- Email: [email protected]
All-optical image transmission through a multimode dielectric waveguide
1 November 2000
Optics Communications 185 (2000) 49±56
www.elsevier.com/locate/optcom
All-optical image transmission through a multimode dielectric waveguide A.R.D. Somervell a,*, C.Y. Wu a, T.G. Haskell b, T.H. Barnes a a
Physics Department, University of Auckland, Private Bag 92019, Auckland, New Zealand b Industrial Research Ltd., Private Bag 31310, Lower Hutt, New Zealand Received 15 February 2000; accepted 6 September 2000
Abstract We show that images can be transmitted directly through optical waveguides by matching image pixels to waveguide modes. We describe how this can be achieved using a dielectric slab waveguide and present initial experimental results showing successful image transmission. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Image transmission; Multimode; Waveguide; Fourier transform
1. Introduction Although ®bre optic data transmission methods are extremely highly developed and widely used by telecommunications companies, there is still considerable interest in new optical methods for encoding and decoding image data for transmission via a single ®bre. Of course, images may be sent all-optically through both single and multimode ®bres by simple scanning techniques. All-optical multiplexing methods based on interferometry have also been developed for sending images via single mode ®bres [1,2]. Multimode ®bres oer the possibility of transmitting several channels of information in parallel fashion by encoding that information onto the ®bre mode structure. However, the eects of modal dispersion must be overcome to achieve this success*
Corresponding author. Fax: +64-9-373-7445. E-mail address: [email protected] (A.R.D. Somervell).
fully. If, for example, an image is focused on the front face of a multimode slab waveguide, many modes are excited. In a dielectric slab waveguide these modes have sinusoidal electric ®eld amplitude distribution across the guide, and the guided modes form a set of orthogonal basis functions into which the image is encoded. The amplitude of each excited mode is therefore determined by contributions from all pixels in the image ± and is commensurate with the expansion of the image into the basis functions represented by the modes. If all modes propagated through the guide at the same speed, they would interfere at the end of the guide to reproduce the input image, but unfortunately this is not the case. Higher order modes travel more slowly than lower order modes, due to modal dispersion, and this introduces phase shifts between the modes at the end of the guide. These phase shifts cause scrambling of the image information and production of a random speckle pattern rather than an image when the modes interfere at the end of the guide.
0030-4018/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 0 9 8 0 - 9
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A.R.D. Somervell et al. / Optics Communications 185 (2000) 49±56
Several techniques have been used in attempts to overcome the modal dispersion problem, including phase conjugation [3,4], holographic ®ltering [5] and several other methods for dispersion compensation [6]. However, most of these only work when the ®bre mode dispersion characteristic remains constant, or if the transmitting ®bre can be divided into two identical sections with phase conjugation occurring between them. The result of this in practice is that the ®bre(s) must remain strictly stationary for successful image transmission. We recently described a dierent technique for sending images through a multimode ®bre with perfectly re¯ecting sidewalls [7], which does not rely on mode dispersion compensation, but rather encodes the input image pixels in such a way that modal dispersion does not aect the ®nal image. Using optical Fourier transforms, we match a single ®bre mode to a single image pixel on the input side, and ensure that each mode contributes to only one pixel at the output side of the ®bre. In this paper, we show that our method may also be applied to dielectric slab waveguides which rely on total internal re¯ection to con®ne the light. We also present the results of initial experiments designed to test these ideas where we were able to successfully demonstrate one-to-one pixel to mode encoding and image transmission. 2. Pixel-mode encoding using optical Fourier transforms Fig. 1 shows the principle of our method. The input image is ®rst sampled in a discrete set of pixels using a pinhole array or a multiple beam diractive beam splitter and mask. The sampled image is then optically Fourier transformed using a lens to produce what is eectively a discrete Fourier transform (because of the sampling) which is then projected onto the input face of the waveguide at appropriate magni®cation. Provided the magni®cation is correct, each input pixel produces a wave at the input to the waveguide which has a phase distribution matching that of one of the even numbered waveguide modes. The amplitude of this mode then depends only on the intensity of light at
Fig. 1. System for image transmission through a slab waveguide.
that input pixel. The modes travel to the end of the waveguide (albeit at dierent velocities), where a second Fourier transform lens reconstructs the sampled input image. Each output pixel arises from only one even numbered mode and so no intermodal interference occurs. Note that dierent pixels in the transmitted image take dierent times to travel through the system, but this is not a problem as arrival times at the output dier by typically only a few nanoseconds. The discrete Fourier transform of the input image at the input of the waveguide also excites odd-numbered modes. Each input pixel in fact excites several odd-numbered modes and their contribution to the output image must therefore be eliminated in order to avoid spurious light in the output image plane.
