Abstract

Modal analysis of an optical field via generalized interferometry (GI) is a novel technique that treats said field as a linear superposition of transverse modes and recovers the amplitudes of modal weighting coefficients. We use phase retrieval by nonlinear optimization to recover the phase of these modal weighting coefficients. Information diversity increases the robustness of the algorithm by better constraining the solution. Additionally, multiple sets of random starting phase values assist the algorithm in overcoming local minima. The algorithm was able to recover nearly all coefficient phases for simulated fields consisting of up to 21 superpositioned Hermite Gaussian modes from simulated data and proved to be resilient to shot noise.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2015 (3)

2014 (1)

2012 (5)

2011 (2)

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Advances in Optics and Photonics 3, 272–365 (2011).
[Crossref]

A. F. Abouraddy, T. M. Yarnall, and B. E. Saleh, “Angular and radial mode analyzer for optical beams,” Opt. Lett. 36, 4683–4685 (2011).
[Crossref] [PubMed]

2009 (1)

2008 (2)

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
[Crossref] [PubMed]

2005 (2)

1990 (1)

1986 (1)

1982 (1)

Abouraddy, A. F.

Acton, D. S.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase retrieval algorithm for JWST flight and testbed telescope,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2006), pp. 626511.

Alieva, T.

Alonso, M. A.

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Advances in Optics and Photonics 3, 272–365 (2011).
[Crossref]

Aronstein, D. L.

D. L. Aronstein, J. S. Smith, T. P. Zielinski, R. Telfer, S. C. Tournois, D. B. Moore, and J. R. Fienup, “Wavefront-error performance characterization for the James Webb Space Telescope (JWST) integrated science instrument module (isim) science instruments,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2016), pp. 990409.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase retrieval algorithm for JWST flight and testbed telescope,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2006), pp. 626511.

Aspelmeyer, M.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Atia, G. K.

Bianco, G.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Bonato, C.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Boyd, R. W.

Calvo, M. L.

Chen, M.

Dainty, J.

C. Fienup and J. Dainty, “Phase retrieval and image reconstruction for astronomy,” Image Recovery: Theory and Application pp. 231–275 (1987).

Dean, B. H.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase retrieval algorithm for JWST flight and testbed telescope,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2006), pp. 626511.

Dong, J.

Fickler, R.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across vienna,” P. Natl. Acad. Sci. USA 112, 14197–14201 (2015).
[Crossref]

Fienup, C.

C. Fienup and J. Dainty, “Phase retrieval and image reconstruction for astronomy,” Image Recovery: Theory and Application pp. 231–275 (1987).

Fienup, J.

Fienup, J. R.

Fink, M.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across vienna,” P. Natl. Acad. Sci. USA 112, 14197–14201 (2015).
[Crossref]

Fink, Y.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[Crossref] [PubMed]

Guizar-Sicairos, M.

Handsteiner, J.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across vienna,” P. Natl. Acad. Sci. USA 112, 14197–14201 (2015).
[Crossref]

Hennelly, B. M.

Jennewein, T.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Joannopoulos, J. D.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[Crossref] [PubMed]

Jones, E.

E. Jones, T. Oliphant, P. Peterson, and et al., “SciPy: Open source scientific tools for Python,” (2001). [Online; accessed March 10, 2016 ].

Jurling, A. S.

Krenn, M.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across vienna,” P. Natl. Acad. Sci. USA 112, 14197–14201 (2015).
[Crossref]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing, 1st ed. (Wiley, 2001).

Lavery, M. P. J.

Leach, J.

Love, J. D.

Luceri, V.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Malik, M.

Mardani, D.

Millane, R. P.

Mirhosseini, M.

Moore, D. B.

D. L. Aronstein, J. S. Smith, T. P. Zielinski, R. Telfer, S. C. Tournois, D. B. Moore, and J. R. Fienup, “Wavefront-error performance characterization for the James Webb Space Telescope (JWST) integrated science instrument module (isim) science instruments,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2016), pp. 990409.

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer, 2006), 2nd ed.

O’Sullivan, M.

O’Sullivan, M. N.

Oliphant, T.

