## All-fiber spectrometer based on speckle pattern reconstruction |

Optics Express, Vol. 21, Issue 5, pp. 6584-6600 (2013)

http://dx.doi.org/10.1364/OE.21.006584

Acrobat PDF (1328 KB)

### 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

## 1. Introduction

1. Z. Xu, Z. Wang, M. E. Sullivan, D. J. Brady, S. H. Foulger, and A. Adibi, “Multimodal multiplex spectroscopy using photonic crystals,” Opt. Express **11**(18), 2126–2133 (2003). [CrossRef] [PubMed]

3. Q. Hang, B. Ung, I. Syed, N. Guo, and M. Skorobogatiy, “Photonic bandgap fiber bundle spectrometer,” Appl. Opt. **49**(25), 4791–4800 (2010). [CrossRef] [PubMed]

1. Z. Xu, Z. Wang, M. E. Sullivan, D. J. Brady, S. H. Foulger, and A. Adibi, “Multimodal multiplex spectroscopy using photonic crystals,” Opt. Express **11**(18), 2126–2133 (2003). [CrossRef] [PubMed]

2. T. W. Kohlgraf-Owens and A. Dogariu, “Transmission matrices of random media: means for spectral polarimetric measurements,” Opt. Lett. **35**(13), 2236–2238 (2010). [CrossRef] [PubMed]

3. Q. Hang, B. Ung, I. Syed, N. Guo, and M. Skorobogatiy, “Photonic bandgap fiber bundle spectrometer,” Appl. Opt. **49**(25), 4791–4800 (2010). [CrossRef] [PubMed]

4. B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. **37**(16), 3384–3386 (2012). [CrossRef] [PubMed]

5. B. Crosignani, B. Diano, and P. D. Porto, “Speckle-pattern visibility of light transmitted through a multimode optical fiber,” J. Opt. Soc. Am. **66**(11), 1312–1313 (1976). [CrossRef]

8. 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]

9. 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]

4. B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. **37**(16), 3384–3386 (2012). [CrossRef] [PubMed]

4. B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. **37**(16), 3384–3386 (2012). [CrossRef] [PubMed]

7. 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]

9. 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]

**37**(16), 3384–3386 (2012). [CrossRef] [PubMed]

## 2. Operation principle of fiber spectrometer

## 3. Effects of fiber geometry on spectral correlation of speckle

*L,*the core diameter

*W*, and the numerical aperture NA are crucial parameters determining the resolution and bandwidth of the fiber-based spectrometer. If we consider a monochromatic input light which excites all the guided modes, then the electric field at the end of a waveguide of length

*L*is:where

*A*and

_{m}*φ*are the amplitude and initial phase of the

_{m}*m*guided mode which has the spatial profile

^{th}**ψ**and propagation constant

_{m}*β*. To simplify the analysis, we considered planar waveguides of width

_{m}*W*and numerical aperture NA, and calculated the mode profile and propagation constant using the method outlined in Ref [10]. We assumed that initially all of the modes in the waveguide were excited equally (

*A*= 1 for all modes) with uncorrleated phases (

_{m}*φ*are random numbers between 0 and 2π). We calculated the spatial distribution of electric field intensity at the end of a waveguide, as a function of the input wavelength, and then computed the spectral correlation function of intensity

_{m}*C*(Δ

*λ*). By repeating this process as we varied

*L, W*, or NA, we were able to obtain the dependence of the spectral correlation width

*δλ*on these parameters. Figure 2(a) plots

*δλ*vs.

*L*for

*W*= 1 mm and NA = 0.22.

*δλ*scales inversely with

*L*, indicating that the spectral resolution can be changed simply by varying the waveguide length. Next we fix

*L*= 1 m, and plot

*δλ*as a function of

*W*in Fig. 2(b) for NA = 0.22. The spectral correlation width initially drops, but quickly saturates. In Fig. 2(c), we varied NA while keeping

*L*= 1 m and

*W*= 1 mm, and observed a reduction of

*δλ*with NA.

