Abstract

Free space propagation and conventional optical systems such as lenses and mirrors all perform spatial unitary transforms. However, the subset of transforms available through these conventional systems is limited in scope. We present here a unitary programmable mode converter (UPMC) capable of performing any spatial unitary transform of the light field. It is based on a succession of reflections on programmable deformable mirrors and free space propagation. We first show theoretically that a UPMC without limitations on resources can perform perfectly any transform. We then build an experimental implementation of the UPMC and show that, even when limited to three reflections on an array of 12 pixels, the UPMC is capable of performing single mode tranforms with an efficiency greater than 80% for the first four modes of the transverse electromagnetic basis.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (2)

A. Preumont, R. Bastaits, and G. Rodrigues, “Scale effects in active optics of large segmented mirrors,” Mechatronics 19, 1286–1293 (2009).
[CrossRef]

J. W. Tay, M. A. Taylor, and W. P. Bowen, “Sagnac-interferometer-based characterization of spatial light modulators,” Appl. Opt. 48, 2236–2242 (2009).
[CrossRef] [PubMed]

2008 (2)

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215–227 (2008).
[CrossRef]

B. Potsaid and J. T.-Y. Wen, “Design of adaptive optics based systems by using mems deformable mirror models,” Int. J. Optomechatronics 2, 104–125 (2008).
[CrossRef]

2007 (2)

M.-L. Hsieh, M.-L. Chen, and C.-J. Cheng, “Improvement of the complex modulated characteristic of cascaded liquid crystal spatial light modulators by using a novel amplitude compensated technique,” Opt. Eng. (Bellingham) 46, 070501 (2007).
[CrossRef]

P. Török and F.-J. Kao, “Optical imaging and microscopy,” Springer Ser. Opt. Sci. 87, 75–85 (2007).

2006 (1)

V. Delaubert, N. Treps, M. Lassen, C. Harb, C. Fabre, P. K. Lam, and H.-A. Bachor, “TEM10 homodyne detection as an optimal small-displacement and tilt-measurement scheme,” Phys. Rev. A 74, 053823 (2006).
[CrossRef]

2004 (1)

2002 (4)

1991 (1)

1986 (1)

A. E. Siegman, Lasers (University Science Books, 1986), p. 1283.

1984 (1)

H. D. Ikramov. Linear Algebra: Problems Book (Victor Kamkin, 1984).

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1981 (1)

Z. I. Borevich and S. L. Krupetskii, “Subgroups of the unitary group that contain the group of diagonal matrices,” J. Math. Sci. (N.Y.) 17, 1951–1959 (1981).
[CrossRef]

1861 (1)

J. F. W. Herschel, “The telescope,” Encyclopedia Britannica (Adam and Charles Black, 1861), p. 190.

Bachor, H.-A.

V. Delaubert, N. Treps, M. Lassen, C. Harb, C. Fabre, P. K. Lam, and H.-A. Bachor, “TEM10 homodyne detection as an optimal small-displacement and tilt-measurement scheme,” Phys. Rev. A 74, 053823 (2006).
[CrossRef]

Bagnoud, V.

Barnett, S.

J. Leach, M. Padgett, S. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Bastaits, R.

A. Preumont, R. Bastaits, and G. Rodrigues, “Scale effects in active optics of large segmented mirrors,” Mechatronics 19, 1286–1293 (2009).
[CrossRef]

Borevich, Z. I.

Z. I. Borevich and S. L. Krupetskii, “Subgroups of the unitary group that contain the group of diagonal matrices,” J. Math. Sci. (N.Y.) 17, 1951–1959 (1981).
[CrossRef]

Bowen, W. P.

Campbell, M.

Chen, M. -L.

M.-L. Hsieh, M.-L. Chen, and C.-J. Cheng, “Improvement of the complex modulated characteristic of cascaded liquid crystal spatial light modulators by using a novel amplitude compensated technique,” Opt. Eng. (Bellingham) 46, 070501 (2007).
[CrossRef]

Cheng, C. -J.

