Abstract

A standard multimode optical fiber can be used as a general purpose spectrometer after calibrating the wavelength dependent speckle patterns produced by interference between the guided modes of the fiber. A transmission matrix was used to store the calibration data and a robust algorithm was developed to reconstruct an arbitrary input spectrum in the presence of experimental noise. We demonstrate that a 20 meter long fiber can resolve two laser lines separated by only 8 pm. At the other extreme, we show that a 2 centimeter long fiber can measure a broadband continuous spectrum generated from a supercontinuum source. We investigate the effect of the fiber geometry on the spectral resolution and bandwidth, and also discuss the additional limitation on the bandwidth imposed by speckle contrast reduction when measuring dense spectra. Finally, we demonstrate a method to reduce the spectrum reconstruction error and increase the bandwidth by separately imaging the speckle patterns of orthogonal polarizations. The multimode fiber spectrometer is compact, lightweight, low cost, and provides high resolution with low loss.

© 2013 OSA

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  19. F. T. S. Yu, J. Zhang, K. Pan, D. Zhao, and P. B. Ruffin, “Fiber vibration sensor that uses the speckle contrast ratio,” Opt. Eng.34(1), 1–236 (1995).
    [CrossRef]
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2012 (3)

2010 (2)

2009 (2)

O. A. Oraby, J. W. Spencer, and G. R. Jones, “Monitoring changes in the speckle field from an optical fibre exposed to low frequency acoustical vibrations,” J. Mod. Opt.56(1), 55–84 (2009).
[CrossRef]

W. Ha, S. Lee, Y. Jung, J. K. Kim, and K. Oh, “Acousto-optic control of speckle contrast in multimode fibers with a cylindrical piezoelectric transducer oscillating in the radial direction,” Opt. Express17(20), 17536–17546 (2009).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(5), 056223 (2002).
[CrossRef] [PubMed]

1997 (1)

1995 (1)

F. T. S. Yu, J. Zhang, K. Pan, D. Zhao, and P. B. Ruffin, “Fiber vibration sensor that uses the speckle contrast ratio,” Opt. Eng.34(1), 1–236 (1995).
[CrossRef]

1994 (2)

P. Hlubina, “Spectral and dispersion analysis of laser sources and multimode fibres via the statistics of the intensity pattern,” J. Mod. Opt.41(5), 1001–1014 (1994).
[CrossRef]

K. Pan, C. M. Uang, F. Cheng, and F. T. Yu, “Multimode fiber sensing by using mean-absolute speckle-intensity variation,” Appl. Opt.33(10), 2095–2098 (1994).
[CrossRef] [PubMed]

1991 (1)

1988 (1)

1986 (1)

W. Freude, G. Fritzsche, G. Grau, and L. Shan-da, “Speckle interferometry for spectral analysis of laser sources and multimode optical waveguides,” J. Lightwave Technol.4(1), 64–72 (1986).
[CrossRef]

1984 (1)

P. F. Steeger, T. Asakura, and A. F. Fercher, “Polarization preservation in circular multimode optical fibers and its measurement by a speckle method,” J. Lightwave Technol.2(4), 435–441 (1984).
[CrossRef]

1980 (2)

M. Imai and Y. Ohtsuka, “Speckle-pattern contrast of semiconductor laser propagating in a multimode optical fiber,” Opt. Commun.33(1), 4–8 (1980).
[CrossRef]

E. G. Rawson, J. W. Goodman, and R. E. Norton, “Frequency dependence of modal noise in multimode optical fibers,” J. Opt. Soc. Am.70(8), 968–976 (1980).
[CrossRef]

1976 (1)

Adibi, A.

Asakura, T.

P. F. Steeger, T. Asakura, and A. F. Fercher, “Polarization preservation in circular multimode optical fibers and its measurement by a speckle method,” J. Lightwave Technol.2(4), 435–441 (1984).
[CrossRef]

Brady, D. J.

Cao, H.

Cheng, F.

Choi, H. S.

Crosignani, B.

Diano, B.

Dogariu, A.

Doya, V.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(5), 056223 (2002).
[CrossRef] [PubMed]

Farahi, S.

