Optoelectronic oscillator stabilized to an intra-loop Fabry-Perot cavity by a dual servo system

Jang Myun Kim; D. Cho

doi:10.1364/OE.18.014905

1. Introduction

Oscillators are basic building blocks and experimental instruments in electronic devices and scientific research. From a Wien-bridge oscillator at radio frequencies (RF) to a laser at optical frequencies, there are a variety of oscillators covering a wide range of the spectrum. For a microwave frequency range, which is important in communications and metrology, there is an atomic clock or a maser with high spectral purity as well as tunable devices like a yittrium-iron-garnet oscillator and a voltage-controlled oscillator. In 1981 Neyer and Voges invented a microwave oscillator based on a photonic principle [1]. A laser beam modulated by an electro-optic modulator (EOM) in a Mach-Zehnder configuration is detected by a fast photodector to produce a beat signal, which is amplified and fed back to the EOM. In 1996 this device was reinvented as a stable microwave oscillator by a group at the Jet Propulsion Laboratory [2]. It is called an optoelectronic oscillator (OEO). As a simple realization of an OEO, see Fig. 1 . With developments of EOMs and photodetectors operating at tens of gigahertz, an OEO has become an important addition to a repertoire of microwave oscillators.

Fig. 1 Optoelectronic oscillator. $f_{1}$ is the laser frequency and $f_{2}$ is the frequency of the sideband produced by the modulator.

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An OEO with a simple loop of Fig. 1 is not a very useful device because of its low Q factor and a broad linewidth. One can add a long optical fiber to increase the Q factor at the expense of bulky size and at the risk of multimode operation due to the reduced mode spacing. One has to add a narrow-band microwave filter in the electronic path to ensure singe mode operation. Alternatively, either an atomic vapor cell with a coherent population trapping (CPT) resonance or a high-Q optical resonator can be included as a part of the OEO loop. Reduced group velocity through the CPT cell can dramatically increase the effective loop length. An OEO with a CPT filter using alkali-metal atoms can serve as an absolute microwave frequency reference [3]. As a high-Q optical resonator, a mode-locked laser in a coupled OEO configuration [4] as well as a Fabry-Perot cavity (FPC) can be used. An OEO with an intra-loop optical resonator can oscillate at an integer multiple of the resonator free spectral range (FSR).

Recently an OEO with a FPC as a free-running microwave filter was demonstrated [5]. It resulted in the OEO frequency stability of 2.6 kHz out of 10.287 GHz over 1 min compared with 5.75 kHz with a RF filter. In their experiment the FPC serves as a passive microwave filter and there was no active stabilization of the laser frequency with respect to the cavity. In this article we report our construction and characterization of an OEO with a FPC, which includes stabilizations of the laser frequency with respect to the cavity mode and the OEO loop length with respect to the cavity FSR. The stabilizations further improved the OEO frequency stability by more than an order of magnitude to 200 Hz out of 3.6 GHz over 1 min.

2. Theory

The theory of an OEO with a FPC has been worked out in detail in Ref [3]. Here we summarize only the details which are relevant to our experiment. We consider a FPC which consists of two identical mirrors with transmission T and negligible loss. They are separated by length $l .$ The effective optical length of the OEO loop excluding the FPC is $L,$ which includes both optical and electronic paths. The boundary condition for an OEO oscillation is

K_{q} L + ϕ_{FPC} = 2 π q,

where

K_{q}

is the wavenumber,

ϕ_{FPC}

is the phase shift due to the FPC and qis an integer. When a laser field

E_{i n}

with frequency ω is incident on the FPC, its output field is given by

E_{o u t} = \frac{T e^{i δ ϕ / 2}}{1 - (1 - T) e^{i δ ϕ}} E_{i n},

where

δ ϕ = 2 l δ ω / c .

δ ω

is a detuning from the longitudinal mode at

ω_{m} = m c / 2 l

with m being an integer. When

δ ϕ ≪ T ≪ 1

, Eq. (2) can be approximated by

E_{o u t} = e^{i δ ϕ / T} E_{i n} .

If the carrier has a detuning

δ ω_{c}

from

ω_{m}

and the sideband has a detuning

δ ω_{s}

from

ω_{m + 1},

ϕ_{FPC} = \frac{2 l}{c} \frac{δ ω_{s} - δ ω_{c}}{T} .

In an ideal case, where both the carrier and the sideband are resonant with the FPC, $ϕ_{FPC} = 0.$ From Eq. (1) the OEO oscillation frequency $F_{q}$ in this case is given by the relation,

F_{q} = q \frac{c}{L} .

