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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 6584–6600
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All-fiber spectrometer based on speckle pattern reconstruction

Brandon Redding, Sebastien M. Popoff, and Hui Cao  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 6584-6600 (2013)
http://dx.doi.org/10.1364/OE.21.006584


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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

While traditional spectrometers are based on one-to-one spectral-to-spatial mapping, spectrometers can also operate on more complex spectral-to-spatial mapping [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]

3

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 these implementations, a transmission matrix is used to store the spatial intensity profile generated by different input wavelengths. A reconstruction algorithm allows an arbitrary input spectrum to be recovered from the measured spatial intensity distribution. While this approach is more complicated than the traditional grating or prism based spectrometers, it affords more flexibility in the choice of dispersive element. For instance, spectrometers based on this approach have been demonstrated using a disordered photonic crystal [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]

], a random scattering medium [2

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]

], and an array of Bragg fibers [3

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]

]. We recently found that a multimode optical fiber is an ideal dispersive element for this type of spectrometer because the long propagation length and the minimal loss enables high spectral resolution and good sensitivity [4

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]

].

In a multimode fiber spectrometer, the interference between the guided modes creates a wavelength-dependent speckle pattern, providing the required spectral-to-spatial mapping. In the past, the contrast of this speckle pattern was found to depend on the spectral width and shape of the optical source [5

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

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]

], allowing researchers to use contrast as a measure of the laser linewidth [9

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]

]. As opposed to using only the statistical property of the speckle such as the contrast, we recently proposed and demonstrated that by recording the entire speckle patterns at different wavelengths, a multimode fiber can be used as a general purpose spectrometer [4

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]

]. The spectral resolution of the device depends on the spectral correlation width of the speckle, which is known to scale inversely with the length of the fiber [4

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

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

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]

]. The advantage of using an optical fiber is that a long propagation length is easily achieved with minimal loss, giving high spectral resolution. Furthermore, the fiber-based spectrometer requires only a multimode fiber and a monochrome CCD camera to record the speckle patterns. Compared to traditional spectrometers, optical fibers are lower cost, lighter weight, and can be coiled into a small volume while providing spectral resolution that is competitive with state-of-the-art grating-based spectrometers.

In this paper, we extend on the proof-of-concept demonstration presented in Ref [4

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]

]. and explore the operational limits of a multimode fiber spectrometer. We provide a theoretical analysis of the effects of the fiber geometry on the spectrometer performance, and then present a reconstruction algorithm combining a truncated inversion technique with a least squares minimization procedure, which enables accurate and robust spectral reconstruction in the presence of experimental noise. We also investigate the effects of spectral and spatial oversampling on the quality of the recovered spectra. Using a 20 meter long fiber, we are able to resolve two laser lines separated by merely 8 pm. A higher spectral resolution is expected for a longer fiber, but we are currently limited by the resolution of the tunable laser source used for calibration. We also discuss the bandwidth limitation when measuring a dense spectra due to speckle contrast reduction. To reduce the reconstruction error and increase the spectral bandwidth, we develop a method based on a polarization-resolved speckle measurement. Finally, we use a 2 centimeter long fiber to measure a continuous broadband spectrum generated by a supercontinuum source.

2. Operation principle of fiber spectrometer

The fiber-based spectrometer consists of a multimode fiber and a monochrome CCD camera that images the speckle pattern at the end of the fiber. The speckle pattern, created by interference among the guided modes in the fiber, is distinct for light at different wavelength, thus providing a fingerprint of the input wavelength. In our experiments, we used commercially available step-index multimode fibers with 105 μm diameter cores (NA = 0.22) and lengths varying from 2 cm to 20 m. A schematic of the experimental configuration is shown in Fig. 1(a)
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.
. A near-IR tunable diode laser (HP 8168F) was used to provide a spectrally controlled input signal for the calibration and initial characterization. A polarization maintaining single-mode fiber was used to couple the laser emission into the multimode fiber through a standard FC/PC mating sleeve. The speckle pattern generated at the exit face of the multimode fiber was imaged onto an InGaAs camera (Xenics Xeva 1.7-320) with a 50 × near-IR microscope objective (NA = 0.55). The integration time of the camera was set to 100 μs. Alternatively, the speckle in the far-field zone may be projected directly onto the camera without the objective.

