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Fidelity of image restoration by partial phase conjugation through multimode fibers
15 January 1995
OPTICS COMMUNICATIONS Optics Communications 114(1995) 50-56
Fidelity of image restoration by partial phase conjugation through multimode fibers Scott Campbell a, Pochi Yeh a,l, Claire Gu b, Q. Byron He ’ a University of California. Department ofElectrical and Computer Engineering, Santa Barbara, CA 93106, USA b Pennsylvania Stale University, Department of Electrical Engineering, University Park, PA 16802, USA ’ Jet Propulsion Laboratory, Calvornia Institute of Technology, Pasadena, CA 91109, USA
Received 10 June 1994; revised manuscript received 18 August 1994
Abstract We investigate the fidelity of image restoration under conditions of partial phase conjugation through a random medium. For the first time, the conjugated image-to-background ratio is experimentally measured as an evaluation of the phase conjugate mirror’s fidelity. Output signal-to-noise ratio and information throughput are also considered as functions of aperture limitations at the various planes of the optical system.
1. Introduction The utility of optical phase conjugation to correct for phase distortions and image aberrations acquired during transmission through a random medium is well known and extensively studied [ l-8 1. However, in cases where the phase conjugate mirror has space and/or bandwidth reflectivity limitations, only a portion of the incident information can be conjugated and the returned output image suffers a loss in fidelity. It is known that when an image carrying wave propagates in free space and a phase conjugate mirror with a finite aperture reflects a portion of the incident wave, the phase conjugated image will be a diffraction-limited replica of the input. However, if there is also a random medium between the input object and the phase conjugate mirror, then the phase conjugated image contains not only the diffraction noise but also a distributed noise of random fashion. ’ Pochi Yeh is also Principle Technical Advisor at the Rockwell International Science Center. 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(94)00522-2
It is therefore important to study phase conjugate systems which contain both random media and limiting apertures. Multi-mode optical fibers are frequently utilized as random media, in which both the spatial and polarization information of the incident field undergo modal scrambling. In the absence of limiting apertures, Beckwith et al. [ 1 ] have presented an information recovery scheme that, in addition to preserving spatial information, also preserves polarization information. As well, Tomita et al. [ 21 have developed a matrix formalism describing the stages of scrambling and recovery in these types of systems. Recently, Gu and Yeh [ 3 ] have shown how reciprocity can be utilized to simplify the analysis of such phase conjugate systems with limited apertures. In this paper, we consider the practical effects on phase conjugated image fidelity due to aperture limitations that act to suppress modes at the various planes of an optical system. For the first time, we experimentally demonstrate the theoretical predictions [ 3 ] analyzing partially phase conjugating systems.
S. Campbellet al. /Optics Communications114 (1995) 50-56
2. Analysis The concept for partial phase conjugation through a random medium is diagrammed in Fig. la. An incident object field E, enters a random medium and propagates through it to exit as field E2. As field E2 propagates towards a ‘perfect’ phase conjugate mirror (PCM), it encounters aperture F, which allows only a fraction of field E2 to reach the PCM. Emanat-
ing from the PCM then is field E3 which, because of limitations due to aperture F, is only the partial phase conjugate of field E2. Field E3 then propagates back through the random medium to exit as field E+ Field E4 is the final output field, containing both image information (the portion of incident field El which is properly phase conjugated) and background noise information (the portion of incident field E, which is improperly returned). Mathematically, we can break field E4 into a linear projection of these two components [ 3 ] E4=aE:+j?Efl,
Medium
OBJ
LFl
PBS,
E3
$
Dl A
LI
PCM
Fig. 1. (a) Generalized experimental setup. (b) Actual experimental setup used to study the effects of aperture limitations on phase conjugate fidelity.
