## Abstract

A novel method for characterizing the amplitude of a coherence function with respect to a delay between two optical waves is proposed and demonstrated by using a distributional Rayleigh speckle analysis based on C-OFDR. This technique allows us to estimate both the coherence time of the laser and that of the spectral profiles from the measured amplitude of the coherence function, if the symmetry of the spectrum can be assumed. The spectral width obtained in the experiment agrees roughly with that obtained using a delayed self-heterodyne method.

© 2011 OSA

## 1. Introduction

Narrow linewidth laser sources are currently being developed for a variety of applications including coherent optical communication and interferometric sensing, and some of them require a linewidth as narrow as kHz level. To evaluate the spectral profile or coherence property of such a narrow-band beam is a rather cumbersome task. A delayed self-heterodyne method (DSHM) [1], in which one arm is delayed to decorrelate the optical fields, is generally used for laser linewidth measurements. Assuming that the spectral profile is Lorentzian or Gaussian, the 3 dB spectral spread is twice or $\sqrt{2}$ the beat frequency width, respectively. However, since perfect decorrelation cannot be achieved unless an infinitely long delay is used, the measurement includes an inherent ambiguity. In practical cases [2,3], we must use a delay of a few ten times the coherence time, and such a long delay is hard to realize when the spectral width is narrow.

In this paper, to allow us measure the spectral profile or the coherence property with such a narrow spectral width, we propose a novel approach for measuring the amplitude of the coherence function as a function of the delay. The method uses a reflectometric measurement of the Rayleigh speckle [4] from a long optical fiber. If we assume that the spectrum is symmetric, the Fourier transform of the coherence function amplitude provides the spectral profile.

## 2. Principle and system configuration

The conceptual configuration of the measurement setup is shown in Fig. 1 . The setup is similar to that used for coherence optical frequency domain reflectometry (C-OFDR) [5,6]. Here, we measure the beam from a laser under test (LUT). The optical frequency of the LUT is swept linearly by an external modulator. This swept beam is used to observe the Rayleigh backscattered distribution in an optical fiber in the C-OFDR setup.

Let *E _{q}(t)* be the electrical amplitude of the measured beam with duration

*T*. Here,

*q*denotes the number of the samples of the lightwave. Consider a single reflection point with the round trip time

*τ*, the back-reflected beam ${E}_{q}^{}(t-{\tau}_{})$ is mixed with the local probe ${E}_{q}^{}(t)$, and the beat signal ${E}_{q}(t){E}_{q}^{*}(t-\tau )$ is yielded in C-OFDR. Consequently the distribution of the reflection is obtained by the amplitude spectrum of the beat signal,

*τ*is the distance from the spectrum center and

_{i}*g*is frequency sweep rates. Note that the coherence function is defined bywhere $\overline{}$ denotes the ensemble average. According to the one-dimensional model of the Rayleigh scattering, the backscattered power ${P}_{q}(\tau )$ observed at the specific delay

*τ*is the sum of those from the scattering centers with random reflectivities [7]:

Since the DC factor of ${f}_{q}({\tau}_{i})$ is the time average of ${E}_{q}(t){E}_{q}^{*}(t-\tau ),$ if the signal can be assumed to be ergodic, it equals the ensemble average, $\overline{{E}_{q}(t){E}_{q}^{*}(t-\tau )}\equiv \gamma (\tau ).$ Thus, ${f}_{q}({\tau}_{i})$ can be decomposed into correlation and decorrelation parts as

where ${n}_{q}({\tau}_{i})$is a random complex number. Substituting (4) into (3) and assuming that $\gamma (\tau )$ is constant ($=\gamma $) over the considered vicinity in $\tau $, and replacing ${n}_{q}({\tau}_{i})={n}_{q,i}$ and $r({\tau}_{i})={r}_{i}$, we obtainIn a similar way, we also obtain another backscattered power ${P}_{s}(\tau )$ measured in the *s* th measurement, and we analyze the correlation between the two measurements.

Let us assume the randomness of ${n}_{q,i}$, $\overline{{n}_{q,i}}=0.$ Also assume that ${n}_{q,i}$ and ${n}_{s,j}$ (*q ≠ s*), and ${r}_{i}$ and ${r}_{j}^{}$ (*i ≠ j*) are statistically independent of each other. Then by using the Kronecker delta expression, we obtain,

By using, $\overline{{P}_{q}(\tau )}=\overline{{P}_{s}(\tau )}={\gamma}_{}^{2}\overline{{r}_{}^{2}}+\overline{{n}_{}^{2}}\overline{{r}_{}^{2}}=\overline{{r}_{}^{2}},$ we obtain,

## 3. Experimental method

Two laser diodes (LD) operating at 1.55 μm, as described in Table 1 , were prepared as LUTs. (I) was NEL (NLK1C6DAAA) and (II) was Rio (RIO0194-1-01-1). We estimated their linewidths by using the DSHM with an 80 km delay. The results are also shown in Table 1.

In our experimental setup shown in Fig. 2
, a single sideband (SSB) modulator (Sumitomo Osaka Cement Co., Ltd) and a frequency-swept RF synthesizer (Agilent E8257D) were used for external frequency sweeping [8]. We use two different frequency sweep rates, *g* = 500 and 125 GHz/s.