*β*) and accumulate different phase decays (

_{m}*β*). The maximum difference occurs between the fundamental mode (

_{m}L*m*= 1) and the highest-order mode (

*m*=

*M*), which we denote

*φ*(

*λ*) =

*β*(

_{1}*λ*)

*L−β*(

_{M}*λ*)

*L*. In order for an input wavelength shift

*δλ*to produce a distinct intensity distribution at the output,

*φ*(

*λ*) should change by approximately π, namely, |

*dφ*(

*λ*)/

*dλ*|

*δλ*~π. By approximating

*β*(

_{1}*λ*) ≈

*k*and

*β*(

_{M}*λ*) ≈

*k*cos(NA) for large

*W*(

*k*= 2π

*n*/

*λ*,

*n*is the refractive index of the waveguide,

*λ*is the vacuum wavelength), we get

*δλ*~(

*λ/n*)

^{2}/(2

*n L*)/[1 - cos(NA)]. For small NA, 1 - cos(NA) ≈(NA)

^{2}/2, and

*δλ ~λ*[

^{2}/*n L*(NA)

^{2}]. This simple expression captures the basic trends in Fig. 2(a-c). In Fig. 2(a), the spectral correlation width scales linearly with

*1/L*. When

*W*is large, the spectral decorrelation does not depend on

*W*[Fig. 2(b)]. Figure 2(c) is a log-log plot showing the calculated

*δλ*fit well with a straight line of slope −2.09, thus confirming the 1/(NA)

^{2}scaling.

*β*-

_{1}*β*. This assumption led to the strong dependence of

_{M}*δλ*on NA, which dictates the highest order mode the waveguide supports. The waveguide width

*W,*which predominantly determines the

*β*spacing of adjacent modes, had little effect on

*δλ*. To validate this assumption, we considered a waveguide with fixed geometry (

*L*= 1 m,

*W =*1 mm, and NA = 0.22) and calculated the spectral correlation function when exciting a subset of the ~300 guided modes (i.e.

*A*= 0 for some

_{m}*m*’s). Figure 2(d) plots

*C(Δλ)*in four cases: (i) all the modes are excited; (ii) only the 100 lowest order modes are excited; (iii) only the 100 highest order modes are excited; and (iv) every 5th mode is excited. The spectral correlation width decreased in (ii) and (iii), but remained virtually unchanged in (iv). These results confirm that the speckle decorrelation depends primarily on the difference between the maximal and minimal

*β*values. Increasing

*W*adds modes in between

*β*and

_{1}*β*, but does not significantly modify the values of

_{M}*β*and

_{1}*β*, thus having little impact on the spectral resolution. However, as will be discussed later, increasing W does increase the spectral bandwidth of the fiber spectrometer.

_{M}11. 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]

## 4. Algorithm for spectral reconstruction

*δλ*= 0.4 nm. We calibrated a transmission matrix from λ = 1450 nm to 1550 nm in steps of 0.2 nm providing

*M*= 500 spectral channels. From the spatial correlation function, we found that the speckle image contained ~600 spatial channels separated by

*δr*, and we sampled 500 of these,

*N*= 500. In section 5, we will discuss these choices in more detail. The transmission matrix was calibrated one column at a time by recording the speckle pattern generated at each sampled wavelength

*λ*in

_{i}*S*. The entire calibration process, consisting of recording speckle images at 500 input wavelengths, was completed in a few minutes. After calibration, we tested the spectrometer operation by measuring the speckle pattern produced by a probe signal and attempted to reconstruct the probe spectrum. An initial test was to recover a spectrum consisting of three narrow lines with unequal spacing and varying height. Since optical signals at different wavelengths do not interfere, we were able to synthesize the probe speckle pattern by adding weighted speckle patterns measured sequentially at the three probe wavelengths.

