OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 25 — Dec. 10, 2007
  • pp: 17401–17409
« Show journal navigation

On fundamental quantum noises of whispering gallery mode electro-optic modulators

Andrey B. Matsko, Anatoliy A. Savchenkov, Vladimir S. Ilchenko, David Seidel, and Lute Maleki  »View Author Affiliations


Optics Express, Vol. 15, Issue 25, pp. 17401-17409 (2007)
http://dx.doi.org/10.1364/OE.15.017401


View Full Text Article

Acrobat PDF (115 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

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

1. Introduction

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 [18

18. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Whispering gallery mode electro-optic modulator and photonic microwave receiver,” J. Opt. Soc. Am. B 20, 333–342 (2003). [CrossRef]

], 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 [27

27. A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002). [CrossRef]

]. 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 [28

28. E. A. Mishkin and D. F. Walls, “Quantum statistics of three interacting boson field modes”, Phys. Rev. 185, 1618–1628 (1969). [CrossRef]

]. 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

29. V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in Fabry-Perot interferometer,” Phys. Lett A 287, 331–338 (2001). [CrossRef]

, 30

30. V. B. Braginsky and S. P. Vyatchanin, “Low quantum noise tranquilizer for Fabry-Perot interferometer”, Phys. Lett A 293, 228–234 (2002). [CrossRef]

] are similar to an unstable EOM [27

27. A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002). [CrossRef]

]. 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

31. B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, “Observation and characterization of an optical spring,” Phys. Rev. A 69, 051801 (2004). [CrossRef]

, 32

32. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005). [CrossRef] [PubMed]

, 33

33. H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13, 5293–5301 (2005). [CrossRef] [PubMed]

, 34

34. T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, and N. Mavalvala, “Measurement of radiation-pressure-induced optomechanical dynamics in a suspended Fabry-Perot cavity,” Phys. Rev. A 74, 021802 (2006). [CrossRef]

].

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 [35

35. E. DAmbrosio and W. Kells, “Considerations on parametric instability in FabryPerot interferometer,” Phys. Lett. A 299, 326–330 (2002). [CrossRef]

]. 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 [18

18. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Whispering gallery mode electro-optic modulator and photonic microwave receiver,” J. Opt. Soc. Am. B 20, 333–342 (2003). [CrossRef]

]. 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

13. D. A. Cohen and A. F. J. Levi, “Microphotonic millimetr-wave receiver architecture,” Electron. Lett. 37, 37–39 (2001). [CrossRef]

, 17

17. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Sub-microWatt photonic microwave receiver,” IEEE Photon. Technol. Lett. 14, 1602–1604 (2002). [CrossRef]

] 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 :

Ĥ0=h¯ωââ+h¯ωb̂b̂+h¯ω+b̂+b̂++h¯ωcĉĉ,
(1)
V̂=h¯g(b̂ĉâ+b̂+ĉâ)+adjoint,
(2)

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

â˙=iωâig*(b̂ĉ+ĉb̂+),
(3)
b̂˙=iωb̂igĉâ,
(4)
b̂˙+=iω+b̂+igĉâ,
(5)
ĉ˙=iωcĉigb̂âig*âb̂+.
(6)

Set (3–4) was solved in the case of undepleted pump in [28

28. E. A. Mishkin and D. F. Walls, “Quantum statistics of three interacting boson field modes”, Phys. Rev. 185, 1618–1628 (1969). [CrossRef]

]. 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

n̂b++n̂b+n̂a=N̂,
(7)
n̂bn̂b+n̂c=N̂S,
(8)
Ŝ+Ŝ++Ŝ+Ŝ+(n̂bn̂b+)2=N̂C,
(9)
Ŝ+Ĉ+ĈŜ+=N̂I;
(10)

where n̂ξξ̂ ξ̂ is the photon number operator.