The sweep range, *ΔF = gT*, was always set at 8 GHz, which was extended with three doublers and band pass filters, a mixer and an oscillator to step the frequency down to a suitable level, corresponding to a theoretical resolution, *Δτ =* 1/ *ΔF,* of 125 ps (round trip time). The sweep range determines the maximum linewidth which can be observed by the system. As described later, since *N* ( = 1000) neighboring samples of the backscattered intensities are used for the statistical analysis to obtain a single value of the coherence function, *γ*(*t*), the interval of the obtained *γ*(*t*) is larger than *NΔτ*. Supposing that the domain of the coherence function is ~1/*Δν*, 1/*Δν*>*NΔτ*, or *ΔF*>*NΔν* is needed to acquire the profile of the coherence function. For example, when *N* = 1000, it is supposed that the linewidth up to a few MHz can be observed with the sweep range of 8 GHz.

The measured beam lengths were 16 and 64 ms, respectively, for each case. The suppression ratio between the + 1 st order sideband and the other sidebands (including the carrier) of the spectrum of the SSB modulated lightwave was about 20 dB. An interferometer was built with two couplers and a circulator that was used to launch the lightwave into the 2-kilometer single mode fiber (20 μs round trip time) and to receive the backscattered light. To prevent acoustic noise, the long fiber was placed in a soundproof box. The polarization effect was disregarded in our experiment since we adopted a polarization diversity scheme. The launching of the lightwave could be divided into the polarization components S and P in the polarization diversity device. Each lightwave was detected with balanced photo detectors (New Focus ^{TM} 1617-AC-FC), then filtered by electrical filters, and acquired with 8 bit A/D converters (NI PXI-5154) with a sampling rate of 1 GHz. These data were sent to a personal computer to analyze the coherence function.

## 4. Experimental results

Figure 3 shows the two backscattered power distribution measurements, ${P}_{q}(\tau )$and ${P}_{s}(\tau )$, with two different delay ranges, only one of which is within the coherence time. The vertical scales of the waveforms are linear, and the plot spacing is 125 ps. As shown Fig. 3(a), the backscattered power pattern of ${P}_{q}(\tau )$ has a strong correlation to ${P}_{s}(\tau )$ in coherence time. Meanwhile, the two patterns are very different in Fig. 3(b). These features are understandable based on the described theory, which states that the correlation value corresponds to ${\left|\gamma \right|}^{4}+1$.

From these speckle distributions, we calculated the correlation of ${P}_{q}(\tau )$and ${P}_{s}(\tau )$, by using a 125 ns section around each τ. We then used Eq. (9) to calculate the amplitude of the coherence function $\left|\gamma (\tau )\right|$. We performed 100 pairs of ${P}_{q}(\tau )$and ${P}_{s}(\tau )$ measurements, and averaged the obtained $\left|\gamma (\tau )\right|$s were over the 100 times measurements. The results are plotted in Fig. 4
. Figure 4(a) and (b) show the results for *T* = 16 ms and *T* = 64 ms, respectively, and there is no clear difference between them. The obtained coherence functions decrease monotonically with respect to τ, and it appears that τ providing $\left|\gamma (\tau )\right|$ ≅ 1/e suggests the coherence time. The results show that LUT (II) has a longer coherence time than LUT (I). This feature agrees well with the DSHM results, which shows that (II) has a narrower linewidth.

Moreover, the obtained profile of the amplitude of the coherence function includes power spectral profile information, and its upwardly convex shape suggests that the spectrum would be Gaussian, rather than Lorentzian. The coherence function $\gamma (\tau )$ lies in the relationship of the Fourier transform pair with respect to the power spectral density *S*(ν) in accordance with the Wiener-Khinchin theorem [9],

Figure 5 shows the Fourier-transform spectra of $\left|\gamma (\tau )\right|$ in Fig. 4(a). If we assume the spectrum is symmetrical, the theorem is modified to

This allows us to estimate the spectral profiles of the lasers from the measured amplitude of the coherence function. The obtained results are shown in Fig. 5. If the power spectral density can have a symmetric profile, the full-widths at half maximum (FWHM) of the *S*(ν)s were 520 and 116 kHz in Fig. 5(a) and (b) after Gaussian fitting, respectively. The spectral width of LUT (I) obtained in the experiment coincides well with that obtained by DSHM assuming that the profile is Gaussian. On the other hand, an approximately 1.7-fold difference is seen between those values in LUT (II). A possible reason for the discrepancy is that the proposed method observes the spectrum of the 16 ~64 ms length signals, while DSHM observes the beat with the 0.4 ms delayed signal (80 km-fiber). If the signal includes a slow frequency drift, the above discrepancy is acceptable.

## 6. Conclusion

In conclusion, we proposed and demonstrated a novel method for characterizing the coherence function amplitude by using distributional Rayleigh speckle analysis based on C-OFDR. This technique provides a unique function to measure the coherence function amplitude over the range of the used delay. Assuming that the power spectrum density can have a symmetric profile, the measured laser linewidths are 520 and 116 kHz in LUT (I) and LUT (II) after Gaussian fitting, respectively. These spectral widths approximately coincide with that obtained by DSHM. The proposed technique would open a new door of characterizing the coherence property and spectral shape of narrow-band beams.

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