Abstract

We develop a method for synthesis of a desired intensity profile at the output of a multimode fiber (MMF) with random mode coupling by controlling the input field distribution using a spatial light modulator (SLM) whose complex reflectance is piecewise constant over a set of disjoint blocks. Depending on the application, the desired intensity profile may be known or unknown a priori. We pose the problem as optimization of an objective function quantifying, and derive a theoretical lower bound on the achievable objective function. We present an adaptive sequential coordinate ascent (SCA) algorithm for controlling the SLM, which does not require characterizing the full transfer characteristic of the MMF, and which converges to near the lower bound after one pass over the SLM blocks. This algorithm is faster than optimizations based on genetic algorithms or random assignment of SLM phases. We present simulated and experimental results applying the algorithm to forming spots of light at a MMF output, and describe how the algorithm can be applied to imaging.

© 2012 OSA

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  1. H. J. Shin, M. C. Pierce, D. Lee, H. Ra, O. Solgaard, and R. Richards-Kortum, “Fiber-optic confocal microscope using a MEMS scanner and miniature objective lens,” Opt. Express 15(15), 9113–9122 (2007).
    [CrossRef] [PubMed]
  2. P. M. Lane, A. L. P. Dlugan, R. Richards-Kortum, and C. E. Macaulay, “Fiber-optic confocal microscopy using a spatial light modulator,” Opt. Lett. 25(24), 1780–1782 (2000).
    [CrossRef] [PubMed]
  3. K. M. Tan, M. Mazilu, T. H. Chow, W. M. Lee, K. Taguichi, B. K. Ng, W. Sibbett, C. S. Herrington, C. T. A. Brown, and K. Dholakia, “In-fiber common-path optical coherence tomography using a conical-tip fiber,” Opt. Express 17(4), 2375–2384 (2009).
    [CrossRef] [PubMed]
  4. N. Sim, D. Bessarab, C. M. Jones, and L. Krivitsky, “Method of targeted delivery of laser beam to isolated retinal rods by fiber optics,” Biomed. Opt. Express 2(11), 2926–2933 (2011).
    [CrossRef] [PubMed]
  5. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2, 1–32 (2008).
  6. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4(6), 388–394 (2010).
    [CrossRef]
  7. G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett. 99(19), 191102 (2011).
    [CrossRef]
  8. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
    [CrossRef] [PubMed]
  9. T. Čižmár and K. Dholakia, “Shaping the light transmission through a multimode optical fibre: complex transformation analysis and applications in biophotonics,” Opt. Express 19(20), 18871–18884 (2011).
    [CrossRef] [PubMed]
  10. M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express 19(2), 933–945 (2011).
    [CrossRef] [PubMed]
  11. R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express 19(1), 247–254 (2011).
    [CrossRef] [PubMed]
  12. I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281(11), 3071–3080 (2008).
    [CrossRef]
  13. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
    [CrossRef] [PubMed]
  14. I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101(12), 120601 (2008).
    [CrossRef] [PubMed]
  15. O. Katz, E. Small, Y. Bomberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
    [CrossRef]
  16. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 2002).
  17. S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, New York, 2004).
  18. B. D. Mangum, C. Mu, and J. M. Gerton, “Resolving single fluorophores within dense ensembles: contrast limits of tip-enhanced fluorescence microscopy,” Opt. Express 16(9), 6183–6193 (2008).
    [CrossRef] [PubMed]
  19. R. A. Panicker and J. M. Kahn, “Algorithms for compensation of multimode fiber dispersion using adaptive optics,” J. Lightwave Technol. 27(24), 5790–5799 (2009).
    [CrossRef]
  20. A. d'Aspremont and S. P. Boyd, “Relaxations and randomized methods for nonconvex QCQPs,” http://www.stanford.edu/class/ee364b/lectures/relaxations.pdf .
  21. B. K. Garside, T. K. Lim, and J. P. Marton, “Propagation characteristics of parabolic-index fiber modes: linearly polarized approximation,” J. Opt. Soc. Am. 70(4), 395–400 (1980).
    [CrossRef]
  22. K. J. Boucher, C. Jan, J. M. Kahn, J. P. Wilde, and O. Solgaard, “Spot formation and scanning microscopy via multimode fibers,” in 2011 IEEE Photonics Conference (PHO) (IEEE, 2011), pp. 713–714.