M.-L. Hsieh, M.-L. Chen, and C.-J. Cheng, “Improvement of the complex modulated characteristic of cascaded liquid crystal spatial light modulators by using a novel amplitude compensated technique,” Opt. Eng. (Bellingham) 46, 070501 (2007).
[CrossRef]

Courtial, J.

J. Leach, M. Padgett, S. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Delaubert, V.

V. Delaubert, N. Treps, M. Lassen, C. Harb, C. Fabre, P. K. Lam, and H.-A. Bachor, “TEM10 homodyne detection as an optimal small-displacement and tilt-measurement scheme,” Phys. Rev. A 74, 053823 (2006).
[CrossRef]

Donnelly, W.

Fabre, C.

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215–227 (2008).
[CrossRef]

V. Delaubert, N. Treps, M. Lassen, C. Harb, C. Fabre, P. K. Lam, and H.-A. Bachor, “TEM10 homodyne detection as an optimal small-displacement and tilt-measurement scheme,” Phys. Rev. A 74, 053823 (2006).
[CrossRef]

Fade, J.

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215–227 (2008).
[CrossRef]

Franke-Arnold, S.

J. Leach, M. Padgett, S. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Harb, C.

V. Delaubert, N. Treps, M. Lassen, C. Harb, C. Fabre, P. K. Lam, and H.-A. Bachor, “TEM10 homodyne detection as an optimal small-displacement and tilt-measurement scheme,” Phys. Rev. A 74, 053823 (2006).
[CrossRef]

Hebert, T.

Herschel, J. F. W.

J. F. W. Herschel, “The telescope,” Encyclopedia Britannica (Adam and Charles Black, 1861), p. 190.

Hsieh, M. -L.

M.-L. Hsieh, M.-L. Chen, and C.-J. Cheng, “Improvement of the complex modulated characteristic of cascaded liquid crystal spatial light modulators by using a novel amplitude compensated technique,” Opt. Eng. (Bellingham) 46, 070501 (2007).
[CrossRef]

Ikramov, H. D.

H. D. Ikramov. Linear Algebra: Problems Book (Victor Kamkin, 1984).

Kao, F. -J.

P. Török and F.-J. Kao, “Optical imaging and microscopy,” Springer Ser. Opt. Sci. 87, 75–85 (2007).

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Krupetskii, S. L.

Z. I. Borevich and S. L. Krupetskii, “Subgroups of the unitary group that contain the group of diagonal matrices,” J. Math. Sci. (N.Y.) 17, 1951–1959 (1981).
[CrossRef]

Lam, P. K.

V. Delaubert, N. Treps, M. Lassen, C. Harb, C. Fabre, P. K. Lam, and H.-A. Bachor, “TEM10 homodyne detection as an optimal small-displacement and tilt-measurement scheme,” Phys. Rev. A 74, 053823 (2006).
[CrossRef]

Lassen, M.

V. Delaubert, N. Treps, M. Lassen, C. Harb, C. Fabre, P. K. Lam, and H.-A. Bachor, “TEM10 homodyne detection as an optimal small-displacement and tilt-measurement scheme,” Phys. Rev. A 74, 053823 (2006).
[CrossRef]

Leach, J.

J. Leach, M. Padgett, S. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Padgett, M.

J. Leach, M. Padgett, S. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Potsaid, B.

B. Potsaid and J. T.-Y. Wen, “Design of adaptive optics based systems by using mems deformable mirror models,” Int. J. Optomechatronics 2, 104–125 (2008).
[CrossRef]

Preumont, A.

A. Preumont, R. Bastaits, and G. Rodrigues, “Scale effects in active optics of large segmented mirrors,” Mechatronics 19, 1286–1293 (2009).
[CrossRef]

Queener, H.

Réfrégier, P.