Figure 1(b) is a movie showing the speckle patterns recorded at the end of a 20 m fiber as the input wavelength is scanned from 1500 nm to 1501 nm in the step of 0.01 nm. The speckle patterns decorrelate for very small changes in wavelength. Such high spatial-spectral diversity gives fine spectral resolution. We calculated the spectral correlation function of the speckle intensity, C(Δλ,x)=I(λ,x)I(λ+Δλ,x)/[I(λ,x)I(λ+Δλ,x)]1,where I(λ, x) is the intensity at a position x for input wavelength λ, represents the average over λ. In Fig. 1(c), we plot the spectral correlation function averaged over many spatial positions across the 105 μm core of a 20 m fiber. The spectral correlation width, δλ is defined as C(δλ/2) = C(0)/2. When the input wavelength shifts by δλ, the output speckle pattern becomes nearly uncorrelated, C(Δλ = δλ) ≈0. In this case, δλ = 10 pm.

To use the fiber as a spectrometer, speckle patterns such as the ones shown in Fig. 1(b) are recorded to construct the transmission matrix, as will be discussed in detail in section 4. After this calibration step, the tunable laser can be replaced by any optical source and the camera will record the speckle pattern. A reconstruction algorithm, to be discussed in section 4, is then applied to recover the spectra of the input. Note that in the experiments presented in this work, a single-mode polarization-maintaining fiber is always used to couple the signal to the multimode fiber. This ensures that the input to the multimode fiber will have the same spatial profile and polarization as the calibration. If the probe signal had a different profile or polarization, it could excite a different combination of fiber modes with different (relative) amplitudes and phases, making the calibration invalid.

3. Effects of fiber geometry on spectral correlation of speckle

The above analysis assumed that the spectral correlation width was determined by the total range of the propagation constants in the waveguide, β1 - βM. This assumption led to the strong dependence of 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. Am = 0 for some 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 β1 and βM, but does not significantly modify the values of β1 and βM, 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.

Finally, we discuss the experimental implication of the above analysis. Figure 2(d) reveals that for a given fiber the fastest speckle decorrelation is reached only when we excite the guided modes that cover the full range of the propagation constants. Experimentally, the measured spectral correlation width of a given fiber changed by as much as ~50% depending on the coupling of the input, which could be varied by adjusting the lateral alignment of the single-mode fiber to the multimode fiber. In the future, chaotic multimode fibers might be used to avoid the alignment issue and ensure all of the modes are excited more or less equally [11

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]

]. The non-circular cross-section of the core, e.g. the D-shaped cross-section, leads to chaotic dynamics of light rays in the fiber. Consequently the majority of the guided modes are spread uniformly over the entire core, and they will be excited no matter where light enters the fiber at the input facet.

4. Algorithm for spectral reconstruction

As an example, we consider the transmission matrix corresponding to a 1 m long fiber with spectral correlation width 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 λi in 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.

In our first attempt to reconstruct the input spectrum, we simply multiplied the measured speckle pattern by the inverse of the transmission matrix: S = T−1 I. In Fig. 3(a)
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.
, 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 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.

To understand why even a small amount of experimental noise can corrupt the reconstruction, we employ the singular value decomposition to factorize the transmission matrix, T = U D VT, where U is a N × Ν unitary matrix, D is a N × Μ diagonal matrix with positive real elements Djj = dj, known as the singular values of T, V is a M × Μ unitary matrix. The rows of V (resp. columns of U) are the input (resp. output) singular vectors and are noted Vj (resp. Uj). To find the inverse of T, we take the reciprocal of each diagonal element of D and then transpose it to obtain a diagonal matrix D’. The inverse of T is given as: T−1 = VD’UT. The issue we confront in the presence of noise is that the small elements of D 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

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 this approach, we compute a truncated version of D’ 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 T is Ttrunc−1 = V D’trunc UT, which we use to reconstruct the input spectrum S = Ttrunc−1 I. 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:μ=1Mλ[Sprobe(λ)Sreconstructed(λ)]2/[1MλSprobe(λ)]. Note that in Ref [4

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]

], we used the signal-to-noise ratio (SNR) as a metric and showed that narrow lines were reconstructed with SNR greater than 1000. In this work, we chose to use the standard deviation because it allowed us to evaluate the ability of the spectrometer to reconstruct complex spectra. In order to optimize the truncated inversion, we reconstructed the input spectra using different threshold values for truncation and calculated the reconstruction error μ. The threshold of truncation was defined as a fraction of the largest element in D. 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 × 10−3 [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 Ttrunc−1 matrix which can then be used to recover any input spectrum with a single matrix multiplication.

