Using the example of whispering gallery mode (WGM) electro-optic modulator (EOM) we show that the majority of phase EOMs, particularly the resonant types, introduce additional quantum noise to the modulated light. The noise power grows quadratically with the optical power and results from the unavoidable spontaneous emission process originating from the strongly nondegenerate parametric interaction. This latter process is the physical basis for modulation.
© 2007 Optical Society of America
Both microwave and optical resonators are used to increase efficiency of electro-optical modulators (EOMs) within limited frequency bands around respective microwave and optical carrier frequencies –. Electro-optically active whispering gallery mode (WGM) resonators are attractive for this application because they feature high quality factors at any optical frequency within the transparency window of the resonator host material –. The modulators generally operate either at the baseband (see, e.g., ) or at bands detuned from the baseband by several free spectral ranges (FSRs) of the WGM resonator. The later versions of the modulator have the highest efficiency at higher microwave (mw) frequencies (see, e.g., [11, 17, 26]).
Modulation in all-resonant EOMs occurs because the microwave field in an externally pumped microwave resonator interacts with modes of a nonlinear optical WGM resonator. All but one optical fields are the modulation sidebands with frequencies equal to a multiple of the microwave pump frequency. The microwave frequency is selected to correspond to the FSR of the WGM resonator so that the optical modulation sidebands coincide with WGMs. The modulation coefficient defined as the ratio of the output power of the first optical harmonic and the optical pump power is proportional to P mw Q 2 QM , where Q and Q M are the loaded quality factors of the optical and microwave modes correspondingly, and P mw is the applied microwave power. Therefore, the higher the quality factors are, the smaller the microwave power has to be for achieving the same modulation efficiency.
It is known that a resonant EOM is stable in the limits of classical physics if the resonator has an equidistant spectrum. A microwave field and optical sidebands are not generated from a single optical pump . However, a quantum analysis of the interaction of three bosonic quantum fields in the presence of a classical pump shows that the process corresponding to the EOM operation is intrinsically unstable . A single optical pump is able to generate both mw and optical harmonics. This quantum parametric instability results in an additional “spontaneous emission” noise of an EOM.
The phenomenon of the induced noise is even more important because multiple physical processes can be described with the same model as the all-resonant EOM. For example, an optical parametric process that converts light into light and mechanical motion of mirrors in long-baseline interferometers [29, 30] are similar to an unstable EOM . Parametric instability in the interferometers arises from ponderomotively mediated coupling between mechanical oscillations of suspended mirrors and the light that is used for detection of the signal shift of the mirrors. The effect is undesirable because it creates an upper limit for the energy stored in the resonator limiting the optical power that can be used to retrieve information on small motion of the mirrors. The sensitivity of the detection increases with the light power and, hence, such a parametric process poses an upper limit on the measurement sensitivity. The opto-mechanical instability has been actually demonstrated in various kinds of resonators [31, 32, 33, 34].
It was shown that, similar to the example of the EOM, the ponderomotive instability is suppressed if the optical resonator has a one-dimensional equidistant spectrum . However, according to the analysis presented in this paper, even then the ponderomotive process would result in an additional noise that can influence the sensitivity of the long-baseline interferometric detectors. It worth note, though, that this noise is small in existing detectors.
In this paper we present a theoretical study of the fundamental quantum noises introduced by all resonant WGM phase modulators . We show that these modulators must posses a specific spontaneous emission noise that results in an additional contamination of the modulated optical signal. The relative intensity noise of the modulated light increases with the optical pump power. Therefore, the signal-to-noise ratio of a receiver based on the EOM [13, 17] is maximized at the optimum optical pump power.
We study the interaction of three optical and a single microwave resonator modes via χ (2) nonlinearity. We initially consider a purely theoretical conservative problem of a nonlinear interaction of modes having infinite Q-factors. After that we focus on a frequency mixing problem involving a lossless interaction of externally pumped optical and microwave modes having finite Q-factors (Q-factors are given by the loading of the resonator, not by the absorption). We then study the generation of Stokes and anti-Stokes sidebands in the other two optical modes occurring as a result of the interaction. The frequencies of the sidebands,ω 0-ω M and ω 0+ω M, are given by frequencies of optical (ω 0) and microwave (ω M) pumping respectively. We assume that neighboring optical modes are resonant with the sidebands.
