Direct image transmission.pdf

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1 December 1998

.Optics Communications 157 1998 1722

Direct image transmissionthrough a multi-mode square optical fiber

C.Y. Wu 1, A.R.D. Somervell, T.H. Barnes

Department of Physics, Uniersity of Auckland, Priate 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 issampled 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, eveneven 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, to-

gether with some new developments can be found in Refs.w x14 .

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 modehas a characteristic amplitude distribution the eigen func-

1E-mail: [email protected]

.tion over the cross-section of the fiber. Usually, eachimage 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 trans-

mission distance and some way of equalising the transmis-

sion 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 corre-

sponding mode in the fiber, by an optical transformation.

At the output end of the fiber, a second optical transforma-

tion 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. .P I I : S 0 0 3 0 - 4 0 1 8 9 8 0 0 4 7 7 - 5


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( )C.Y. Wu et al.r Optics Communications 157 1998 172218

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-di-

mensional 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 dis-

posed about the origin with appropriate signs, is a sinu-

w xsoidal function 5 . It is therefore reasonable to expect thatif 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 timewould 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 ambigu-

ity, 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 its

applications are very well known, and can be found in

w xRefs. 1,6 . The propagation modes of a rectangular wave-

guide with perfectly reflecting walls and those of slab

waveguides have also been thoroughly studied in the litera-w xture 7 12 . The rectangular optical waveguides are widely

w xused in integrated optics applications 8 . Although mathe-

matical difficulties have prevented the exact analysis of

rectangular dielectric waveguides, many valuable approxi-mations have been developed see, for example Refs.

w x.810 .

For simplicity, in this paper we restrict ourselves to thediscussion 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 L1

and L are Fourier transform lenses with focal length L ,4 f

and L and L are lens systems of magnification h. The2 3optical fiber has a square cross-section with side size a

and length L, and the cladding is a perfect reflector. . .E x , y where i s 1 6 represent the electrical field TMi i i

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 exp yi vt suppressed.

The field inside the fiber can be expressed by using thew xGreens theorem 11 as

E x ,y ,z .


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( )C.Y. Wu et al.r Optics Communications 157 1998 1722 19

< .the plane z s 0. The Greens function G x,y,z x ,y ,z0 0 0 0

satisfies the Helmholtz equation,

2 2


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ev od . . . .Fig. 2. DFT of the fiber mode functions for m s5: a I m for an even mode and b I m for an odd mode.m 1 m 1

while for the odd modes, m s 2 m y 1, we have

amq1od w xI s y1 sinc m y m q 1r2 . m1 1

2

qsinc m q m y 1r2 . 15 . . .1ev odI and I as discrete functions of m , for the case m s 5m1 m1 1

. .are shown in Fig. 2 a and 2 b respectively.This figure confirms our statement in the last section

that each even mode relates to two symmetrically located

pixels at m and ym while each odd mode may relate to1 1

several neighbuoring pixels. For I ,I and I we havem6 n1 n6similar expressions. By using these expressions we can

write

F x ,y ,x ,y s Iev qIod Iev qIod . . .mn 1 1 6 6 m1 m1 n1 n1

= Iev qIod Iev qIod 16 . . .m6 m6 n6 n6

and

E m , n .6 6 6

Nr2 Nr22 ` `4As E m , n . 1 1 12a ms 1 ns1 m syNr2q1 n syNr2q11 1

= Iev qIod Iev qIod . .m1 m1 n1 n1

= Iev qIod Iev qIod e igm , nL. 17 . . .m6 m6 n6 n6

In the special cases when both m and n are even and both

are odd we have respectively

E m , n .6 6 6

Nr2 Nr22 2 ` `a As E m ,n . 1 1 1

4ms 1 ns1 m syNr2q1 n syNr2q11 1

= d y d d y d . .m, m m ,ym m , m m ,y m1 1 6 6

= ig L2 m ,2 nd y d d y d e 18 . . .n , n n ,yn n , n n ,yn1 1 6 6

and

E m , n .6 6 6

Nr2 Nr22 2 ` `a As E m , n . 1 1 1


= 1 1sinc m y m q q sinc m q m y . .1 12 2

= 1 1sinc m y m q q sinc m q m y . .6 62 2

= 1 1sinc n y n q q sinc n q n y . .1 12 2

= 1 1sinc n y n q q sinc n q n y . .6 62 2

= ig L2 my 1, 2 ny 1e . 19 .