51
(1)
where (Yis the fraction of properly returned object field, /3 is the fraction of improperly returned object field, and E;, is a field orthogonal to E:. The tidelity of the returned image, as controlled by F, is determined by how well E4 conjugates E,. Since the PCM is modeled here to be perfect, the entire cause of the imperfections during the phase conjugate process are attributed to aperture F. The combination of the ‘perfect’ PCM and aperture F create, then, a ‘realistic’ PCM. Fig. lb then goes on to show how such a system was implemented in the work presented here. In Fig. 1b, the original input object information passes through a standard beam splitter, followed by a polarizing beam splitter (PBS) and a quarter-wave plate (l/4) which act to null reflections from the front surface of the fiber. All other (non-PCM) reflections beyond this plane can not be so nullified, and are thus characterized by the variable R. R is the (integrated) back-scattered power reflectivity for the random medium for the light which returns to the input plane. Complementing R is T, which is the forward power transmissivity of the random medium for the light which reaches aperture F. In the case of absorptive or scattering losses (the general case), R + T< 1. Injection lens Li with focal lengthfand aperture of diameter D is then utilized to inject the input information into the front surface of a multi-mode fiber (MMF) of diameter d and numerical aperture NA. This information is then scrambled by the MMF as it propagates to the fiber’s output face. Upon exiting the tiber, this randomly polarized, diverging field is collected by lens Lc which collimates it and sends it to pass through the aperture F. The measure of F is a fraction varying from zero to one, given by the ratio of the power passing through it divided by the power
52
S. Campbellet al. /Optics Communications114 (1995) 50-56
entering it (as determined by the opening, closing, or moving of an iris). The light emanating from this aperture then passes through another polarizing beam splitter ( PBS2) which sends the p polarized light one way and the s polarized light another. In the path of the s light is a half-wave plate (A/2) which converts it to p light. This is necessary since the conjugator works primarily with p light. Reduction lens LR then slowly funnels these beams through a glass slide (a pick-off beam splitter that sends beams to detectors D, and D2 for system analysis) and finally into a phase conjugate mirror (PCM) with amplitude reflection coefficient p. The conjugated light then retraces the optical system, producing a resultant image at the system’s output plane. Aperture limitations due to D, 2fNA, d, and F then act to reduce how well the output information conjugates the input information. Notice that the aperture limitations due to D,2jNA, and d lead to diffraction noise in the conjugated image, while the aperture limitation due to F leads to a randomly distributed noise. We begin our analysis by considering effects at the fiber’s output end due to aperture F. As this aperture is closed down or moved around, various fiberscrambled modes are blocked and partial phase conjugation of the incident field ensues. Such partial phase conjugation results in a decrease in image plane phase conjugate fidelity. (The PCM, in the absence of F, is assumed to have an infinite aperture and signal-to-noise ratio, and uniform phase conjugate reflectivity for all incident angles.) It has been shown [ 31 that the loss in image fidelity can be characterized by a measure of the image-to-background ratio, IBR, where the numerator measures the properly-returned input information and the denominator measures the improperly-returned input information (the total information returned minus the properlyreturned information). This ratio has been shown to be given by [ 3 ]
where a: is the fraction of properly returned object field, as in Eq. ( 1), and is given by
(2b) where x, y are the coordinates at the input plane, x’,
y’ are the coordinates at the PCM plane, and F represents the finite PCM aperture. Notice that JIE412 dx dy contains all of the power at the output plane, which includes the properly returned image, the background noise due to partial phase conjugation, and the background noise due to backscatter reflections from the random medium. As well, the derivations in Ref. [ 3 ] include the utilization of reciprocity to simplify the analysis of such partial phase conjugate systems. Notice that the reciprocity theorem holds as long as the optical system is in steady state, and is thus independent of the light source’s coherence length. In this context, reciprocity states an equivalence between two power ratios: the ratio of the power received by the PCM to that emanating from the input object and the ratio of the power in the phase conjugate image to that reflected by the PCM. Considering Fig. 1a, this can be represented in terms of integral equations as JdEz12d-Wf= ll& 12bdy
SlaE:12dxdy JIE3 12dx’dy’ ’
(2c)
It can be noted that the image-to-background ratio (IBR) is similar to the well known signal-to-noise ratio (SNR). The difference between the two is that the IBR considers all of the image (signal) power and all of the background (noise) power, while the SNR considers all of the signal (image) power but only a selective amount of noise (background) power. This means that the SNR allows for selective filtering of the noise spectrum (typically considering only that bandpass of noise which overlaps the signal’s bandpass) while the IBR does not. It has been shown [ 21 that the conjugate image’s SNR can be maximized if the ratio of the area it occupies to the area within the projection of the total background noise is minimized. We will call this ratio 0, since it represents the fraction of the background noise’s bandpass which is allowed to overlap with the signal’s (0 ranges from zero to one). The implications are that the back-scattered and improperly-conjugated noises from this system are spread fairly uniformly across the output plane, while the properly-conjugated signal (the image) may only occupy a fraction of that projection, thereby optimizing the possible SNR within the image region for small @ (the bandpass of noise measured is only that which overlaps the signal’s band-
S. Campbellet al. /Optics Communications114 (1995) 50-56
pass). We therefore obtain the expression for the output signal-to-noise ratio as SNR=
!f!! @.
