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We describe and verify the dynamic behavior of a novel technique to optimize and actively control the optical impedance matching condition of a coupled resonator system. The technique employs radio frequency modulation and demodulation to interrogate the reflection amplitude response of the coupled cavity system. The sign and magnitude of the demodulated signal is used in a closed loop feedback system which controls the coupling condition of a three-mirror resonator. This was done by actuating on the spacing between two of mirrors, effectively using the pair as a variable reflectivity compound mirror. We propose that this technique can be used for controlling the signal bandwidth of next-generation gravitational wave detectors, as well as optimizing circulating optical carrier power in the instrument.
©2010 Optical Society of America
1. Introduction
Modern gravitational wave detectors use long-baseline high resolution laser interferometry to measure minute distortions in space-time. Most ground-based instruments are based on variations of the Michelson interferometer, with enhancements in its optical configuration to improve signal sensitivity [1, 2]. In a next-generation detector employing a dual-recycled Michelson interferometer, as illustrated in Fig. 1, one mirror is placed at the output of the Michelson interferometer to form a signal recycling cavity, and another is placed at the input of the Michelson interferometer to form a power recycling cavity [3, 4]. The signal recycling cavity resonates the gravitational wave signals to enhance the instrument sensitivity by its resonance Q factor. The power recycling cavity, on the other hand, is used to increase the circulating laser carrier power in the instrument to improve its shot noise performance.
If either the signal or power recycling mirror reflectivity can be varied by an external actuation signal, the coupling condition for each cavity can be varied with this actuation signal. Additionally, with an appropriate impedance matching sensing readout, each recycling cavity can then be actively controlled and optimized.
For example, by tuning the reflectivity of the signal recycling mirror, the finesse of the signal recycling cavity can be changed. This provides a means to vary the instrument sensitivity bandwidth [5–7]. On the other hand, a feedback control loop can be implemented to control the reflectivity of the power recycling mirror, to keep the power recycling cavity impedance matched. For a given input laser optical power, this minimizes the reflected carrier from the instrument, thereby maximizing its circulating power, hence improving the shot noise performance of the instrument.
Recently, a technique using radio-frequency (RF) laser amplitude modulation (AM) was used to actively sense and control the coupling condition of a fiber ring resonator [8, 9]. Aside from an experimental demonstration, Ref. [8] described various applications including the optimization of power coupling into coupled resonator systems. For interferometric gravitational wave detectors, the ability to actively control the impedance condition for a given resonator system offers a novel shot-noise limited, closed loop feedback technique to optimize interferometer circulating power. Furthermore, there has been recent developments in high resolution intensity fluctuation diagnostics [10] and stabilization [11] of laser sources for gravitational wave instruments, where precise control of resonator impedance would be an enabling technology.
In this paper we demonstrate the coherent readout and control of the impedance matching condition for a three mirror coupled resonator. Two of the mirrors act as a variable reflectivity compound reflector by actuating on their relative spacing. The third mirror has a fixed reflectivity and emulates a locked Michelson interferometer. This experiment serves as a proof of concept for using amplitude modulation to either control the detection bandwidth of the signal recycling cavity, or maximizing the circulating power in the power recycling cavity.
2. Reflection response of an optical cavity on resonance
Consider a simple two mirror, lossless optical cavity as shown in Fig. 2, where R 1 and R 2 are the reflectivities for the input and output couplers respectively, while T 1 is the transmissivity of the input mirror. The mirrors have reflection coefficients r 1 and r 2, such that r 1 2 = R 1, and r 2 2 = R 2; and transmission coefficients t 1 and t 2, such that t 1 2 = T 1 and t 2 2 = T 2.
A laser beam with electric field E inc is incident on the cavity. The reflected electric field from the cavity is E refl, while the transmitted field is E trans. On resonance, the transfer function of the reflected electric field is purely real, given by
Figure 3 shows the calculated reflected intensity and amplitude response of a simple, resonant Fabry Perot resonator as a function of the input coupler transmission, as described by Eq. (1). In the model the mirrors are considered lossless and thus r 1 2+t 1 2 = 1. The output coupler reflectivity is fixed at R 2 = 0.9, and T 2 = 0.1. In Fig. 3(a), we plot the reflected power transfer function |𝓡res|2 as the input coupler transmissivity is varied. As the input coupler transmissivity T 1 is increased from T 1 ≪ T 2 to approach the point where T 1 = T 2, the reflected optical power decreases until it reaches zero at T 1 = T 2. As T 1 continues to increase above T 2, the reflected light increases and approaches unity when T 1 becomes much larger than T 2.