*S**=*

*T*

^{−1}**. In Fig. 3(a), we plot the original spectrum with a red dotted line and the reconstructed spectrum with a blue solid line. This simple approach failed because the inversion of matrix**

*I**T*is ill-conditioned in the presence of experimental noise. There are several sources of noise in our spectrometer: (i) discrete intensity resolution of a CCD array, (ii) ambient light, (iii) mechanical instability of the fiber and fluctuation of experimental environment, (iv) intensity fluctuation and wavelength drift of the laser source. Among them we believe (iii) has the dominant contribution.

*T = U D V**where*

^{T},**is a**

*U**N × Ν*unitary matrix,

**is a**

*D**N × Μ*diagonal matrix with positive real elements

*D*, known as the singular values of

_{jj}= d_{j}**,**

*T***is a**

*V**M × Μ*unitary matrix. The rows of

**(resp. columns of**

*V***) are the input (resp. output) singular vectors and are noted**

*U*

*V**(resp.*

_{j}

*U**). To find the inverse of*

_{j}**, we take the reciprocal of each diagonal element of**

*T***and then transpose it to obtain a diagonal matrix**

*D***. The inverse of**

*D’**T*is given as:

*T*

^{−1}=

*VD’U**. The issue we confront in the presence of noise is that the small elements of*

^{T}**are the elements most corrupted by experimental noise, and they are effectively amplified in**

*D***when we take their reciprocal. To overcome this issue, we adopted a “truncated inversion” technique, similar to the approach in Ref [3**

*D’*3. Q. Hang, B. Ung, I. Syed, N. Guo, and M. Skorobogatiy, “Photonic bandgap fiber bundle spectrometer,” Appl. Opt. **49**(25), 4791–4800 (2010). [CrossRef] [PubMed]

**in which we only take the reciprocal of the elements of**

*D’***above a threshold value and set the remaining elements to zero. The truncated inverse of**

*D***is**

*T*

*T*

_{trunc}^{−1}= V

*D’*

_{trunc}

*U**, which we use to reconstruct the input spectrum*

^{T}

*S**=*

*T*

_{trunc}^{−1}**. Using this truncated inversion technique, we obtained a far superior reconstruction of the input spectrum, as shown in Fig. 3(b). Clearly the truncated inversion technique was able to recover the input spectrum. To provide a quantitative measure of the quality of spectral reconstruction, we calculated the spectrum reconstruction error, defined as the standard deviation between the probe spectrum and the reconstructed spectrum:**

*I***37**(16), 3384–3386 (2012). [CrossRef] [PubMed]

*μ*. The threshold of truncation was defined as a fraction of the largest element in

**. For example, at a threshold of 0.01, any element in**

*D***with amplitude less than 1% of the maximal element in**

*D***would be discarded in the inversion process. In Fig. 3(c), we show the spectrum reconstruction error as a function of the truncation threshold. We found that the optimal threshold value was ~2 × 10**

*D*^{−3}[this threshold value was used to reconstruct the spectrum shown in Fig. 3(b)]. After optimizing the truncation threshold, the singular value decomposition is only performed once for a given fiber, providing a

*T**matrix which can then be used to recover any input spectrum with a single matrix multiplication.*

_{trunc}^{−1}**, is the one that minimizes the energy**

*S**E*= ||

**−**

*I***||**

*T S*^{2}[1

1. Z. Xu, Z. Wang, M. E. Sullivan, D. J. Brady, S. H. Foulger, and A. Adibi, “Multimodal multiplex spectroscopy using photonic crystals,” Opt. Express **11**(18), 2126–2133 (2003). [CrossRef] [PubMed]

**, we developed a simulated annealing algorithm. At each step in the optimization routine, we changed one element in**

*S***at a time by multiplying it by a random number between 0.5 and 2. This provided us with a new spectrum,**

*S***. We then calculated the change in energy Δ**

*S*’*E*= ||

**−**

*I***||**

*T S*’^{2}− ||

**−**

*I***||**

*T S*^{2}and kept the change (i.e.