IBRx
1 Xd(IW.)
Xm,(F.T.)z
X
experimental results, we must replace the integrals in Eq. (2) with system parameters. Doing so gives the following equations:
(3)
With Eq. (3), one can determine and/or optimize the SNR for a given IBR by minimizing 0. Practically, however, minimizing 0 to maximize the SNR may work to also diminish the space and/or bandwidth information throughput due to coupling losses, leading to a net reduction in image resolution. Reproducing well a low resolution input is often no better than reproducing poorly a high resolution input. Such information throughput losses result from the mismatch of the confined modes of an optical fiber and the input optical field [ lo]. In this context, these losses in space and bandwidth information (which must be considered separately) are determined by comparing the input object’s information extent with the limiting apertures 2fNA, D or d at the input end of the MMF. Consider, for example, an input SLM device with pixel dimensions x. For injection into an optical fiber there exist two injection regime extremes: one where the (demagnified) image of the SLM’s object is injected into the fiber’s entrance face (with an absolute magnification ]M] +Z 1), and the other where the Fourier transform of the SLM’s object is injected into the fiber’s entrance face. In these two cases, the minimum SLM pixel sizes that the MMF can resolve, Xmin(Img.) and Xmin(F.T.), respectively, are given as ,M,
(NA)
,
y,
where 1 is the wavelength of light utilized. One can now see that a loss in bandwidth information throughput will result if 0 is reduced via the utilization of a reduced-size SLM with x< x,~. As well, for an SLM with x > Xmin,0 may be reduced by blocking the pixels around the perimeter of the SLM, thereby causing a loss in space information throughput. Of course, the SLM’s pixel size, x, can be synthetically altered by magnifying or demagnifying it to the input object plane. In order to compare the theoretical predictions with
53
a=pTF,
lp12T2F2 R+ IplZT2F(
1 -F)
’
(5a) (5b)
where, again, p is the amplitude reflectivity of the PCM, T is the power transmissivity of the random medium, R is the power reflectivity of the random medium, and F is the PCM’s effective aperture given as the ratio of the power passing through it to that arriving at it. Eq. (5a) is a direct result of Eq. (2a), even when utilizing short coherence length sources, because reciprocity does not rely on the source’s coherence length. The approximation in Eq. (Sa) is due to the backscattered reflectivity term R. An exact representation of Eq. (5a) from Eq. (2a) would require the inclusion of numerous higher order series terms from multiple reflections within the fiber, between the fiber and the PCM, and various combinations thereof. As well, the analysis of which portions of R return light into the conjugated image and which return light into the background is nontrivial. However, for the parameters of our system, as described shortly, these higher order reflection terms accounted for less than 0.1% of R. We therefore neglected them in Eq. (5a). Finally, before discussing our experimental results, we wish to note the effects the light source’s coherence length, L,, has on the throughput IBR and SNR ifF-1 and ]p12- 1, through the terms T and R. If the system path lengths are notably longer than L,, then coherent subtraction of the scattered noise terms by the conjugate beams will not take place upon phase conjugation and T and R will be unaffected. Image restoration via phase conjugation is still valid in such a system because the phase conjugate wave unravels the aberrations acquired during forward transit. If L, is long compared to system path lengths, then Twill increase while R decreases upon phase conjugation due to coherent subtraction of the scattered noise terms by the phase conjugate beams. If in fact ]p] ‘= 1 for all modes, F= 1, and L, is notably greater than numerous round trip system path lengths, then Twill go to one while R goes to zero as the phase conjugation process saturates.
S. Campbell et al. /Optics Communications I14 (1995) 50-56
54
3. Experimentalresults To experimentally investigate IBR, SNR, and reciprocity, we utilized the setup given in Fig. 1. The PCM was a barium titanate semilinear double phase conjugate mirror, SDPCM [ 9 1, with a power reflectivity of about 22%. The other parameters were f= 38 mm, 0~25 mm, d=lOO pm, NA=0.3, T-50%, R-l%, 1~514.5 nm, L,-100 mm, and an MMF length of 50 m. At the PCM end of the system, the collimating lens’ focal length was 38 mm, the PBS aperture was 2.5 cm, the half-wave plate’s aperture was 2 cm, and the reduction lens’ focal length was 50 cm
-FF-0.5, 10
-
o
11lq 0
0.2
F~0.8,
Theory Experlment
-I
;..._ ._.__ _.. .... ..___..___ 4 . 0.4
0.6
0.8
1
CJ
Fig. 3. Experimental points and the corresponding theoretical curves for SNR versus @for F=0.8 and 0.5.
0.6
0.4
0.2
-_
0
0
0.2
0.4
P IntoPCM
0.6
0.8
1
’ ~%l,.cl
Fig. 2. (a) Experimental points and the corresponding theoretical curve for IBR versus F for 0= 1. (b ) Experimental points for reciprocity, comparing the ratio of the power in the phase conjugate signal to the power reflected by the phase conjugate mirror with the ratio of the power captured by the phase conjugate mirror to the power emanating from the input object. The straight line in (b) represents perfect reciprocity.
