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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 20 — Sep. 24, 2012
  • pp: 21977–21991
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Entangled photon generation using four-wave mixing in azimuthally symmetric microresonators

Ryan M. Camacho  »View Author Affiliations


Optics Express, Vol. 20, Issue 20, pp. 21977-21991 (2012)
http://dx.doi.org/10.1364/OE.20.021977


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Abstract

A novel quantum mechanical formulation of the bi-photon wavefunction and spectra resulting from four-wave mixing is developed for azimuthally symmetric systems. Numerical calculations are performed verifying the use of the angular group velocity and angular group velocity dispersion in such systems, as opposed their commonly used linear counterparts. The dispersion profile and bi-photon spectra of two illustrative examples are given, emphasizing the physical origin of the effects leading to the conditions for angular momentum and energy conservation. A scheme is proposed in which widely spaced narrowband entangled photons may be produced through a four-wave mixing process in a chip-scale ring resonator. The entangled photon pairs are found to conserve energy and momentum in the four-wave mixing interaction, even though both photon modes lie in spectral regions of steep angular group velocity dispersion.

© 2012 OSA

1. Introduction

Entangled photon generation is an essential ingredient in quantum information processing and quantum communication. In many instances, entangled photons with narrow spectral bandwidths are desirable in order to couple to long-lived quantum memories or repeaters [1

1. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997). [CrossRef]

, 2

2. T. C. Ralph and G. J. Pryde, “Optical quantum computation,” in Prog. Optics,vol. 54, E. Wolf, ed. (Elsevier Science2010), pp. 209–269. [CrossRef]

]. Methods to produce narrowband entangled photons have traditionally focused on bulk table-top systems, including using spontaneous raman scattering in atomic systems [3

3. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003). [CrossRef] [PubMed]

], narrowband filtering of the output of spontaneous parametric downconversion in a crystal [4

4. A. Haase, N. Piro, J. Eschner, and M. W. Mitchell, “Tunable narrowband entangled photon pair source for resonant single-photon single-atom interaction,” Opt. Lett. 34, 55–57 (2009). [CrossRef]

, 5

5. K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. 11, 013049 (2009). [CrossRef]

], or using a cavity to enchance narrowband spontaneous parametric downconversion in a crystal [6

6. Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. 83, 2556–2559 (1999). [CrossRef]

]. Recently, there has also been interest in producing narrowband entangled photons using four-wave mixing (FWM) in chip-scale micro-resonators [7

7. S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, P. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558–16570 (2009). [CrossRef] [PubMed]

9

9. J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a Silicon-on-Insulator micro-resonator,” Opt. Express 19, 1470–1483 (2011). [CrossRef] [PubMed]

], which would reduce the cost and power of bi-photon production, as well as offer scalability.

FWM in microresonators is usually performed in systems with azimuthal symmetry, since cylindrical devices most easily allow for near-equal mode spacing, small mode volumes, and high optical quality factors. While spontaneous FWM is used to produce entangled photons in fibers and microstructures [10

10. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002). [CrossRef]

12

12. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. I. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007). [CrossRef]

] an identical FWM process can be driven above the parametric oscillation threshhold to produce optical frequency combs [16

16. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007). [CrossRef]

22

22. A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100GHz repetition rates,” Opt. Lett. 37, 875–877 (2012). [CrossRef] [PubMed]

]. Stimulated FWM processes can also enable frequency conversion in micro-resonator based systems [23

23. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000). [CrossRef]

, 24

24. A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16, 4881–4887 (2008). [CrossRef] [PubMed]

]. In all these cases, the underlying electronic nonlinearity is degenerate FWM occurring via 2ωp = ωs + ωi, where ωj(j = p, s, i) are the frequencies of the pump, signal and idler waves, respectively.

While several efforts have been made to theoretically model the spectrum resulting from spontaneous and driven FWM in azimuthally symmetric systems, most have approximated cylindrical resonators as unfolded straight waveguides rather than working in cylindrical coordinates, which are more naturally suited to systems with azimuthal symmetry. A notable exception is the work of Chembo et al. [25

25. Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010). [CrossRef]

, 26

26. Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. 104, 103902 (2010). [CrossRef] [PubMed]

], who performed the first fully-vectorial treatment of cascaded FWM in a spherical resonator, taking into account all components of the electric field, spatial mode overlaps, and resonator dispersion profiles. While their results only apply to spherical resonators and do not take into account material dispersion, they successfully track the spectral evolution resulting from the FWM interaction.