E. Jones, T. Oliphant, P. Peterson, and et al., “SciPy: Open source scientific tools for Python,” (2001). [Online; accessed March 10, 2016 ].

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing, 1st ed. (Wiley, 2001).

Padgett, M. J.

Pernechele, C.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Peterson, P.

E. Jones, T. Oliphant, P. Peterson, and et al., “SciPy: Open source scientific tools for Python,” (2001). [Online; accessed March 10, 2016 ].

Riesen, N.

Rodenburg, B.

Rodrigo, J. A.

Saleh, B. E.

Saleh, B. E. A.

Shapira, O.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[Crossref] [PubMed]

Sheridan, J. T.

Shiri, R.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase retrieval algorithm for JWST flight and testbed telescope,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2006), pp. 626511.

Siegman, A.

A. Siegman, Lasers (University Science Books, 1986).

Smith, J. S.

D. L. Aronstein, J. S. Smith, T. P. Zielinski, R. Telfer, S. C. Tournois, D. B. Moore, and J. R. Fienup, “Wavefront-error performance characterization for the James Webb Space Telescope (JWST) integrated science instrument module (isim) science instruments,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2016), pp. 990409.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase retrieval algorithm for JWST flight and testbed telescope,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2006), pp. 626511.

Soltanolkotabi, M.

Tamburini, F.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Tang, G.

Telfer, R.

D. L. Aronstein, J. S. Smith, T. P. Zielinski, R. Telfer, S. C. Tournois, D. B. Moore, and J. R. Fienup, “Wavefront-error performance characterization for the James Webb Space Telescope (JWST) integrated science instrument module (isim) science instruments,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2016), pp. 990409.

Tian, L.

Tournois, S. C.

D. L. Aronstein, J. S. Smith, T. P. Zielinski, R. Telfer, S. C. Tournois, D. B. Moore, and J. R. Fienup, “Wavefront-error performance characterization for the James Webb Space Telescope (JWST) integrated science instrument module (isim) science instruments,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2016), pp. 990409.

Ursin, R.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Villoresi, P.

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Wackerman, C.

Waller, L.

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer, 2006), 2nd ed.

Yarnall, T. M.

Yeh, L.-H.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing, 1st ed. (Wiley, 2001).

Zeilinger, A.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across vienna,” P. Natl. Acad. Sci. USA 112, 14197–14201 (2015).
[Crossref]

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Zhong, J.

Zielinski, T. P.

D. L. Aronstein, J. S. Smith, T. P. Zielinski, R. Telfer, S. C. Tournois, D. B. Moore, and J. R. Fienup, “Wavefront-error performance characterization for the James Webb Space Telescope (JWST) integrated science instrument module (isim) science instruments,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2016), pp. 990409.

Advances in Optics and Photonics (1)

M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Advances in Optics and Photonics 3, 272–365 (2011).
[Crossref]

Appl. Opt. (2)

J. Opt. Soc. Am. A (5)

New J. Phys. (1)

P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, and A. Zeilinger, “Experimental verification of the feasibility of a quantum channel between space and earth,” New J. Phys. 10, 033038 (2008).
[Crossref]

Opt. Express (6)

Opt. Lett. (2)

P. Natl. Acad. Sci. USA (1)

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across vienna,” P. Natl. Acad. Sci. USA 112, 14197–14201 (2015).
[Crossref]

Phys. Rev. Lett. (1)

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[Crossref] [PubMed]

Other (7)

C. Fienup and J. Dainty, “Phase retrieval and image reconstruction for astronomy,” Image Recovery: Theory and Application pp. 231–275 (1987).

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase retrieval algorithm for JWST flight and testbed telescope,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2006), pp. 626511.

D. L. Aronstein, J. S. Smith, T. P. Zielinski, R. Telfer, S. C. Tournois, D. B. Moore, and J. R. Fienup, “Wavefront-error performance characterization for the James Webb Space Telescope (JWST) integrated science instrument module (isim) science instruments,” in “SPIE Astronomical Telescopes+ Instrumentation,” (International Society for Optics and Photonics, 2016), pp. 990409.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing, 1st ed. (Wiley, 2001).

A. Siegman, Lasers (University Science Books, 1986).

J. Nocedal and S. J. Wright, Numerical Optimization (Springer, 2006), 2nd ed.

E. Jones, T. Oliphant, P. Peterson, and et al., “SciPy: Open source scientific tools for Python,” (2001). [Online; accessed March 10, 2016 ].

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Figures (6)

Fig. 1
Fig. 1 Mach-Zehnder generalized interferometer configuration used for phase retrieval simulations. The array detector placed at the upper port of the figure also functions as a bucket detector for GOI amplitude recovery by integrating over all pixels
Fig. 2
Fig. 2 Chart compares percent of successful retrievals for single-plane, two-plane, two-plane targeted, three-plane, and three-plane targeted phase retrieval techniques using only a single set of ϕ ^ init starting values.
Fig. 3
Fig. 3 Heatmap of successful retrieval rates as a function of α, β values for two-plane GI phase retrieval with 10 modes (4th order).
Fig. 4
Fig. 4 Heatmaps detail success of three-plane phase retrieval of a 6th order superposition with 22 equally-spaced values of α in the third plane.
Fig. 5
Fig. 5 Percent successful retrievals when three targeted α planes and 5 random starting sets of phase values are permitted (noiseless).
Fig. 6
Fig. 6 Variability in percentage success with final algorithm when in the presence of limited photon budget and Poisson noise.

Tables (2)

Tables Icon

Table 1 Success of single α = β = 0 plane GI phase retrieval

Tables Icon

Table 2 Success rates of two α, β plane GI phase retrieval

Equations (12)

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

U ( x , y ) = n c n ψ n ( x , y ) ,
Λ α { U ( x , y ) } = m c m e im π α 2 ψ m ( x , y ) ,
HG m n ( x , y ) = H m ( 2 x w 0 ) H n ( 2 y w 0 ) exp [ ( x 2 + y 2 ) w 0 2 ] ,
U out ( x , y ; α , β ) = 1 2 [ Λ α + Λ β ] U in ( x , y ) = 1 2 [ Λ α + Λ β ] m , n c m n HG m n ( x , y ) = 1 2 m , n c m n [ e im π α 2 HG m n ( x , y ) + e in π β 2 HG m n ( x , y ) ] = 1 2 m , n c m n HG m n ( x , y ) ( e im π α 2 + e in π β 2 ) .
I out ( x , y ; α , β ) = | 1 2 m , n c m n HG m n ( x , y ) ( e im π α 2 + e in π β 2 ) | 2 = 1 2 { m , n | c m n | 2 H G m n 2 ( x , y ) [ 2 + 2 cos ( m π α 2 n π β 2 ) ] + [ m , n ; m , n m , n m , n c m n c m , n * HG m n ( x , y ) HG m n ( x , y ) ] × ( e im π α 2 + e in π β 2 ) ( e i m π α 2 + e i n π β 2 ) } .
d x d y HG m n ( x , y ) HG m n ( x , y ) = S ( m , n ) δ m m δ n n ,
P out ( α , β ) = d x d y I out ( x , y ; α , β ) = m , n | c m n | 2 [ 1 + cos ( m π α 2 n π β 2 ) ] S ( m , n ) .
P out ( α , β ) = P out ( α , β ) m n | c m n | 2 = m n | c m n | 2 cos ( m π α 2 n π β 2 ) .
P ˜ out ( m , n ) = { P out ( α , β ) } { α , β } { m , n } = m , n | c m n | 2 { cos ( m π α 2 n π β 2 ) } = m , n | c m n | 2 [ δ ( m m 4 , n + n 4 ) + δ ( m + m 4 , n n 4 ) ] .
E = x , y | I est ( x , y ; α , β ; ϕ ^ ) I d ( x , y ; α , β ) | 2 ,
I est ( x , y ; α , β ; ϕ ^ ) = 1 2 | m n | c m n | HG m n ( x , y ) e i ϕ ^ m n ( e im π α 2 + e in π β 2 ) | 2 .
E agg = { α , β } E ( { α , β } ) = { α , β } x , y | I est ( x , y ; α , β ; ϕ ^ ) I d ( x , y ; α , β ) | 2 .

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