2. Closed system
The Hamiltonian describing the interaction of three optical (â, b̂-, and b̂+) and one microwave (ĉ) modes is presented as a sum of free (Ĥ0) and interaction (V̂) parts Ĥ=Ĥ0+V̂ :
where ω and ω ± are the eigenfrequencies of the optical modes, ω c is the eigenfrequency of the microwave mode, â, b̂±, and ĉ are the annihilation operators for these modes respectively, g is the coupling constant given by the system geometry and nonlinearity.
We derive equations of motion for the field operators using the Hamiltonian
Set (3–4) was solved in the case of undepleted pump in . However, such a solution does not show the maximum number of microwave photons that could be generated in the absence of the microwave signal.
Set (3–4) is integrable in the case of equidistant modes. Assuming that ω-ω-=ω +-ω=ω c, and introducing operators Ŝ+=(gb̂†-â+g*â† b̂+)exp(-iωct)/|g| and Ĉ=ĉexp(-iωct) we obtain several integrals of motion
where n̂ξ≡ξ̂† ξ̂ is the photon number operator.
It is easy to obtain a closed set of two equations
which is transformed to a second order nonlinear equation with respect to operator Ĉ
and an equation with respect to the operator n̂c
Eq. (14) has a solution in terms of elliptic integrals. For simplicity reasons we do not discuss this solution here, and only give an approximate solution for the expectation value of n̂c if the pump mode a is in the coherent state and 〈N̂〉≫1 and sideband modes b ± are initially in the vacuum state
This equation correctly describes the maximum of the photon number as well as the oscillation period, however it gives √2 faster growth at t→0. The correct expression is
This solution results in
The basic message of this idealized analysis is that the optical sidebands as well as the microwave field can be excited from the ground state if some photons are present in the pump mode. This leads us to the conclusion that the presence of an optical drive will result in additional noise of the Stokes and anti-Stokes sidebands in an EOM. To estimate the significance of this effect we have to study an open system.
3. Open system
In reality, the optical and microwave resonators are open systems, pumped externally, and the pumping and decay terms do not follow from the Hamiltonian approach. For an open symmetric system Eqs. (3–6) transform to
where Â, B̂±, and Ĉ are the slowly-varying amplitudes of the operators â, b̂, and ĉ respectively; γ and γ M are the optical and microwave decay rates respectively (we assume that the modes are overloaded so the intrinsic losses can be neglected and γ and γ M are the decay rates of the loaded modes); F̂A, F̂B±, and F̂C are the Langevin forces with properties given by
where 〈…〉 stands for the reservoir averaging, P is the power of the external optical pump of mode A, P mw is the power of the external microwave pump of mode C, n̄th=[exp(h̄ω c/k B T)-1]-1 is the average number of thermal photons in the microwave mode, k B is the Boltzmann constant, and T is the temperature. The other expectation values, quadratic deviations, and correlations are equal to zero.
Let us solve the linearized set (22–25) in steady state assuming an undepleted classical pump Â→A=〈F̂A〉/γ=const. We present the force and the field operators as a sum of an expectation and fluctuational parts like
where 〈f̂A(ω)f̂A(ω)†〉=〈f̂B ±(ω)f̂B ±(ω)†〉=4πγδ(ω-ω ′) and 〈f̂C(ω)f̂C(ω)†〉=4πγ M(n̄th+1)δ(ω-ω ′); and
The solution of the linearized set (22–25) is given by
where we use notations Γ=γ-iω̃ and ΓM=γ M-iω̃.
It is easy to see now that in the absence of microwave pumping (〈F̂C〉=0) and in the steady state, the expectation values of the photon number in the optical modes and the photon number in the microwave mode are
Let us find the output signal in the system if the microwave pumping is present. Neglecting the optical saturation of the microwave as well as the pump field, we obtain the expectation values for the fields from (22–25)
We next derive the expression for the expectation values
as well as fluctuations of the optical fields
We used expressions , , , is the resonator round trip time τ=2πR/c, where R is the radius of the resonator. We assume, for the sake of simplicity, that the resonator is phase matched (or empty,), so that τ=τ M.
