Very long range quasi-Fourier spectroscopy for narrowband lasers

Masaaki Inoue; Fumihiko Ito; Xinyu Fan; Yusuke Koshikiya

doi:10.1364/OE.20.00B566

1. Introduction

Fourier spectroscopy [1–3] is a powerful tool particularly in relation to infrared spectroscopy. As with any types of spectrometric diagnosis, the resolution is determined by the maximum delay, hence, the size of the interferometer. This fact imposes a large burden when deploying the technique with narrow bandwidth lasers. Here, we present a novel Fourier spectroscopic approach to evaluate the modulus of the degree of coherence (DOC) over a very long range by using a speckle analysis in coherent fiber reflectometry. The term “quasi-“ in the title is used to indicate the missing phase modulation of the DOC, with the result that only the even-functional information of the spectral profile is observed. We have already provided a preliminary result and intuitive theoretical description [4], and here we confirm the validity of the technique in a delay of hundreds of microseconds (several tens of km in a vacuum), which enables kHz-level resolution.

The laser speckle is a well-known phenomenon closely related to the coherence of light [5–8], and it has been intensively investigated in such areas as imaging applications, metrology for detecting displacement, deformation, and vibration. Generally, since the ability to form speckle patterns stems from a coherent interaction, an appropriate optics set-up can measure the DOC by monitoring the speckle’s contrast [9, 10]. In silica optical fibres, a micro-size fluctuation in refractive index originates from such amorphous nature features as “rough surfaces”. In particular, our interest here is focused on a reflectometric measurement of such scattered lights (Rayleigh scattering) in single-mode fiber (SMF) [11]. With coherent fibre reflectometry, the reflected intensity is jagged along the propagation direction [12]. It is known that observing the alternation of the speckle pattern with respect to the reference pattern allows us to measure local changes in the optical length of the fibre caused by strain and/or temperature [13].

2. Theory of principle

The system set-up is shown in Fig. 1 . The beam emitted from the laser under the test (LUT) is used as a light source for coherent optical frequency domain reflectometry (C-OFDR) [13–16], which observes the Rayleigh backscattered beam in the fiber as a function of the distance or roundtrip delay to the scattering point in a SMF. The linear frequency sweep of the LUT is accomplished by sweeping the modulation sideband of an external single-sideband (SSB) modulator. Let $a_{q} (t) e^{j 2 π (ν_{q} + g t / 2) t}$ be the modulated beam, i.e., the input into the SMF, where g is the sweep rate (Hz/s), $ν_{q}$ is the optical frequency, and $a_{q} (t) = e^{j θ_{q} (t)}$ is the random fluctuation of the phase of the laser beam, which degrades the coherence. A single point reflection at the roundtrip time τ' in the fiber is detected as the spectrum (Fourier transfer), $F_{q, τ'} (g τ),$ of the beat signal between the local beam and the reflected beam (replica of the local beam delayed by τ), $a {}_{q}{(τ')} a_{q}^{*} (t - τ') e^{j (2 π g τ t + 2 π ν_{q} τ' - π g τ'^{2})} .$ The amplitude of the speckle, $e_{q} (τ)$ observed at τ is the superposition of the spectrum, $F_{q, τ'} (g τ),$ weighted by the random but time-invariant scattering coefficient $r (τ'),$ where $τ'$ is a scattering point located in the vicinity of τ [12].

Fig. 1 Coherent-optical frequency domain reflectometry (C-OFDR) based setup for Rayleigh speckle measurement. The optical frequency of the laser under test (LUT) at 1.55 μm is swept linearly by the single side band (SSB) modulator by using the generated + 1st order sideband. A/D, Analog-to-digital converter ; FFT, fast Fourier transform.