Scholz et al. [27

27. M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518–3523 (2009). [CrossRef]

] analytically studied the quantum correlations resulting from spontaneous FWM in a linear micro-cavity, and later Chen et al. [9

9. J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a Silicon-on-Insulator micro-resonator,” Opt. Express 19, 1470–1483 (2011). [CrossRef] [PubMed]

] applied a similar formalism to the case of azimuthally symmetric ring resonators, predicting the generation of a frequency-bin entangled comb of photon pairs. However, the quantum formalism of Ref. [9

9. J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a Silicon-on-Insulator micro-resonator,” Opt. Express 19, 1470–1483 (2011). [CrossRef] [PubMed]

] employs an effective linear group velocity and linear k-vector, which are integrated over an unfolded cavity of linear length L to enforce phase matching. In a cylindrical system, however, linear momentum is zero and the linear wavevector and group velocities are not well defined, especially as the radius of the resonator is reduced. This is because no translational symmetry exists in the system, meaning that attempting to model the propagating waves using a linear wavevector (a vector that describes linear translation of the optical waves) is at best an approximation. Thus, a more accurate approach would involve the angular wavevector and angular group velocities rather than their linear counterparts.

In this paper, we develop a fully vectorial quantum model in cylindrical coordinates to describe spontaneous FWM in axially symmetric systems, and use it to calculate the spectra of entangled bi-photons generated in two representative geometries: a thick silicon nitride disk and a larger silicon nitride ring. In the second device, we describe a special situation in which the dispersion in the ring resonator allows for generation of entangled pairs with extremely wide spectral separation, while suppressing pair generation for closely spaced modes.

In systems with azimuthal symmetry, the optical modes may be labeled by an integer angular wavenumber m: Em(r, t) = Em(ρ, z) ei(ωmt+). The basic geometry considered is similar to the ring resonator shown in Fig. 1, along with examples of the radial field profiles Eρ (ρ, z) for three TE-like resonant modes with differing angular wavenumbers and radial orders. In the numerical calculations performed in this paper, we only consider the fundamental radial TE-like modes. If the ring is constructed of a material with a third order nonlinear suceptiblity χ(3) (ρ, z), then when a strong pump beam with frequency ωp is coupled into mode m of the ring, two pump photons can be annihilated to produce entangled signal and idler photons at ωs and ωi via spontaneous FWM, so long as the process conserves both energy and momentum. The conservation of energy requires that ωs + ωi = 2ωp, while the conservation of momentum requires that ms + mi = 2mp.

Fig. 1 Ring resonator geometry and example radial field profiles (Eρ) of the first three (3) radial modes with odd vector symmetry about the z-axis (TE-like modes). As can be seen from each of the field profiles, the circumnavigating field near the outside of the mode travels a much greater linear distance than the field near the inside of the mode, leading to linear group and phase velocities that depend on the coordinate ρ. The angular group and phase velocities are therefore better suited to describing light propagation in this system.

The conservation of angular momentum in this system can always be fulfilled since m is an integer; for every integer Δm there will be a pair of phase matched modes at m+ = mp + Δm and m = mp − Δm. The conservation of energy, however, is not guaranteed, since ωm, while discrete, can take on a continuum of values which are not necessarily equally spaced. Thus, the dispersion of the optical modes (i.e their spectral spacing) is the most important determinant as to whether spontaneous FWM will occur and produce entangled photon pairs.

2. Calculation of entangled photon spectra

To calculate the spectra of the entangled photon pairs in this system, we begin by quantizing the electromagnetic field in the ring. The quantized electric field operator for a mode labeled by the mode index k in may be written
E^k(r,t)=ih¯ωkvgk2ε0vpk[a^k(t)a^k(t)]uk(r),
(1)
where uk(r) is spatial function describing the mode profile, normalized such that
Vε(r)uk1(r)uk2(r)=δk1,k2
(2)
Note the explicit inclusion of the group and phase velocities in the normalization of the electric field operator, essential to include if the total energy of of the system is to be accounted for in a Hamiltonian formulation (see Appendix 1).