2011

2010

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4(6), 388–394 (2010).
[CrossRef]

2009

2008

B. D. Mangum, C. Mu, and J. M. Gerton, “Resolving single fluorophores within dense ensembles: contrast limits of tip-enhanced fluorescence microscopy,” Opt. Express 16(9), 6183–6193 (2008).
[CrossRef] [PubMed]

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2, 1–32 (2008).

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101(12), 120601 (2008).
[CrossRef] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281(11), 3071–3080 (2008).
[CrossRef]

2007

2000

1980

Baumgartl, J.

Bessarab, D.

Bianchi, S.

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Bomberg, Y.

O. Katz, E. Small, Y. Bomberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[CrossRef]

Brown, C. T. A.

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Chow, T. H.

Cižmár, T.

Dholakia, K.

Di Leonardo, R.

Dienerowitz, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2, 1–32 (2008).

Dlugan, A. L. P.

Fink, M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Garside, B. K.

Gerton, J. M.

Gigan, S.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Harris, J.

G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett. 99(19), 191102 (2011).
[CrossRef]

Herrington, C. S.

Jones, C. M.

Kahn, J. M.

Katz, O.

O. Katz, E. Small, Y. Bomberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[CrossRef]

Kosmeier, S.

Krivitsky, L.

Lane, P. M.

Lee, D.

Lee, W. M.

Lerosey, G.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Lim, T. K.

Macaulay, C. E.

Mangum, B. D.

Marton, J. P.

Mazilu, M.

Mosk, A. P.

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281(11), 3071–3080 (2008).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101(12), 120601 (2008).
[CrossRef] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
[CrossRef] [PubMed]

Mu, C.

Ng, B. K.

Panicker, R. A.

Pierce, M. C.

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

Provine, J.

G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett. 99(19), 191102 (2011).
[CrossRef]

Ra, H.

Richards-Kortum, R.

Riviore, K.

G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett. 99(19), 191102 (2011).
[CrossRef]

Sarmiento, T.

G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett. 99(19), 191102 (2011).
[CrossRef]

Shambat, G.

G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett. 99(19), 191102 (2011).
[CrossRef]

Shin, H. J.

Sibbett, W.

Silberberg, Y.

O. Katz, E. Small, Y. Bomberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[CrossRef]

Sim, N.

Small, E.

O. Katz, E. Small, Y. Bomberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[CrossRef]

Solgaard, O.

Taguichi, K.

Tan, K. M.

Vellekoop, I. M.

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281(11), 3071–3080 (2008).
[CrossRef]

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101(12), 120601 (2008).
[CrossRef] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
[CrossRef] [PubMed]

Vuckovic, J.

G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett. 99(19), 191102 (2011).
[CrossRef]

Appl. Phys. Lett.

G. Shambat, J. Provine, K. Riviore, T. Sarmiento, J. Harris, and J. Vučković, “Optical fiber tips functionalized with semiconductor photonic crystal cavities,” Appl. Phys. Lett. 99(19), 191102 (2011).
[CrossRef]

Biomed. Opt. Express

J. Lightwave Technol.

J. Nanophotonics

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2, 1–32 (2008).

J. Opt. Soc. Am.

Nat. Photonics

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4(6), 388–394 (2010).
[CrossRef]

O. Katz, E. Small, Y. Bomberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5(6), 372–377 (2011).
[CrossRef]

Opt. Commun.

I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281(11), 3071–3080 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[CrossRef] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101(12), 120601 (2008).
[CrossRef] [PubMed]

Other

A. d'Aspremont and S. P. Boyd, “Relaxations and randomized methods for nonconvex QCQPs,” http://www.stanford.edu/class/ee364b/lectures/relaxations.pdf .

K. J. Boucher, C. Jan, J. M. Kahn, J. P. Wilde, and O. Solgaard, “Spot formation and scanning microscopy via multimode fibers,” in 2011 IEEE Photonics Conference (PHO) (IEEE, 2011), pp. 713–714.

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 2002).

S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, New York, 2004).

