March 15, 2011 / Vol. 36, No. 6 / OPTICS LETTERS 933 Balancing interferometers with slow-light elements Alireza Marandi,* Brian T. Lantz, and Robert L. Byer Ginzton Laboratory, Stanford University, 450 Via Palou, Stanford, California 94305-4088, USA *Corresponding author: [email protected] Received December 7, 2010; revised February 9, 2011; accepted February 10, 2011; posted February 18, 2011 (Doc. ID 139284); published March 10, 2011 In this Letter we show that interferometers with unbalanced arm lengths can be balanced using optical elements with appropriate group delays. For matched group delays of the arms, the balanced interferometer becomes insen- sitive to the frequency noise of the source. For experimental illustration, a ring resonator is employed as a slow-light element to compensate the arm-length mismatch of a Mach–Zehnder interferometer. An arm-length mismatch of 9:4 m is compensated by a ring resonator with a finesse of 70 and a perimeter of 42 cm. © 2011 Optical Society of America OCIS codes: 120.3180, 260.2030, 120.5050. Measurements with interferometric sensors can be very Figure 1 shows a schematic diagram of an MZI that has sensitive to the frequency noise of the source if the arm a lossless dispersive element with length of L3 in one lengths are not matched. This can happen in many appli- arm. The output current of the balanced detector is given cations where matching the arm lengths physically is not by easily achievable. Stabilizing the frequency of the source ¼ ð Δ − þ ϕ Þ ð Þ by locking it to a stable reference can improve the noise Iout I0 cos k0 L k3L3 0 ; 1 performance of the sensor to some extent [1]. To achieve where k0 ¼ ω=c is the free-space wavenumber, ΔL ¼ balanced arm lengths optical delay lines or high-index L1 − L2 is the arm-length mismatch, k3 ¼ ωn3ðωÞ=c, and materials can be used if feasible. For the Laser Inter- ϕ0 accounts for other phase differences of the paths. ferometer Space Antenna (LISA), different methods are The sensitivity of the output to the frequency deviation proposed to cancel the effects of phase noise of the can be calculated by source including time delay interferometry [2] and lock- ing the laser frequency to the LISA arm [3]. Our approach dIout ΔL dk3 for dealing with this issue is matching the group delays of ¼ I0 − þ L3 sinðk0ΔL − k3L3 þ ϕ0Þ: ð2Þ dω c dω the arms. We show that, by using a slow-light element in – the shorter arm of a Mach Zehnder Interferometer (MZI), This gives two options for making the output insensi- the group delays in the arms can be matched to achieve a dI tive to frequency changes, i.e., out ¼ 0. The first option is balanced interferometer. dω to make the argument of the sine equal to zero, which is Recently, slow-light elements have been used in diff- equivalent to matching the optical path lengths in the erent types of interferometric systems. It was demon- arms for white-light interferometry. The second option strated that an atomic vapor cell, as an element with is to make the coefficient of the sine equal to zero, which negative dispersion in a resonator, can broaden the cav- ity bandwidth while keeping the cavity buildup high [4]. A will give us slow-light element in one arm of an interferometer has Δ ω L ¼ dk3 ⇒ gr ≡ d ¼ L3 ð Þ been used to improve the spectral sensitivity [5]. It is also L3 v3 c ; 3 c dω dk3 ΔL demonstrated that a variable slow-light medium can re- gr place the moving arm of a conventional Fourier trans- where v3 is the group velocity of light in the dispersive gr form interferometer [6]. Moreover, slow-light elements element. For ΔL ≫ L3, v3 has to be slower than the free- have been studied for a wide range of applications includ- space speed of light by a factor of L3=ΔL to make the MZI ing optical gyroscopes [7] and optical communication insensitive to the frequency deviations. This condition is systems [8], and group delay matching techniques have the same as matching the group delays in the arms; been employed for optical coherence tomography [9]. that is, In this Letter we first show that a slow-light element in the shorter arm of an unbalanced interferometer can make the output insensitive to frequency deviations of the source. This is conceptually different from [5], where, in a similar scheme, it is shown that the spectral sensi- tivity of a spectroscopic interferometer is proportional to the group index of the slow-light medium. We also pro- pose using a ring resonator as a slow-light element, which is limited to the cavity bandwidth, and derive the resona- tor parameters required to compensate the arm-length mismatch. In the experiment, we show that the effects Fig. 1. (Color online) Schematic diagram of an MZI with a of the intentional frequency modulation of the source dispersive element in one arm. M1, M2, M3, interferometer can be eliminated by this method. mirrors; BS1, BS2, beam splitters; PD1, PD2, photodetectors. 