**=**

*S***) with a probability equal to exp[−Δ**

*S*’*E*/

*T*], where

_{0}*T*is the “temperature”. After performing this process once for every element in

_{0}**, the temperature was reduced and the process was repeated. At high temperature the algorithm is more likely to accept “bad” choices (which increase the energy) in order to search broadly for a global minimum. Later in the algorithm, at lower temperatures, the algorithm is less likely to accept a “bad” choice. Note that changes which reduce the energy (Δ**

*S**E*< 0

*)*are always accepted. The algorithm stopped after a fixed number of steps or if the energy dropped below a threshold value. In Fig. 3(d), we show the reconstructed spectrum obtained using the simulated annealing algorithm. The background noise is largely suppressed and the spectrum reconstruction error is reduced. The simulated annealing algorithm typically required a few hundred iterations to reach the optimal solution; however, the truncated inversion technique provides a good initial guess of

**, which dramatically reduced the simulation time.**

*S**λ*used in calibration. The reconstructed spectrum (blue solid line) clearly identifies the two peaks even though they are separated by merely 8 pm. Even higher spectral resolution is possible by using a longer fiber; however we would need a tunable laser with finer spectral resolution in order to calibrate such a fiber. This level of spectral resolution is competitive with the top commercially available grating based spectrometers.

_{i}## 5. Spatial and spectral sampling

*B,*in the fiber if all modes are more or less equally excited.

*B*is determined by the fiber core diameter,

*W*, the NA, and the wavelength of light

*λ*,

*W*= 105 μm, NA = 0.22) support ~1000 modes at an operating wavelength of 1500 nm. In principle, increasing

*W*or NA will increase the number of available channels for transmitting spectral information, thus increasing the bandwidth of the fiber spectrometer. In reality, however, the experimental noise may reduce the number of spectral channels that can be recovered simultaneously due to degradation of the spectral reconstruction. In this section, we investigate how the spectrum reconstruction error is affected by spectral and spatial sampling rates.

*δλ*= 0.4 nm and a spatial correlation width

*δr*= 4.1 μm. Our tunable laser covers the wavelength range from 1450 nm to 1550 nm, proving a maximum operating bandwidth of 100 nm. The speckle image had a radius of ~52.5 μm, providing ~600 spatial channels separated by

*δr*. This number is less than the total number of guided modes in the fiber, because we did not excite all the modes [13

13. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express **20**(10), 10583–10590 (2012). [CrossRef] [PubMed]

## 6. Fiber-spectrometer bandwidth

*Δλ*=

*B δλ*, where

*B*is the total number of guided modes that the fiber supports, and

*δλ*is the spectral correlation width of the speckle pattern. For a given fiber core diameter and numerical aperture,

*B*is fixed, while

*δλ*varies with the fiber length. A longer fiber gives better spectral resolution (smaller

*δλ*), but narrower bandwidth. Such a tradeoff between the resolution and the bandwidth is similar to that of a traditional spectrometer. What is different from a traditional spectrometer is that the spectral coverage in a single measurement does not need to be continuous. If it is known

*a priori*that the probe signal does not have components within certain spectral regions, such regions can be excluded from the spectrum reconstruction, thus the finite number of spectral channels (

*B*) may cover an even broader spectral range.

*N*= 500 spatial channels and

*M*= 500 spectral channels. The spacing of spectral channels was 0.2 nm, and the bandwidth was 100 nm (from 1450 nm to 1550 nm). In Fig. 6, we examine the ability of the fiber spectrometer to reconstruct Lorentzian shaped spectra with varying width. These probes were synthesized as described above using a weighted sum of sequentially measured speckle patterns. In Fig. 6(a), the probe spectra are shown in red and the reconstructed spectra in blue. While the fiber spectrometer very accurately reconstructs the narrow spectrum, the reconstruction degrades for broader spectra and the reconstruction error increases. The difficulty that the fiber spectrometer has in dealing with broad spectra lies in the reduction of speckle contrast. This can be seen clearly in Fig. 6(b), which shows the speckle patterns corresponding to the three spectra in Fig. 6(a). These speckle patterns are synthesized by the summation of the measured speckle patterns at different wavelengths scaled by the Lorentzian functions with different bandwidth (