3. Theoretical analysis If the electric ®eld across the input image is denoted E1 x1 as shown in Fig. 1, then the electric ®eld at the front face of the waveguide, E2 x2 is the Fourier transform of the input image which is given by: Z x2 1 E2 x2 E1 x1 exp ÿ i2p x1 dx1 fk where: f is the focal length of the Fourier transform lens k is the wavelength of the light used. In order to ®nd the electric ®eld at the waveguide output it is necessary to know the GreenÕs function for a slab waveguide. Then the electric
A.R.D. Somervell et al. / Optics Communications 185 (2000) 49±56
®eld after propagating a distance L along the waveguide is given by the following: Z ÿ1 oG x3 ; zjx2 ; z0 E2 x 2 ; 0 dx2 E3 x 3 ; z 4p on0 z0 0 2 where: G x3 ; zjx2 ; z0 is the GreenÕs function for the slab waveguide n0 is the outward normal of the plane z0 0 E2 x2 ; 0 is the electric ®eld distribution across the front face of the waveguide. The GreenÕs function for the waveguide is found by solving the non-homogeneous Helmholtz equation: ÿ 2 r k 2 x G x; zjx0 ; z0 ÿ4pd x ÿ x0 d z ÿ z0
3
where: G x; 0jx0 ; z0 0, k x k1 for j xj 6 a=2 and k x k2 for j xj P a=2 and the boundary conditions are given by the continuity of the tangential components of the electric and magnetic ®elds across the boundaries at x a=2. We assume there is no cladding layer so re¯ection takes place from a glass±air interface. Solving this equation gives: Z w B x< wT x> G x; zjx0 ; z0 ÿ2 1 ÿ R2 exp i2n1 a
sin cz0 exp in1 a exp icz dc n1 4
where: x< is the lesser of x and x0 and x> is the greater of x and x0 , n1 is the component of the wave vector in the xdirection, R is the re¯ection coecient for a wave re¯ected from the glass±air interface with an x-component of the wave vector given by n1 , a x exp in x ÿ w 1 T 2 a R exp ÿ in1 x ÿ 2 and
51
a
w B x exp ÿ in1 x 2a R exp in1 x 2 Substituting the expression for the GreenÕs function into Eq. (2) and solving gives the following for the electric ®eld at the output of the waveguide after propagating a distance z L: M sin mp x3 1 exp icm L 2 am 2X E3 x3 ; L 0 a m0 1 ÿ 2 U2m Z a=2 x2 1 dx2 E2 x2 sin mp am 2 ÿa=2 5 where: cm is the wave number for the mth mode U0m is the derivative of the phase change on re¯ection experienced by each mode on re¯ection from the glass±air interface with respect to the component of the wave vector in the x-direction. am is the eective width of the waveguide for the mth mode and is given by: a 6 am m 1 2U mp where Um is the phase change on re¯ection from the glass±air interface (the Goos±Hanchen shift) for the mth mode. Combining Eqs. (1), (4) and (5) leads to the following expression for the electric ®eld of the mth mode: x3 1 7 exp icm L E3m x3 Am sin mp am 2 where the amplitude of the mth mode, Am , is given by: Am
a 1 0 2 1 ÿ 2Um a Z x1 a m Um E1 x1 sinc ÿ ÿ p fk 2 x1 a m U m m dx1 ÿ ÿ 1 sinc p fk 2
8
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A.R.D. Somervell et al. / Optics Communications 185 (2000) 49±56
Eq. (9) shows that the amplitude of the even modes is determined by the whole of the input image multiplied by a weighting function given by the sum of two sinc functions whose position is dependent on the mode number. At positions where the weighting functions for an even numbered mode equal one or negative one, the weighting functions for the other even modes are very close to zero. This is illustrated in Fig. 2 for the even modes m 10, 12 and 14. By sampling at these positions each even mode has a contribution from only two points in the input plane, one on either side of the optic axis. In fact, the amplitude of the electric ®eld is proportional to the electric
®eld at the pixel in the positive half plane minus the electric ®eld in the negative half plane. By placing the image in the positive half plane only, ensuring the electric ®eld is zero over all the negative half plane, the amplitude of the even modes is equal to the amplitude of just one of the input pixels. The eect of having dierent values for the phase shift on re¯ection, Um for dierent modes is to make the sample spacing slightly non-linear. It also causes the position where the weighting function for a given mode is one to be slightly oset from the zeros of the weighting functions corresponding to other modes. This allows a contribution to each mode from more than one image pixel. However, the contribution from unwanted modes is negligible. The contribution of each of the image pixels to one of the odd numbered modes is shown in Fig. 3. As the image is to place in the positive half plane only the sampling positions in that half plane are shown. Examining Fig. 3 shows the sampling positions do not match up with the positions where the weighting function is zero and one and there is therefore a contribution to the amplitude of each mode from many image pixels. The electric ®eld at the output of the waveguide is Fourier transformed giving an electric ®eld distribution E4 x4 . The contribution to the electric ®eld from each waveguide mode is described by the following equation:
Fig. 2. Even mode sampling positions.
Fig. 3. Odd mode amplitude.