Fercher, A. F.

P. F. Steeger, T. Asakura, and A. F. Fercher, “Polarization preservation in circular multimode optical fibers and its measurement by a speckle method,” J. Lightwave Technol.2(4), 435–441 (1984).
[CrossRef]

Foulger, S. H.

Freude, W.

W. Freude, G. Fritzsche, G. Grau, and L. Shan-da, “Speckle interferometry for spectral analysis of laser sources and multimode optical waveguides,” J. Lightwave Technol.4(1), 64–72 (1986).
[CrossRef]

Fritzsche, G.

W. Freude, G. Fritzsche, G. Grau, and L. Shan-da, “Speckle interferometry for spectral analysis of laser sources and multimode optical waveguides,” J. Lightwave Technol.4(1), 64–72 (1986).
[CrossRef]

Fujiwara, E.

E. Fujiwara, Y. T. Wu, and C. K. Suzuki, “Vibration-based specklegram fiber sensor for measurement of properties of liquids,” Opt. Lasers Eng.50(12), 1726–1730 (2012).
[CrossRef]

Goodman, J. W.

Grau, G.

W. Freude, G. Fritzsche, G. Grau, and L. Shan-da, “Speckle interferometry for spectral analysis of laser sources and multimode optical waveguides,” J. Lightwave Technol.4(1), 64–72 (1986).
[CrossRef]

Guo, N.

Ha, W.

Hang, Q.

Hlubina, P.

P. Hlubina, “Spectral and dispersion analysis of laser sources and multimode fibres via the statistics of the intensity pattern,” J. Mod. Opt.41(5), 1001–1014 (1994).
[CrossRef]

Imai, M.

M. Imai and Y. Ohtsuka, “Speckle-pattern contrast of semiconductor laser propagating in a multimode optical fiber,” Opt. Commun.33(1), 4–8 (1980).
[CrossRef]

Jones, G. R.

O. A. Oraby, J. W. Spencer, and G. R. Jones, “Monitoring changes in the speckle field from an optical fibre exposed to low frequency acoustical vibrations,” J. Mod. Opt.56(1), 55–84 (2009).
[CrossRef]

Jung, Y.

Kim, J. K.

Kohlgraf-Owens, T. W.

Lee, C. E.

Lee, S.

Legrand, O.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(5), 056223 (2002).
[CrossRef] [PubMed]

Miniatura, C.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(5), 056223 (2002).
[CrossRef] [PubMed]

Mortessagne, F.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(5), 056223 (2002).
[CrossRef] [PubMed]

Moser, C.

Norton, R. E.

Oh, K.

Ohtsuka, Y.

M. Imai and Y. Ohtsuka, “Speckle-pattern contrast of semiconductor laser propagating in a multimode optical fiber,” Opt. Commun.33(1), 4–8 (1980).
[CrossRef]

Okamoto, T.

Oraby, O. A.

O. A. Oraby, J. W. Spencer, and G. R. Jones, “Monitoring changes in the speckle field from an optical fibre exposed to low frequency acoustical vibrations,” J. Mod. Opt.56(1), 55–84 (2009).
[CrossRef]

Pan, K.

F. T. S. Yu, J. Zhang, K. Pan, D. Zhao, and P. B. Ruffin, “Fiber vibration sensor that uses the speckle contrast ratio,” Opt. Eng.34(1), 1–236 (1995).
[CrossRef]

K. Pan, C. M. Uang, F. Cheng, and F. T. Yu, “Multimode fiber sensing by using mean-absolute speckle-intensity variation,” Appl. Opt.33(10), 2095–2098 (1994).
[CrossRef] [PubMed]

Papadopoulos, I. N.

Porto, P. D.

Psaltis, D.

Rawson, E. G.

Redding, B.

Ruffin, P. B.

F. T. S. Yu, J. Zhang, K. Pan, D. Zhao, and P. B. Ruffin, “Fiber vibration sensor that uses the speckle contrast ratio,” Opt. Eng.34(1), 1–236 (1995).
[CrossRef]

Shan-da, L.