*Δλ*). Quantitatively, the speckle contrast is calculated as

_{L}*σ*is the standard deviation of pixel intensity from the mean value

*C~*0.1 for the bandwidth of

*Δλ*~50 nm. The number of independent speckle patterns that constitute the speckle image of the Lorentzian signal is approximately

_{L}*Δλ*, and

_{L}/δλ*C*scales as [

*Δλ*]

_{L}/δλ^{-1/2}.

*C ~*[

*Δλ/δλ*]

^{-1/2}>

*σ*where

_{exp}*Δλ*is the operation bandwidth and

*σ*is the experimental error. We estimated

_{exp}*σ*from the standard deviation between two subsequently measured transmission matrices for a 1 m fiber, and obtained

_{exp}*σ*~5 × 10

_{exp}^{−3}. In our implementation, this limits the number of independent spectral channels to a few hundred. Of course, if the spectrometer is used to measure sparse spectra, consisting of only a few narrow spectral lines, this limitation does not apply and much larger bandwidth is possible.

## 7. Polarization-resolved detection

*N*= 2000 spatial channels. In the case of polarized detection, the number of spatial channels was doubled to take advantage of the second image. Hence the polarization-resolved detection scheme provides two advantages: the speckle contrast is increased by

*μ*. We repeated this experiment for similar probe spectra with varying Lorentzian widths and plot the spectrum reconstruction error

*μ*in Fig. 7(c). As the width of the Lorentzian dip decreased the input spectrum became denser, yielding lower speckle contrast, and

*μ*increased. However, in all cases the polarization-resolved detection gave smaller reconstruction error.

## 8. Measurement of broadband continuous spectra

## 9. Discussion and summary

15. T. Okamoto and I. Yamaguchi, “Multimode fiber-optic Mach-Zehnder interferometer and its use in temperature measurement,” Appl. Opt. **27**(15), 3085–3087 (1988). [CrossRef] [PubMed]

16. 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]

17. 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]

18. 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]

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]

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

*n*, as d

*n/n*d

*T*= 7 × 10

^{−6}°C

^{−1}and the fiber length,

*L*, as d

*L/L*d

*T =*5 × 10

^{−7}°C

^{−1}[22

22. H. S. Choi, H. F. Taylor, and C. E. Lee, “High-performance fiber-optic temperature sensor using low-coherence interferometry,” Opt. Lett. **22**(23), 1814–1816 (1997). [CrossRef] [PubMed]

*W*= 1 mm and NA = 0.22 as a function of temperature (i.e. changes in

*n*and

*L*in our model). We considered waveguides with length

*L*= 0.1, 1, or 10 m. We fixed the input wavelength at 1500 nm and calculated the intensity distributions at the output of the waveguide at different temperatures. By correlating the speckle patterns at different temperatures, we obtained the temperature correlation functions, shown in Fig. 9(a). As expected, the correlation function falls off more rapidly for longer fibers. Figure 9(b) presents the temperature correlation widths, defined as twice of the change in temperature required to produce a 50% decorrelation of the speckle pattern. We see that the temperature correlation width scales inversely with the fiber length, confirming that longer fibers are more susceptible to temperature induced changes in the speckle pattern. For a 1 m long fiber, the temperature would need to change by ~8°C to decorrelate the speckle pattern. The effect of the temperature change on the fiber spectrometer performance depends on the spectra it reconstructs, and a detailed study will be presented in the future. One approach to mitigate the temperature sensitivity would be to calibrate a bank of transmission matrices at different temperatures and then use the transmission matrix corresponding to the ambient temperature during a given measurement.