0146-9592/11/060933-03$15.00/0 © 2011 Optical Society of America 934 OPTICS LETTERS / Vol. 36, No. 6 / March 15, 2011 Δ L3 ¼ L ð Þ gr : 4 v3 c One way of realizing a slow group velocity in the short- er arm is using an optical resonator. The required param- eters for a lossless three-mirror ring resonator can be calculated using the cavity transmission equation [10]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − T T expð−jωL =cÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiin out in=out T r ¼ ; ð5Þ 1 − RinRoutR3 expð−jωpm=cÞ where T in, T out and Rin, Rout, R3 are power transmissions and reflections of the mirrors, respectively; Lin=out is the Fig. 2. (Color online) Schematic diagram of an MZI with a re- distance between the input and the output mirrors; and sonator in one arm to compensate the arm-length mismatch. pm is the perimeter of the cavity. Hence the phase Laser frequency is locked to the cavity using a PDH scheme. response is HW, half-wave plate; PBS1, polarizing beam splitter; EOM, electro-optic modulator; LO, local oscillator; BS1, BS2, beam ω splitters, PD1, PD2, PD3, photodetectors; M1, M2, M3, inter- Lin=out −1 grt sinðωpm=cÞ ∠T r ¼ π − − tan ; ð6Þ ferometer mirrors; PZT, piezoelectric transducer. c 1 − grt cosðωpm=cÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cavity where the group delay obeys Eq. (7). On the other where grt ¼ RinRoutR3, and 1 − grt is the round-trip field ω ¼ 2πcq hand, frequency locking can suppress the frequency loss. At resonant frequencies, q p (q is an integer), the group delay of the cavity is givenm by noise of the source over the bandwidth of the controller. This is usually in the kilohertz range, and is set by the frequency response of the actuator. The amount of noise d Lin=out pm grt tgr ¼ − ∠T rjω¼ω ¼ þ : ð7Þ rejection provided by the PDH locking depends on the dω q c c 1 − g rt gain of the controller. Increasing the gain can improve the noise rejection; however, the rejection strength is lim- Assuming that ΔL ¼ L1 − L2 − L , to have equal in=out ited since a very high gain can increase the amount of group delays in the arms, the grt should satisfy the follow- ing condition: electronically induced noise and lead to stability issues. In contrast, with our slow-light method, high gain is not ΔL required, and one can make the interferometer insensi- g ¼ : ð8Þ tive to the frequency noise within the transmission band rt p þ ΔL m of the cavity, which can be in megahertz or gigahertz range. As the frequency departs from the center of the For ΔL ≫ pm, the group delay matching condition in terms of the cavity finesse is cavity response, the suppression strength decreases. A slow-light element, in this case a ring resonator, can ΔL be beneficial over the conventional multi-bounce delay F ≃ π : ð9Þ lines [12] from the practical point of view. As the arm- pm length mismatch increases, realizing a delay line becomes more challenging compared to a high-finesse This condition can also be expressed in terms of the optical resonator on a stable substrate. cavity storage time and effective path length. For a high- One concern in using a resonator as a slow-light ele- finesse cavity, the storage time and the effective path ment is the effects of mechanical noises, such as thermal length are given by expansion of the cavity. A small deviation in the peri- meter causes a large deviation in the relative phase of Fp Fp τ ¼ m ¼ m ð Þ the arms because of the steep phase response at the re- s π ;Leff π ; 10 c sonance. A cavity length deviation within the frequency respectively [10]. Substitution into Eq. (9) shows that the band of the PDH system results in a deviation in the laser group delay matching is the same as matching the storage frequency. The propagation of frequency deviated light Δ time of the cavity to the time delay difference of the arms. over the distance of L causes the deviation in the rela- σ This is equivalent to setting the effective path length of tive phases of the arms, ϕ, which is given by the cavity equal to the arm-length mismatch. The experimental setup is depicted in Fig. 2. The cavity ω ΔL σϕ ¼ σpm; ð11Þ has a finesse of 70 and a perimeter of 42 cm correspond- c pm ing to an arm-length mismatch of 9:4 m based on Eq.