with the crystal placed about 45 cm from that lens. As well, the optics were arranged as compactly as possible to minimize path length disturbances. The test object utilized for studying IBR and reciprocity was a semi-circle illuminating half of the field-of-view of the fiber’s full numerical aperture, with injection powers in the 1 to 20 mW range. Such an input allowed us to measure image-plus-background across one half of the output plane while measuring just background across the other half of the output plane. Fig. 2a compares our experimental data with the theoretical predictions for IBR versus F (utilizing Eq. ( 5a) ). For small F, the experimental results are in excellent agreement with theory. As Fapproaches one, imperfections in the (real) PCM begin appearing and the data diverges from the theory. The rapid appearance of these imperfections is perhaps due to the fact that we were imaging a diffuse source (the 100 pm wide fiber tip) rather than collimating a point source, as well as the more dominant contribution from R in Eq. (5a)‘s denominator. Fig. 2b then demonstrates the reciprocity of our system by comparing experimental values of power ratios for beams going one way vs. the other, as mentioned above and defined by Eq. (2~) (the straight line in Fig. 2b indicates a perfectly reciprocal system). We then turned our attention to varying 0 (by demagnifying the input semicircle utilized in studying IBR and reciprocity) when F= 0.8 and 0.5 to measure our system’s SNR versus @.Results for this are given in Fig. 3, where the curves are the theoretical predictions and the dots are the
S. Campbellet al. /Optics Communications114 11995) 50-56
Fig. 4. Images of an Air Force resolution target. (a) corresponds to imaging via the injection optics with the 100 urn diameter optical fiber replaced by a 100 pm pinhole. (b, c, d, e) correspond to investigations with the set up given in Fig. 1 for @= 1 and F= 1,0.5,0.2, and 0, respectively. (f ) then corresponds to investigations with the set up given in Fig. 1 for @=O.l and F= 1. Comparing (b) to (f ) shows an obvious increase in SNR but decrease in resolution.
experimental results. As can be seen, experimental values for the SNR exceed theoretical values as ip gets small. This is because much of the noise existed near
the perimeter of the output plane, possibly as a result of fiber core surface damage during cleaving. Furthermore, in order to give a practical, visual demon-
56
S. Campbell et al. / Optics Communications 114 (1995) 50-56
stration of partial phase conjugation due to aperture limitations at both the input and output ends of the MMF, we injected an Air Force resolution target into our system. These results are shown in Fig. 4. First, Fig. 4a shows the resolution target as imaged through a 100 ym diameter pinhole (simulating the maximum object information that can enter our 100 urn diameter optical fiber for this system). Then, Figs. 4b-e show results from our whole system as given in Fig. 1 for 0= 1 and F= 1,0.5,0.2, and 0, respectively (the non-zero output in Fig. 4e shows the ever-present back reflections from the optical fiber due to R ) . In these images, it is clear that the injection apertures limit primarily the space information entering the fiber system. Finally, Fig. 4f corresponds to @=O. 1 with F= 1. In this image, it is clear that the injection apertures noticeably limit the bandwidth information entering the fiber. In order to not overexpose Fig. 4f, its exposure time was less than those in Figs. 4ae since the same object power is concentrated in a smaller area. Such a shortening of the exposure time brings the noise background below the threshold for film response. Clearly, limiting the aperture of the PCM (F) degrades the quality of the conjugate image in a global sense, as seen through Figs. 4b-e. As well, limiting the portion of NA that the input object information occupies (@) significantly improves the SNR, but unfortunately acts to reduce the image resolution due to information coupling losses, as seen in Fig. 4f.
4. Summary In summary, we have, for the first time, experimentally investigated a reciprocity-based theoretical model for partial phase conjugate fidelity, showing good agreement between theory and experiment. We have also demonstrated the trade-off between returned image resolution and SNR in such systems.
Acknowledgements This work was supported by the U.S. Office of Naval Research and Air Force Office of Scientific Research.
References [ 1] P. Beckwith, I. McMichael and P. Yeh, Optics Lett. 12 (1987) 510.
[ 2 ] Y. Tomita, R. Yahalom, K. Kyuma, A. Yariv and N. Kwong, IEEE J. Quantum Electron. 25 (1989) 315.
[ 31 C. Gu and P. Yeh, Optics Comm. 107 ( 1994) 353. [4] J.W. Goodman, W.H. Huntley Jr., D.W. Jackson and M. Lehmann, Appl. Phys. Lett. 8 ( 1966) 3 11. [ 51 A. Yariv, Optics Comm. 21 (1977) 49. [6] D.M. Pepper and A. Yariv, Optics Lett. 5 ( 1980) 59. [ 71 J. Falk, Optics Lett. 7 (1982) 620. [ 81 A. Volyar, A. Gnatovskii, N. Kukhtarev and S. Lapaeva, Appl. Phys. B 52 ( 1991) 400. [9] Q. He and J. Duthie, Optics Comm. 75 (1990) 311. [ 10 ] See, for example, J.P. Powers, An introduction to fiber optic systems (Aksen Ass., 1993).