## Acknowledgments

## References and links

1. | Z. Xu, Z. Wang, M. E. Sullivan, D. J. Brady, S. H. Foulger, and A. Adibi, “Multimodal multiplex spectroscopy using photonic crystals,” Opt. Express |

2. | T. W. Kohlgraf-Owens and A. Dogariu, “Transmission matrices of random media: means for spectral polarimetric measurements,” Opt. Lett. |

3. | Q. Hang, B. Ung, I. Syed, N. Guo, and M. Skorobogatiy, “Photonic bandgap fiber bundle spectrometer,” Appl. Opt. |

4. | B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett. |

5. | B. Crosignani, B. Diano, and P. D. Porto, “Speckle-pattern visibility of light transmitted through a multimode optical fiber,” J. Opt. Soc. Am. |

6. | M. Imai and Y. Ohtsuka, “Speckle-pattern contrast of semiconductor laser propagating in a multimode optical fiber,” Opt. Commun. |

7. | E. G. Rawson, J. W. Goodman, and R. E. Norton, “Frequency dependence of modal noise in multimode optical fibers,” J. Opt. Soc. Am. |

8. | P. Hlubina, “Spectral and dispersion analysis of laser sources and multimode fibres via the statistics of the intensity pattern,” J. Mod. Opt. |

9. | 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. |

10. | K. Okamoto, |

11. | V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Speckle statistics in a chaotic multimode fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

12. | J. W. Goodman, |

13. | I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express |

14. | 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. |

15. | T. Okamoto and I. Yamaguchi, “Multimode fiber-optic Mach-Zehnder interferometer and its use in temperature measurement,” Appl. Opt. |

16. | K. Pan, C. M. Uang, F. Cheng, and F. T. Yu, “Multimode fiber sensing by using mean-absolute speckle-intensity variation,” Appl. Opt. |

17. | 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. |

18. | E. Fujiwara, Y. T. Wu, and C. K. Suzuki, “Vibration-based specklegram fiber sensor for measurement of properties of liquids,” Opt. Lasers Eng. |

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. |

20. | 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. Express |

21. | S. Wu, S. Yin, and F. T. S. Yu, “Sensing with fiber specklegrams,” Appl. Opt. |

22. | H. S. Choi, H. F. Taylor, and C. E. Lee, “High-performance fiber-optic temperature sensor using low-coherence interferometry,” Opt. Lett. |

**OCIS Codes**

(060.2370) Fiber optics and optical communications : Fiber optics sensors

(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

(300.6190) Spectroscopy : Spectrometers

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: December 20, 2012

Revised Manuscript: February 28, 2013

Manuscript Accepted: March 1, 2013

Published: March 8, 2013

**Virtual Issues**

Vol. 8, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Brandon Redding, Sebastien M. Popoff, and Hui Cao, "All-fiber spectrometer based on speckle pattern reconstruction," Opt. Express **21**, 6584-6600 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-6584

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### References

- Z. Xu, Z. Wang, M. E. Sullivan, D. J. Brady, S. H. Foulger, and A. Adibi, “Multimodal multiplex spectroscopy using photonic crystals,” Opt. Express11(18), 2126–2133 (2003). [CrossRef] [PubMed]
- T. W. Kohlgraf-Owens and A. Dogariu, “Transmission matrices of random media: means for spectral polarimetric measurements,” Opt. Lett.35(13), 2236–2238 (2010). [CrossRef] [PubMed]
- Q. Hang, B. Ung, I. Syed, N. Guo, and M. Skorobogatiy, “Photonic bandgap fiber bundle spectrometer,” Appl. Opt.49(25), 4791–4800 (2010). [CrossRef] [PubMed]
- B. Redding and H. Cao, “Using a multimode fiber as a high-resolution, low-loss spectrometer,” Opt. Lett.37(16), 3384–3386 (2012). [CrossRef] [PubMed]
- B. Crosignani, B. Diano, and P. D. Porto, “Speckle-pattern visibility of light transmitted through a multimode optical fiber,” J. Opt. Soc. Am.66(11), 1312–1313 (1976). [CrossRef]
- 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]
- 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]
- 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]
- K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2006).
- 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]
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