.Expanding the right-hand side terms in Eq. 18 andcarrying out the summation we can see that carried through

by eveneven modes, m s 2 m , n s 2 n , each input pixel1 1 .E m ,n will be seen on the output screen as four

1 1 1

. . .images, E m , n , E ym , n , E m ,y n and6 1 1 6 1 1 6 1 1

. .E ym ,y n meanwhile each output image E m , n6 1 1 6 6 6

will be the superposition of the images of four symmetrical . . .input pixels, E m ,n , E ym , n , E m ,y n and

1 6 6 1 6 6 1 6 6

.E ym ,y n with the same phase velocity. This ambigu-1 6 6

ity can be resolved by restricting the input image inside .one quadrant as shown in Fig. 3 a . Through this treat-

ment, we will have four unambiguous images in the whole .output screen as shown in Fig. 3 b which is numerically

. .calculated according to Eqs. 5 7 for a sampling matrix .of N=N,Ns 64. From Eq. 18 it is also expected that if

the effects of the odd numbered modes are suppressed itshould be possible to recover the input image perfectly

.without crosstalk, which is shown in Fig. 3 c when onlythe eveneven modes are accounted for. Finally, as a

.comparison we show in Fig. 3 d the output image directlytransmitted through the fiber without using the Fourier

lenses, L and L .1 4

Now we show that the crosstalk the effect of odd ..numbered modes as shown in Fig. 3 b can be removed


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( )C.Y. Wu et al.r Optics Communications 157 1998 1722 21

. . .Fig. 3. Numerical calculation of the amplitude of the images: a Imput image; b output image with both even and odd modes included; c .output image when even modes only are included and d out put image without using the Fourier lenses, L and L .1 4

theoretically by properly combining the four images in the evoutput screen. In fact, if we expand the product I qm6

od . ev od . .I I qI in Eq. 17 referring to the expressions ofm6 n6 n6Iev and Iod in the formm1 m1

Iev qIod Iev qIod . .m6 m6 n6 n6sIev Iev qIev Iod qIodIev qIodIodm6 n6 m6 n6 m6 n6 m6 n6

2s a r4 y d y d d y d . . .m , m m ,ym n , n n ,yn6 6 6 6q i d y d sinc n y n q 1r2 . .m , m m ,ym 66 6qsinc n q n y 1r2 q i sinc m y m q 1r2 . .. 6 6

qsinc m q m y 1r2 d y d .. .6 n , n n ,yn6 6q sinc m y m q 1r2 q sinc m q m y 1r2 . . .6 6

= sinc n y n q 1r2 q sinc n q n y 1r2 , . . . 46 620 .

then it is easy to see that1

E m , n yE ym , n yE m ,y nw . . .6 6 6 6 6 6 6 6 64qE ym ,y n x .6 6 6

Nr2 Nr22 2 ` `a As E m ,n . 1 1 1


= d y d i d y d . .m, m m ,ym m , m m ,ym1 1 6 6= ig L2 m ,2 nd y d d y d e . .n , n n ,yn n , n n ,yn1 1 6 6

s a2A2E m ,n e ig2 m6 ,2 n 6L. 21 . .1 6 6

This combination can be achieved by changing the phase . .of E ym , n and E m ,y n by p, and adding the

6 6 6 6 6 6

images interferometrically. This has also been confirmed

numerically by adding up the four terms in the left-hand . .side of Eq. 21 calculated from Eq. 7 . This leads to an

.output image exactly the same as that shown in Fig. 3 cexcept for an increase of the intensity by a factor of 4.

3. Discussion

.1 When the finite length of the fiber is taken into .account, the Greens function, Eq. 3 , will be slightly

igmn L .altered and the factor, e , in Eq. 8 will be replacedw . i2gmn L.x igmn Lcorrespondingly by 1 qR r 1 qRe e . Since

.the normal reflection coefficient R amplitude is about0.2 from the glassrair interface the terms in square brack-

ets have been treated as unity in this analysis. Moreover,

because its influence is restricted to single pixels, it willnot have significant effect on the overall picture.

.2 Ordinary round section fibers do not have the imagetransmission characteristics described above. This is mainly

because of the angular ambiguity or azimuthal uniformityw xindicated in Ref. 3 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.


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.3 This approach can, in principle, be used for animperfect reflecting square fiber as long as it supports

sufficient guided wave modes to carry the pixels.

4. Conclusion

The possibility of direct image transmission through a

perfect reflecting square fiber is demonstrated theoreti-

cally. The key point of the new approach is the introduc-tion 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. To-

mohiro Shirai, Optical Engineering Division, Mechanical

Engineering Laboratory, Japan, for interesting discussionsand to Dr. D.D. Wu for help with preparation of the

manuscript. Finally, we acknowledge the financial support

from the New Zealand Institute for Industrial Research and

Development.

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.Appl. Optics 35 1996 273.w x5 R.N. Bracewell, The Fourier Transform and its Applications,

2nd ed., McGraw-Hill, 1986.w x6 F.T.S. Yu and Suganda Jutamulia, Optical Signal Processing,

Computing, and Neural Networks, Wiley, New York, 1992.w x .7 R. Ulrich, W. Prettl, Appl. Phys. 1 1973 55.w x .8 T. Tamir Ed. , Guided-wave Optoelectronics, 2nd ed.,

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