OPTICS COMMUNICATIONS Optics Communications 114(1995) 50-56
Fidelity of image restoration by partial phase conjugation through multimode fibers Scott Campbell a, Pochi Yeh a,l, Claire Gu b, Q. Byron He ’ a University of California. Department ofElectrical and Computer Engineering, Santa Barbara, CA 93106, USA b Pennsylvania Stale University, Department of Electrical Engineering, University Park, PA 16802, USA ’ Jet Propulsion Laboratory, Calvornia Institute of Technology, Pasadena, CA 91109, USA
Received 10 June 1994; revised manuscript received 18 August 1994
Abstract We investigate the fidelity of image restoration under conditions of partial phase conjugation through a random medium. For the first time, the conjugated image-to-background ratio is experimentally measured as an evaluation of the phase conjugate mirror’s fidelity. Output signal-to-noise ratio and information throughput are also considered as functions of aperture limitations at the various planes of the optical system.
1. Introduction The utility of optical phase conjugation to correct for phase distortions and image aberrations acquired during transmission through a random medium is well known and extensively studied [ l-8 1. However, in cases where the phase conjugate mirror has space and/or bandwidth reflectivity limitations, only a portion of the incident information can be conjugated and the returned output image suffers a loss in fidelity. It is known that when an image carrying wave propagates in free space and a phase conjugate mirror with a finite aperture reflects a portion of the incident wave, the phase conjugated image will be a diffraction-limited replica of the input. However, if there is also a random medium between the input object and the phase conjugate mirror, then the phase conjugated image contains not only the diffraction noise but also a distributed noise of random fashion. ’ Pochi Yeh is also Principle Technical Advisor at the Rockwell International Science Center. 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(94)00522-2
It is therefore important to study phase conjugate systems which contain both random media and limiting apertures. Multi-mode optical fibers are frequently utilized as random media, in which both the spatial and polarization information of the incident field undergo modal scrambling. In the absence of limiting apertures, Beckwith et al. [ 1 ] have presented an information recovery scheme that, in addition to preserving spatial information, also preserves polarization information. As well, Tomita et al. [ 21 have developed a matrix formalism describing the stages of scrambling and recovery in these types of systems. Recently, Gu and Yeh [ 3 ] have shown how reciprocity can be utilized to simplify the analysis of such phase conjugate systems with limited apertures. In this paper, we consider the practical effects on phase conjugated image fidelity due to aperture limitations that act to suppress modes at the various planes of an optical system. For the first time, we experimentally demonstrate the theoretical predictions [ 3 ] analyzing partially phase conjugating systems.
S. Campbellet al. /Optics Communications114 (1995) 50-56
2. Analysis The concept for partial phase conjugation through a random medium is diagrammed in Fig. la. An incident object field E, enters a random medium and propagates through it to exit as field E2. As field E2 propagates towards a ‘perfect’ phase conjugate mirror (PCM), it encounters aperture F, which allows only a fraction of field E2 to reach the PCM. Emanat-
ing from the PCM then is field E3 which, because of limitations due to aperture F, is only the partial phase conjugate of field E2. Field E3 then propagates back through the random medium to exit as field E+ Field E4 is the final output field, containing both image information (the portion of incident field El which is properly phase conjugated) and background noise information (the portion of incident field E, which is improperly returned). Mathematically, we can break field E4 into a linear projection of these two components [ 3 ] E4=aE:+j?Efl,
Medium
OBJ
LFl
PBS,
E3
$
Dl A
LI
PCM
Fig. 1. (a) Generalized experimental setup. (b) Actual experimental setup used to study the effects of aperture limitations on phase conjugate fidelity.