The amplitude transfer function 𝓡res is displayed in Fig. 3(b). We note that it has a zero-crossing at T 1 = T 2, with a positive value when T 1 < T 2, and negative value when T 1 > T 2.
For an optical cavity illustrated in Fig. 2, its impedance coupling conditions can be categorized as [12]
The interrogation of impedance coupling relies on the ability to sense the amplitude and polarity of the electric field reflected from a resonator.
Fig. 3. Reflected intensity and amplitude response as a function of the input coupler transmission. (a) Normalized reflected intensity as a function of the input coupler transmission; (b) Amplitude response as a function of input coupler transmission. In both plots the output mirror is set to r 2 = √0.9.
3. Amplitude modulation and the cavity coupling condition
We now interrogate a simple linear cavity as described in Section 2 with an amplitude modulated laser as shown in Fig. 4(a). We assume that the modulation frequency is well outside the resonance full-width half-maximum as illustrated by Fig. 4(b). We further assume that the laser carrier is kept on resonance by active feedback control using a technique such as the Pound-Drever-Hall (PDH) [13] frequency locking. The reflected light is received by a photodetector, whose electronic signal is then demodulated with a local oscillator. Following the derivations as described in detail in Ref. [8], we obtain the demodulated error signal voltage for a lossless cavity:
where ρ is the photodetector responsivity (Amps/Watt); R pd is the trans-impedance gain of the photodetector (Ω); β is the modulation depth, such that β ≪ 1; P opt is the incident laser power; and are the amplitude reflection coefficients of the two cavity mirrors.
Fig. 4. A two-mirror resonant cavity interrogated by an amplitude modulated laser. (a) the schematic of the interrogation and readout setup; (b) the operating spectral condition of the AM sidebands, showing the AM sidebands well outside the resonance FWHM of the cavity, such that they are mostly reflected.
Near the impedance matching condition, r 1 ≈ r 2, and Eq. (2) can be approximated by
Equation (3) implies that the magnitude and sign of the demodulated AM voltage is a direct measure of the coupling condition of the cavity on resonance. This impedance matching error signal can be used both as a readout, as well as a feedback signal in a control loop to actively impedance match the cavity, provided there is a mechanism to vary the reflectivity of one of the mirrors.
4. Impedance Matching with a Variable Reflectivity Mirror
The input coupler reflectivity R 1 in Fig. 4 can be varied if it is replaced with an optical etalon, as illustrated by Fig. 5. In Fig. 5(a), the reflectivity R 1(L 1) of the etalon, formed by the two closely spaced mirrors, is dependent on their separation L 1. We note, however, that by varying L 1, the effective main cavity length will be changed as a result of the phase response of the etalon, thus changing the resonance frequency of the main cavity. To minimize coupling between detuning of the etalon and this frequency pulling of the main cavity, L 1 is chosen to be much smaller than the main cavity length.
For high reflection, the laser carrier is nearly anti-resonant with the etalon, as illustrated by Fig. 5(b). For the coupled resonator system to be impedance matched, L 1 must be detuned from the etalon resonance such that R 1(L 1) = R 2, as shown in Fig. 5(b). The RF interrogation technique described in Section 3 can be used to control the laser detuning from the etalon resonance, by actuating on its spacing L 1. This in turn controls the reflectivity of R 1.
Fig. 5. A three mirror coupled cavity. (a) the two closely spaced mirrors form an etalon, whose reflectivity can be varied by their spacing; (b) if the output coupler reflectivity R 2 is high, then the laser must be well away from resonance of the etalon for the three mirror system to be impedance matched.
5. Experimental Technique
Figure 6 shows a simplified schematic of the optical layout and control system of our impedance matching experiment. The optical system is a three mirror coupled resonator [5, 14], where the output mirror emulates the reflection from a locked Michelson interferometer. For the remainder of this paper we refer to the m 1a-m 1b cavity as the etalon, with reflectivity R 1, while the m 1b-m 2 cavity is referred to as the main cavity. In order to actuate on the operating conditions of the etalon and main cavity, the mirrors were mounted on piezo-electric transducers.
Fig. 6. Simplified experimental layout of the impedance matching experiment. The reflected and transmitted signals are measured on Rx and Tx respectively. The carrier laser has two modulation frequencies. One set of 190 MHz PM sidebands and one set of 30 MHz AM sidebands. The signal laser is used to create a single sideband providing a signal frequency to map the frequency response of the cavity system.
