Abstract

We experimentally demonstrate endoscopic imaging through a multi-mode fiber (MMF) in which the number of resolvable image features approaches four times the number of spatial modes per polarization propagating in the fiber. In our method, a sequence of random field patterns is input to the fiber, generating a sequence of random intensity patterns at the output, which are used to sample an object. Reflected power values are returned through the fiber and linear optimization is used to reconstruct an image. The factor-of-four resolution enhancement is due to mixing of modes by the squaring inherent in field-to-intensity conversion. The incoherent point-spread function (PSF) at the center of the fiber output plane is an Airy disk equivalent to the coherent PSF of a conventional diffraction-limited imaging system having a numerical aperture twice that of the fiber. All previous methods for imaging through MMF can only resolve a number of features equal to the number of modes. Most of these methods use localized intensity patterns for sampling the object and use local image reconstruction.

© 2013 OSA

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  1. A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett.28(2), 88–89 (1976).
    [CrossRef]
  2. B. Fischer and S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett.46(2), 113–114 (1985).
    [CrossRef]
  3. K. K. Tsia, K. Goda, and B. Jalali, “Simultaneous mechanical-scan-free confocal microscopy and laser microsurgery,” Opt. Lett.34, 2099–2101 (2009).
    [CrossRef] [PubMed]
  4. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express20(10), 10583–10590 (2012).
    [CrossRef] [PubMed]
  5. S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip12(3), 635–639 (2012).
    [CrossRef] [PubMed]
  6. T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fiber-based imaging,” Nat. Commun.3, 1027 (2012).
    [CrossRef]
  7. L. Yang, A. Raighne, E. M. McCabe, L. A. Dunbar, and T. Scharf, “Confocal microscopy using variable-focal-length microlenses and an optical fiber bundle,” Appl. Opt.44(28), 5928–5936 (2005).
    [CrossRef] [PubMed]
  8. 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]
  9. M. T. Myaing, D. J. MacDonald, and X. Li, “Fiber-optic scanning two-photon fluorescence endoscope,” Opt. Lett.31(8), 1076–1078 (2006).
    [CrossRef] [PubMed]
  10. L. Fu, X. Gan, and M. Gu, “Nonlinear optical microscopy based on double-clad photonic crystal fibers,” Opt. Express13(14), 5528–5534 (2005).
    [CrossRef] [PubMed]
  11. D. Bird and M. Gu, “Two-photon fluorescence endoscopy with a micro-optic scanning head,” Opt. Lett.28(17), 1552–1554 (2003).
    [CrossRef] [PubMed]
  12. 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. Express17(4), 2375–2384 (2009).
    [CrossRef] [PubMed]
  13. Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
    [CrossRef] [PubMed]
  14. B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods2(12), 941–950 (2005).
    [CrossRef] [PubMed]
  15. R. N. Mahalati, D. Askarov, J. P. Wilde, and J. M. Kahn, “Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling,” Opt. Express20(13), 14321–14337 (2012).
    [CrossRef] [PubMed]
  16. I. M. Vellekoop and A. P. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun.281(11), 3071–3080 (2008).
    [CrossRef]
  17. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett.32(16), 2309–2311 (2007).
    [CrossRef] [PubMed]
  18. R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express19(1), 247–254 (2011).
    [CrossRef] [PubMed]
  19. 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]
  20. T. Čižmár and K. Dholakia, “Shaping the light transmission through a multimode optical fibre: complex transformation analysis and applications in biophotonics,” Opt. Express19(20), 18871–18884 (2011).
    [CrossRef] [PubMed]
  21. I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett.101(12), 120601 (2008).
    [CrossRef] [PubMed]
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Science/Engineering/Math, 1996).
  23. J. A. Buck, Fundamentals of Optical Fibers, 2nd Ed. (John Wiley & Sons, Inc., New Jersey, 2004).
  24. S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, New York, 2004).
  25. D. Marcuse, “Excitation of parabolic-index fibers with incoherent sources,” Bell Syst. Tech. J.54, 1507–1530 (1975).
  26. D. Gloge, “Weakly guiding fibers,” Appl. Opt.10(10), 2252–2258 (1971).
    [CrossRef] [PubMed]
  27. G. Szegö, Orthogonal Polynomials (American Mathematical Society, Rhode Island, 1939).
  28. P. Ferraro, A. Wax, and Z. Zalevsky, Coherent Light Microscopy (Springer, New York, 2011).
  29. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).
  30. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, Upper Saddle River, New Jersey, 1999).
  31. T. Lauer, “Deconvolution with a spatially-variant PSF,” in Proceedings of SPIE Conference on Astronomical Data Analysis II (International Society for Optics and Photonics, Waikoloa, Hawaii, 2002) 167–173.
  32. X. Shen, J. M. Kahn, and M. A. Horowitz, “Compensation for multimode fiber dispersion by adaptive optics,” Opt. Lett.30(22), 2985–2987 (2005).
    [CrossRef] [PubMed]