51
(1)
where (Yis the fraction of properly returned object field, /3 is the fraction of improperly returned object field, and E;, is a field orthogonal to E:. The tidelity of the returned image, as controlled by F, is determined by how well E4 conjugates E,. Since the PCM is modeled here to be perfect, the entire cause of the imperfections during the phase conjugate process are attributed to aperture F. The combination of the ‘perfect’ PCM and aperture F create, then, a ‘realistic’ PCM. Fig. lb then goes on to show how such a system was implemented in the work presented here. In Fig. 1b, the original input object information passes through a standard beam splitter, followed by a polarizing beam splitter (PBS) and a quarter-wave plate (l/4) which act to null reflections from the front surface of the fiber. All other (non-PCM) reflections beyond this plane can not be so nullified, and are thus characterized by the variable R. R is the (integrated) back-scattered power reflectivity for the random medium for the light which returns to the input plane. Complementing R is T, which is the forward power transmissivity of the random medium for the light which reaches aperture F. In the case of absorptive or scattering losses (the general case), R + T< 1. Injection lens Li with focal lengthfand aperture of diameter D is then utilized to inject the input information into the front surface of a multi-mode fiber (MMF) of diameter d and numerical aperture NA. This information is then scrambled by the MMF as it propagates to the fiber’s output face. Upon exiting the tiber, this randomly polarized, diverging field is collected by lens Lc which collimates it and sends it to pass through the aperture F. The measure of F is a fraction varying from zero to one, given by the ratio of the power passing through it divided by the power
52
S. Campbellet al. /Optics Communications114 (1995) 50-56
entering it (as determined by the opening, closing, or moving of an iris). The light emanating from this aperture then passes through another polarizing beam splitter ( PBS2) which sends the p polarized light one way and the s polarized light another. In the path of the s light is a half-wave plate (A/2) which converts it to p light. This is necessary since the conjugator works primarily with p light. Reduction lens LR then slowly funnels these beams through a glass slide (a pick-off beam splitter that sends beams to detectors D, and D2 for system analysis) and finally into a phase conjugate mirror (PCM) with amplitude reflection coefficient p. The conjugated light then retraces the optical system, producing a resultant image at the system’s output plane. Aperture limitations due to D, 2fNA, d, and F then act to reduce how well the output information conjugates the input information. Notice that the aperture limitations due to D,2jNA, and d lead to diffraction noise in the conjugated image, while the aperture limitation due to F leads to a randomly distributed noise. We begin our analysis by considering effects at the fiber’s output end due to aperture F. As this aperture is closed down or moved around, various fiberscrambled modes are blocked and partial phase conjugation of the incident field ensues. Such partial phase conjugation results in a decrease in image plane phase conjugate fidelity. (The PCM, in the absence of F, is assumed to have an infinite aperture and signal-to-noise ratio, and uniform phase conjugate reflectivity for all incident angles.) It has been shown [ 31 that the loss in image fidelity can be characterized by a measure of the image-to-background ratio, IBR, where the numerator measures the properly-returned input information and the denominator measures the improperly-returned input information (the total information returned minus the properlyreturned information). This ratio has been shown to be given by [ 3 ]
where a: is the fraction of properly returned object field, as in Eq. ( 1), and is given by
(2b) where x, y are the coordinates at the input plane, x’,
y’ are the coordinates at the PCM plane, and F represents the finite PCM aperture. Notice that JIE412 dx dy contains all of the power at the output plane, which includes the properly returned image, the background noise due to partial phase conjugation, and the background noise due to backscatter reflections from the random medium. As well, the derivations in Ref. [ 3 ] include the utilization of reciprocity to simplify the analysis of such partial phase conjugate systems. Notice that the reciprocity theorem holds as long as the optical system is in steady state, and is thus independent of the light source’s coherence length. In this context, reciprocity states an equivalence between two power ratios: the ratio of the power received by the PCM to that emanating from the input object and the ratio of the power in the phase conjugate image to that reflected by the PCM. Considering Fig. 1a, this can be represented in terms of integral equations as JdEz12d-Wf= ll& 12bdy
SlaE:12dxdy JIE3 12dx’dy’ ’
(2c)
It can be noted that the image-to-background ratio (IBR) is similar to the well known signal-to-noise ratio (SNR). The difference between the two is that the IBR considers all of the image (signal) power and all of the background (noise) power, while the SNR considers all of the signal (image) power but only a selective amount of noise (background) power. This means that the SNR allows for selective filtering of the noise spectrum (typically considering only that bandpass of noise which overlaps the signal’s bandpass) while the IBR does not. It has been shown [ 21 that the conjugate image’s SNR can be maximized if the ratio of the area it occupies to the area within the projection of the total background noise is minimized. We will call this ratio 0, since it represents the fraction of the background noise’s bandpass which is allowed to overlap with the signal’s (0 ranges from zero to one). The implications are that the back-scattered and improperly-conjugated noises from this system are spread fairly uniformly across the output plane, while the properly-conjugated signal (the image) may only occupy a fraction of that projection, thereby optimizing the possible SNR within the image region for small @ (the bandpass of noise measured is only that which overlaps the signal’s band-
S. Campbellet al. /Optics Communications114 (1995) 50-56
pass). We therefore obtain the expression for the output signal-to-noise ratio as SNR=
!f!! @.
