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A new and simple technique is proposed and demonstrated for measuring the free spectral range (FSR) and bandwidth of optical resonators. For a broadband light input the resonator output forms an incoherent frequency comb with the spacing and linewidth corresponding to the FSR and bandwidth of the resonator, respectively. Photodetection of the resonator output produces heterodyne beat signals between the comb lines, from which the above two parameters can be estimated by spectrum analysis. The proposed technique overcomes the difficulties of conventional methods base on frequency-swept lasers. As demonstrations, fiber-optic Fabry-Perot and ring resonators are successfully characterized with the bandwidths as small as 10 kHz.
©2012 Optical Society of America
1. Introduction
Optical resonators have been widely used for scientific and industrial applications such as mode selection in laser cavities [1–3], references for laser frequency stabilization [4], optical frequency comb generation [5], signal processing in optical communication systems [6–8], etc. Since the performance of optical resonators can be characterized in terms of a free spectral range (FSR) and a bandwidth, it is a fundamental and important task to precisely evaluate these two parameters in order to meet the demand from the above applications.
A conventional method for evaluating these parameters is based on a frequency-swept laser measuring the frequency dependence of the transmitted or reflected light intensity [9]. This technique suffers from inaccuracies associated with 1) the finite laser linewidth, 2) instabilities of the laser and the resonator caused by the environmental disturbances, and 3) laser frequency calibration during the sweep. There have been several reports [10–16] for FSR and/or bandwidth measurements to overcome the above difficulties. All of these methods employed narrow linewidth lasers to mitigate the first inaccuracies. The influences of the environmental disturbances can be reduced by negative feedback control of the laser frequency or the resonant frequency of the resonator [11–14,16]. The third inaccuracy associated with the laser frequency calibration can be removed by the use of optical frequency comb [13], phase modulation frequency tuning [14–16], measurement of the decay time [10,11] and the frequency response function [11,12] instead of sweeping the laser frequency.
This paper reports a new and simple technique for characterizing optical resonators, which overcomes the above difficulties without the use of narrow linewidth lasers and negative feedback control of the laser and resonator frequency. The technique is based on spectral analysis of heterodyne beat signals produced by a broadband incoherent light that is transmitted through the resonator. The FSR and bandwidth of the resonator can be directly measured from the center frequency and half width at half maximum (HWHM) of the beat signal spectra, respectively. First, the principle of the measurement is described in Sections 2. Sections 3 and 4 describe results of measurements for fiber-optic Fabry-Perot and ring resonators, respectively. Finally, a brief summary is given in Section 5.
2. Principle of measurement
Figure 1 shows a schematic of the measurement apparatus, which consists of an incoherent light source, a resonator under test, a photodetector (PD), and an RF spectrum analyzer. The incoherent light source can be realized by the combination of an erbium-doped fiber amplifier (EDFA) and an optical bandpass filter (OBPF). An amplified spontaneous emission (ASE) from the EDFA is input to the resonator after passing through the OBPF and the transmitted light is detected with the PD. As shown in Fig. 1, the transmitted light forms an incoherent frequency comb with the spacing and linewidth corresponding to the FSR and bandwidth of the resonator, respectively. Detection of the resonator output with the PD produces multiple heterodyne beat signals and individual spectrum is separately measured with the RF spectrum analyzer. The center frequency and full width at half maximum (FWHM) of each spectrum are equal to the integer multiple of the FSR and twice the bandwidth of the resonator, respectively. Therefore, these two parameters can be directly measured from the spectra.