The term outside the integral varies with m because the ®bre modes do not quite form an orthogonal set, however its variation with mode number is very small and for all practical purposes it can be considered constant. In order to show how the image should be sampled to ensure one pixel excites one mode it is easiest to consider the odd and even numbered modes separately. For even numbered modes, i.e. m 2; 4; 6 . . . ; the amplitude Am (ignoring the ®rst term in Eq. (8)) is approximately: Z a x1 a m U m E1 x1 sinc ÿ ÿ Am p 2 fk 2 x1 a m U m 9 dx1 ÿ sinc fk 2 p
A.R.D. Somervell et al. / Optics Communications 185 (2000) 49±56
x4 a m U m E4m x4 Am exp icm L sinc ÿ ÿ p fk 2 x4 a m Um m ÿ ÿ 1 sinc p fk 2
10
The total electric ®eld output is found from the sum of the electric ®elds from each waveguide mode. Again it is easiest to think about even and odd numbered modes separately. For even numbered modes, at the positions where the amplitude due to one of the modes is Am (where the term in square brackets is 1) the output from other modes is zero. By sampling at these positions output from the even modes gives a reconstruction of the input image. This is because the amplitude of each mode is the same as one of the image pixels. Note that there is no intermodal interference between the output due to even waveguide modes because each output pixel is located in a dierent region of space. However each pixel arrives at the output at a dierent time. Also, there are two versions of the input image, one in either half plane [7]. Although there is a contribution from just one even mode at each output pixel, there is an extra contribution from many odd numbered modes. This needs to be removed in order to see the reconstructed image. Examination of Eq. (10) reveals that the output ®eld due to even mode numbers is an odd function, E4even x4 ÿE4even ÿx4 g ± i.e. there are two copies of the output from even numbered mode on either side of the optic axis which are out of phase with each other. The output from odd numbered modes is an even function, E4odd x4 E4odd ÿx4 ± i.e. the two copies of the output from odd numbered modes are in phase with one another. The contribution from the odd modes can be removed by ®rst creating two copies of the output plane, E4 x4 . Inverting one copy gives an image, E4 ÿx4 which can be subtracted from the non-inverted image. This results in destructive interference between from the odd the outputs odd numbered modes, E4 x4 ÿ E4odd ÿx4 0 and constructive interference between the even numbered modes E4even x4 ÿ E4even ÿx4 2E4even x4 . Only the output from the even numbered modes is left giving a reconstruction of the original input image at
53
the ®nal output. In principle at least this procedure can be done optically using an interferometer. Also there is no reason for this image subtraction procedure to be performed at the output of the system. In fact because the system is linear the image subtraction can be performed at any point along it including positions before the waveguide.
4. Experimental system Experiments were performed to show that images could indeed be transmitted through a slab waveguide using the techniques described above. A diagram of the experimental set-up is shown in Fig. 4. A TM polarized He±Ne laser beam is expanded and collimated, and then passes through a diractive optic beam splitter giving 10 plane waves each separated by an angle of 0.4°. These beams then pass through a lens giving an array of 10 spots at its focal plane. The beams on one side of the optic axis are blocked to ensure the input image lies on one half of the optic axis only leaving an array of ®ve spots. The spacing between the spots is adjusted using a variable magni®cation telescope consisting of a zoom lens together with a ®xed focal length lens. The output from this system is used as the sampled image to be transmitted. The image is Fourier transformed using a lens and the Fourier transform is incident on the front face of a dielectric slab waveguide. Light propagates to the back face of the waveguide where it is Fourier transformed again and the intensity distribution is measured using CCD camera.
Fig. 4. Experimental system used for transmitting images.