IBRx
1 Xd(IW.)
Xm,(F.T.)z
X
experimental results, we must replace the integrals in Eq. (2) with system parameters. Doing so gives the following equations:
(3)
With Eq. (3), one can determine and/or optimize the SNR for a given IBR by minimizing 0. Practically, however, minimizing 0 to maximize the SNR may work to also diminish the space and/or bandwidth information throughput due to coupling losses, leading to a net reduction in image resolution. Reproducing well a low resolution input is often no better than reproducing poorly a high resolution input. Such information throughput losses result from the mismatch of the confined modes of an optical fiber and the input optical field [ lo]. In this context, these losses in space and bandwidth information (which must be considered separately) are determined by comparing the input object’s information extent with the limiting apertures 2fNA, D or d at the input end of the MMF. Consider, for example, an input SLM device with pixel dimensions x. For injection into an optical fiber there exist two injection regime extremes: one where the (demagnified) image of the SLM’s object is injected into the fiber’s entrance face (with an absolute magnification ]M] +Z 1), and the other where the Fourier transform of the SLM’s object is injected into the fiber’s entrance face. In these two cases, the minimum SLM pixel sizes that the MMF can resolve, Xmin(Img.) and Xmin(F.T.), respectively, are given as ,M,
(NA)
,
y,
where 1 is the wavelength of light utilized. One can now see that a loss in bandwidth information throughput will result if 0 is reduced via the utilization of a reduced-size SLM with x< x,~. As well, for an SLM with x > Xmin,0 may be reduced by blocking the pixels around the perimeter of the SLM, thereby causing a loss in space information throughput. Of course, the SLM’s pixel size, x, can be synthetically altered by magnifying or demagnifying it to the input object plane. In order to compare the theoretical predictions with
53
a=pTF,
lp12T2F2 R+ IplZT2F(
1 -F)
’
(5a) (5b)
where, again, p is the amplitude reflectivity of the PCM, T is the power transmissivity of the random medium, R is the power reflectivity of the random medium, and F is the PCM’s effective aperture given as the ratio of the power passing through it to that arriving at it. Eq. (5a) is a direct result of Eq. (2a), even when utilizing short coherence length sources, because reciprocity does not rely on the source’s coherence length. The approximation in Eq. (Sa) is due to the backscattered reflectivity term R. An exact representation of Eq. (5a) from Eq. (2a) would require the inclusion of numerous higher order series terms from multiple reflections within the fiber, between the fiber and the PCM, and various combinations thereof. As well, the analysis of which portions of R return light into the conjugated image and which return light into the background is nontrivial. However, for the parameters of our system, as described shortly, these higher order reflection terms accounted for less than 0.1% of R. We therefore neglected them in Eq. (5a). Finally, before discussing our experimental results, we wish to note the effects the light source’s coherence length, L,, has on the throughput IBR and SNR ifF-1 and ]p12- 1, through the terms T and R. If the system path lengths are notably longer than L,, then coherent subtraction of the scattered noise terms by the conjugate beams will not take place upon phase conjugation and T and R will be unaffected. Image restoration via phase conjugation is still valid in such a system because the phase conjugate wave unravels the aberrations acquired during forward transit. If L, is long compared to system path lengths, then Twill increase while R decreases upon phase conjugation due to coherent subtraction of the scattered noise terms by the phase conjugate beams. If in fact ]p] ‘= 1 for all modes, F= 1, and L, is notably greater than numerous round trip system path lengths, then Twill go to one while R goes to zero as the phase conjugation process saturates.
S. Campbell et al. /Optics Communications I14 (1995) 50-56
54
3. Experimentalresults To experimentally investigate IBR, SNR, and reciprocity, we utilized the setup given in Fig. 1. The PCM was a barium titanate semilinear double phase conjugate mirror, SDPCM [ 9 1, with a power reflectivity of about 22%. The other parameters were f= 38 mm, 0~25 mm, d=lOO pm, NA=0.3, T-50%, R-l%, 1~514.5 nm, L,-100 mm, and an MMF length of 50 m. At the PCM end of the system, the collimating lens’ focal length was 38 mm, the PBS aperture was 2.5 cm, the half-wave plate’s aperture was 2 cm, and the reduction lens’ focal length was 50 cm
-FF-0.5, 10
-
o
11lq 0
0.2
F~0.8,
Theory Experlment
-I
;..._ ._.__ _.. .... ..___..___ 4 . 0.4
0.6
0.8
1
CJ
Fig. 3. Experimental points and the corresponding theoretical curves for SNR versus @for F=0.8 and 0.5.
0.6
0.4
0.2
-_
0
0
0.2
0.4
P IntoPCM
0.6
0.8
1
’ ~%l,.cl
Fig. 2. (a) Experimental points and the corresponding theoretical curve for IBR versus F for 0= 1. (b ) Experimental points for reciprocity, comparing the ratio of the power in the phase conjugate signal to the power reflected by the phase conjugate mirror with the ratio of the power captured by the phase conjugate mirror to the power emanating from the input object. The straight line in (b) represents perfect reciprocity.