In practice, these two reconstruction algorithms could be used in tandem. The truncated inversion technique provides a decent near-instantaneous reconstruction, which operates in real-time, since it requires a single matrix multiplication. The simulated annealing algorithm could be used to obtain a more accurate spectrum in applications where fast temporal response is less critical, or after the measurement is concluded. Of course, in both cases, the reconstruction will be improved by reducing the noise of the experimental measurements. In the current setup, the main source of noise is expected to be the stability of the multimode fiber, which was secured to an optical table during testing. Improved methods to rigidly stabilize the multimode fiber are expected to reduce the noise and provide more accurate spectral reconstructions.

Using the reconstruction algorithm that combines truncated inversion and simulated annealing, we characterized the spectral resolution of the fiber spectrometer by testing its ability to discriminate between two closely spaced spectral lines. In order to synthesize the probe speckle pattern, we separately recorded speckle patterns at the two probe wavelengths and then added them in intensity. In 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.
, we show the reconstructed spectrum measured using a 20 m long fiber. The spectral positions of the probe lines (red dotted line) are in between the sampled wavelengths λi 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.

5. Spatial and spectral sampling

In contrast to the conventional spectrometers which map spectral information to one spatial dimension, our fiber spectrometer maps to two-dimensions (2D). This 2D spatial-spectral mapping fully utilizes the large detection area of modern 2D cameras to achieve large bandwidth of operation. The maximal number of independent spectral channels that can be measured in parallel is limited by the number of independent spatial channels that encode the spectral information. The number of independent spatial channels is equal to the number of speckles in the intensity distribution at the output end of the fiber, and it is also equal to the total number of guided modes, 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, Bπ2(NA)2W2/2λ2 [12

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

]. The fibers considered in this work (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.

We first investigated the effect of the operating bandwidth on the ability of the spectrometer to reconstruct the Lorentzian spectrum. In this case, we fixed the spectral channel spacing at δλ/2 = 0.2 nm and the spatial channel spacing at δr = 4.1 μm, and selected N = 500 spatial channels. We then constructed transmission matrices using 125, 250, or 500 spectral channels, corresponding to an operating bandwidth (Δλ = M δλ/2) of 25, 50 or 100 nm. In Fig. 5(a)
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.
, we present the spectrum reconstruction error μ for the three transmission matrices. We found that the error decreased with bandwidth. This is expected since less bandwidth (with fixed channel spacing) essentially means there are fewer spectral channels to recover. In other words, the number of variables (M) to solve for during the reconstruction is less than the number of equations given by the number of measured spatial channels (N). The redundant equations reduce the uncertainty in determining the intensity in each spectral channel. The larger the number of redundant equations (N−M), the lower the reconstruction error. Of course, reducing the bandwidth (M) also limits the utility of the spectrometer, so this may not always be practical. However, increasing the fiber core radius W would increase the number of available spatial channels so that M < N would still be possible even for a large bandwidth.

We then considered the effect of changing the spectral channel spacing for a transmission matrix with a fixed bandwidth. First, we set the bandwidth Δλ = 100 nm and kept the same number of spatial channels N = 500, but varied the spectral channel spacing . We constructed matrices with = 0.2, 0.4, and 0.8 nm, corresponding to M = 500, 250, and 125. As shown in Fig. 5(b), the spectrum reconstruction error decreased for larger spectral channel separation. Again, this corresponds to fewer spectral channels to recover. However, increasing the spectral channel spacing much beyond the spectral correlation width could lead to errors in the reconstruction. If the spectral channels are separated too far, an optical signal falling between the discretized wavelengths will generate a speckle pattern with minimal correlation to the speckle patterns stored in the transmission matrix, and its wavelength could not be recovered. For example, in Fig. 4 the probe lines were set at wavelengths in between the discretized wavelengths used in the transmission matrix. Since = δλ/2, the speckle pattern at the probe wavelengths are still partially correlated with the speckle patterns generated at the nearby discretized wavelengths in T. Hence, the reconstruction algorithm is able to identify the wavelength of the probe signal. If the discretized spectral channels were separated by much more than the spectral correlation width, >> δλ, optical signals falling between these channels would have speckle patterns distinct from any speckle pattern recorded in the transmission matrix, thus recovery of their wavelengths becomes impossible.