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A.R.D. Somervell et al. / Optics Communications 185 (2000) 49±56
It should be noted that there is no system for removing the output which occurs due to the odd waveguide modes and therefore the reconstructed output image will not be an exact copy of the input image. Also, the image pixels are not positioned accurately enough so that just one even numbered mode is excited by each pixel as it has been shown is necessary for exact reconstruction of the input image. This is because in the set-up used for this experiment only fairly course adjustment of the pixel spacing is achievable using the variable magni®cation telescope system. Adjustment of the pixel spacing is made even more dicult because the magni®cation system and therefore the sample spacing cannot be adjusted without causing the positions all of the following components to need readjusting. Due to inaccuracies in the sampling positions each pixel excites a combination of many even and odd numbered modes. The waveguide used in this experiment was a 50 20 0:5 mm3 waveguide custom built from BK7 glass. Light entered and exited the 0:5 20 mm2 faces which were ¯at to better than a wavelength over the entire surface and antire¯ection coated at 633 nm. The 0:5 20 mm2 faces were perpendicular to the 50 20 mm2 faces to better than 10 arcs and the 50 20 mm2 surfaces were parallel to better than 15 arcs. It is important that the 0:5 20 mm2 faces are optically ¯at and perpendicular to the 50 20 mm2 faces to ensure the waves entering and exiting the waveguide are aberration free so as not to alter the amplitude of the modes are excited by each pixel. The ¯atness of the 50 20 mm2 faces ensures that there is no coupling between modes as light propagates through the waveguide. The waveguide was mounted on a tilt/rotation stage accurate to 3 arcs together with an x±y translation stage to enable precise positioning of the waveguide. 5. Results Fig. 5 shows four images before and after transmission, each with a dierent number of image pixels. Each input image is shown above its
corresponding output image. The input images were obtained by blocking one or more of the pixels in the ®ve pixel input array. Note that for each input image pixel there are two corresponding output pixels as expected theoretically. Each output pixel is spread out both horizontally and vertically. The vertical spreading occurs because the guiding eect only occurs in one direction in the slab waveguide and light is free to diract in the other direction. The horizontal spreading occurs because several waveguide modes are excited by each input pixel as mentioned above. This also increases the spacing needed between the pixels in the image to be transmitted in order to form separate output pixels, thus the total number of pixels which can be transmitted is reduced when both odd and even modes are excited. The total number of pixels which can be sent if only even numbered modes were excited is equal to the number of even numbered modes supported by the waveguide. For the waveguide used in this experiment this is about 1200 guided even numbered modes. When the pixels are not positioned extremely accurately and the odd modes not removed as is the case in this experiment each input pixel excites about 20±30 modes (both odd and even modes included) with signi®cant amplitude (greater than 10% of the maximum mode amplitude). Thus the number of pixels which can be sent through the image is reduced by a factor of 10±15 when odd modes are not removed. Also, as the spacing between the image pixels increases, the cross-talk between the dierent output pixels reduces. It is also seen from Fig. 5 that although each image pixel is of nearly the same intensity, the intensity of the output pixels is not. For example, in Fig. 5c, the output pixels on the right-hand side underneath the two input image pixels show markedly dierent intensities. This can be explained as follows. Each output pixel arises from the interference between the individual components which make up that pixel ± i.e. the components due to the several waveguide modes excited by a single image pixel. Each of the components in the set which makes up a single output pixel has a particular phase which arises
A.R.D. Somervell et al. / Optics Communications 185 (2000) 49±56
55
Fig. 5. Input and output images: (a) (top) four input pixels shown above the corresponding output image, (b) (second from top) the output obtained from three input pixels, (c) (second from bottom) output due to two input pixels, (d) (bottom) output from a single input pixel.
from modal dispersion. The phase dierences between each of the components in the set of modes making a given output pixel is not the same as those in the set making up a dierent pixel. Interference between the components therefore results in dierent intensities for dierent pixels. From Fig. 5d it can also be seen that the two output pixels which arise from the same image pixel are not the same intensity even though the phase dierence due to modal dispersion and amplitude of the waveguide modes which make up those output pixels is the same. This is because in the left-hand side of the output plane there is additional p phase shift between the output components arising from odd modes and those arising from even modes. This phase shift is not present in the right-hand side output plane so the intensities of either output pixel is dierent. It should be noted that this eect would disappear if odd mode cancellation was applied.
6. Conclusion We have described a technique for transmitting one dimensional images through a dielectric slab waveguide. In principle this involves matching even numbered waveguide modes to image pixels by sampling the image appropriately, then illuminate the front face of the ®bre with the Fourier transform of the sampled image. The image is then reconstructed by Fourier transforming the waveguide output and sampling again. In order to get an exact reconstruction of the original image the contribution to the output from odd numbered modes should be removed by subtracting the electric ®eld distribution in one side of the output plane from that in the opposite side. We have also described initial experiments showing transmission of images through a slab waveguide. The output image was not an exact replica of the input image because the output due to odd numbered modes was not removed and the
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A.R.D. Somervell et al. / Optics Communications 185 (2000) 49±56
input pixels were not positioned accurately enough to excite only one even mode each. This resulted in spreading of the output pixels. Acknowledgements We would like to express our thanks to the Royal Society of New Zealand for ®nancial assistance provided through the Marsden fund. We are also grateful to Industrial Research Limited for their help in this work. References [1] P. Naulleau, M. Brown, C. Chen, E. Leith, Direct threedimensional image transmission through single-mode
[2]
[3]
[4] [5] [6] [7]
®bers with monochromatic light, Opt. Lett. 21 (1996) 36±38. A.R.D. Somervell, C.Y. Wu, T.G. Haskell, T.H. Barnes, All-optical binary phase encoding/decoding for image transmission through apertures smaller than the Rayleigh limit, Opt. Commun. 162 (1999) 291±298. P.H. Beckwith, I. McMichael, P. Yeh, Image distortion in multimode ®bers and restoration by polarization preserving phase conjugation, Optics Lett. 12 (1987) 510±512. B. Fischer, S. Sternklar, Image transmission and interferometry with multimode ®bres using self-pumped phase conjugation, Appl. Phys. Lett. 46 (1985) 113±114. U. Levy, A.A. Friesem, Direct picture transmission in a single optical ®bre with holographic ®lters, Opt. Commun. 30 (1979) 163±165. R. Ulrich, Image formation by phase coincidences in optical waveguides, Opt. Commun. 13 (1975) 259±264. C.Y. Wu, A.R.D. Somervell, T.H. Barnes, Direct image transmission through a square multi-mode optical ®bre, Opt. Commun. 157 (1998) 17±22.