We note that $F_{q}$ is also the beat frequency between the carrier and the sideband. The OEO oscillates at the cavity FSR. If the OEO loop length L changes due to either environmental perturbation or electronic noise, $F_{q}$ is not the same as the FSR anymore and the laser fields become detuned. When L changes by $δ L$ , we use Eq. (1) to find the OEO frequency shift,

δ F_{q} = - F_{q} \frac{δ L}{L + 2 l / T} .

We note that the effective OEO loop length is $L + 2 l / T .$ When T is small, the intra-loop FPC can increase the loop length significantly. If the cavity length l changes by $δ l$ due to, for example, thermal expansion, the OEO frequency changes by

δ F_{q} = - F_{q} \frac{2 δ l / T}{L + 2 l / T} .

When

L ≪ 2 l / T,

Eqs. (6) and (7) reduce to

δ F_{q} ≅ - T F_{q} δ L / 2 l

and

δ F_{q} ≅ - F_{q} δ l / l,

respectively.

F_{q}

is much more susceptible to

δ l

than

δ L

and we may conclude that in order to obtain stable oscillation frequency it is most important to keep the FPC stable. In practice, while the cavity can be placed in an evacuated chamber with temperature stabilization, the remaining part of the OEO with L is inevitably open to the environment. To suppress

δ F_{q}

due to

δ L

, one has to use a cavity with a high enough finesse.

3. Apparatus

Our experimental setup is shown in Fig. 2 . The apparatus consists of three major parts: (i) The laser frequency is first stabilized to a cesium $D_{2}$ transition by using modulation transfer spectroscopy (MTS) [6]. (ii) An acousto-optic modulator (AOM) shifts the stabilized laser frequency to lock it to the FPC by using Pound-Drever-Hall (PDH) technique. We employ the double locking scheme with an intention to relate the OEO frequency to an atomic transition as an absolute frequency standard. We will further discuss this possibility in the conclusions section. (iii) The laser beam is then phase modulated by a 3.6-GHz EOM and both the carrier and the sidebands couple to Gaussian modes of the FPC. The cavity transmission is detected by a fast photodetector to generate a 3.6-GHz beat signal, which is amplified and fed back to the EOM to close an OEO loop. In the following section we will describe an additional servo system to adjust the OEO loop length by using a phase shifter to further improve the OEO stability.

Fig. 2 Experimental setup. ECDL: extended-cavity diode laser; PD: photodetector; LO: local oscillator; PS: power splitter; QWP: quarterwave plate; PMF: polarization-maintaining optical fiber; $Δ ϕ :$ phase shifter; AM: amplitude modulation input; VCO: voltage controlled oscillator.

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As a light source we use a standard extended-cavity diode laser with a Littrow configuration. It delivers 38 mW at 852 nm. After a 65-dB optical isolator and an anamorphic prism pair, the laser beam goes through a 10-MHz EOM (New Focus Model 4001). It imprints phase modulation for the MTS and the PDH systems. When the laser is locked to the Cs transition by using MTS, its root-mean-squared frequency jitter is less than 200 kHz.

The frequency-stabilized laser beam then makes a double pass through the AOM so that we can tune its frequency without steering it. Double-pass deflection efficiency is 45%. The 100-MHz AOM tunable over ±12 MHz is driven by a voltage-controlled oscillator, which acts as a transducer for the PDH locking system. Output from the AOM is coupled to an optical fiber for spatial filtering. Coupling efficiency is 55%. The fiber is a polarization-maintaining type because the modulation efficiency of the downstream 3.6-GHz EOM (New Focus Model 4431) depends critically on the light polarization.

The EOM is of a resonant type with a 3-dB bandwidth of only a few MHz, and it serves as a narrow-band microwave filer as well. Output power from the EOM is monitored and stabilized by adjusting the RF power to the AOM. It is also mode matched to the FPC. 50% of the power is coupled to the Gaussian mode of the cavity. Length of the FPC is 41.5 mm so that its FSR is 3.61 GHz. The full width at half maximum (FWHM) of the cavity resonance is only 540 kHz. The 10-MHz sidebands are reflected by the input mirror and do not interfere with the OEO operation. From the cavity linewidth we estimate the mirror transmission T to be 470 ppm and the effective cavity length $2 l / T$ to be 180 m. The cavity is installed in an octagonal chamber [7]. The chamber is kept under vacuum and its temperature is actively stabilized. The cavity can be temperature tuned so that one of its longitudinal modes is within the AOM bandwidth of the laser frequency. When the EOM modulation frequency matches the FSR of the cavity, the plus and minus first-order sidebands as well as the carrier transmit through the cavity. Because the pair of sidebands generated by phase modulation are $180^{\circ}$ out of phase, photodetector output from the cavity transmission has a very small 3.61-GHz beat signal. We remove one of the sidebands by reflecting the cavity-transmitted beam off a solid etalon. Free spectral range of the etalon is 12.2 GHz and the FWHM of its resonance is 700 MHz. Its resonance frequency is temperature tuned to the sideband, which is to be removed. The system could have been simplified much if we used an intensity modulator instead of an EOM for phase modulation. We could not use an intensity modulator because the modulation bandwidth and operating wavelength of commercially available ones do not match our laser and cavity systems. The reflected beam is detected by a 9-GHz photodetector (Hamamatsu Model G4176). Its output is amplified and applied to the EOM, via a voltage-controlled attenuator (VCA, UMCC Model AT-E000) and a phase shifter (Mini-Circuits Model JSPHS-2484), to form an OEO.