The normalized beat signal spectrum Sn(f) is expressed as
where n is a positive integer representing the order of the beat signals, ΔνFSR and Δν1/2 are the FSR and bandwidth of the resonator, respectively. The finesse Φ is given by ΔνFSR/Δν1/2. As shown by Eq. (1), each spectrum has the same shape except for the center frequency. It is advantageous to use a higher order spectrum for the FSR measurement due to the n-times enhancement in the resolution. The maximum value of ΔνFSR measured with this technique is limited by the frequency response of the PD. In the next sections the measured spectra are fitted to Eq. (1) to determine the values of ΔνFSR and Δν1/2.The amplitude Vn of the n-th order beat signal is roughly expressed as
Here, PASE represents the power density of the ASE expressed in the unit of W/Hz and ΔνOBPF is the OBPF bandwidth. The factors PASEΔν1/2 and ΔνOBP/nΔνFSR correspond to the intensity of each comb line and the number of the comb lines that contribute to beat signal generation, respectively. It should be noticed that the amplitude Vn is proportional to Δν1/2 and to the reciprocal of ΔνFSR. Therefore, the beat signal amplitude decreases for resonators with narrower bandwidth and/or larger FSR. It can be seen from Eq. (2) that the beat signal amplitude can be enhanced with increasing the ASE power and/or OBPF bandwidth.It is worth noting that the proposed technique overcomes all the difficulties explained in the Introduction. First, the technique does not require laser sources with the linewidth smaller than the bandwidth of the resonators. The resolution of the measurement is dependent on that of the RF spectral analysis and the resolution below 10 kHz is readily realized as will be shown in the next sections. Second, the accuracy of the technique is less influenced by the frequency fluctuations of light sources and resonators, because the frequency differences between the comb lines are detected with a broadband light, which is less affected by environmental disturbances. Therefore, the technique is applicable without the use of negative feedback control. Finally, the technique is free from the inaccuracy associated with the laser frequency calibration, because the measurement is performed on referring to the frequency of electrical signals.
3. Evaluation of fiber-optic Fabry-Perot resonators
To demonstrate the validity of the technique experiments were carried out using several types of resonators. This section describes results of measurements for fiber-optic Fabry-Perot resonators (Micron Optics, FFP-SI) with the FSRs of 5 GHz and 2 MHz, respectively. These resonators consist of a single mode fiber with highly reflective mirrors directly deposited on the both end faces.
The first sample is the one with the 5-GHz FSR. The values of the FSR and bandwidth are 5.04 GHz and 8.53 MHz, respectively, as evaluated by the manufacturer. As the incoherent light source two cascaded pairs of the EDFA and OBPF were used for increasing the input power. The bandwidths of the first and second stage OBPFs are 5 and 3 nm, respectively. The output light from the resonator is detected with a 12-GHz photoreceiver (New Focus, 1544A).
Figure 2(a) shows the optical spectra of the input and output lights observed with the resolution bandwidth (RBW) of 10 pm. The total input and output powers are 17.4 and - 12.1 dBm, respectively. The inset represents the magnified view of the output spectrum for 1550.0 ± 0.2 nm revealing the frequency comb lines. Figure 2(b) shows the beat signal spectrum for 0 – 20 GHz frequency range. The observed sharp peaks at about 5, 10, and 15 GHz correspond to the first, second, and third order beat signals, respectively. The center frequencies of these peaks measured with the RBW of 300 kHz are 5.041, 10.082, and 15.121 GHz, respectively. Figure 2(c) shows the normalized spectra of the first order beat signal for 400-MHz span and 300-kHz RBW. The trace with a grey line represents the noise floor, which is dominated by that of the photoreceiver. A relatively small signal-to-noise ratio is attributed to the large FSR value as indicated by Eq. (2). Curve fitting was performed with the additional noise floor term in Eq. (1). The FSR and bandwidth obtained are 5041.01 ± 0.11 and 8.52 ± 0.12 MHz, respectively, which agree well with the manufacturer’s data.
Fig. 2 Experimental results for the 5-GHz Fabry-Perot resonator. (a) Optical spectra for the input and output lights. (b) Beat signal spectrum for 0 – 20 GHz frequency range. The RBW and number of averaging are 1 MHz, and 64, respectively. (c) Normalized beat signal spectra for 400-MHz span, where the center frequency and number of averaging are 5.041 GHz and 64, respectively.
The second sample is the fiber-optic Fabry-Perot resonator with the 2-MHz FSR. Due to the limitation in the availability of narrow linewidth lasers, the manufacturer provided only the target specifications for the FSR and bandwidth as 2 MHz and 10 kHz, respectively. As an incoherent light source, two pairs of the EDFA and OBP were cascaded. The bandwidths of the first and second stage OBPFs are 3 and 1 nm, respectively. The output light from the resonator is detected with a 125-MHz photoreceiver (New Focus, 1811).
Figure 3(a) shows the beat signal spectrum for 0 – 20 MHz frequency range. The center wavelength of the incoherent light is 1550.0 nm and the total input and output powers for the resonator are 3.78 and - 20.1 dBm, respectively. Figure 3(b) shows the normalized spectra of the first order beat signal for 800-kHz span and 300-Hz RBW. The FSR and bandwidth obtained from the fitted curve are 2001.52747 ± 0.00004 and 9.773 ± 0.013 kHz, respectively, which are close to the target specification.