It is easy to obtain a closed set of two equations

Ĉ˙=igŜ+
(11)
Ŝ˙+=ig[n̂c+N̂S]Ĉ,
(12)

which is transformed to a second order nonlinear equation with respect to operator Ĉ

Ĉ̈+g2[n̂c(t)+N̂S]Ĉ=0.
(13)

and an equation with respect to the operator n̂c

n̂̈c+g2[n̂c2+n̂cN̂CN̂S2+N̂S]=0.
(14)

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

n̂c6N̂sin2[gt(N̂3)1/4].
(15)

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

n̂c|t0N̂g2t2.
(16)

Eq. (16) corresponds to the solution of a linearized problem for the photon number [28

28. E. A. Mishkin and D. F. Walls, “Quantum statistics of three interacting boson field modes”, Phys. Rev. 185, 1618–1628 (1969). [CrossRef]

]. The general linearized solution is

b̂(t)=b̂(0)+ig*a*(ĉ(0)t+c̃t2/2),
(17)
b̂+(t)=b̂+(0)iga(ĉ(0)t+c̃t2/2),
(18)
ĉ=ĉ(0)+c˜t,c˜=i(gab̂(0)+g*a*b̂+(0)).
(19)

This solution results in

n̂b|t0N̂g2t2+N̂2g4t4/4,
(20)
n̂b+t0N̂2g4t4/4.
(21)

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

Â˙=γÂig*(B̂Ĉ+ĈB̂+)+F̂A,
(22)
B̂˙=γB̂igĈÂ+F̂B,
(23)
B̂˙+=γB̂+igĈÂ+F̂B+,
(24)
Ĉ˙=γMĈigB̂Âig*ÂB̂++F̂C.
(25)

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

F̂A=2Pγh¯ω,
(26)
F̂C=2PmwγMh¯ωc,
(27)
F̂A(t)F̂A(t)=2γδ(tt),
(28)
F̂B±(t)F̂B±(t)=2γδ(tt),
(29)
F̂C(t)F̂C(t)=2γM(n¯th+1)δ(tt),
(30)
F̂C(t)F̂C(t)=2γMn¯thδ(tt),
(31)

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

F̂B±=f̂B±(ω˜)eiω˜tdω˜2π,
(32)
F̂C=F̂C+f̂C(ω˜)eiω˜tdω˜2π,
(33)

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

B̂±=δB̂±(ω˜)eiω˜tdω˜2π,
(34)
Ĉ=C+δĈ(ω˜)eiω˜tdω˜2π.
(35)

The solution of the linearized set (22–25) is given by

δB̂(ω)=(1+g2A2ΓΓM)f̂B(ω˜)Γ+g2A2ΓΓMf̂B+(ω˜)ΓigAΓf̂C(ω˜)ΓM,
(36)
δB̂+(ω˜)=(1g2A2ΓΓM)f̂B+(ω˜)Γg2A2ΓΓMf̂B(ω˜)ΓigAΓf̂C(ω˜)ΓM,
(37)

where we use notations Γ=γ-̃ and ΓM=γ M-̃.

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

n̂B=(2γ+γM)γM2γ2[g2A2γM(γM+γ)]2+g2A2γ(γM+γ)(n¯th+1),
(38)
n̂B+=(2γ+γM)γM2γ2[g2A2γM(γM+γ)]2+g2A2γ(γM+γ)n¯th,
(39)
n̂C=n¯th+g2A2γM(γM+γ).
(40)

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)

AF̂Aγ,CF̂CγM,BigAγC*,B+igAγC.
(41)

EoutEin,Eout=Ein2igγC*,Eout+=Ein2igγC.
(42)

as well as fluctuations of the optical fields

êout(ω˜)=(Γ*Γ+2γg2A2Γ2ΓM)êin(ω˜)+2γg2A2Γ2ΓMêin+(ω˜)2igAγγMΓΓMêC(ω˜),
(43)
êout+(ω˜)=(Γ*Γ2γg2A2Γ2ΓM)êin+(ω˜)2γg2A2Γ2ΓMêin(ω˜)2igAγγMΓΓMêC(ω˜).
(44)

We used expressions A2γτ=2Ein, FA=Ein2γτ, Eout±=B±2γτ, 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

Pout+=Pin4g2γ22(Pmw+ΔPmw)h¯ωcγM.
(45)

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

Pmw=Pinmw+S(ω˜)dω˜2π=Pinmw+γγMγ+γMkBT2+Pg2(γ+γM)2ωcω(1+γM2γ),
(46)
S(ω˜)=kBTγ2γM2(γ2+ω˜2)(γM2+ω˜2)+Pωcω2g2γ2γM(γ2+ω˜2)2(γM2+ω˜2),
(47)

and the noise

ΔPmw2=2n¯th(n¯th+1)[h¯ωc2γγMγ+γM]2+Pinmw[h¯ωγ2PγγM4g2h¯ωc+2S(ω˜)]dω˜2π.