4. OEO Operations

In order to initiate and maintain a stable OEO oscillation, we have to control two parameters: one is the loop gain and the other is the loop length. The loop gain is adjusted by the VCA, which controls the microwave power applied to the EOM. The OEO loop length, which includes both optical and electronic paths, should be such that the FSR of the loop is commensurate with that of the FPC. We put the fast photodetector and its upstream lens on a translation stage to make a coarse adjustment of the optical path length. For a fine adjustment, we use the phase shifter on the electronic path. Unfortunately, the optimal bandwidths of both the VCA (4-8 GHz) and the phase shifter (2.6 GHz) were outside 3.6 GHz and introduced, respectively, 2-10 dB and 10 dB power loss. We had to add another power amplifier of 20-dB gain at the expense of unnecessary amplifier noise. The combined gain of the two cascaded amplifiers added up to 70 dB.

While the carrier is locked to the FPC, as we increase the loop gain and adjust the loop length the OEO begins to oscillate. A small part of the microwave power is taken off by a power splitter (Mini-Circuits Model ZAPD-4-S+) and it is used as an OEO output. The output is monitored by a spectrum analyzer and a frequency counter. We also take a part of the laser beam right before the FPC and analyze its spectrum by using a confocal spectrum analyzer. It lets us know the relative size of the carrier and the sidebands generated by the 3.6-GHz EOM under an OEO oscillation. At an optimal loop gain, approximately 1 W of microwave power is applied to the EOM with a modulation index of 0.88. In order to put maximum power to the first-order sidebands, 2.5 W is required. When more than 1.5 W is applied, however, the output laser beam from the EOM is steered off the straight path due a thermal lensing effect.

To evaluate the OEO performance, we measure the Allan deviation and the microwave spectrum. The OEO output is mixed with a frequency synthesizer signal at 3.64 GHz to produce a mixer output at around 30 MHz. The output frequency is counted to measure the Allan deviation. Both the frequency synthesizer and the counter are phase locked to a rubidium clock (Stanford Research Model PRS10). We first studied performance of the OEO without the FPC. Its output spectrum has a carrier at 3.61 GHz and sidebands at ±16 MHz offset. It implies that the loop length of the OEO without the FPC is 19 m. The Allan deviation is shown as an uppermost trace (■) in Fig. 3(a) . When the FPC is added, we expect the loop length is increased to 200 m and its FSR is reduced to 1.5 MHz. However, we cannot observe any sidebands due to the strong mode selection by the cavity. The Allan deviation with the FPC is shown as a middle trace (●). Its long term frequency fluctuation is 1 kHz. Although it is a significant improvement over an OEO without the FPC, it is problematic that the fluctuation tends to become larger as the integration time τ increases.

Fig. 3 (a) Allan deviation of the OEO frequency without the Fabry-Perot cavity (■) and with the cavity when it is free running (●) and when its loop length is corrected with respect to the cavity free spectral range (○). (b) Microwave spectrum of the OEO output with the loop-length correction. Resolution bandwidth was 1 kHz.

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We may analyze the situation from the perspectives of Eqs. (6) and (7): (i) Although the chamber that houses the FPC is carefully temperature stabilized and insulated, the cavity resonance frequency still shows thermal drift of 1 MHz during 10 minutes. It represents, however, $δ l$ of only $2 \times 10^{- 11}$ m or $δ F = 15$ Hz from Eq. (7). (ii) Although the carrier is locked to the FPC, the PDH servo bandwidth is limited to 33 kHz by the propagation delay in the AOM. From the error signal we estimate that the root-mean-squared frequency offset from the cavity mode is 30 kHz. The fluctuation, however, is of high frequency and we expect its effect on the long-term OEO stability is not large. (iii) If we interpret the long term drift of 1 kHz in terms of drift in $L,$ it corresponds to $δ L = 50$ μm from Eq. (6). It is of an expected order of magnitude considering the open optical path and thermal drift in the high power amplifiers.