W. Freude, G. Fritzsche, G. Grau, and L. Shan-da, “Speckle interferometry for spectral analysis of laser sources and multimode optical waveguides,” J. Lightwave Technol.4(1), 64–72 (1986).
[CrossRef]

Skorobogatiy, M.

Spencer, J. W.

O. A. Oraby, J. W. Spencer, and G. R. Jones, “Monitoring changes in the speckle field from an optical fibre exposed to low frequency acoustical vibrations,” J. Mod. Opt.56(1), 55–84 (2009).
[CrossRef]

Steeger, P. F.

P. F. Steeger, T. Asakura, and A. F. Fercher, “Polarization preservation in circular multimode optical fibers and its measurement by a speckle method,” J. Lightwave Technol.2(4), 435–441 (1984).
[CrossRef]

Sullivan, M. E.

Suzuki, C. K.

E. Fujiwara, Y. T. Wu, and C. K. Suzuki, “Vibration-based specklegram fiber sensor for measurement of properties of liquids,” Opt. Lasers Eng.50(12), 1726–1730 (2012).
[CrossRef]

Syed, I.

Taylor, H. F.

Uang, C. M.

Ung, B.

Wang, Z.

Wu, S.

Wu, Y. T.

E. Fujiwara, Y. T. Wu, and C. K. Suzuki, “Vibration-based specklegram fiber sensor for measurement of properties of liquids,” Opt. Lasers Eng.50(12), 1726–1730 (2012).
[CrossRef]

Xu, Z.

Yamaguchi, I.

Yin, S.

Yu, F. T.

Yu, F. T. S.

F. T. S. Yu, J. Zhang, K. Pan, D. Zhao, and P. B. Ruffin, “Fiber vibration sensor that uses the speckle contrast ratio,” Opt. Eng.34(1), 1–236 (1995).
[CrossRef]

S. Wu, S. Yin, and F. T. S. Yu, “Sensing with fiber specklegrams,” Appl. Opt.30(31), 4468–4470 (1991).
[CrossRef] [PubMed]

Zhang, J.

F. T. S. Yu, J. Zhang, K. Pan, D. Zhao, and P. B. Ruffin, “Fiber vibration sensor that uses the speckle contrast ratio,” Opt. Eng.34(1), 1–236 (1995).
[CrossRef]

Zhao, D.

F. T. S. Yu, J. Zhang, K. Pan, D. Zhao, and P. B. Ruffin, “Fiber vibration sensor that uses the speckle contrast ratio,” Opt. Eng.34(1), 1–236 (1995).
[CrossRef]

Appl. Opt. (4)

J. Lightwave Technol. (2)

W. Freude, G. Fritzsche, G. Grau, and L. Shan-da, “Speckle interferometry for spectral analysis of laser sources and multimode optical waveguides,” J. Lightwave Technol.4(1), 64–72 (1986).
[CrossRef]

P. F. Steeger, T. Asakura, and A. F. Fercher, “Polarization preservation in circular multimode optical fibers and its measurement by a speckle method,” J. Lightwave Technol.2(4), 435–441 (1984).
[CrossRef]

J. Mod. Opt. (2)

O. A. Oraby, J. W. Spencer, and G. R. Jones, “Monitoring changes in the speckle field from an optical fibre exposed to low frequency acoustical vibrations,” J. Mod. Opt.56(1), 55–84 (2009).
[CrossRef]

P. Hlubina, “Spectral and dispersion analysis of laser sources and multimode fibres via the statistics of the intensity pattern,” J. Mod. Opt.41(5), 1001–1014 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

M. Imai and Y. Ohtsuka, “Speckle-pattern contrast of semiconductor laser propagating in a multimode optical fiber,” Opt. Commun.33(1), 4–8 (1980).
[CrossRef]

Opt. Eng. (1)

F. T. S. Yu, J. Zhang, K. Pan, D. Zhao, and P. B. Ruffin, “Fiber vibration sensor that uses the speckle contrast ratio,” Opt. Eng.34(1), 1–236 (1995).
[CrossRef]

Opt. Express (3)

Opt. Lasers Eng. (1)

E. Fujiwara, Y. T. Wu, and C. K. Suzuki, “Vibration-based specklegram fiber sensor for measurement of properties of liquids,” Opt. Lasers Eng.50(12), 1726–1730 (2012).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(5), 056223 (2002).
[CrossRef] [PubMed]

Other (2)

J. W. Goodman, Speckle Phenomena in Optics (Ben Roberts & Co., 2007).

K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2006).