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Figures (9)

Fig. 1
Fig. 1

System for synthesis of a known intensity pattern. Light from a laser illuminates an SLM, and light reflected from the SLM is focused into an MMF. At the MMF output, the intensity distribution is measured using a microscope and camera. The goal is to find an SLM pattern such that the output intensity distribution approximates a desired distribution. The inset shows the two regions R1 and R2 at the MMF output used in defining the objective function for synthesizing a known intensity profile.

Fig. 2
Fig. 2

Characteristics of the spots formed using CPSCA and APSCA in known locations at different distances from the center of the fiber, in simulation and experiment: (a) longitudinal spot size, (b) transverse spot size, (c) centroid location, (d) peak sidelobe ratio and (e) integrated sidelobe ratio. Note that CPSCA and APSCA yield higher peak and integrated sidelobe ratios than a backpropagated delta function sampled at the same resolution.

Fig. 3
Fig. 3

Normalized objective function convergence curve for a spot in a known location 5 µm from the center of the fiber, for simulated (solid) and experimental (dashed) CPSCA. In both cases, after one pass over the SLM, the objective function converges to a value close to the theoretical lower bound.

Fig. 4
Fig. 4

Experimental setup for forming spots in known locations and for imaging. Light from a 1550-nm laser is directed onto the SLM. Light reflected from the SLM is focused into the MMF. A camera measures the intensity profile at the MMF output and sends the data to a PC that controls the SLM phases. The inset shows an imaging mode of operation, where a test object is placed in front of the fiber. Previously saved patterns are loaded on the SLM to generate spots of light at different locations on the fiber output. The spots sample the test object and the reflected intensity is measured by the power meter and used to reconstruct the image.

Fig. 5
Fig. 5

Intensity distributions formed in known locations at the output of a 50-μm parabolic-index MMF. A phase-only SLM with 16 × 16 blocks is adapted using CPSCA. White circles show the fiber core boundary. Spot focused at center of core: (a) simulated, (b) experimental. Spot focused 20 µm away from center of core: (c) simulated, (d) experimental.

Fig. 6
Fig. 6

Simulated intensity distributions formed in unknown locations at the output of a 50-μm parabolic-index MMF. Fluorophore distributions having 4-μm FWHM are centered: (a) at center of core, (b) 10 µm away from center of core and (c) 20 µm away from center of core. No a priori knowledge of the fluorophore location is assumed. Adaptive CPSCA is used on a phase-only SLM with 16 × 16 blocks to maximize the total back-reflected fluorescent light intensity. White circles show the fiber core boundary.

Fig. 7
Fig. 7

Normalized objective function convergence curve for targeted light delivery to a fluorophore distribution having 4-μm FWHM and centered 10 µm away from center of core, for adaptation by CPSCA. After one pass over the SLM, the objective function converges to a value close to the theoretical lower bound.

Fig. 8
Fig. 8

Simulated imaging of an infinite checkerboard using spots formed by (a-c) backpropagated delta functions sampled by an infinite-resolution amplitude-and-phase SLM (a-c) and (d-f) by adaptive CPSCA using a 16 × 16-block SLM. The square size is (a),(d) 3.5 µm, (b),(e) 4.5 µm, and (c),(f) 5.5 µm. The fiber core boundary is indicated by the white circles.

Fig. 9
Fig. 9

Characteristics of spots formed by backpropagation of delta functions at different distances from the center of the core, for different sampling resolutions and using phase-only or amplitude-and-phase sampling: (a) longitudinal spot size, (b) transverse spot size, (c) centroid location, (d) peak sidelobe ratio and (e) integrated sidelobe ratio.

Equations (46)