For full operation of the impedance matching control technique, the system requires two feedback control loops. Firstly, the main cavity must be kept resonant with the laser. This is implemented with a Pound-Drever-Hall frequency locking feedback control loop [13], by using phase modulation (PM) sidebands at 190 MHz. The second control loop is the impedance matching readout itself, and employs AM sidebands at 30 MHz.
The etalon mirrors, m 1a and m 1b in Fig. 6, had a measured power transmissivity of t 1a 2 = 0.09 and t 1b 2 = 0.03 respectively, resulting in a theoretical etalon finesse of ℱ ≈ 50. The spacing between the etalon mirrors was set to 3.7 mm, resulting in a full-width half-maximum (FWHM) bandwidth of ≈ 800 MHz. The length of the main cavity was designed so that its free spectral range (FSR) was twice the PM sideband frequency. This corresponds to a main cavity length of 39.47 cm for an FSR of 380 MHz. Mirror m 2 had a measured power transmissivity of t 2 2 = 0.0045. The bandwidth of the main cavity was determined by its finesse, and hence the reflectivity of the etalon. The etalon reflectivity, in turn, was determined by the frequency detuning between its resonance and the laser.
The optical signal reflected from the coupled cavity was observed with photodetector Rx, while the transmission was monitored on photodetector Tx. Each photodetector provided a DC and AC coupled output. The AC coupled output of Rx was split into three separate paths to distribute the measured RF signals to the appropriate demodulators for the PDH and impedance matching control loops. The demodulation electronics of each error signal included a phase shifter operating on the appropriate RF local oscillator. After demodulation, the AM and PM error signals were filtered and amplified to use as feedback signals for mirrors m 1a and m 2 respectively.
We note that the design of the three mirror cavity is such that the PM sidebands produce orthogonal frequency error signals between the main cavity and the etalon when the laser is detuned. The PM sideband frequency was chosen to be half the main cavity FSR, so that the sidebands are anti-resonant with the main cavity when locked. Furthermore, due to the small etalon spacing, the PM sideband frequency is much smaller than the FWHM of the etalon resonance. The result is that the sidebands experience primarily (1) phase rotation relative to the carrier due to the main cavity; but (2) differential amplitude attenuation due to the etalon. This implies that any frequency detuning of the laser relative to the main cavity would produce an error signal when demodulated in-quadrature (Q). In contrast, for the etalon, the frequency error signal is observed when demodulated in-phase (I). Hence, the PDH frequency error signal for locking the main cavity, extracted in-quadrature, is largely immune to variations in etalon spacing. The concept of I and Q locking was discussed in some detail in Ref. [15].
6. Experimental Results
In order to observe the demodulated AM signal for different impedance conditions, a PZT ramp was applied to mirror m 2 to tune the main cavity resonance frequency. The spacing of the etalon was then stepped by applying a voltage offset to the PZT of mirror m 1a. Figure 7 shows the resulting coupled cavity transmission and demodulated AM signals. Figure 7(a) was obtained by recording the voltage from photodetector Tx with an oscilloscope, while m 2 was scanned. It shows the cavity transmitted power for three cases. The three plots are normalized to the transmitted peak of the impedance matched spectrum, and displays sample spectra for three conditions: (a) blue dotted trace for an over-coupled cavity; (b) solid black trace for an under-coupled cavity; and (c) red dashed trace for an impedance matched cavity. These three cases were obtained by applying three different voltages to the PZT of mirror m 1a, thereby choosing three different effective etalon reflectivities. We can see that, on resonance, the transmitted power is highest for the impedance matched cavity as expected, since we expect its circulating power to be maximized.
Figure 7(b) shows the oscilloscope traces for the corresponding demodulated AM error signals for these three cases while the main cavity resonance frequency was scanned. The region of interest in these error signals is where the carrier is exactly on resonance with the main cavity. For the case of an over coupled cavity, the AM error signal on resonance was negative, as expected by Eq. (3). Furthermore, we see that the AM error signal on resonance was positive when the cavity was under coupled; and zero when impedance matched. Thus as predicted by Eq. (3), over and under coupled cavities yield error signals with opposing polarities. This demodulated AM error signal can thus be used for cavity coupling feedback control to lock the cavity to impedance matching, where the error signal is zero.
Fig. 7. (a) The normalised transmitted intensity for three different etalon phase offsets; (b) The impedance locking error signal for three different etalon phase offsets. As the etalon phase is varied the coupling condition of the cavity changes from over coupled to impedance matched to under coupled.
From Fig. 7 we can also infer that dynamic tuning of the coupled cavity bandwidth was possible by adding or subtracting a DC voltage from the impedance matching locking error signal. In this way, the system behaves much like a cavity with a variable reflectivity mirror [5].
