E1Ž m1 ,n1 .
ms 1 ns1 m 1 syNr2q1 n 1 syNr2q1
= Ž dm, m 1 y dm ,y m 1 .Ž dm , m 6 y dm ,y m 6 . = Ž dn, n 1 y dn ,y n 1 .Ž dn , n 6 y dn ,y n 6 . e ig 2 m ,2 n L
Ž 18 .
Nr2
Nr2
Ý
Ý
E1Ž m1 ,n1 .
ms 1 ns1 m 1 syNr2q1 n 1 syNr2q1
= sinc Ž m1 y m q 12 . q sinc Ž m1 q m y 12 . = sinc Ž m 6 y m q 12 . q sinc Ž m 6 q m y 12 . = sinc Ž n1 y n q 12 . q sinc Ž n1 q n y 12 . = sinc Ž n 6 y n q 12 . q sinc Ž n 6 q n y 12 . =e ig 2 my 1 ,2 ny 1 L .
ev od q Im1 Fmn Ž x 1 , y 1 , x 6 , y6 . s Ž Im1 .Ž In1ev q In1od .
s
s
Ž 19 .
Expanding the right-hand side terms in Eq. Ž18. and carrying out the summation we can see that carried through by even–even modes, m s 2 m1, n s 2 n1, each input pixel E1Ž m1,n1 . will be seen on the output screen as four images, E6 Ž m1,n1 ., E6 Žym1, n1 ., E6 Ž m1,y n1 . and E6Žym1,y n1 . meanwhile each output image E6Ž m 6 ,n 6 . will be the superposition of the images of four symmetrical input pixels, E1Ž m 6 ,n 6 ., E1Žym 6 ,n 6 ., E1Ž m 6 ,y n 6 . and E1Žym 6 ,y n 6 . with the same phase velocity. This ambiguity can be resolved by restricting the input image inside one quadrant as shown in Fig. 3Ža.. Through this treatment, we will have four unambiguous images in the whole output screen as shown in Fig. 3Žb. which is numerically calculated according to Eqs. Ž5. – Ž7. for a sampling matrix of N = N, N s 64. From Eq. Ž18. it is also expected that if the effects of the odd numbered modes are suppressed it should be possible to recover the input image perfectly without crosstalk, which is shown in Fig. 3Žc. when only the even–even modes are accounted for. Finally, as a comparison we show in Fig. 3Žd. the output image directly transmitted through the fiber without using the Fourier lenses, L 1 and L 4 . Now we show that the crosstalk Žthe effect of odd numbered modes as shown in Fig. 3Žb.. can be removed
C.Y. Wu et al.r Optics Communications 157 (1998) 17–22
21
Fig. 3. Numerical calculation of the amplitude of the images: Ža. Imput image; Žb. output image with both even and odd modes included; Žc. output image when even modes only are included and Žd. out put image without using the Fourier lenses, L 1 and L 4 .
theoretically by properly combining the four images in the ev output screen. In fact, if we expand the product Ž Im6 q od .Ž ev od . Im6 In6 q In6 in Eq. Ž17. referring to the expressions of ev od Im1 and Im1 in the form ev od q Im6 Ž Im6 .Ž In6ev q In6od . ev ev ev od od ev od od s Im6 In6 q Im6 In6 q Im6 In6 q Im6 In6
s Ž a 2r4 .
yŽ d
m, m 6 y d m ,y m 6
This combination can be achieved by changing the phase of E6Žym 6 ,n 6 . and E6Ž m 6 ,y n 6 . by p , and adding the images interferometrically. This has also been confirmed numerically by adding up the four terms in the left-hand side of Eq. Ž21. calculated from Eq. Ž7.. This leads to an output image exactly the same as that shown in Fig. 3Žc. except for an increase of the intensity by a factor of 4.
.Ž dn , n y dn ,y n . 6
6
q i Ž dm, m 6 y dm ,y m 6 . Ž sinc Ž n 6 y n q 1r2 . qsinc Ž n 6 q n y 1r2 . . q i Ž sinc Ž m 6 y m q 1r2 . qsinc Ž m 6 q m y 1r2 . . Ž dn , n 6 y dn ,y n 6 .