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V(x,y)= k=1 N υ k s k (x,y) ,
s k (x,y)={ 1, if(x,y)intheinteriorofthe k th block 0, otherwise ,
E SLM,out (x,y)=V(x,y) E SLM,in (x,y)= k=1 N υ k s k (x,y) E SLM,in (x,y).
E fiber,in (x,y)=L[ E SLM,out (x,y) ]=L[ k=1 N υ k s k (x,y) E SLM,in (x,y) ]= k=1 N υ k E k (x,y) ,
E fiber,in (x,y)= i c i E M,i (x,y)+radiation modes ,
c i = [ E fiber,in (x,y)× H M,i (x,y) ] z ^ dxdy = [ ( k=1 N υ k E k (x,y) )× H M,i (x,y) ] z ^ dxdy = k=1 N υ k [ E k (x,y)× H M,i (x,y) ] z ^ dxdy = a i T υ,
a ik = [ E k (x,y)× H M,i (x,y) ] z ^ dxdy ,
a ik = E k (x,y) E M,i (x,y)dxdy .
[ E M,k (x,y)× H M,i (x,y) ] z ^ dxdy= δ ki ,
E M,k (x,y) E M,i (x,y)dxdy = δ ki .
E coupled,in =c=Aυ,
A=( a 1 T a 2 T ).
E coupled,out =U E coupled,in =UAυ.
E fiber,out (x,y)= i E coupled,out,i E M,i (x,y) = M T (x,y) E coupled,out ,
M(x,y)=( E M,1 T (x,y) E M,2 T (x,y) ).
E fiber,out (x,y)= M T (x,y) E coupled,out = M T (x,y)UAυ= w H (x,y)υ.
I fiber,out (x,y)= | E fiber,out (x,y) | 2 = υ H w(x,y) w H (x,y)υ.
F(υ)= R 2 I fiber,out (x,y)dxdy +κ R 1 | I fiber,out (x,y) I des (x,y) |dxdy ,
minimizeF(υ) subjectto| υ |=1
F(υ)=κ I fiber,out n (x,y)P(x,y)dxdy ,
F( φ i )= a i sin φ i + b i cos φ i + c i ,
F i,1 =F( φ i =0) F i,2 =F( φ i =2π/3) F i,3 =F( φ i =4π/3),
[ a i b i c i ]= [ 0 1 1 3 /2 1/2 1 3 /2 1/2 1 ] 1 [ F i,1 F i,2 F i,3 ].
φ i,opt ={ φ i if a i sin φ i + b i cos φ i <0 φ i +πotherwise .
F( r i , φ i )= d i r i 2 + a i r i sin φ i + b i r i cos φ i + c i ,
F i,1 =F( υ i =0) F i,2 =F( υ i =1) F i,3 =F( υ i = e j2π/3 ) F i,4 =F( υ i = e j4π/3 ),
c i = F i,1 [ d i a i b i ]= [ 1 0 1 1 3 /2 1/2 1 3 /2 1/2 ] 1 [ F i,2 F i,1 F i,3 F i,1 F i,4 F i,1 ].
r i,opt = a i sin φ i,opt b i cos φ i,opt 2 d i .
r i,opt =1, φ i,opt = φ i if d i + a i sin φ i + b i cos φ i <0 r i,opt =1, φ i,opt = φ i +πif d i a i sin φ i b i cos φ i <0 r i,opt =0, φ i,opt =0otherwise.
υ= 1 max(1, r i,opt ) υ.
W(x,y)=w(x,y) w H (x,y).
V=υ υ H .
I fiber,out (x,y)= υ H w(x,y) w H (x,y)υ=tr( W(x,y)V ),
F(V)= R 2 tr( W(x,y)V )dxdy +κ R 1 | tr( W(x,y)V ) I des (x,y) |dxdy ,
F(V)=κ tr( W(x,y)V )P(x,y)dxdy .
minimizeF(V) subjecttoV=υ υ H diag(V)=1
minimizeF(V) subjectto[ V υ υ H 1 ]0 diag(V)=1
I des, SG (x,y)= I 0 exp[ a ( (x x 0 ) 2 + (y y 0 ) 2 ) m ],
FWHM=2 ( ln2 a ) 1/2m .
E des,SG (x,y)= E des,SG (x,y)e= [ I des,SG (x,y) ] 1/2 e,
E des,δ (x,y)=δ(x x 0 ,y y 0 )e.
E coupled,out,i δ = δ(x x 0 ,y y 0 )e E M,i (x,y)dxdy = E M,i ( x 0 , y 0 ).
E coupled,out δ =( E M,1 ( x 0 , y 0 ) E M,2 ( x 0 , y 0 ) ),
E coupled,in δ = U H E coupled,out δ ,
E fiber,in δ (x,y)= i E coupled,in,i δ E M,i (x,y) = M T (x,y) E coupled,in δ = M T (x,y) U H E coupled,out δ .
E SLM,out δ (x,y)= L 1 [ E fiber,in δ (x,y) ]= L 1 [ M T (x,y) U H E coupled,out δ ].

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