Finally, we considered the effect of spatial oversampling. In this case, we used 500 fixed spectral channels separated by = δλ/2 = 0.2 nm, providing 100 nm bandwidth. Since the speckle image contained ~600 speckles of average size δr = 6 pixel of the CCD camera and the total number of pixels in the image was ~17,000, there was plenty of room for oversampling. We therefore generated transmission matrices with 500, 1000, and 2000 spatial channels, corresponding to channel spacing dr = 6, 4, and 3 pixel. As shown in Fig. 5(c), increasing the number of spatial channels improves the spectral reconstruction. Even though the additional spatial channels are correlated, they provide redundancy which aids the reconstruction in the presence of noise. Furthermore, the number of available independent spatial channels can be increased by using a fiber with a larger core. As discussed in Section 2, the core diameter W will not significantly affect the spectral resolution and could be adjusted to support a desired number of spatial channels. The downside to spatial oversampling is that the transmission matrix is larger, slowing down the reconstruction algorithm.

6. Fiber-spectrometer bandwidth

Thus far, we have used the fiber spectrometer to reconstruct spectrally sparse, narrowband signals. If all the guided modes in the fiber are excited by the probe signal, the bandwidth for a sparse spectrum (most of the spectral channels carrying no signal) is 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.

Next, we investigate the ability of the fiber spectrometer to recover spectrally dense, broadband signals. The bandwidth for the dense spectra (in which optical signals are present at many spectral channels) is further limited by the speckle contrast. When the input signal is spectrally broad, different wavelengths produce distinct speckle patterns which add in intensity causing the speckle contrast to decrease. Since the speckle contrast is effectively the “signal” used to reconstruct the input spectrum, the reconstruction degrades as the contrast is reduced. In particular, as the speckle contrast approaches the level of the noise in the measurement, it is hard to tell if an intensity variation is caused by the signal or the noise.

We illustrated this problem using a 1 m long fiber with spectral correlation width of 0.4 nm. We constructed a transmission matrix with 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
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.
, 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 (ΔλL). Quantitatively, the speckle contrast is calculated as C=σ/I, where σ is the standard deviation of pixel intensity from the mean value I, and is an average over all pixels across the speckle image. When the input spectrum gets broader, the speckle contrast decreases such that C~0.1 for the bandwidth of ΔλL ~50 nm. The number of independent speckle patterns that constitute the speckle image of the Lorentzian signal is approximately ΔλL/δλ, and C scales as [ΔλL/δλ]-1/2.

The speckle contrast places a practical limit on the bandwidth of the fiber spectrometer in cases when it will be used to measure dense optical spectra. The spectrum reconstruction requires that the speckle contrast be larger than the uncertainty of the experimental measurement. That is: C ~ [Δλ/δλ]-1/2 > σexp where Δλ is the operation bandwidth and σexp is the experimental error. We estimated σexp from the standard deviation between two subsequently measured transmission matrices for a 1 m fiber, and obtained σexp ~5 × 10−3. 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

To reduce the spectrum reconstruction error and increase the bandwidth, we shall increase the speckle contrast, which can be done by performing a polarization-resolved measurement. Although the input signal to the multimode fiber had a fixed linear polarization, the polarization after propagating through the fiber was scrambled [14

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. 2(4), 435–441 (1984). [CrossRef]

]. Since the two orthogonal polarization components produced distinct speckle patterns, the speckle contrast was reduced even for a single input wavelength. We therefore adapted our experimental setup to separate the two linear polarization components, effectively increasing the speckle contrast by a factor of 2. As sketched in Fig. 7(a)
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).
, two polarizing beam splitters and two mirrors were used to produce spatially displaced images of the speckle patterns with orthogonal linear polarizations on the camera. The inset of Fig. 7(a) shows the speckle images taken at λ = 1500 nm with the polarizing beam splitters labeled “Polarized Detection” and, as a reference, we removed both beam splitters and compared the speckle image using “Unpolarized Detection”. The contrast of the speckle images collected using polarized detection was 0.86, whereas the contrast using unpolarized detection was only 0.6. Ideally a linearly polarized speckle has a contrast of 1, but the finite pixel size of the CCD camera reduces the contrast. The two speckle patterns of orthogonal polarizations taken at the same wavelength are uncorrelated, and their sum gives the unpolarized speckle pattern with a factor of 2 lower contrast.