Optics Communications 185 (2000) 49±56
www.elsevier.com/locate/optcom
All-optical image transmission through a multimode dielectric waveguide A.R.D. Somervell a,*, C.Y. Wu a, T.G. Haskell b, T.H. Barnes a a
Physics Department, University of Auckland, Private Bag 92019, Auckland, New Zealand b Industrial Research Ltd., Private Bag 31310, Lower Hutt, New Zealand Received 15 February 2000; accepted 6 September 2000
Abstract We show that images can be transmitted directly through optical waveguides by matching image pixels to waveguide modes. We describe how this can be achieved using a dielectric slab waveguide and present initial experimental results showing successful image transmission. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Image transmission; Multimode; Waveguide; Fourier transform
1. Introduction Although ®bre optic data transmission methods are extremely highly developed and widely used by telecommunications companies, there is still considerable interest in new optical methods for encoding and decoding image data for transmission via a single ®bre. Of course, images may be sent all-optically through both single and multimode ®bres by simple scanning techniques. All-optical multiplexing methods based on interferometry have also been developed for sending images via single mode ®bres [1,2]. Multimode ®bres oer the possibility of transmitting several channels of information in parallel fashion by encoding that information onto the ®bre mode structure. However, the eects of modal dispersion must be overcome to achieve this success*
Corresponding author. Fax: +64-9-373-7445. E-mail address: [email protected] (A.R.D. Somervell).
fully. If, for example, an image is focused on the front face of a multimode slab waveguide, many modes are excited. In a dielectric slab waveguide these modes have sinusoidal electric ®eld amplitude distribution across the guide, and the guided modes form a set of orthogonal basis functions into which the image is encoded. The amplitude of each excited mode is therefore determined by contributions from all pixels in the image ± and is commensurate with the expansion of the image into the basis functions represented by the modes. If all modes propagated through the guide at the same speed, they would interfere at the end of the guide to reproduce the input image, but unfortunately this is not the case. Higher order modes travel more slowly than lower order modes, due to modal dispersion, and this introduces phase shifts between the modes at the end of the guide. These phase shifts cause scrambling of the image information and production of a random speckle pattern rather than an image when the modes interfere at the end of the guide.
0030-4018/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 0 9 8 0 - 9
50
A.R.D. Somervell et al. / Optics Communications 185 (2000) 49±56
Several techniques have been used in attempts to overcome the modal dispersion problem, including phase conjugation [3,4], holographic ®ltering [5] and several other methods for dispersion compensation [6]. However, most of these only work when the ®bre mode dispersion characteristic remains constant, or if the transmitting ®bre can be divided into two identical sections with phase conjugation occurring between them. The result of this in practice is that the ®bre(s) must remain strictly stationary for successful image transmission. We recently described a dierent technique for sending images through a multimode ®bre with perfectly re¯ecting sidewalls [7], which does not rely on mode dispersion compensation, but rather encodes the input image pixels in such a way that modal dispersion does not aect the ®nal image. Using optical Fourier transforms, we match a single ®bre mode to a single image pixel on the input side, and ensure that each mode contributes to only one pixel at the output side of the ®bre. In this paper, we show that our method may also be applied to dielectric slab waveguides which rely on total internal re¯ection to con®ne the light. We also present the results of initial experiments designed to test these ideas where we were able to successfully demonstrate one-to-one pixel to mode encoding and image transmission. 2. Pixel-mode encoding using optical Fourier transforms Fig. 1 shows the principle of our method. The input image is ®rst sampled in a discrete set of pixels using a pinhole array or a multiple beam diractive beam splitter and mask. The sampled image is then optically Fourier transformed using a lens to produce what is eectively a discrete Fourier transform (because of the sampling) which is then projected onto the input face of the waveguide at appropriate magni®cation. Provided the magni®cation is correct, each input pixel produces a wave at the input to the waveguide which has a phase distribution matching that of one of the even numbered waveguide modes. The amplitude of this mode then depends only on the intensity of light at
Fig. 1. System for image transmission through a slab waveguide.
that input pixel. The modes travel to the end of the waveguide (albeit at dierent velocities), where a second Fourier transform lens reconstructs the sampled input image. Each output pixel arises from only one even numbered mode and so no intermodal interference occurs. Note that dierent pixels in the transmitted image take dierent times to travel through the system, but this is not a problem as arrival times at the output dier by typically only a few nanoseconds. The discrete Fourier transform of the input image at the input of the waveguide also excites odd-numbered modes. Each input pixel in fact excites several odd-numbered modes and their contribution to the output image must therefore be eliminated in order to avoid spurious light in the output image plane.