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215–227 (2008).
[CrossRef]

Rodrigues, G.

A. Preumont, R. Bastaits, and G. Rodrigues, “Scale effects in active optics of large segmented mirrors,” Mechatronics 19, 1286–1293 (2009).
[CrossRef]

Romero-Borja, F.

Roorda, A.

Serre, D.

D. Serre, Matrices: Theory and Applications (Springer, 2002), p. 202.

Shirai, T.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986), p. 1283.

Tay, J. W.

Taylor, M. A.

Török, P.

P. Török and F.-J. Kao, “Optical imaging and microscopy,” Springer Ser. Opt. Sci. 87, 75–85 (2007).

Treps, N.

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215–227 (2008).
[CrossRef]

V. Delaubert, N. Treps, M. Lassen, C. Harb, C. Fabre, P. K. Lam, and H.-A. Bachor, “TEM10 homodyne detection as an optimal small-displacement and tilt-measurement scheme,” Phys. Rev. A 74, 053823 (2006).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Wen, J. T.-Y.

B. Potsaid and J. T.-Y. Wen, “Design of adaptive optics based systems by using mems deformable mirror models,” Int. J. Optomechatronics 2, 104–125 (2008).
[CrossRef]

Wyrowski, F.

Zuegel, J.

Appl. Opt. (2)

Eur. Phys. J. D (1)

J. Fade, N. Treps, C. Fabre, and P. Réfrégier, “Optimal precision of parameter estimation in images with local sub-Poissonian quantum fluctuations,” Eur. Phys. J. D 50, 215–227 (2008).
[CrossRef]

Int. J. Optomechatronics (1)

B. Potsaid and J. T.-Y. Wen, “Design of adaptive optics based systems by using mems deformable mirror models,” Int. J. Optomechatronics 2, 104–125 (2008).
[CrossRef]

J. Math. Sci. (N.Y.) (1)

Z. I. Borevich and S. L. Krupetskii, “Subgroups of the unitary group that contain the group of diagonal matrices,” J. Math. Sci. (N.Y.) 17, 1951–1959 (1981).
[CrossRef]

Mechatronics (1)

A. Preumont, R. Bastaits, and G. Rodrigues, “Scale effects in active optics of large segmented mirrors,” Mechatronics 19, 1286–1293 (2009).
[CrossRef]

Opt. Eng. (Bellingham) (1)

M.-L. Hsieh, M.-L. Chen, and C.-J. Cheng, “Improvement of the complex modulated characteristic of cascaded liquid crystal spatial light modulators by using a novel amplitude compensated technique,” Opt. Eng. (Bellingham) 46, 070501 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

V. Delaubert, N. Treps, M. Lassen, C. Harb, C. Fabre, P. K. Lam, and H.-A. Bachor, “TEM10 homodyne detection as an optimal small-displacement and tilt-measurement scheme,” Phys. Rev. A 74, 053823 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

J. Leach, M. Padgett, S. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Springer Ser. Opt. Sci. (1)

P. Török and F.-J. Kao, “Optical imaging and microscopy,” Springer Ser. Opt. Sci. 87, 75–85 (2007).

Other (4)

J. F. W. Herschel, “The telescope,” Encyclopedia Britannica (Adam and Charles Black, 1861), p. 190.

A. E. Siegman, Lasers (University Science Books, 1986), p. 1283.

D. Serre, Matrices: Theory and Applications (Springer, 2002), p. 202.

H. D. Ikramov. Linear Algebra: Problems Book (Victor Kamkin, 1984).

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Figures (6)

Fig. 1
Fig. 1

Beam is coupled into the UPMC by a reflection on the polarizing beam splitter PBS1. It is then focused on to the deformable mirror DM. The beam first undergoes a two-dimensional (2D) FT through the 2 spherical lens SL (focal length f SL = 300   mm ), followed by a vertical FT going through the cylindrical lens CL ( f CL = 50   mm ) , and finally another 2D FT going back through SL. The optical path length between DM and SL is 300 mm and between SL and CL 350 mm. The sliding half-wave plates HWP1 and HWP2 are used to choose the number of reflections on the DM before being coupled out on PBS1. The angle θ is 0.16° to allow for three reflections on the deformable mirror. The inset shows the pixel layout and the measured positions and sizes of the beam for the three reflections, when the beam is a simple Gaussian mode. Manipulation of the beam makes the spatial profile bigger than the simple Gaussian, hence the small footprint of the mode compared to the size of the deformable mirror.