Fig. 8. The coupled cavity reflection response compared with the impedance locking error signal when the etalon spacing was scanned. (a) the reflection response of the etalon, when m 2 was blocked; (b) the reflection response of the three mirror system while the laser was locked to resonance; (c) the demodulated AM error signal corresponding to (b).
To demonstrate the relationship between the etalon reflectivity, R 1, and the coupling condition of the three-mirror resonator system, we scanned the etalon spacing by applying a 10 Hz, 60 V P-P ramp to the PZT of mirror m 1a. Figure 8(a) displays the reflection response of the etalon, which was first described by the cartoon displayed in Fig. 5(b). The oscilloscope trace was obtained with photodetector Rx while mirror m 2 was blocked, so that only the optical response of the etalon was measured.
The corresponding response of the three-mirror coupled resonator system as measured by Rx, after mirror m 2 was unblocked, is shown in Fig. 8(b). This plot was obtained while the coupled cavity system was locked to the laser, using PDH active feedback to the PZT of mirror m 2, to keep the laser resonant. The coupled resonator system was impedance matched at the two minima of the plot in Fig. 8(b). This is where the reflected carrier intensity was zero, and the only reflected power observed by photodetector Rx was due to the RF sidebands. The impedance matched operating points are marked by the red dashed vertical lines in Fig. 8. We can deduce that in the central region of Figs. 8(a) and 8(b), in the neighborhood of the etalon resonance, the main cavity was over coupled, since r 1 < r 2. On either side of this central region far from etalon resonance, R 1 should be high, such that the cavity becomes under coupled. When the laser is resonant with the etalon, we expect its reflected power as shown in Fig. 8(a) to be minimum, and the etalon is mostly transmissive. When this happens, the reflected power of the three mirror system, as seen in Fig. 8(b), is at maximum, as the compound cavity becomes severely over coupled. This is when the carrier power experience mainly the mirror reflection of m 2.
With the compound cavity still kept on resonance using PDH frequency locking, the corresponding demodulated AM error signal of the coupled cavity system was observed as displayed in Fig. 8(c). This error signal was obtained using the technique described by Section 3, by demodulating the RF electronic signal from photodetector Rx, while the etalon spacing was continuously scanned. As predicted by Eq. (3), the voltage crosses zero (at 34 nm) on either side of main cavity resonance, when it was impedance matched. This voltage was positive when under coupled and negative when over coupled, consistent with Fig. 8(b) and Eq. (3).
We note that using the impedance matching error signal, a wide range of R 1 values can be accessed, by offsetting the lock point from impedance matched with an adder/subtractor. From Fig. 8(c), we see that using this technique, an etalon spacing tuning range of ±25 nm around the impedance matching point can be reasonably controlled, which corresponds to over 70 percent of possible R 1 values, from about 0.2 up to 0.95.
Fig. 9. Magnitude of the coupled cavity signal responses. ‘×’ Signal response fit of an over coupled cavity with a bandwidth γ = 8.8 MHz. ‘○’ Signal response fit of an impedance matched cavity with a bandwidth γ= 2.27 MHz. ‘◇’ Signal response fit of an under coupled cavity with a bandwidth γ = 1.33 MHz.
7. Signal Response of the Coupled Cavity
The signal response of the coupled cavity system was measured while the main cavity was frequency locked using PDH reflection locking, and the etalon spacing locked using the impedance matching locking technique. To measure the signal response, a second laser was injected through the back mirror of the interferometer [16]. The frequency of this signal laser was changed by tuning its temperature. While changing the signal laser frequency, the beat signal at the frequency difference of the two lasers was measured on photodetector Rx and captured on a spectrum analyzer. Signal responses were measured using a 0V, +14mV, and -9mV DC voltage offset on the impedance matching error signal. Figure 9 shows the results for the over-coupled, impedance matched and under-coupled signal responses. The signal responses clearly show the change in signal enhancement when the etalon is detuned from impedance matching. In this experiment the signal response bandwidth was varied by a factor of six. Of specific interest is that the signal response of the interferometer is optimised when the coupled cavity is impedance matched. Increasing the reflectivity of the etalon any further reduces signal response, degrades overall sensitivity and also reduces the resulting signal bandwidth.
8. Conclusions
In conclusion, we have examined the dynamic behaviour of a new shot-noise limited closed loop feedback technique to actively sense the impedance matching condition of a three mirror coupled cavity. We have shown that it can be used to optimize the circulating power in the coupled cavity and provides an error signal that is suitable for tuning signal bandwidth of interferometric measurements over a large range of values.
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