2012

I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express20(10), 10583–10590 (2012).
[CrossRef] [PubMed]

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip12(3), 635–639 (2012).
[CrossRef] [PubMed]

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fiber-based imaging,” Nat. Commun.3, 1027 (2012).
[CrossRef]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

R. N. Mahalati, D. Askarov, J. P. Wilde, and J. M. Kahn, “Adaptive control of input field to achieve desired output intensity profile in multimode fiber with random mode coupling,” Opt. Express20(13), 14321–14337 (2012).
[CrossRef] [PubMed]

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]

2009

2008

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]

2007

2006

2005

2003

2000

1985

B. Fischer and S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett.46(2), 113–114 (1985).
[CrossRef]

1976

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett.28(2), 88–89 (1976).
[CrossRef]

1975

D. Marcuse, “Excitation of parabolic-index fibers with incoherent sources,” Bell Syst. Tech. J.54, 1507–1530 (1975).

1971

Askarov, D.

Bianchi, S.

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip12(3), 635–639 (2012).
[CrossRef] [PubMed]

R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express19(1), 247–254 (2011).
[CrossRef] [PubMed]

Bird, D.

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]

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]

Cheung, E. L. M.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods2(12), 941–950 (2005).
[CrossRef] [PubMed]

Choi, W.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Choi, Y.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Chow, T. H.

Cižmár, T.

Cocker, E. D.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods2(12), 941–950 (2005).
[CrossRef] [PubMed]

Dasari, R. R.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Dholakia, K.

Di Leonardo, R.

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip12(3), 635–639 (2012).
[CrossRef] [PubMed]

R. Di Leonardo and S. Bianchi, “Hologram transmission through multi-mode optical fibers,” Opt. Express19(1), 247–254 (2011).
[CrossRef] [PubMed]

Dlugan, A. L. P.

Dunbar, L. A.

Fang-Yen, C.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Farahi, S.

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]

Fischer, B.

B. Fischer and S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett.46(2), 113–114 (1985).
[CrossRef]

Flusberg, B. A.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods2(12), 941–950 (2005).
[CrossRef] [PubMed]

Fu, L.

Gan, X.

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]

Gloge, D.

Goda, K.

Gu, M.

Herrington, C. S.

Horowitz, M. A.

Jalali, B.

Jung, J. C.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods2(12), 941–950 (2005).
[CrossRef] [PubMed]

Kahn, J. M.

Kim, M.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Lane, P. M.

Lee, K. J.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

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]

Li, X.

Macaulay, C. E.

MacDonald, D. J.

Mahalati, R. N.

Marcuse, D.

D. Marcuse, “Excitation of parabolic-index fibers with incoherent sources,” Bell Syst. Tech. J.54, 1507–1530 (1975).

Mazilu, M.

McCabe, E. M.

Moser, C.

Mosk, A. P.

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]

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

Myaing, M. T.

Ng, B. K.

Papadopoulos, I. N.

Piyawattanametha, W.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods2(12), 941–950 (2005).
[CrossRef] [PubMed]

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]

Psaltis, D.

Raighne, A.

Richards-Kortum, R.

Scharf, T.

Schnitzer, M. J.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods2(12), 941–950 (2005).
[CrossRef] [PubMed]

Shen, X.

Sibbett, W.

Sternklar, S.

B. Fischer and S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett.46(2), 113–114 (1985).
[CrossRef]

Taguichi, K.

Tan, K. M.

Tsia, K. K.

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]

Wilde, J. P.

Yang, L.

Yang, T. D.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Yariv, A.

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett.28(2), 88–89 (1976).
[CrossRef]

Yoon, C.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett.28(2), 88–89 (1976).
[CrossRef]

B. Fischer and S. Sternklar, “Image transmission and interferometry with multimode fibers using self-pumped phase conjugation,” Appl. Phys. Lett.46(2), 113–114 (1985).
[CrossRef]

Bell Syst. Tech. J.

D. Marcuse, “Excitation of parabolic-index fibers with incoherent sources,” Bell Syst. Tech. J.54, 1507–1530 (1975).

Lab Chip

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip12(3), 635–639 (2012).
[CrossRef] [PubMed]

Nat. Commun.

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fiber-based imaging,” Nat. Commun.3, 1027 (2012).
[CrossRef]

Nat. Methods

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods2(12), 941–950 (2005).
[CrossRef] [PubMed]

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]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett.109(20), 203901 (2012).
[CrossRef] [PubMed]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Science/Engineering/Math, 1996).

J. A. Buck, Fundamentals of Optical Fibers, 2nd Ed. (John Wiley & Sons, Inc., New Jersey, 2004).

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

G. Szegö, Orthogonal Polynomials (American Mathematical Society, Rhode Island, 1939).

P. Ferraro, A. Wax, and Z. Zalevsky, Coherent Light Microscopy (Springer, New York, 2011).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice Hall, Upper Saddle River, New Jersey, 1999).