3. Theoretical analysis If the electric ®eld across the input image is denoted E1 x1 as shown in Fig. 1, then the electric ®eld at the front face of the waveguide, E2 x2 is the Fourier transform of the input image which is given by: Z x2 1 E2 x2 E1 x1 exp ÿ i2p x1 dx1 fk where: f is the focal length of the Fourier transform lens k is the wavelength of the light used. In order to ®nd the electric ®eld at the waveguide output it is necessary to know the GreenÕs function for a slab waveguide. Then the electric
A.R.D. Somervell et al. / Optics Communications 185 (2000) 49±56
®eld after propagating a distance L along the waveguide is given by the following: Z ÿ1 oG x3 ; zjx2 ; z0 E2 x 2 ; 0 dx2 E3 x 3 ; z 4p on0 z0 0 2 where: G x3 ; zjx2 ; z0 is the GreenÕs function for the slab waveguide n0 is the outward normal of the plane z0 0 E2 x2 ; 0 is the electric ®eld distribution across the front face of the waveguide. The GreenÕs function for the waveguide is found by solving the non-homogeneous Helmholtz equation: ÿ 2 r k 2 x G x; zjx0 ; z0 ÿ4pd x ÿ x0 d z ÿ z0
3
where: G x; 0jx0 ; z0 0, k x k1 for j xj 6 a=2 and k x k2 for j xj P a=2 and the boundary conditions are given by the continuity of the tangential components of the electric and magnetic ®elds across the boundaries at x a=2. We assume there is no cladding layer so re¯ection takes place from a glass±air interface. Solving this equation gives: Z w B x< wT x> G x; zjx0 ; z0 ÿ2 1 ÿ R2 exp i2n1 a
sin cz0 exp in1 a exp icz dc n1 4
where: x< is the lesser of x and x0 and x> is the greater of x and x0 , n1 is the component of the wave vector in the xdirection, R is the re¯ection coecient for a wave re¯ected from the glass±air interface with an x-component of the wave vector given by n1 , a x exp in x ÿ w 1 T 2 a R exp ÿ in1 x ÿ 2 and
51
a
w B x exp ÿ in1 x 2a R exp in1 x 2 Substituting the expression for the GreenÕs function into Eq. (2) and solving gives the following for the electric ®eld at the output of the waveguide after propagating a distance z L: M sin mp x3 1 exp icm L 2 am 2X E3 x3 ; L 0 a m0 1 ÿ 2 U2m Z a=2 x2 1 dx2 E2 x2 sin mp am 2 ÿa=2 5 where: cm is the wave number for the mth mode U0m is the derivative of the phase change on re¯ection experienced by each mode on re¯ection from the glass±air interface with respect to the component of the wave vector in the x-direction. am is the eective width of the waveguide for the mth mode and is given by: a 6 am m 1 2U mp where Um is the phase change on re¯ection from the glass±air interface (the Goos±Hanchen shift) for the mth mode. Combining Eqs. (1), (4) and (5) leads to the following expression for the electric ®eld of the mth mode: x3 1 7 exp icm L E3m x3 Am sin mp am 2 where the amplitude of the mth mode, Am , is given by: Am
a 1 0 2 1 ÿ 2Um a Z x1 a m Um E1 x1 sinc ÿ ÿ p fk 2 x1 a m U m m dx1 ÿ ÿ 1 sinc p fk 2
8
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A.R.D. Somervell et al. / Optics Communications 185 (2000) 49±56
Eq. (9) shows that the amplitude of the even modes is determined by the whole of the input image multiplied by a weighting function given by the sum of two sinc functions whose position is dependent on the mode number. At positions where the weighting functions for an even numbered mode equal one or negative one, the weighting functions for the other even modes are very close to zero. This is illustrated in Fig. 2 for the even modes m 10, 12 and 14. By sampling at these positions each even mode has a contribution from only two points in the input plane, one on either side of the optic axis. In fact, the amplitude of the electric ®eld is proportional to the electric
®eld at the pixel in the positive half plane minus the electric ®eld in the negative half plane. By placing the image in the positive half plane only, ensuring the electric ®eld is zero over all the negative half plane, the amplitude of the even modes is equal to the amplitude of just one of the input pixels. The eect of having dierent values for the phase shift on re¯ection, Um for dierent modes is to make the sample spacing slightly non-linear. It also causes the position where the weighting function for a given mode is one to be slightly oset from the zeros of the weighting functions corresponding to other modes. This allows a contribution to each mode from more than one image pixel. However, the contribution from unwanted modes is negligible. The contribution of each of the image pixels to one of the odd numbered modes is shown in Fig. 3. As the image is to place in the positive half plane only the sampling positions in that half plane are shown. Examining Fig. 3 shows the sampling positions do not match up with the positions where the weighting function is zero and one and there is therefore a contribution to the amplitude of each mode from many image pixels. The electric ®eld at the output of the waveguide is Fourier transformed giving an electric ®eld distribution E4 x4 . The contribution to the electric ®eld from each waveguide mode is described by the following equation:
Fig. 2. Even mode sampling positions.
Fig. 3. Odd mode amplitude.