Supplementary Material (1)

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Figures (9)

Fig. 1
Fig. 1

(a) A schematic of the fiber spectrometer setup. A near-IR laser, wavelength tunable from 1435 nm to 1590 nm, is used for calibration and testing. Emission from the laser is coupled via a single-mode polarization-maintaining fiber (SMF) to the multimode fiber (MMF), with a standard FC/PC mating sleeve. A 50 × objective lens is used to image the speckle pattern generated at the end facet of the fiber to the monochrome CCD camera. (b) (Media 1) Movie showing the speckle pattern generated at the end of a 20 m multimode fiber as the input wavelength varies from 1500 nm to 1501 nm in the step of 0.01 nm. The wavelength is written on the top and also marked by the red line in the bottom scale. The speckle pattern decorrelates rapidly with wavelength, illustrating high spatial-spectral diversity. (c) Spectral correlation function of speckle intensity obtained from (b) exhibits a correlation width δλ = 0.01 nm, which enables fine spectral resolution.

Fig. 2
Fig. 2

(a) Calculated spectral correlation width δλ as a function of the length L of a planar waveguide with a fixed width W = 1 mm and NA = 0.22. The wavelength of the input light is 1500 nm. δλ scales linearly with 1/L (blue line), indicating that a longer waveguide provides finer spectral resolution. (b) Calculated spectral correlation width δλ as a function of the width W of a planar waveguide with a fixed length L = 1 m and NA = 0.22. The wavelength of the input light is 1500 nm. δλ decreases quickly at small W and then saturates at large W. (c) Calculated spectral correlation width δλ as a function of the numerical aperture NA of a planar waveguide with L = 1 m and W = 1 mm. The wavelength of the input light is 1500 nm. Red dotted line is a linear fit in the log-log plot of the calculated δλ (blue dots) vs. NA, it has a slope of −2.09, indicating δλ scales as 1/NA2. (d) Calculated spectral correlation function of the intensity at the end of a planar waveguide (L = 1 m, W = 1 mm, NA = 0.22), when a subset of the waveguide modes were excited. The waveguide supports ~300 modes at λ = 1500 nm. (i) All the modes were excited (black solid line), (ii) only the 100 lowest order modes were excited (blue dash-dotted line), (iii) only the 100 highest order modes were excited (green dashed line), (iv) every fifth mode was excited (red dotted line). The spectral correlation width decreased in the cases of (ii) and (iii), but remained virtually unchanged in (iv), indicating the speckle decorrelation is determined primarily by the total range of β values of the modes that are excited by the input signal.

Fig. 3
Fig. 3

(a) Initial attempt to reconstruct a probe spectrum, consisting of three narrow lines centered at 1484, 1500, and 1508 nm and with relative amplitude of 0.4, 1, and 0.6 (indicated by the red dotted line), simply by inversion of the transmission matrix: S = T−1 I fails, because this inversion process is ill-conditioned in the presence of experimental noise. (b) The same probe spectrum reconstructed using the truncated inversion technique described in the text. The truncation threshold was set to 2 × 10−3, and the spectrum reconstruction error μ = 0.028. The truncated inversion technique was able to recover the input spectrum, although background noise is still evident. (c) The spectrum reconstruction error, μ, of the reconstructed spectrum as a function of the truncation threshold. The minimal μ gives the optimal threshold value. (d) A further improved reconstruction was achieved using a nonlinear optimization procedure. The spectrum obtained from the truncated inversion technique was used as a starting guess to reduce the computation time, and the spectrum reconstruction error μ = 0.004.

Fig. 4
Fig. 4

Reconstructed spectrum (blue solid line with circular dots marking the calibrated wavelengths λi) of two spectral lines separated by 8 pm. It is obtained with a 20 m long fiber spectrometer. The red dotted lines mark the center wavelengths of the input lines.