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:

ΔP(ω˜)=h¯ωc[h¯ωγ2PγγM4g2+122Ph¯ωγ4g2γγMγ4γM2Γ4ΓM2]+2kBTγ2γM2Γ2ΓM2.
(48)

This noise power density determines the signal-to-noise ratio of the receiver S/N=P inmw/[∫ΔP(ω̃)̃/(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

ΔP(ω˜)min=2h¯ωc+2kBT,
(49)

if

2Popth¯ωγ4g2γγM=2.
(50)

Let us represent the optimal power in measurable units. We introduce a directly measurable modulation coefficient m

m=Pout+Pout=8g2γ2Pmwh¯ωcγM.
(51)

It is easy to measure experimentally the input microwave power P mwsat corresponding to m≃1. The optimum optical power is given then by

Popt=2PmwmωωcPmwsatωωc.
(52)

The value of the optimal power shows if the nonlinear noise is important. In case of small optical power PP opt the noise is primarily given by the photon shot noise and ΔP(ω̃) equals to

ΔP(ω˜)Pmw2h¯ωPout++2kBTPmw.
(53)

In the case of large optical power P>P opt the nonlinear noise exceeds the shot noise.

4. Discussion

According to theoretical analysis of WGM modulators [18

18. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Whispering gallery mode electro-optic modulator and photonic microwave receiver,” J. Opt. Soc. Am. B 20, 333–342 (2003). [CrossRef]

] the saturation power can be found from

Pmwsath¯ω2ωc216g2Q2Qm.
(54)

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 [18

18. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Whispering gallery mode electro-optic modulator and photonic microwave receiver,” J. Opt. Soc. Am. B 20, 333–342 (2003). [CrossRef]

]). 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 [27

27. A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002). [CrossRef]

], 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

g=ωLh¯2MωM.
(55)

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 [29

29. V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in Fabry-Perot interferometer,” Phys. Lett A 287, 331–338 (2001). [CrossRef]

]

Pth=γ4MωM2L2QQM,
(56)

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 [35

35. E. DAmbrosio and W. Kells, “Considerations on parametric instability in FabryPerot interferometer,” Phys. Lett. A 299, 326–330 (2002). [CrossRef]

]. 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 γγ M)

n̂M=n¯th+PPth
(57)

even in a classically stable system. According to the estimations presented in [29

29. V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in Fabry-Perot interferometer,” Phys. Lett A 287, 331–338 (2001). [CrossRef]

], 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.

5. Conclusion

Acknowledgement

The research described in this paper was supported by DARPA.

References and links

1.

E. I. Gordon and J. D. Rigden, “Fabry-Perot electro-optic modulator,” Bell System Tech. J. 42, 155–179 (1963).

2.

R. C. Alferness, “Waveguide electro-optic modulators,” IEEE Trans. Microwave Theor. and Techniques 30, 1121–1137 (1982). [CrossRef]

3.

K. P. Ho and J. M. Kahn, “Optical frequency comb generator using phase modulation in amplified circulating loop”, IEEE Photon. Technol. Lett. 5, 721–725 (1993). [CrossRef]

4.

T. Kawanishi, S. Oikawa, K. Higuma, Y. Matsuo, and M. Izutsu, “LiNbO3 resonant-type optical modulator with double-stub structure,” Electron. Lett. 37, 12441246 (2001).

5.

I. L. Gheorma and R. M. Osgood, “Fundamental limitations of optical resonator based high-speed EO modulators,” IEEE Photon. Technol. Lett. 14, 795–797 (2002). [CrossRef]

6.

M. Kato, K. Fujiura, and T. Kurihara, “Generation of super-stable 40 GHz pulses from Fabry-Perot resonator integrated with electro-optic phase modulator,” Electron. Lett. 40, 299–301 (2001). [CrossRef]

7.

N. Benter, R. P. Bertram, E. Soergel, K. Buse, D. Apitz, L. B. Jacobsen, and P. M. Johansen, “Large-area Fabry-Perot modulator based on electro-optic polymers,” Appl. Opt. 44, 6235–6239 (2005). [CrossRef] [PubMed]

8.