T. Lauer, “Deconvolution with a spatially-variant PSF,” in Proceedings of SPIE Conference on Astronomical Data Analysis II (International Society for Optics and Photonics, Waikoloa, Hawaii, 2002) 167–173.

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Figures (6)

Fig. 1
Fig. 1

Experimental setup for imaging through a multi-mode fiber (MMF). (a) In calibration, a 65 × objective lens and camera are placed at the MMF output to record intensity patterns generated there. (b) In imaging, an object is placed at the MMF output and illuminated by intensity patterns, and the reflected power coupled back into the MMF is recorded. The setup can use localized spots or random patterns at the MMF output for sampling the object.

Fig. 2
Fig. 2

Spot (a) and random pattern (b) generated at the MMF output by the setup shown in Fig. 1 and used for sampling an object. The white circle denotes the area in which spots or random patterns can be generated. A gamma adjustment of 1.5 is used in displaying the images.

Fig. 3
Fig. 3

Point-spread functions (PSFs) at center of fiber output plane for two imaging methods. Dashed and solid blue curves (squares) show theoretically optimal and experimental PSFs for local sampling with local reconstruction. Dashed and solid red curves (circles) show theoretically optimal and experimental PSFs for random sampling and optimization-based reconstruction.

Fig. 4
Fig. 4

Images formed by random-pattern sampling and optimization-based reconstruction: (a) numeral 2, (b) bars of 9.8-µm pitch, (c) bars of 6.2-µm pitch, (d) bars of 4.4-µm pitch. Using the spot-scanning method with local reconstruction, (a) and (b) are easily resolved, (c) is barely resolved, while (d) is not resolved. Features are from groups 6 and 7 of 1951 USAF chrome-on-glass resolution target. Portions of other features can be seen in images (c) and (d). The white circle denotes the area in which spots and random patterns can be generated. A gamma adjustment of 1.5 is used in displaying the images.

Fig. 5
Fig. 5

Singular values (SVs) for (a) electric field patterns and (b) intensity patterns at the output of a 45-mode graded-index MMF. Red circles and blue squares show SVs of random patterns and spots, respectively, simulated using the exact LP modes of finite-core MMF. Green diamonds denote SVs of random patterns measured experimentally. 500 patterns are used in each matrix.

Fig. 6
Fig. 6

Peak-to-zero widths of PSFs for optimization-based reconstruction at different radial positions in the fiber output plane. Dashed and solid blue lines (squares) show transverse and longitudinal widths for the graded-index fiber. Dashed and solid red lines (circles) show transverse and longitudinal widths for the step-index fiber. PSFs were obtained from simulation of a noiseless random-pattern imaging system with optimization-based reconstruction using exact LP modes of fibers having NA = 0.19 and core diameter d = 50 μm. The dashed black line corresponds to idealized PSF given by Eq. (7).

Equations (18)

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

p i κ I out,i (x,y) R obj (x,y)dxdy ,
W(x,y)= i=1 M p i s i (x,y) ,
I A (ηr)= I 0 ( 2 J 1 (ηr) ηr ) 2 ,
p i κ ˜ k=1 L I out,i ( x k , y k ) R obj ( x k , y k ) ,
w ^ = argmin w p I ˜ w 2 ,
w ^ =V D 1 U T p,
E A (2ηr)= E 0 2 J 1 (2ηr) 2ηr ,
E lm (r,φ)= c lm w 0 ( 2 r w 0 ) | l | e r 2 w 0 2 L m | l | ( 2 r 2 w 0 2 ) e ilφ ,
E out (r,φ)= 02m+| l | n max a lm E lm (r,φ) = e r 2 w 0 2 02m+| l | n max a ˜ lm ( 2 r w 0 ) | l | ( 2 r 2 w 0 2 ) m e ilφ ,
I out (r,φ)= e 2 r 2 w 0 2 02m+| l |2 n max ( 2 r w 0 ) | l | ( 2 r 2 w 0 2 ) m ( b lm e ilφ + b lm e ilφ ) = 02m+| l |2 n max b ˜ lm E ˜ lm (r,φ) ,
w ^ =V D 1 U T I ˜ r,
w ^ =V V T r.
E δ (r,φ)= 02m+| l |2 n max E ˜ lm (0,0) E ˜ lm (r,φ) .
E δ (r)= m=0 n max E ˜ 0m (0,0) E ˜ 0m (r,φ) = 2 π w 0 2 e 2 r 2 w 0 2 m=0 n max L m 0 ( 4 r 2 w 0 2 ) .
L n α (r) n α/2 = e r/2 r α/2 J α ( 2 nr ).
2 J 1 (2ηr) 2ηr = e η 2 r 2 /2n n L n 1 ( η 2 r 2 n ).
2 J 1 (2ηr) 2ηr = e η 2 r 2 /2n n m=0 n L m 0 ( η 2 r 2 n ) .
E δ (r)= 2 π w 0 2 e 2 r 2 w 0 2 m=0 n max L m 0 ( 4 r 2 w 0 2 ) = E 0 2 J 1 (2ηr) 2ηr = E A (2ηr).

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