Fig. 2
Fig. 2

Beams coming from the laser go through Gaussian mode selectors GMS locked to the desired input and output modes (here I = T E M 00 and O = T E M 20 ). The desired output is phase modulated using the electroactuator piezoelectric transducer (PZT) and overlapped with the output of the UPMC. The interference signal is then measured on a photodiode, and the intensity overlap is derived. Using the measured overlap, the stochastic optimization algorithm changes the control signal to all the pixels of the deformable mirror.

Fig. 3
Fig. 3

Measured mode conversion efficiency α 2 for three different transformations (light, green online), compared to simulated results (dark, blue online). The number of reflections on the UPMC is varied; the screenshots below the plots represent stills from the CCD camera, capturing the output of the UPMC. For repeated optimization procedures of the same transform, with the same number of reflections, the membrane topography was found to differ greatly, while the mode conversion efficiency was consistent. That is, the fraction of power in the desired mode remains constant while the remaining power follows a random distribution. This can be explained by the high number of remaining degrees of freedom. When the maximum mode conversion efficiency was low, i.e., for small number of reflections, the shape of the optimized output mode differed from one optimization to another.

Fig. 4
Fig. 4

Presentation of the mode conversion efficiency α 2 as a function of the transform considered and the number of reflections N ref allowed. The transforms are a. T E M 00 T E M 10 , b. T E M 00 T E M 20 , c. T E M 00 T E M 30 , d. T E M 10 T E M 30 , and e. T E M 10 to flip mode.

Fig. 5
Fig. 5

Transverse profiles of the magnitude of the field when undergoing a succession of FTs and reflections on deformable mirrors DM. The transverse profile of the magnitude remains constant at the reflection surfaces, while the phase has a sharp discontinuity. The transverse axis is renormalized for all planes to keep the profile of T E M 00 constant throughout propagation.

Fig. 6
Fig. 6

Transform quality α n 2 when the realistic UPMC is optimized to perform (a) a beam splitter U BS and (b) a phase operator U P . They are plotted as functions of their respective parameters r and ϕ and the number of reflections. The black curve represents the single reflection theoretical maximum. The overlap is perfect for α n 2 = 1 . Additional simulations with the same number of trials for a constant transform presented similar small fluctuations in the case of three and four reflections. These are artifacts of the optimization process.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

E ( x , y ) = m N , n N a m n T E M m n ( x , y ) .
U T = ( 1 2 1 2 0 0 1 2 1 2 0 0 0 0 1 0 0 0 0 1 )
E ( x , y ) e i ϕ DM ( x , y ) E ( x , y ) ,
U DM ( ϕ ) = ( e i ϕ 1 0 0 0 0 e i ϕ 2 0 0 0 0 e i ϕ 3 0 0 0 0 e i ϕ n ) .
T i j ( θ ) = ( 1 0 0 0 0 0 1 0 0 0 0 0 cos ( θ ) sin ( θ ) 0 1 0 0 sin ( θ ) cos ( θ ) 0 0 0 0 0 0 0 0 0 0 1 ) ,
α = | i K ( x , y ) O ¯ i ( x , y ) O i ( x , y ) | .
α = | i K m N , n N o ¯ i , m , n o i , m , n | .
U BS ( r ) = ( r t t r ) ,
U P ( ϕ ) = ( 1 0 0 e i ϕ )

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