with the crystal placed about 45 cm from that lens. As well, the optics were arranged as compactly as possible to minimize path length disturbances. The test object utilized for studying IBR and reciprocity was a semi-circle illuminating half of the field-of-view of the fiber’s full numerical aperture, with injection powers in the 1 to 20 mW range. Such an input allowed us to measure image-plus-background across one half of the output plane while measuring just background across the other half of the output plane. Fig. 2a compares our experimental data with the theoretical predictions for IBR versus F (utilizing Eq. ( 5a) ). For small F, the experimental results are in excellent agreement with theory. As Fapproaches one, imperfections in the (real) PCM begin appearing and the data diverges from the theory. The rapid appearance of these imperfections is perhaps due to the fact that we were imaging a diffuse source (the 100 pm wide fiber tip) rather than collimating a point source, as well as the more dominant contribution from R in Eq. (5a)‘s denominator. Fig. 2b then demonstrates the reciprocity of our system by comparing experimental values of power ratios for beams going one way vs. the other, as mentioned above and defined by Eq. (2~) (the straight line in Fig. 2b indicates a perfectly reciprocal system). We then turned our attention to varying 0 (by demagnifying the input semicircle utilized in studying IBR and reciprocity) when F= 0.8 and 0.5 to measure our system’s SNR versus @.Results for this are given in Fig. 3, where the curves are the theoretical predictions and the dots are the
S. Campbellet al. /Optics Communications114 11995) 50-56
Fig. 4. Images of an Air Force resolution target. (a) corresponds to imaging via the injection optics with the 100 urn diameter optical fiber replaced by a 100 pm pinhole. (b, c, d, e) correspond to investigations with the set up given in Fig. 1 for @= 1 and F= 1,0.5,0.2, and 0, respectively. (f ) then corresponds to investigations with the set up given in Fig. 1 for @=O.l and F= 1. Comparing (b) to (f ) shows an obvious increase in SNR but decrease in resolution.
experimental results. As can be seen, experimental values for the SNR exceed theoretical values as ip gets small. This is because much of the noise existed near
the perimeter of the output plane, possibly as a result of fiber core surface damage during cleaving. Furthermore, in order to give a practical, visual demon-
56
S. Campbell et al. / Optics Communications 114 (1995) 50-56
stration of partial phase conjugation due to aperture limitations at both the input and output ends of the MMF, we injected an Air Force resolution target into our system. These results are shown in Fig. 4. First, Fig. 4a shows the resolution target as imaged through a 100 ym diameter pinhole (simulating the maximum object information that can enter our 100 urn diameter optical fiber for this system). Then, Figs. 4b-e show results from our whole system as given in Fig. 1 for 0= 1 and F= 1,0.5,0.2, and 0, respectively (the non-zero output in Fig. 4e shows the ever-present back reflections from the optical fiber due to R ) . In these images, it is clear that the injection apertures limit primarily the space information entering the fiber system. Finally, Fig. 4f corresponds to @=O. 1 with F= 1. In this image, it is clear that the injection apertures noticeably limit the bandwidth information entering the fiber. In order to not overexpose Fig. 4f, its exposure time was less than those in Figs. 4ae since the same object power is concentrated in a smaller area. Such a shortening of the exposure time brings the noise background below the threshold for film response. Clearly, limiting the aperture of the PCM (F) degrades the quality of the conjugate image in a global sense, as seen through Figs. 4b-e. As well, limiting the portion of NA that the input object information occupies (@) significantly improves the SNR, but unfortunately acts to reduce the image resolution due to information coupling losses, as seen in Fig. 4f.
4. Summary In summary, we have, for the first time, experimentally investigated a reciprocity-based theoretical model for partial phase conjugate fidelity, showing good agreement between theory and experiment. We have also demonstrated the trade-off between returned image resolution and SNR in such systems.
Acknowledgements This work was supported by the U.S. Office of Naval Research and Air Force Office of Scientific Research.
References [ 1] P. Beckwith, I. McMichael and P. Yeh, Optics Lett. 12 (1987) 510.
[ 2 ] Y. Tomita, R. Yahalom, K. Kyuma, A. Yariv and N. Kwong, IEEE J. Quantum Electron. 25 (1989) 315.
[ 31 C. Gu and P. Yeh, Optics Comm. 107 ( 1994) 353. [4] J.W. Goodman, W.H. Huntley Jr., D.W. Jackson and M. Lehmann, Appl. Phys. Lett. 8 ( 1966) 3 11. [ 51 A. Yariv, Optics Comm. 21 (1977) 49. [6] D.M. Pepper and A. Yariv, Optics Lett. 5 ( 1980) 59. [ 71 J. Falk, Optics Lett. 7 (1982) 620. [ 81 A. Volyar, A. Gnatovskii, N. Kukhtarev and S. Lapaeva, Appl. Phys. B 52 ( 1991) 400. [9] Q. He and J. Duthie, Optics Comm. 75 (1990) 311. [ 10 ] See, for example, J.P. Powers, An introduction to fiber optic systems (Aksen Ass., 1993).