Ž sinc Ž m6 y m q 1r2. q sinc Ž m6 q m y 1r2. . = Ž sinc Ž n 6 y n q 1r2 . q sinc Ž n 6 q n y 1r2 . . 4 ,
q
Ž 20. then it is easy to see that 1 4 w E6 Ž m 6 ,n 6 . y E6 Ž ym 6 ,n 6 . y E6 Ž m 6 ,y n 6 . qE6 Ž ym 6 ,y n 6 . x a 2A2 s 4
`
`
Ý Ý
Nr2
Nr2
Ý
Ý
E1Ž m1 ,n1 .
ms 1 ns1 m 1 syNr2q1 n 1 syNr2q1
= Ž dm, m 1 y dm ,y m 1 . i Ž dm , m 6 y dm ,y m 6 . = Ž dn, n 1 y dn ,y n 1 .Ž dn , n 6 y dn ,y n 6 . e ig 2 m ,2 n L s a2A2 E1Ž m 6 ,n 6 . e ig 2 m 6 ,2 n 6 L .
Ž 21.
3. Discussion Ž1. When the finite length of the fiber is taken into account, the Green’s function, Eq. Ž3., will be slightly altered and the factor, e ig m n L, in Eq. Ž8. will be replaced correspondingly by wŽ1 q R .rŽ1 q Re i2g m n L .xe ig m n L. Since the normal reflection coefficient R Žamplitude. is about 0.2 from the glassrair interface the terms in square brackets have been treated as unity in this analysis. Moreover, because its influence is restricted to single pixels, it will not have significant effect on the overall picture. Ž2. Ordinary round section fibers do not have the image transmission characteristics described above. This is mainly because of the angular ambiguity or ‘azimuthal uniformity’ indicated in Ref. w3x as mentioned before. Besides, the spatial distribution of the zero points of the mode functions is also different for different modes. This causes additional crosstalk when a fixed sampling rate is used.
22
C.Y. Wu et al.r Optics Communications 157 (1998) 17–22
Ž3. This approach can, in principle, be used for an imperfect reflecting square fiber as long as it supports sufficient guided wave modes to carry the pixels.
from the New Zealand Institute for Industrial Research and Development. References
4. Conclusion The possibility of direct image transmission through a perfect reflecting square fiber is demonstrated theoretically. The key point of the new approach is the introduction of a pair of Fourier lenses and the use of a square fiber.
Acknowledgements We wish to thank Professor A.C. Kibblewhite and Professor G.L. Austin, both of the Physics Department, University of Auckland, for their help and support during this work. We are also indebted to Dr. S.M. Tan of the Physics Department, University of Auckland, and Dr. Tomohiro Shirai, Optical Engineering Division, Mechanical Engineering Laboratory, Japan, for interesting discussions and to Dr. D.D. Wu for help with preparation of the manuscript. Finally, we acknowledge the financial support
w1x A. Yariv, Optical Electronics in Modern Communications, 5th ed., Oxford University Press, 1997. w2x A. Yariv, J. Opt. Soc. Am. 66 Ž1976. 306. w3x A.A. Friesem, U. Levy, Y. Silberberg, Proc. IEEE 71 Ž1983. 208. w4x J.-Y. Son, V.I. Bobrinev, H.-W. Jeon, Y.-H. Cho, Y.-S. Eom, Appl. Optics 35 Ž1996. 273. w5x R.N. Bracewell, The Fourier Transform and its Applications, 2nd ed., McGraw-Hill, 1986. w6x F.T.S. Yu and Suganda Jutamulia, Optical Signal Processing, Computing, and Neural Networks, Wiley, New York, 1992. w7x R. Ulrich, W. Prettl, Appl. Phys. 1 Ž1973. 55. w8x T. Tamir ŽEd.., Guided-wave Optoelectronics, 2nd ed., Springer, Berlin, 1990. w9x M.J. Adams, An Introduction to Optical Waveguides, Wiley, New York, 1981. w10x R.E. Collin, Field Theory of Guided Waves, 2nd ed., IEEE Press, 1991. w11x P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953. w12x L.B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, Prentice-Hall, Englewood Cliffs, NJ, 1973.