M. Kato, K. Fujiura, and T. Kurihara, “Generation of a superstable Lorentzian pulse train with a high repetition frequency based on a Fabry-Perot resonator integrated with an electro-optic phase modulator,” Appl. Opt. 44, 1263–1269 (2005). [CrossRef] [PubMed]

9.

V. S. Ilchenko, X. S. Yao, and L. Maleki, “Microsphere integration in active and passive photonics devices,” Proc. SPIE 3930, 154–162 (2000). [CrossRef]

10.

V. S. Ilchenko and L. Maleki, “Novel whispering-gallery resonators for lasers, modulators, and sensors,” Proc. SPIE 4270, 120–130 (2001). [CrossRef]

11.

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “Microphotonic modulator for microwave receiver,” Electron. Lett. 37, 300–301 (2001). [CrossRef]

12.

D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, “High-Q microphotonic electro-optic modulator,” Solid State Electron. 45, 1577–1589 (2001). [CrossRef]

13.

D. A. Cohen and A. F. J. Levi, “Microphotonic millimetr-wave receiver architecture,” Electron. Lett. 37, 37–39 (2001). [CrossRef]

14.

D. A. Cohen and A. F. J. Levi, “Microphotonic components for a mm-wave receiver,” Solid State Electron. 45, 495–505 (2001). [CrossRef]

15.

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer micro-ring filters and modulators,” J. Lightwave Technol. 20, 1968–1975 (2002). [CrossRef]

16.

L. Maleki, A. F. J. Levi, S. Yao, and V. Ilchenko, “Light modulation in whispering-gallery-mode resonators,” US papent 6,473,218 (2002).

17.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Sub-microWatt photonic microwave receiver,” IEEE Photon. Technol. Lett. 14, 1602–1604 (2002). [CrossRef]

18.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Whispering gallery mode electro-optic modulator and photonic microwave receiver,” J. Opt. Soc. Am. B 20, 333–342 (2003). [CrossRef]

19.

M. A. J. Weldon, S. V. Hum, R. J. Davies, and M. Okoniewski, “Traveling-wave ring circuit for resonant enhancement of electrooptic modulators,” IEEE Photon. Technol. Lett. 161295–1297 (2004). [CrossRef]

20.

M. Hossein-Zadeh and A. F. J. Levi, “Self-homodyne RF-optical LiNbO3 microdisk receiver,” Solid State Electron. 49, 1428–1434 (2005). [CrossRef]

21.

H. Tazawa and W. H. Steier, “Linearity of ring resonator-based electrooptic polymer modulator,” Electron. Lett. 41, 12971298 (2005). [CrossRef]

22.

M. Hossein-Zadeh and A. F. J. Levi, “14.6-GHz LiNbO3 microdisk photonic self-homodyne RF receiver,” IEEE Trans. Microwave Theory Tech. 54, 821–831 (2006). [CrossRef]

23.

H. Tazawa and W. H. Steier “Analysis of ring resonator-based traveling-wave modulators,” IEEE Photon. Technol. Lett. 18, 211–213 (2006). [CrossRef]

24.

H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, “Ring resonator-based electrooptic polymer traveling-wave modulator,” J. Lightwave Technology 24, 3514–3519 (2006). [CrossRef]

25.

S. Li, F. Yi, X. M. Zhang, and S. L. Zheng, “Optimized electrode structure for a high-Q electro-optic microdisk-based optical phase modulator,” Microwave Opt. Technol. Lett. 49, 313–316 (2007). [CrossRef]

26.

B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, “Electrooptic polymer ring resonator modulation up to 165 GHz,” IEEE J. Sel. Top. Quantum Electron. 13, 104–110 (2007). [CrossRef]

27.

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002). [CrossRef]

28.

E. A. Mishkin and D. F. Walls, “Quantum statistics of three interacting boson field modes”, Phys. Rev. 185, 1618–1628 (1969). [CrossRef]

29.

V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, “Parametric oscillatory instability in Fabry-Perot interferometer,” Phys. Lett A 287, 331–338 (2001). [CrossRef]

30.

V. B. Braginsky and S. P. Vyatchanin, “Low quantum noise tranquilizer for Fabry-Perot interferometer”, Phys. Lett A 293, 228–234 (2002). [CrossRef]

31.