The term outside the integral varies with m because the ®bre modes do not quite form an orthogonal set, however its variation with mode number is very small and for all practical purposes it can be considered constant. In order to show how the image should be sampled to ensure one pixel excites one mode it is easiest to consider the odd and even numbered modes separately. For even numbered modes, i.e. m 2; 4; 6 . . . ; the amplitude Am (ignoring the ®rst term in Eq. (8)) is approximately: Z a x1 a m U m E1 x1 sinc ÿ ÿ Am p 2 fk 2 x1 a m U m 9 dx1 ÿ sinc fk 2 p
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x4 a m U m E4m x4 Am exp icm L sinc ÿ ÿ p fk 2 x4 a m Um m ÿ ÿ 1 sinc p fk 2
10
The total electric ®eld output is found from the sum of the electric ®elds from each waveguide mode. Again it is easiest to think about even and odd numbered modes separately. For even numbered modes, at the positions where the amplitude due to one of the modes is Am (where the term in square brackets is 1) the output from other modes is zero. By sampling at these positions output from the even modes gives a reconstruction of the input image. This is because the amplitude of each mode is the same as one of the image pixels. Note that there is no intermodal interference between the output due to even waveguide modes because each output pixel is located in a dierent region of space. However each pixel arrives at the output at a dierent time. Also, there are two versions of the input image, one in either half plane [7]. Although there is a contribution from just one even mode at each output pixel, there is an extra contribution from many odd numbered modes. This needs to be removed in order to see the reconstructed image. Examination of Eq. (10) reveals that the output ®eld due to even mode numbers is an odd function, E4even x4 ÿE4even ÿx4 g ± i.e. there are two copies of the output from even numbered mode on either side of the optic axis which are out of phase with each other. The output from odd numbered modes is an even function, E4odd x4 E4odd ÿx4 ± i.e. the two copies of the output from odd numbered modes are in phase with one another. The contribution from the odd modes can be removed by ®rst creating two copies of the output plane, E4 x4 . Inverting one copy gives an image, E4 ÿx4 which can be subtracted from the non-inverted image. This results in destructive interference between from the odd the outputs odd numbered modes, E4 x4 ÿ E4odd ÿx4 0 and constructive interference between the even numbered modes E4even x4 ÿ E4even ÿx4 2E4even x4 . Only the output from the even numbered modes is left giving a reconstruction of the original input image at
53
the ®nal output. In principle at least this procedure can be done optically using an interferometer. Also there is no reason for this image subtraction procedure to be performed at the output of the system. In fact because the system is linear the image subtraction can be performed at any point along it including positions before the waveguide.
4. Experimental system Experiments were performed to show that images could indeed be transmitted through a slab waveguide using the techniques described above. A diagram of the experimental set-up is shown in Fig. 4. A TM polarized He±Ne laser beam is expanded and collimated, and then passes through a diractive optic beam splitter giving 10 plane waves each separated by an angle of 0.4°. These beams then pass through a lens giving an array of 10 spots at its focal plane. The beams on one side of the optic axis are blocked to ensure the input image lies on one half of the optic axis only leaving an array of ®ve spots. The spacing between the spots is adjusted using a variable magni®cation telescope consisting of a zoom lens together with a ®xed focal length lens. The output from this system is used as the sampled image to be transmitted. The image is Fourier transformed using a lens and the Fourier transform is incident on the front face of a dielectric slab waveguide. Light propagates to the back face of the waveguide where it is Fourier transformed again and the intensity distribution is measured using CCD camera.
Fig. 4. Experimental system used for transmitting images.