Let us assume that one measures Ê† out+Êout + to detect a monochromatic microwave signal with averaged power P inmw. The optical power that corresponds to photon number in the “+” mode is given by expression
The microwave power P mw is the averaged microwave signal and ΔP mw is the noise part of the signal we calculate using Eq. (44). We find the following expressions for the averaged signal
and the noise
It is worth noting that the detection bandwidth for the noise floor (calculated for P inmw→0) is given by the geometrical average of the bandwidths of the microwave and optical resonators (γ -1+γ -1 M)-1.
It is possible to introduce the noise power density for the case P inmw≠0:
This noise power density determines the signal-to-noise ratio of the receiver S/N=P inmw/[∫ΔP(ω̃)dω̃/(2π)]. Eq. (48) contains three terms. The first term, inversely proportional to P, comes from the photon shot noise. The second term, proportional to P, comes from the spontaneous emission noise. Finally, the third term, independent on P, comes from the thermal noise.
We are interested in case |ω̃|<γ,γ M. The receiver has the smallest noise power density
Let us represent the optimal power in measurable units. We introduce a directly measurable modulation coefficient m
It is easy to measure experimentally the input microwave power P mwsat corresponding to m≃1. The optimum optical power is given then by
The value of the optimal power shows if the nonlinear noise is important. In case of small optical power P≪P opt the noise is primarily given by the photon shot noise and ΔP(ω̃) equals to
In the case of large optical power P>P opt the nonlinear noise exceeds the shot noise.
According to theoretical analysis of WGM modulators  the saturation power can be found from
In practice P mwsat exceeds 0.01 mW and ω/ω c>103 so that we always have P opt>10 mW (P mwsat=10mW, ω/ω c=2×104, P=5mW, and P opt=200Win ). Therefore, it is safe to assume that the modulator does not introduce any significant nonlinear noise to the modulated signal in majority of the present as well as future experiments and applications. The noise is determined by the shot noise. Nevertheless, the spontaneous emission noise can be significant and must be taken into account in applications involving very efficient modulators and/or large optical powers.
The spontaneous emission noise is present in any kind of EOM based on parametric interactions. In particular, the expression (52) can also be used for transient (not resonant) types of modulators because it does not directly contain the information about the resonator quality factors. However, the influence of the spontaneous emission is rather small for small optical powers. On the other hand, in modulators based on high-Q WGM resonators this noise can become important because P mwsat~(Q 2 Q M)-1, and increasing the quality factors will result in buildup of the nonlinear noises. This issue has to be studied in more details.
Let us now estimate the importance of the noise for long base optical interferometers. Following discussion in , we consider a Fabry-Perot interferometer with one movable mirror that has mass M and mechanical resonance frequency ω M. The distance between mirrors of the resonator is equal to L. The coupling constant between mechanical degrees of freedom of the mirror and optical modes is
The description of the behavior of the interferometer corresponds to the behavior of the modulator (the same Hamiltonian), where mechanical mode substitutes microwave mode. The interferometer becomes unstable if scattering to the red-shifted optical mode is more preferable (pump photon decays to the red shifted mode and generates a single quantum in the mechanical oscillator, while scattering to the blue-shifted optical mode results in absorption of the mechanical energy). If the scattering to the blue-shifted optical mode is forbidden because of the morphology of the system, the instability is characterized by threshold optical power 
where Q and Q M are the optical and mechanical quality factors respectively. If scattering to both red- and blue-shifted modes are equally allowed, the system is classically stable . However, according to the calculations reported in this paper, the average photon number in the mechanical mode is equal to (follows from Eq. (40) when γ≫γ M)
even in a classically stable system. According to the estimations presented in , P/P th≃300 for the planned pump power in the LIGO interferometers. This value is small compared with the number of thermal quanta in the mode n̄th. Hence, the nonlinear noise is not important in realistic LIGO interferometers.
We have shown that parametrically induced spontaneous emission introduces additional specific type of noise in the phase-type electro-optic modulators. The noise has rather fundamental importance. It becomes more important in all-resonant modulators (e.g. whispering gallery mode modulators) characterized with high-Q optical as well as high-Q microwave resonances, or in any modulators operating at high enough optical power. However, it is rather negligible compared with other noises in the majority of existing realistic systems.
The research described in this paper was supported by DARPA.
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