B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, “Observation and characterization of an optical spring,” Phys. Rev. A 69, 051801 (2004). [CrossRef]

32.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95, 033901 (2005). [CrossRef] [PubMed]

33.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13, 5293–5301 (2005). [CrossRef] [PubMed]

34.

T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, and N. Mavalvala, “Measurement of radiation-pressure-induced optomechanical dynamics in a suspended Fabry-Perot cavity,” Phys. Rev. A 74, 021802 (2006). [CrossRef]

35.

E. DAmbrosio and W. Kells, “Considerations on parametric instability in FabryPerot interferometer,” Phys. Lett. A 299, 326–330 (2002). [CrossRef]

OCIS Codes
(190.4360) Nonlinear optics : Nonlinear optics, devices
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(230.2090) Optical devices : Electro-optical devices
(230.4110) Optical devices : Modulators
(230.5750) Optical devices : Resonators

ToC Category:
Novel Concepts and Theory

History
Original Manuscript: September 25, 2007
Revised Manuscript: November 13, 2007
Manuscript Accepted: November 13, 2007
Published: December 10, 2007

Virtual Issues
Physics and Applications of Microresonators (2007) Optics Express

Citation
Andrey B. Matsko, Anatoliy A. Savchenkov, Vladimir S. Ilchenko, David Seidel, and Lute Maleki, "On fundamental quantum noises of whispering gallery mode electro-optic modulators," Opt. Express 15, 17401-17409 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-17401