We calibrated transmission matrices using a 1 m long fiber with spectral correlation width of 0.4 nm. The bandwidth was set to 100 nm and the spectral channel spacing was 0.2 nm. In the case of unpolarized detection, we used 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 2 and the number of spatial channels is doubled, although the total intensity of each polarized speckle pattern is half of the unpolarized one. In Fig. 7(b), we compare spectra reconstructed using the two detection schemes. In both cases, we use a Lorentzian shaped dip in an otherwise flat spectrum as a probe. The probe spectrum is shown in red and the reconstructed spectrum in blue. The polarization-resolved detection clearly provides a more accurate reconstruction, as confirmed by the reconstruction error 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

The studies presented above demonstrate that multimode optical fibers can be used as high-resolution, general purpose spectrometers. The spectral-to-spatial mapping is provided by interference of the guided modes which generates a wavelength dependent speckle pattern. Of course, this speckle pattern is also sensitive to the spatial profile and polarization of the incident optical signal as well as to the environment surrounding the fiber. In fact the speckle of multimode fibers has been used for a variety of environmental sensing applications. For instance, researchers have monitored the speckle to track the changes to the surrounding environmental such as temperature [15

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]

], pressure [16

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]

], acoustic vibrations [17

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]

], or properties of a liquid the fiber was immersed in [18

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]

]. The speckle patterns have also been used to sense tiny displacements or vibrations of the fiber position [19

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

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

]. In these environmental sensing applications, the optical probe signal is typically provided by a laser source with a fixed wavelength. In our approach of using the fiber as a spectrometer, the opposite constraint is required: in order to measure changes in the incident optical signal (i.e. wavelength) the environment around the fiber must remain constant.

To provide an estimate of the sensitivity of the multimode fiber spectrometer to changes in the environment, we used the modeling technique presented in Section 3 to study the effect of ambient temperature. For glass fibers, changes in temperature affect both the refractive index, n, as dn/ndT = 7 × 10−6 °C−1 and the fiber length, L, as dL/LdT = 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]

]. Using these two expressions, we calculated the speckle patterns at the end of a waveguide with width 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)
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.
. 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.

Fiber-based spectrometers can have much lighter weight, smaller size, and lower cost than traditional grating based spectrometers, while potentially providing state of the art performance in terms of spectral resolution and efficiency (throughput). Large bandwidth can be achieved, especially for the detection of sparse spectra; and, in general, the resolution can be traded for bandwidth, as in a traditional spectrometer. Using a 1 m long fiber, we demonstrated 100 nm bandwidth with a spectral correlation width of 0.4 nm. Using a 20 m long fiber, we demonstrated 8 pm spectral resolution with a bandwidth of 1 nm. A single multi-mode fiber can also operate at varying spectral regions similar to the way traditional spectrometers do by rotating the grating. For a fiber spectrometer, the analog of rotating the grating is achieved by switching the transmission matrix to the one calibrated for the spectral region of interest. Of course, the fiber-based spectrometers also have limitations. They require a calibration step and after the calibration the fiber cannot be twisted or bent further. In addition, the spatial distribution and polarization of the input signal must be identical to that used in the calibration. In our implementation, this was realized by using a polarization maintaining single mode fiber to couple the probe signal to the multimode fiber. If the probe signal has spectral components outside the operating bandwidth, this will introduce additional noise, since the transmission matrix was not calibrated for these wavelengths. Finally, software is required to reconstruct the actual spectrum from the probe signal. Nonetheless, as we showed, the reconstruction algorithm can be fast and robust to experimental noise. The dramatically reduced size and cost of the fiber spectrometers could enable a range of new spectroscopy applications.

Acknowledgments

We are grateful to Prof. Hong Tang and Dr. Carlson Schuck for lending us the optical spectrum analyzer. We also thank Profs. Paul Fleury, Allard Mosk, Willem Vos, Doug Stone, and Michael Choma for useful discussions. This work was supported in part by the National Science Foundation under the Grant No. ECCS-1128542.

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 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]

6.

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]

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]

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]

10.

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

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]

12.

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

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]

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. 2(4), 435–441 (1984). [CrossRef]

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]

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 17(20), 17536–17546 (2009). [CrossRef] [PubMed]

21.

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

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]

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

  1. 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]
  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]
  6. 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]
  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]
  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]
  10. K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2006).
  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]
  12. J. W. Goodman, Speckle Phenomena in Optics (Ben Roberts & Co., 2007).
  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. Express20(10), 10583–10590 (2012). [CrossRef] [PubMed]
  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.2(4), 435–441 (1984). [CrossRef]
  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]
  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. Express17(20), 17536–17546 (2009). [CrossRef] [PubMed]
  21. S. Wu, S. Yin, and F. T. S. Yu, “Sensing with fiber specklegrams,” Appl. Opt.30(31), 4468–4470 (1991). [CrossRef] [PubMed]
  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]

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