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Suppose that another separate but identical measurement is accomplished, represented by the suffix “s”, with respect to “q”. The sole difference is the fluctuation of the tested beam represented by $a_{q} (t) = e^{j θ_{q} (t)}$ and $a_{s} (t) = e^{j θ_{s} (t)} :$ the two beams are modulated with an identical waveform, and the speckle amplitudes $e_{(q, s)} (τ)$ are superposition of $F_{(q, s), τ'} (g τ)$ with an identical random coefficient, $r (τ')$ . Therefore, their correlation $\bar{e_{q} (τ) e_{s}^{*} (τ)}$ should directly reflect that between $F_{q, τ'} (g τ)$ and $F_{s, τ'} (g τ),$ or between their inverse Fourier transforms (Parceval’s theorem). Thus, we obtain

\bar{e_{q} (τ) e_{s}^{*} (τ)} = e^{j 2 π (ν_{q} - ν_{s}) τ} \cdot \int_{0}^{T} a {}_{q}{(τ)} a_{q}^{*} (t - τ) a_{s}^{*} (τ) a_{s} (t - τ) d t .

At the final stage, the exchange of $τ'$ for τ is based on the assumption that the correlation is constant in the vicinity of $τ'$ . If the integration time T is sufficiently long compared with the coherence time so that the time average equals the ensemble average, and the average product of the independent q-th and s-th processes equals the product of each average. Hence, we obtain

\bar{e_{q} (τ) e_{s}^{*} (τ)} = e^{j 2 π (ν_{q} - ν_{s}) τ} \cdot {| γ (τ) |}^{2},

and find that

\bar{e_{q} (τ) e_{s}^{*} (τ)}

must be real. Note that

γ (τ) = \int_{0}^{T} a_{q, s} (t) a_{q, s}^{*} (t - τ) d t

is the DOC [17].

3. Experimental set-up and laser under test

Two fiber lasers and a laser diode emitting at 1.55 μm were prepared as LUTs. LUT#1 was an NKT Photonics Koheras E15, LUT#2 was an Orbits Lightwave Ethernal and LUT#3 was a RIO PLANEX (RIO0194-1-01-1). The linewidth (full width at half maximum, FWHM) estimated by the self-delayed heterodyne method with a 160 km delay was about 110 kHz in LUT#3.

The frequency of the LUT was swept by using a single sideband (SSB) modulator. A frequency synthesizer with a ramp function (Agilent E8257D) generated the sinusoidal RF signal whose frequency was linearly swept. The modulation sideband of the optical beam was swept accordingly.

The sweep rate g and integration time T, which determine the sweep range ΔF = gT, can be selected very flexibly, but the following considerations are necessary. First, T must be sufficiently large compared with the expected coherence time so that it involves every random fluctuation. To realize this, we tested several different Ts, and looked for one that seemed to provide sufficient randomisation. The second issue stems from the requirement that the DOC should be constant over the range in which the speckle correlation is obtained. The reflectometer's resolution is Δτ = 1/ΔF. If we use N samples, or the NΔτ range to obtain the correlation, the spectral width Δν of the LUT must be smaller than 1/ NΔτ so that the DOC is constant in the range. Hence, ΔF > NΔν is required. Therefore, if the sweep is accomplished with an electrical modulator, its bandwidth restricts the maximum spectral width that can be measured. The parameters used in the experiment are shown in the figure captions.

The output powers of the LUTs were typically 10 dBm. The SSB modulator imposed a total loss of ~-19 dB, so an optical amplifier was used to obtain a power of ~6 dBm at the entrance of the SMF. The C-OFDR provided a signal to noise ratio of about 25 dB even at a 40 km range.

The measurements were performed every 0.2 s, and a succeeding pair of obtained speckle patterns was used for the correlation analysis. The x- and y- polarizations of the scattering were obtained with polarization diversity, and the speckle correlation was analysed by using the inner product of the Jones vectors,

\bar{{\vec{e}}_{q} (τ) \cdot {\vec{e}}_{s}^{*} (τ)} = \bar{e_{q}^{(x)} (τ) e_{s}^{(x) *} (τ) + e_{q}^{(y)} (τ) e_{s}^{(y) *} (τ)} .

4. Experimental results

We tested three LUTs experimentally. Figure 2 shows an example of the observed reflection speckle at delays both shorter and longer than the laser's expected coherence time (τ_c). Since the SMF we used was a non-polarization-hold type, the polarization diversity was incorporated in the coherent receiver. The intensity plot of the scattering is the square sum of the x- and y-polarizations. The blue plots show the results of the first (q-th) measurement, and red dashed plots shows those of the second (s-th) measurement that was performed successively with the same LUT. As shown in the figure, when τ <τ_c, these two results exhibit a strong correlation, whereas when τ >τ_c, the correlation diminishes. This tendency was observed for all the LUTs used in the experiment, and was well predicted by the theoretical analysis.

Fig. 2 OFDR traces of e_q(τ) (blue) and e_s(τ) (red) in τ < τ _c (left side) and τ > τ _c (right side) for LUT #1. The top row shows the total intensities of x- and y-polarizations. The bottom row shows the phases of the x-polarization.

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The results of the correlation $\bar{e_{q} (τ) e_{s}^{*} (τ)}$ for the three LUTs are shown in Fig. 3 . The real and imaginary parts of the correlation are displayed in Fig. 3(a). Their correlation value should theoretically be real, but the practical value was complex and includes oscillation, as shown in the figures. The oscillation may be attributed to the bias of the laser frequency or a change in the fiber temperature imposed between the two measurements $e_{q, s} (τ)$ . The modulus of the complex numbers is calculated from these real and imaginary parts, and shown in Fig. 3(b), where the modulus shows the squared DOC. Three typical measurement results for LUT#1 (light-green, blue and brown) and #2 (red, green and purple) are plotted, and the results for LUT#3 are plotted in orange in the same figure (also shown separately on a larger scale). These DOC curves decrease monotonically with respect to the delay, but we see that the rates of decrease are unique to each LUT.

Fig. 3 (a) The real and imaginary parts of the correlation of $\bar{e_{q} (τ) e_{s}^{*} (τ)}$ observed for the three LUTs. N = 1000 samples of amplitude values in the vicinity of the specific delay were used to obtain a single correlation value. (b) The squared modulus of DOCs obtained from the real and imaginary parts.

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By taking the squared-root of $\bar{e_{q} (τ) e_{s}^{*} (τ)},$ we obtain the modulus of the DOC. From the Wiener-Khinchin relationship, the even-functional information of the power spectrum can be obtained, while the lack of the phase of the DOC veils the odd-functional information. Spectral profiles thus obtained are depicted in Fig. 4 . The measurements were performed every 0.2 s, and the succeeding pair of speckle charts were processed to examine their correlation. We found no significant deviation of the observed spectra for the same LUT throughout the measurement. The full-widths at half-maximum (FWHM) were about 6, 10, and 120 kHz, for LUT#1 to #3, respectively.

Fig. 4 Measured spectral profiles of LUTs. The power spectral densities are obtained with Wiener-Khinchin's theorem by using the DOCs shown in Fig. 3.

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To examine the dependence of the above results on the sampling time T, we collected data for several Ts, as shown in Fig. 5 . The observed tendency in LUT#1 in particular, which has the longest expected coherence time of the three, is that the obtained $\bar{e_{q} (τ) e_{s}^{*} (τ)}$ curves exhibited the curvature perturbation in every experiment when the integration time T was short, while the perturbation disappeared when T was long. As described in the theoretical consideration, we presumed that the observed perturbation is a result of the fact that there is insufficient sampling time to contain all possible random events. In such cases, the light sometimes appears coherent even with very large delays, while the interference may diminish at a specific delay, and the cases are dependent on each individual event.

Fig. 5 The observed squared modulus of DOCs in the setting of different sweep times.

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Finally, to examine the validity of the results, we conducted a reference experiment, in which we used a Mach-Zehnder interferometer with a few fixed delays. Figure 6(a) shows the reference measurement set-up we used to characterize the phase noise of a laser using a Mach-Zehnder interferometer with a fixed delay. The continuous light produced from the laser under test (LUT) was divided into two paths. One path was delayed by time τ through a single-mode fiber (SMF), and frequency-shifted by an acousto-optical frequency shifter (AOF) $Δ ω_{0}$ to allow the use of an ac-coupled photodetector. The two beams were mixed by a 90-degree hybrid to produce the constellation diagram. After removing the constant rotation, we evaluated the phase variance $σ {(τ)}^{2}$ at a few fixed delays (0 ~200 μs). The DOC was estimated by

Fig. 6 Common reference experiment design based on a Mach-Zehnder interferometer with a few fixed delays. (a) We used an acousto-optic frequency shifter to enable us to employ ac-coupled photo detectors, and this frequency bias was removed by signal processing. The delay fiber was placed in the same soundproof box as in Fig. 1. AOF: acousto-optical frequency shifter. (b) By obtaining the autocorrelation of the phase noise at delays of 0, ~25, ~50, ~100 and ~200 ms in LUT#1, we analysed the DOC with ${| γ (τ) |}^{2} = \exp (- σ (τ) {}^{2})$ ; that is, ${| γ (τ) |}^{2}$ = 1.0, 0.85, 0.7, 0.4 and 0.08, respectively. The squared modulus DOCs of LUT#1 and #2 are indicated within the bar as in Fig. 3(b). The integration time T was 64 ms.

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{| γ (τ) |}^{2} = \exp (- σ {(τ)}^{2}) .

By obtaining the phase-error variance at delays of 0, ~25, ~50, ~100 and ~200 μs, we were able to analyze the DOC as indicated within the red and grey bars, for LUT#1 and #2, respectively, for each delay in Fig. 3(b). We can clearly see that there is very good agreement with the results obtained with the proposed method. In addition, we describe the effect of the mechanical vibration of the SMF, which could lead to an underestimation of the coherence. The amplitude of the sound pressure density was typically 58 dB (16 mPa) outside the soundproof box, and 54 dB (10 mPa) inside it. SMFs with identical 40 km lengths (~200 μs delay) were inserted in both paths of the interferometer. The two SMFs were placed apart from each other in different soundproof boxes, so we believe that their vibrations were uncorrelated. The observed phase variance was much smaller than that observed when the path length was different and laser’s phase noise was incorporated.

We examined the phase variance caused by the mechanical vibration of SMFs with identical 40 km lengths (~200 μs delay) in soundproof boxes (Fig. 7(b) ). The amplitude of the sound pressure density was approximately 54 dB (10 mPa) in the soundproof box. Clearly, the broadening of the constellation is much smaller than that caused by the LUT’s phase fluctuation. On the assumption that the vibrations of the two SMFs are uncorrelated, we estimate the variance caused by a single SMF to be ~0.1 rad²; this corresponds to ${| γ (τ) |}^{2}$ ≅ 0.95, whereas the observed DOC at the corresponding delay was almost eliminated. Thus, we conclude that the DOC was slightly underestimated, but the effect was not significant.

Fig. 7 Experimental set-up for measuring the effect of the mechanical vibration of SMF. (a) SMFs with identical 40 km lengths (~200 μs delay) were inserted in both paths of the interferometer. The two SMFs were placed apart from each other in different soundproof boxes. (b) Observed constellation diagram. The corresponding ${| γ (τ) |}^{2}$ was 0.95. (c) Sound pressure densities in the soundproof box.

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5. Conclusion

In summary, we have described a Fourier spectroscopic measurement with a range of several hundreds of microseconds with the aim of characterizing the coherence of narrowband laser beams, and have successfully determined the spectral profile of kHz-level lasers. The unique feature of this new technique makes it attractive for characterizing narrowband beams. This is because the existing technique using a delayed self-heterodyne interferometer [18, 19], which only observes a beat spectrum with a finitely delayed portion, includes a fundamental error when it is adapted for such narrowband sources.

References and links

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Very long range quasi-Fourier spectroscopy for narrowband lasers

Abstract

1. Introduction

2. Theory of principle

3. Experimental set-up and laser under test

4. Experimental results

5. Conclusion

References and links

Cited By

Figures (7)

Equations (5)

Optics Express