Equation (1) is valid in any coordinate system, but specializing to the case of azimuthal symmetry yields some simplifications and insight. Importantly, it is the ratio of the angular phase and group velocities that are relevant in this case, since the unfolded linear phase and group velocities are not well-defined (i.e. they depend on r). The angular phase and group velocities (in units of rad/s) may be written
vpm=ωmm
(3a)
vgm=ωmmΔωmΔm=2πfm,
(3b)
where fm is the free spectral range (FSR), in Hz, near mode m. Using first order perturbation theory, we may also write angular group velocity in terms of the fields:
vgmϕ=S|ϕ^[Em(r)×Hm*(r)]|12Vε0d[ε(r)ω]dω|ωm|Em(r)|2+μ0|Hm(r)|2,
(4)
which is simply the integrated energy flux passing through a cross-sectional area of the loop divided by the total field energy in the loop. Equation (4) is especially useful when numerically calculating the dispersion of the system, and to the author’s knowledge does not seem to have appeared in the literature. It gives the FSR directly from the fields, avoiding taking a finite difference of the unknown quantity ωm. In addition, it gives an exact value for vgm rather than the approximate value Δωmm, which becomes less accurate when the FSR is large.

The spectrum of the signal photon is
ψ|a^(ωs)a^(ωs)|ψ=4π4Δm4κ12m+mfm+fm|κtoti(ωsωm+)|2|κtot+i(ωsωmΔ0)|2×[ρ,zχ(3)(ρ,z)ρ|Ep(ρ,z)|2um+um]2,
(9)
and the idler spectrum may be found by replacing ωs with ωi. We now proceed to calculate the spectra of entangled bi-photons for two illustrative examples.

3. Numerical examples

Consider first a silicon nitride disk in air with a radius of 20 μm and a thickness of 250 nm. Silicon nitride is an important optical material whose spectral bandgap lies above the energy required for two-photon absorption at 1550 nm. The field profiles um(r) and eigenfrequencies ωm are calculated using a fully vectorial 2-D axially symmetric weighted residual formulation of Maxwell’s equations implemented in Comsol Multiphysics software. An iterative approach is used to incorporate material dispersion, which is expressed using a Sellmeier equation for the wavelength dependent dielctric constant [29

29. T. Bååk, “Silicon oxynitride; a material for GRIN optics,” Appl. Optics 21, 1069–1072 (1982). [CrossRef]

]:
εSi3N4=1+C1λ2λ2C22,
(10)
with C1 = 2.8939 and C2 = 139.67×10−3. From the field profiles and eigenfrequencies, we calculate the local FSR fm for each eigenmode, the spectral walk-off Δ0, and the overlap integrals necessary to calculate the spectra of the signal and idler photons. The result is shown in Fig. 2.

Fig. 2 Dispersion profile and bi-photon generation spectra of a 20 μm × 250 nm Si3N4 disk in air. (a) Mode frequency vs. angular wavenumber m, showing a transition from the air light line to the Si3N4 light line with increasing mode frequency. Sub-panels show the spatial profiles um(ρ, z) for two wavenumbers, showing the transition from the air to the dielectric. (b) FSR (fm) calculated using Eq. (3b)(blue line), and Eq. (4)(circles) and discrete derivative of the the FSR (Δfm)(red), all vs. angular wavenumber m. The dashed line indicates the location of the pump beam for the spectra shown in panel (c). (c) Calculated bi-photon signal spectra from Eq. (9),assuming a cavity Q of 105. The FSR walk-off Δfm of approximately −2 GHz/m causes nearly a complete linewidth mismatch in energy conservation by the second sideband, as shown in the zoomed in sub-panels.

Figure 2(a) plots the frequency 2πωm of the fundamental TE-like mode (with odd vector symmetry about the z-axis) of the disk as a function of the angular wavevector m. Physically, m represents the number of wavelengths that fit around the disk for a given mode. Hence, at small values of m, the optical wavelength is large (low frequency), and thus spills out into the surrounding air. At larger values of m, the wavelength is much smaller and the optical mode is pulled into the dielectric. As a reference, light lines fl(m)=mc/2πreffε for air and silicon nitride are also plotted, indicating the (ficticious) frequency each mode would have were it contained completely within the silicon nitride or the air, while possesing the same mode profile. Qualitatively, the mode must be bounded by these two lines, and asymptotically approach them as m tends to zero or infinity. Mode dispersion occurs as a result of the transition of the mode frequency from the air line to the dielectric line.