Fig. 3 Experimental results for the 2-MHz Fabry-Perot resonator. (a) Beat signal spectrum for 0 – 20 MHz frequency range, where the RBW and number of averaging are 10 kHz and 128, respectively. (b) Normalized beat signal spectra for 400-kHz span, where the center frequency and number of averaging are 2.00 MHz 128, respectively. (c) Normalized beat signal spectra for different wavelengths, where the center frequency, RBW, and number of averaging are 160.125 MHz, 300 Hz, and 128, respectively. (d) Wavelength dependence of the measured FSR and bandwidth.
Since the resonator contains a relatively long fiber (about 50 m), the FSR is expected to be wavelength dependent due to the group velocity dispersion of the fiber. Using the typical value of 17 ps/nm/km for single mode fibers the wavelength dependence of the FSR is calculated to be - 6.8 Hz/nm. Figure 3(c) shows the normalized beat signal spectra for the input lights at 1530, 1540, 1550, and 1560 nm, respectively. To increase the sensitivity the 80-th beat signals at about 160 MHz were detected. Wavelength dependent shifts of the center frequency were clearly observed. Figure 3(d) shows the measured FSR and bandwidth plotted as a function of the wavelength. The wavelength dependence of the FSR is estimated to be - 6.60 ± 0.17 Hz/nm, which is consistent with the prediction. The measured bandwidths are almost independent of the wavelength.
4. Evaluation of a 1-km fiber-optic ring resonator
This section describes results of measurements for a fiber-optic ring resonator with a very long optical path. As shown in Fig. 4(a) , the resonator is composed of 98:2 and 95:5 directional couplers and a 1-km single mode fiber. The incoherent light source and photoreceiver used in the measurement are the same as those used for the 2-MHz Fabry-Perot resonator. The center wavelength of the incoherent light is fixed at 1550.0 nm and the total input and output powers for the resonator are 9.97 and - 15.7 dBm, respectively. Taking account of the coupling ratio, excess loss of the couplers and the propagation loss of the fiber, the finesse of the resonator is estimated to be 21.8.
Fig. 4 Experimental results for the 1-km fiber ring resonator. (a) Structure of the resonator. (b) Beat signal spectrum for 2-MHz span, where the RBW and number of averaging are 1 kHz and 128, respectively. (c) Normalized beat signal spectra for 200-kHz span, where the center frequency and number of averaging are 199.67 kHz and 128, respectively. (d) Dependence of the measured FSR and bandwidth on the order of the beat signal.
Figure 4(b) shows the beat signal spectral for 2-MHz frequency span, where the center frequencies for the upper, middle, and lower traces are 0.5, 10, and 100 MHz, respectively. It can be seen that the beat signals are generated over a wide frequency range. Figure 4(c) shows the normalized spectra of the first order beat signal for 200-kHz span and 100-Hz RBW. The FSR and bandwidth obtained from the fitted curve are 199.656 ± 0.016 kHz and 9.675 ± 0.004 kHz, respectively. The corresponding finesse is 20.7, which is consistent with the prediction. The discrepancy can be attributed to the loss of connectors between the couplers and the fiber. Similar measurements were performed for higher order beat signals. Figure 4(d) shows the dependence of the FSR and bandwidth on the order of the beat signal. The values of the FSR are almost constant above the fifth order, which is attributed to the increased resolution for higher order beat signals. The measured bandwidths are almost independent of the beat signal order. The amount of scatter in the bandwidth is slightly larger for the lower order partly attributed to the larger excess noise of the photoreceiver at low frequencies. It is worth noting that the proposed technique enabled us to characterize resonators with a long optical path that is susceptible to environmental disturbances without the use of negative feedback control.
5. Summary
In summary a new technique was proposed and demonstrated for characterizing optical resonators. The technique is based on the spectral analysis of the beat signals produced by a incoherent light transmitted through the resonator and does not rely on the use of narrow linewidth lasers or negative feedback control of the laser and resonator frequencies. As demonstrations, the FSRs and bandwidths of the fiber Fabry-Perot and ring resonators were evaluated. As compared with the conventional method based on frequency-swept lasers, the proposed technique offers a higher resolution (< 10 kHz) with a simpler configuration. It is necessary to evaluate the accuracy of the proposed technique on referring to more precise techniques such as those based on femtosecond frequency combs [13].
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