We incorporate a negative feedback loop in order to correct the drift in $L .$ Because the carrier is locked to the cavity mode independently of the OEO loop, change in L causes a detuning for the sideband. The detuning $δ ω_{s}$ produces phase shift $ϕ_{FPC} = 2 l δ ω_{s} / c T$ relative to the carrier. We measure the phase shift by comparing the carrier-sideband beat signals before and after the FPC. The beat signal from the laser beam taken downstream of the FPC is mixed with that from the upstream in quadrature. The mixer output is low-pass filtered to produce a signal proportional to $\sin ϕ_{FPC}$ . The error signal is used to correct the OEO loop length via the phase shifter. We show the sideband transmission and the error signal as a function of the sideband detuning in Fig. 4 . For this measurement the EOM is driven directly by a frequency synthesizer near 3.6 GHz while the carrier stays locked. The sideband transmission in the figure is on top of the carrier transmission. The lower trace (○) in Fig. 3 (a) shows the Allan deviation measured with the negative feedback loop. It is smaller than that of the free-running case, and especially the frequency fluctuation becomes smaller for larger $τ .$ The long-term frequency drift is on the order of 200 Hz, which is $4 \times 10^{- 4}$ of the cavity linewidth. The microwave spectrum of the OEO output is shown in Fig. 3 (b). The carrier frequency is 3.6105 GHz and the resolution bandwidth of the spectrum analyzer is set to 1 kHz. The fractional power spectral density of the phase noise to the carrier power is −80 dBc/Hz. This relatively large phase noise is mainly due to the large amplification factor.

Fig. 4 The cavity transmission in mW and the error signal in arbitrary units vs. the sideband detuning.

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5. Conclusions

We have constructed and characterized an optoelectronic oscillator with an intra-loop Fabry-Perot cavity. The cavity serves as a frequency reference as well as providing mode selection and increasing the effective loop length. We built a dual servo system in our OEO so that it was self-stabilizing to the free spectral range of the cavity. The carrier is locked to the cavity mode by the Pound-Drever-Hall method, and the OEO loop length is stabilized with reference to the FSR of the cavity. With additional servo systems for the seed laser frequency and power we achieved long term frequency stability of 200 Hz for the OEO frequency, which represents more than an order of magnitude improvement over the previous design [5].

The OEO with a FPC described in this article can be improved in a few respects: (i) We used an EOM built for a phase modulation, and we had to remove one of the sidebands to obtain a beating. An EOM in a Mach-Zehnder configuration produces an intensity modulation [8] simplifying the setup greatly. (ii) In addition, while the half-wave voltage of our EOM is 36 V, that of a commercial intensity modulator is only a few volts and it requires far less driving power. As mentioned above, the operating bandwidths of the VCA and the phase shifter in our setup did not match the OEO frequency and we had to provide extra amplification. The consequent amplifier noise can be avoided by using suitable microwave devices. (iii) Recently there has been progress in both fabrication and mounting of a FPC as a stable frequency reference [9]. Use of a higher finesse cavity with better environmental isolation should improve the OEO frequency stability.

With our double locking scheme of the laser frequency first to the cesium transition and then to the cavity mode by using an AOM, the OEO frequency $F_{q}$ can be related to the atomic frequency $f_{C s}$ by

f_{C s} = 2 f_{A O M} + q F_{q},

where

f_{A O M}

is the RF frequency to the AOM and

q ≅ 10^{5}

in our case. Resonance frequencies of the cesium D₂ transitions are known to within 5.5 kHz [10] and the RF frequency can be measured with high precision. It implies that the OEO frequency can be absolutely defined within 55 mHz. Although with the current frequency stability of 200 Hz it is not very meaningful to specify the absolute frequency of the OEO in terms of the atomic transition, with further improvements this scheme can be applied to use the OEO as an absolute frequency standard.

Acknowledgments

This work was supported by the National Research Foundation of Korea (Grant No. 2009-0080091). J.M.K. acknowledges support from Seoul R&BD program (Grant No. 10543).

References and links

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8. T. R. Ranganath and S. Wang, “Ti-diffused LiNbO₃ branched-waveguide modulators: Performance and design,” IEEE J. Quantum Electron. 13(4), 290–295 (1977). [CrossRef]

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10. V. Gerginov, C. E. Tanner, S. Diddams, A. Bartels, and L. Hollberg, “Optical frequency measurements of 6s²S_1/2 - 6p²P_3/2 transition in a ¹³³Cs atomic beam using a femtosecond laser frequency comb,” Phys. Rev. A 70(4), 042505 (2004). [CrossRef]

Optoelectronic oscillator stabilized to an intra-loop Fabry-Perot cavity by a dual servo system

Abstract

1. Introduction

2. Theory

3. Apparatus

4. OEO Operations

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (8)

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