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It should be noted that there is no system for removing the output which occurs due to the odd waveguide modes and therefore the reconstructed output image will not be an exact copy of the input image. Also, the image pixels are not positioned accurately enough so that just one even numbered mode is excited by each pixel as it has been shown is necessary for exact reconstruction of the input image. This is because in the set-up used for this experiment only fairly course adjustment of the pixel spacing is achievable using the variable magni®cation telescope system. Adjustment of the pixel spacing is made even more dicult because the magni®cation system and therefore the sample spacing cannot be adjusted without causing the positions all of the following components to need readjusting. Due to inaccuracies in the sampling positions each pixel excites a combination of many even and odd numbered modes. The waveguide used in this experiment was a 50 20 0:5 mm3 waveguide custom built from BK7 glass. Light entered and exited the 0:5 20 mm2 faces which were ¯at to better than a wavelength over the entire surface and antire¯ection coated at 633 nm. The 0:5 20 mm2 faces were perpendicular to the 50 20 mm2 faces to better than 10 arcs and the 50 20 mm2 surfaces were parallel to better than 15 arcs. It is important that the 0:5 20 mm2 faces are optically ¯at and perpendicular to the 50 20 mm2 faces to ensure the waves entering and exiting the waveguide are aberration free so as not to alter the amplitude of the modes are excited by each pixel. The ¯atness of the 50 20 mm2 faces ensures that there is no coupling between modes as light propagates through the waveguide. The waveguide was mounted on a tilt/rotation stage accurate to 3 arcs together with an x±y translation stage to enable precise positioning of the waveguide. 5. Results Fig. 5 shows four images before and after transmission, each with a dierent number of image pixels. Each input image is shown above its
corresponding output image. The input images were obtained by blocking one or more of the pixels in the ®ve pixel input array. Note that for each input image pixel there are two corresponding output pixels as expected theoretically. Each output pixel is spread out both horizontally and vertically. The vertical spreading occurs because the guiding eect only occurs in one direction in the slab waveguide and light is free to diract in the other direction. The horizontal spreading occurs because several waveguide modes are excited by each input pixel as mentioned above. This also increases the spacing needed between the pixels in the image to be transmitted in order to form separate output pixels, thus the total number of pixels which can be transmitted is reduced when both odd and even modes are excited. The total number of pixels which can be sent if only even numbered modes were excited is equal to the number of even numbered modes supported by the waveguide. For the waveguide used in this experiment this is about 1200 guided even numbered modes. When the pixels are not positioned extremely accurately and the odd modes not removed as is the case in this experiment each input pixel excites about 20±30 modes (both odd and even modes included) with signi®cant amplitude (greater than 10% of the maximum mode amplitude). Thus the number of pixels which can be sent through the image is reduced by a factor of 10±15 when odd modes are not removed. Also, as the spacing between the image pixels increases, the cross-talk between the dierent output pixels reduces. It is also seen from Fig. 5 that although each image pixel is of nearly the same intensity, the intensity of the output pixels is not. For example, in Fig. 5c, the output pixels on the right-hand side underneath the two input image pixels show markedly dierent intensities. This can be explained as follows. Each output pixel arises from the interference between the individual components which make up that pixel ± i.e. the components due to the several waveguide modes excited by a single image pixel. Each of the components in the set which makes up a single output pixel has a particular phase which arises
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Fig. 5. Input and output images: (a) (top) four input pixels shown above the corresponding output image, (b) (second from top) the output obtained from three input pixels, (c) (second from bottom) output due to two input pixels, (d) (bottom) output from a single input pixel.
from modal dispersion. The phase dierences between each of the components in the set of modes making a given output pixel is not the same as those in the set making up a dierent pixel. Interference between the components therefore results in dierent intensities for dierent pixels. From Fig. 5d it can also be seen that the two output pixels which arise from the same image pixel are not the same intensity even though the phase dierence due to modal dispersion and amplitude of the waveguide modes which make up those output pixels is the same. This is because in the left-hand side of the output plane there is additional p phase shift between the output components arising from odd modes and those arising from even modes. This phase shift is not present in the right-hand side output plane so the intensities of either output pixel is dierent. It should be noted that this eect would disappear if odd mode cancellation was applied.
6. Conclusion We have described a technique for transmitting one dimensional images through a dielectric slab waveguide. In principle this involves matching even numbered waveguide modes to image pixels by sampling the image appropriately, then illuminate the front face of the ®bre with the Fourier transform of the sampled image. The image is then reconstructed by Fourier transforming the waveguide output and sampling again. In order to get an exact reconstruction of the original image the contribution to the output from odd numbered modes should be removed by subtracting the electric ®eld distribution in one side of the output plane from that in the opposite side. We have also described initial experiments showing transmission of images through a slab waveguide. The output image was not an exact replica of the input image because the output due to odd numbered modes was not removed and the
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input pixels were not positioned accurately enough to excite only one even mode each. This resulted in spreading of the output pixels. Acknowledgements We would like to express our thanks to the Royal Society of New Zealand for ®nancial assistance provided through the Marsden fund. We are also grateful to Industrial Research Limited for their help in this work. References [1] P. Naulleau, M. Brown, C. Chen, E. Leith, Direct threedimensional image transmission through single-mode
[2]
[3]
[4] [5] [6] [7]
®bers with monochromatic light, Opt. Lett. 21 (1996) 36±38. A.R.D. Somervell, C.Y. Wu, T.G. Haskell, T.H. Barnes, All-optical binary phase encoding/decoding for image transmission through apertures smaller than the Rayleigh limit, Opt. Commun. 162 (1999) 291±298. P.H. Beckwith, I. McMichael, P. Yeh, Image distortion in multimode ®bers and restoration by polarization preserving phase conjugation, Optics Lett. 12 (1987) 510±512. B. Fischer, S. Sternklar, Image transmission and interferometry with multimode ®bres using self-pumped phase conjugation, Appl. Phys. Lett. 46 (1985) 113±114. U. Levy, A.A. Friesem, Direct picture transmission in a single optical ®bre with holographic ®lters, Opt. Commun. 30 (1979) 163±165. R. Ulrich, Image formation by phase coincidences in optical waveguides, Opt. Commun. 13 (1975) 259±264. C.Y. Wu, A.R.D. Somervell, T.H. Barnes, Direct image transmission through a square multi-mode optical ®bre, Opt. Commun. 157 (1998) 17±22.