Fig. 5
Fig. 5

The data points represent the spectrum reconstruction error, μ, for a Lorentzian spectrum of FWHM = 1 nm using different transmission matrices, which were constructed for a 1 m long fiber with spectral correlation width δλ = 0.4 nm. (a) With a fixed spectral channel spacing dλ = 0.2 nm and N = 500 spatial channels, we compare the spectrum reconstruction error using the transmission matrices with varying bandwidth, Δλ. By reducing the bandwidth, the accuracy of the spectrum reconstruction is improved. (b) With a fixed 100 nm bandwidth and N = 500 spatial channels, we varied the spectral channel spacing, . By increasing the channel spacing, the error of spectrum reconstruction reduced. (c) With a fixed bandwidth of 100 nm and spectral channel spacing of 0.2 nm, we varied the number of spatial channels extracted from the speckle pattern by reducing their spacing dr. Increasing the number of spatial samples, even though they were correlated (dr < δr), improved the accuracy of spectrum reconstruction.

Fig. 6
Fig. 6

(a) Reconstructed spectra (blue line) for the probe signals (red line) with Lorentzian spectra of varying width (FHWM) ΔλL. The spectrum reconstruction error μ increases with the bandwidth of the probe spectrum. (b) Speckle images corresponding to the three spectra in (a). They are synthesized by summation of sequentially measured speckle patterns where the Lorentzian signals in (a) were used to weight the speckle patterns. The speckle contrast decreases with the bandwidth of the probe signal (ΔλL), leading to less accurate spectral reconstructions.

Fig. 7
Fig. 7

(a) Schematic of the setup used to separately record the speckle patterns of orthogonal linear polarizations. Two polarizing beam splitters (PBS) and two mirrors are used to separate the two polarization components of the speckle pattern and project both of them to the camera. Right inset: an image of two polarized speckle patterns taken at λ = 1500 nm. The contrast, C, is 0.86, lower than 1 due to the finite pixel size of the CCD camera. Left inset: an unpolarized speckle image taken at the same λ with the two polarized beamsplitters removed. The contrast is 0.6, about a factor of 2 lower than the polarized speckle, because the speckle patterns of the two polarizations add in intensity. (b) Original (red line) and reconstructed (blue line) flat spectra with a Lorentzian dip. The flat spectrum extended from 1450 nm to 1550 nm, the Lorentizn dip is centered at 1500 nm with a FWHM of 10 nm. The bottom blue curve was obtained with the polarized detection scheme, and the top with unpolarized. The noise in the reconstructed spectra, characterized by the reconstruction error μ, is more than a factor of 2 lower in the polarization-resolved measurement. (c) Error to reconstruct the flat spectrum with a Lorentzian dip of varying width. The broader the Lorentzian dip corresponded to less “dense” spectra, and thus the reconstructed spectra exhibited lower μ. In all cases, the polarized detection scheme (green circles connected by dotted line) provided superior reconstruction than the unpolarized detection (blue squares connected by solid line).

Fig. 8
Fig. 8

(a) Spectral correlation function for a 2 cm long multimode fiber. The correlation width δλ is 4 nm. (b) The 2 cm long multimode fiber was used to measure the spectrum of a supercontinuum source passed through an interference filter. The filter was tilted to provide three probe spectra, centered at 1510, 1505, and 1500 nm. The probe spectra were measured separately using an optical spectrum analyzer (solid lines) and the corresponding spectra reconstructed using the multimode fiber are shown in the dotted lines.

Fig. 9
Fig. 9

(a) Calculated temperature correlation function for waveguides with W = 1 mm, NA = 0.22, and L = 0.1, 1, or 10 m. (b) The calculated temperature correlation width scales inversely with the waveguide length. For L = 1 m, the temperature correlation width ~8°C.

Equations (3)

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E(r,θ,λ,L)= m A m ψ m ( r,θ,λ )exp[ i( β m ( λ )Lωt+ φ m ) ]
I( r )= S( λ ) T( r,λ )dλ.
I=TS,

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