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. E. I. Gordon and J. D. Rigden, "Fabry-Perot electro-optic modulator," Bell System Tech. J. 42, 155-179 (1963).
  2. R. C. Alferness, "Waveguide electro-optic modulators," IEEE Trans. Microwave Theor. and Techniques 30, 1121-1137 (1982). [CrossRef]
  3. K. P. Ho and J. M. Kahn, "Optical frequency comb generator using phase modulation in amplified circulating loop", IEEE Photon. Technol. Lett. 5, 721-725 (1993). [CrossRef]
  4. T. Kawanishi, S. Oikawa, K. Higuma, Y. Matsuo, and M. Izutsu, "LiNbO3 resonant-type optical modulator with double-stub structure," Electron. Lett. 37, 12441246 (2001).
  5. I. L. Gheorma and R. M. Osgood, "Fundamental limitations of optical resonator based high-speed EO modulators," IEEE Photon. Technol. Lett. 14, 795-797 (2002). [CrossRef]
  6. M. Kato, K. Fujiura, and T. Kurihara, "Generation of super-stable 40 GHz pulses from Fabry-Perot resonator integrated with electro-optic phase modulator," Electron. Lett. 40, 299-301 (2001). [CrossRef]
  7. N. Benter, R. P. Bertram, E. Soergel, K. Buse, D. Apitz, L. B. Jacobsen, and P. M. Johansen, "Large-area Fabry-Perot modulator based on electro-optic polymers," Appl. Opt. 44, 6235-6239 (2005). [CrossRef] [PubMed]
  8. M. Kato, K. Fujiura, T. Kurihara, "Generation of a superstable Lorentzian pulse train with a high repetition frequency based on a Fabry-Perot resonator integrated with an electro-optic phase modulator," Appl. Opt. 44, 1263-1269 (2005). [CrossRef] [PubMed]
  9. V. S. Ilchenko, X. S. Yao, and L. Maleki, "Microsphere integration in active and passive photonics devices," Proc. SPIE 3930, 154-162 (2000). [CrossRef]
  10. V. S. Ilchenko and L. Maleki, "Novel whispering-gallery resonators for lasers, modulators, and sensors," Proc. SPIE 4270, 120-130 (2001). [CrossRef]
  11. D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, "Microphotonic modulator for microwave receiver," Electron. Lett. 37, 300-301 (2001). [CrossRef]
  12. D. A. Cohen, M. Hossein-Zadeh, and A. F. J. Levi, "High-Q microphotonic electro-optic modulator," Solid State Electron. 45, 1577-1589 (2001). [CrossRef]
  13. D. A. Cohen and A. F. J. Levi, "Microphotonic millimetr-wave receiver architecture," Electron. Lett. 37, 37-39 (2001). [CrossRef]
  14. D. A. Cohen and A. F. J. Levi, "Microphotonic components for a mm-wave receiver," Solid State Electron. 45, 495-505 (2001). [CrossRef]
  15. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, "Polymer micro-ring filters and modulators," J. Lightwave Technol. 20, 1968-1975 (2002). [CrossRef]
  16. L. Maleki, A. F. J. Levi, S. Yao, and V. Ilchenko, "Light modulation in whispering-gallery-mode resonators," US papent 6,473,218 (2002).
  17. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Sub-microWatt photonic microwave receiver," IEEE Photon. Technol. Lett. 14, 1602-1604 (2002). [CrossRef]
  18. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, "Whispering gallery mode electro-optic modulator and photonic microwave receiver," J. Opt. Soc. Am. B 20, 333-342 (2003). [CrossRef]
  19. M. A. J. Weldon, S. V. Hum, R. J. Davies, and M. Okoniewski, "Traveling-wave ring circuit for resonant enhancement of electrooptic modulators," IEEE Photon. Technol. Lett. 161295-1297 (2004). [CrossRef]
  20. M. Hossein-Zadeh, and A. F. J. Levi, "Self-homodyne RF-optical LiNbO3 microdisk receiver," Solid State Electron. 49, 1428-1434 (2005). [CrossRef]
  21. H. Tazawa and W. H. Steier, "Linearity of ring resonator-based electrooptic polymer modulator," Electron. Lett. 41, 12971298 (2005). [CrossRef]
  22. M. Hossein-Zadeh, and A. F. J. Levi, "14.6-GHz LiNbO3 microdisk photonic self-homodyne RF receiver," IEEE Trans. Microwave Theory Tech. 54, 821-831 (2006). [CrossRef]
  23. H. Tazawa, W. H. Steier "Analysis of ring resonator-based traveling-wave modulators," IEEE Photon. Technol. Lett. 18, 211-213 (2006). [CrossRef]
  24. H. Tazawa, Y. H. Kuo, I. Dunayevskiy, J. D. Luo, A. K.-Y. Jen, H. R. Fetterman, and W. H. Steier, "Ring resonator-based electrooptic polymer traveling-wave modulator," J. Lightwave Technology 24, 3514-3519 (2006). [CrossRef]
  25. S. Li, F. Yi, X. M. Zhang, and S. L. Zheng, "Optimized electrode structure for a high-Q electro-optic microdisk-based optical phase modulator," Microwave Opt. Technol. Lett. 49, 313-316 (2007). [CrossRef]
  26. B. Bortnik, Y.-C. Hung, H. Tazawa, B.-J. Seo, J. D. Luo, A. K.-Y. Jen, W. H. Steier, and H. R. Fetterman, "Electrooptic polymer ring resonator modulation up to 165 GHz," IEEE J. Sel. Top. Quantum Electron. 13, 104-110 (2007). [CrossRef]
  27. A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, "Highly nondegenerate all-resonant optical parametric oscillator," Phys. Rev. A 66, 043814 (2002). [CrossRef]
  28. E. A. Mishkin and D. F. Walls, "Quantum statistics of three interacting boson field modes," Phys. Rev. 185, 1618-1628 (1969). [CrossRef]
  29. V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett A 287, 331-338 (2001). [CrossRef]
  30. V. B. Braginsky and S. P. Vyatchanin, "Low quantum noise tranquilizer for Fabry-Perot interferometer", Phys. Lett A 293, 228-234 (2002). [CrossRef]
  31. B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, "Observation and characterization of an optical spring," Phys. Rev. A 69, 051801 (2004). [CrossRef]
  32. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity," Phys. Rev. Lett. 95, 033901 (2005). [CrossRef] [PubMed]
  33. H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Radiation-pressure-driven micro-mechanical oscillator," Opt. Express 13, 5293-5301 (2005). [CrossRef] [PubMed]
  34. T. Corbitt, D. Ottaway, E. Innerhofer, J. Pelc, N. Mavalvala, "Measurement of radiation-pressure-induced opto-mechanica dynamics in a suspended Fabry-Perot cavity," Phys. Rev. A 74, 021802 (2006). [CrossRef]
  35. E. DAmbrosio and W. Kells, "Considerations on parametric instability in Fabry Perot interferometer," Phys. Lett. A 299, 326-330 (2002). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited