Abstract

We explore the effect of noise on the energy convergence for extremely weak signals in the object field of a holographic experiment. The impact of noise for the energy-on-target in the iterative, bootstrapping process of a holographic phase conjugator system is theoretically derived to obtain a recursive analytical solution. Theoretical results are compared with numerical simulations for a weak-signal holographic conjugator.

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References

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  1. M. A. Vorontsov, V. V. Kolosov, and A. Kohnle, “Adaptive laser beam projection on an extended target: phase- and field-conjugate precompensation,” J. Opt. Soc. Am. A 24, 1975–1993 (2007).
    [Crossref]
  2. R. J. Grasso and E. A. Stappaerts, “Linear phase conjugation for atmospheric aberration compensation,” Proc. SPIE 3219, 124–132 (1998).
    [Crossref]
  3. A. T. Watnik and P. S. Lebow, “Dynamic holography for extended object beam shaping,” Proc. SPIE 8843, 88430E (2013).
    [Crossref]
  4. T. Takenaka, K. Tanaka, and O. Fukumitsu, “Signal-to-noise ratio in optical heterodyne-detection for Gaussian fields,” Appl. Opt. 17, 3466–3471 (1978).
    [Crossref]
  5. K. Si, R. Fiolka, and M. Cui, “Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy,” Sci. Rep. 2, 748 (2012).
    [Crossref]
  6. M. O’Donnell and S. W. Flax, “Adaptive coherent energy beam formation using iterative phase conjugation,” U.S. patent4,989,143 (29January1991).
  7. H. C. Song, W. A. Kuperman, W. S. Hodgkiss, T. Akal, and C. Ferla, “Iterative time reversal in the ocean,” J. Acoust. Soc. Am. 105, 3176–3184 (1999).
    [Crossref]
  8. A. T. Watnik and P. S. Lebow, “Adaptive digital holography for gain-enhanced imaging,” in Imaging and Applied Optics, J. Christou and D. Miller, eds. (Optical Society of America, 2013), paper ITu3E.5.
  9. R. W. Gerchberg and W. O. Saxton, “Practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

2013 (1)

A. T. Watnik and P. S. Lebow, “Dynamic holography for extended object beam shaping,” Proc. SPIE 8843, 88430E (2013).
[Crossref]

2012 (1)

K. Si, R. Fiolka, and M. Cui, “Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy,” Sci. Rep. 2, 748 (2012).
[Crossref]

2007 (1)

1999 (1)

H. C. Song, W. A. Kuperman, W. S. Hodgkiss, T. Akal, and C. Ferla, “Iterative time reversal in the ocean,” J. Acoust. Soc. Am. 105, 3176–3184 (1999).
[Crossref]

1998 (1)

R. J. Grasso and E. A. Stappaerts, “Linear phase conjugation for atmospheric aberration compensation,” Proc. SPIE 3219, 124–132 (1998).
[Crossref]

1978 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “Practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Akal, T.

H. C. Song, W. A. Kuperman, W. S. Hodgkiss, T. Akal, and C. Ferla, “Iterative time reversal in the ocean,” J. Acoust. Soc. Am. 105, 3176–3184 (1999).
[Crossref]

Cui, M.

K. Si, R. Fiolka, and M. Cui, “Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy,” Sci. Rep. 2, 748 (2012).
[Crossref]

Ferla, C.

H. C. Song, W. A. Kuperman, W. S. Hodgkiss, T. Akal, and C. Ferla, “Iterative time reversal in the ocean,” J. Acoust. Soc. Am. 105, 3176–3184 (1999).
[Crossref]

Fiolka, R.

K. Si, R. Fiolka, and M. Cui, “Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy,” Sci. Rep. 2, 748 (2012).
[Crossref]

Flax, S. W.

M. O’Donnell and S. W. Flax, “Adaptive coherent energy beam formation using iterative phase conjugation,” U.S. patent4,989,143 (29January1991).

Fukumitsu, O.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “Practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Grasso, R. J.

R. J. Grasso and E. A. Stappaerts, “Linear phase conjugation for atmospheric aberration compensation,” Proc. SPIE 3219, 124–132 (1998).
[Crossref]

Hodgkiss, W. S.

H. C. Song, W. A. Kuperman, W. S. Hodgkiss, T. Akal, and C. Ferla, “Iterative time reversal in the ocean,” J. Acoust. Soc. Am. 105, 3176–3184 (1999).
[Crossref]

Kohnle, A.

Kolosov, V. V.

Kuperman, W. A.

H. C. Song, W. A. Kuperman, W. S. Hodgkiss, T. Akal, and C. Ferla, “Iterative time reversal in the ocean,” J. Acoust. Soc. Am. 105, 3176–3184 (1999).
[Crossref]

Lebow, P. S.

A. T. Watnik and P. S. Lebow, “Dynamic holography for extended object beam shaping,” Proc. SPIE 8843, 88430E (2013).
[Crossref]

A. T. Watnik and P. S. Lebow, “Adaptive digital holography for gain-enhanced imaging,” in Imaging and Applied Optics, J. Christou and D. Miller, eds. (Optical Society of America, 2013), paper ITu3E.5.

O’Donnell, M.

M. O’Donnell and S. W. Flax, “Adaptive coherent energy beam formation using iterative phase conjugation,” U.S. patent4,989,143 (29January1991).

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “Practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Si, K.

K. Si, R. Fiolka, and M. Cui, “Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy,” Sci. Rep. 2, 748 (2012).
[Crossref]

Song, H. C.

H. C. Song, W. A. Kuperman, W. S. Hodgkiss, T. Akal, and C. Ferla, “Iterative time reversal in the ocean,” J. Acoust. Soc. Am. 105, 3176–3184 (1999).
[Crossref]

Stappaerts, E. A.

R. J. Grasso and E. A. Stappaerts, “Linear phase conjugation for atmospheric aberration compensation,” Proc. SPIE 3219, 124–132 (1998).
[Crossref]

Takenaka, T.

Tanaka, K.

Vorontsov, M. A.

Watnik, A. T.

A. T. Watnik and P. S. Lebow, “Dynamic holography for extended object beam shaping,” Proc. SPIE 8843, 88430E (2013).
[Crossref]

A. T. Watnik and P. S. Lebow, “Adaptive digital holography for gain-enhanced imaging,” in Imaging and Applied Optics, J. Christou and D. Miller, eds. (Optical Society of America, 2013), paper ITu3E.5.

Appl. Opt. (1)

J. Acoust. Soc. Am. (1)

H. C. Song, W. A. Kuperman, W. S. Hodgkiss, T. Akal, and C. Ferla, “Iterative time reversal in the ocean,” J. Acoust. Soc. Am. 105, 3176–3184 (1999).
[Crossref]

J. Opt. Soc. Am. A (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “Practical algorithm for determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. SPIE (2)

R. J. Grasso and E. A. Stappaerts, “Linear phase conjugation for atmospheric aberration compensation,” Proc. SPIE 3219, 124–132 (1998).
[Crossref]

A. T. Watnik and P. S. Lebow, “Dynamic holography for extended object beam shaping,” Proc. SPIE 8843, 88430E (2013).
[Crossref]

Sci. Rep. (1)

K. Si, R. Fiolka, and M. Cui, “Breaking the spatial resolution barrier via iterative sound-light interaction in deep tissue microscopy,” Sci. Rep. 2, 748 (2012).
[Crossref]

Other (2)

M. O’Donnell and S. W. Flax, “Adaptive coherent energy beam formation using iterative phase conjugation,” U.S. patent4,989,143 (29January1991).

A. T. Watnik and P. S. Lebow, “Adaptive digital holography for gain-enhanced imaging,” in Imaging and Applied Optics, J. Christou and D. Miller, eds. (Optical Society of America, 2013), paper ITu3E.5.

Supplementary Material (2)

» Media 1: AVI (416 KB)     
» Media 2: AVI (417 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

Schematic to illustrate both the physical holographic recording and computational processing steps in an iterative, holographic conjugator. Numbers correspond with equations in text. SLM, spatial light modulator; B/S, beamsplitter; FFT, fast Fourier transform; IFFT, inverse fast Fourier transform.

Fig. 2.
Fig. 2.

EOT versus iteration for different object beam illumination energies of 400, 800, 1200, and 2400e. Blue (+) = simulated, Red (o) = theoretical.

Fig. 3.
Fig. 3.

Simulation of image plane in an iterative holographic bootstrapping experiment with a single resolution element target and total object beam illumination energy of 400e. (a) Iteration i=1, (b) i=3, (c) i=5, (d) i=8, (e) i=12, and (f) i=16. 40×40 pixel area is shown out of entire 1024×1024 array, displayed as amplitude raised to the 0.25 power (Media 1).

Fig. 4.
Fig. 4.

Simulation of image plane in an iterative holographic bootstrapping experiment with a single resolution element target and total object beam illumination energy of 1200e. (a) Iteration i=1, (b) i=3, (c) i=5, (d) i=8, (e) i=12, and (f) i=16. 40×40 pixel area is shown out of entire 1024×1024 array, displayed as amplitude raised to the 0.25 power (Media 2).

Equations (34)

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H=|R+F|2+|R+F|2N01,1+σrN01,2,
H=HGA/D+112N01,3,
H=|R+F|2GA/D+(|R+F|2+σr2(GA/D)2+112)N01,4.
H=|R+F|2GA/D+|R|GA/DN01,4.
H=1GA/D(|R|2+|F|2+FR*+F*R)+1GA/D|R|N01,4.
gc=AtotAW,
Hdf=1GA/D(F*R+12gc|R|Nc01,1),
Hdf,ideal=F*RGA/D
Ndf=|R|Nc01,1GA/D2gc,
|Ndf|2=(RoGA/D2gc)2|Nc01,1|2=(PRoGA/Dgc)2,
|Hdf|2|Hdf,ideal|2+|Ndf|2.
Etot=|Fout,i|2=|Fo|2=P2Fo.
Fout,i=FoHdf|Hdf|2.
EOTi=|M(x,y)fi(x,y)|2|fi(x,y)|2,
EOTi=|Hdf,ideal,i|2|Hdf,i|2
EOTi=|Hdf,ideal,i|2|Hdf,ideal,i|2+|Ndf,i|2.
|Fi+1|2=EOTi×|Fout,i|2=EOTi×|Fo|2.
|Fi+1|2=|Fi+1R*R|2=|Fi+1R*|2|Ro|2.
|Fi+1|2=(GA/DRo)2|Hdf,ideal,i+1|2.
|Fi+1|2=(GA/DRo)2|Hdf,ideal,i+1|2=EOTi×|Fo|2.
|Hdf,ideal,i+1|2=(RoGA/D)2|Fo|2×EOTi=α×EOTi.
EOTi+1=|Hdf,ideal,i+1|2|Hdf,ideal,i+1|2+|Ndf,i+1|2EOTi+1=α×EOTiα×EOTi+|Ndf,i+1|2.
EOTi=EOTo(α|Ndf,i|2)[(1EOTo)α|Ndf,i|2](|Ndf,i|2α)i+EOTo×α.
EOTo=Total area of targetTotal illumination area|Fo|2.
EOTc=EOTi=EOTi+1.
EOTc=α×EOTcα×EOTc+|Ndf,c|2,
EOTc=α|Ndf,c|2α=1|Ndf,c|2α.
|Ndf,i|2α=[(PRo)/(GA/Dgc)]2(Ro/GA/D)2|Fo|2=P2gc1|Fo|2.
EOTc=EOTo(α|Ndf,c|2)EOTo×α=α|Ndf,c|2α,
EOTc=1(PRoGA/Dgc)2(RoGA/D)2|Fo|2=1P2gc|Fo|2.
|Fi+1|2=EOTi×|rtFout,i|2=EOTi×|rtFo|2,
|Fi+1|2=(GA/DRo)2|Hdf,ideal,i+1|2=EOTi×|rtFo|2.
αr=(rtGA/DRo)2×|Fo|2.
EOTc=1(PRoGA/Dgc)21αr=1P2gc×rt2|Fo|2.

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