Figure 2(b) shows the first and second derivatives of 2πωm, which physically correspond to FSR fm (red) and the change in the FSR per mode Δfm(blue). The FSR is calculated two ways: using a direct difference of two eigenfrequenies [Eq. (3b)] and using the field profile from a single solution [Eq. (4)], and the two methods are found to have excellent agreement. As one might expect, the FSR decreases as the mode transitions from the air line to the dielectric line, eventually leveling off to a value near 1.2 THz. An important observation is that the FSR decreases monotonically for this range in m, as can most easily be seen by looking at its derivative, which is always negative. As a result, there is no point of zero angular group velocity dispersion, and hence no set of modes available that can perfectly conserve energy in a FWM interaction. Figure 2(c) shows an example of the entangled photon spectra calculated using Eq. (9) that results when the disk is pumped near 1.55 μm (m = 109). An optical quality factor of 105 (i.e. κ1 = κtot = ωm/105) has been assumed for the disk, a value easily obtainable experimentally for silicon nitride disks with this geometry. The spectra has been normalized such that peaks have a value of unity when the mode pair ω+ and ω corresponding to that peak exactly conserve energy (Δ0 = 0), have identical angular group velocities (f+ = f), have a perfect spatial overlap [u+(r) = u(r) = up(r)], and the approximation m+mmp is valid. Any reduction in the amplitude of a peak in the spectra results from one of these criteria not being fulfilled, and is dominated by nonzero values of Δ0. The dotted reference line in Figure 2(b) shows the location of the pump beam on the dispersion walk-off curve Δfm. As can be seen, the change in the FSR for each successive mode is approximately 2 GHz, which corresponds to approximately a full linewidth at these frequencies (i.e. 200 THz/105 = 2 GHz). As a result, the first order side-bands in the spectra are reduced by almost half from the ideal case, and the second order sidebands and beyond are strongly suppressed owing to the walk-off.

Fig. 3 Dispersion profile and bi-photon generation spectra of a Si3N4 ring in SiO2. The ring has a geometry of r = 20 μm, h = 750 nm, and w = 1.1 μm. (a) Mode frequency vs. angular wavenumber m, showing a transition from the low index light line to the high index light line with increasing mode frequency. Sub-panels show the spatial profiles um(ρ, z) for two wavenumbers, showing the transition. (b) FSR (fm) calculated using Eq. (3b)(blue line), and Eq. (4)(circles) and discrete derivative of the the FSR (Δfm)(red), all vs. angular wavenumber m. The two arrows indicates the location of the pump beam for the spectra shown in panels (c) and (d). The shaded yellow region indicates an area proportional to the spectral spacing between the pump mode and the signal and idler modes. Equal areas on either side of the pump indicate energy conservation. Two “special” modes which conserve energy far outside the flat dispersion bandwidth of the ring are highlighted with blue circles. (c) Bi-photon generation spectra when the pump beam is tuned to the mode closest to zero Δfm (m = 137)(d) Bi-photon generation spectra when the pump beam is tuned to the mode with slightly higher angular wavenumber (m = 145) than that with the minimum Δfm. A cavity Q of 105 is assumed.

Figure 3(d) shows the resulting signal spectra when the ring is pumped slightly above the zero Δfm point. The number of lines in the comb-like spectra is reduced owing to the faster dispersion walk-off (nonzero Δ0), but two large peaks in the spectra appear at large values of Δm. The peaks originate from an accidental degeneracy in the energy separation of the signal and idler modes from the pump, and may be physically understood as follows: The total frequency separation between the pump mode ωp and any other mode ωm is given by the discrete sum of the FSRs of each mode between ωp and ωm. This sum is shown graphically as the area under the fm curve in Fig. 3(b). When the pump beam is slightly detuned from the minimum dispersion point to the side with a shallow slope (higher wavenumber), there exists a unique Δm at which the area under the curve to the left and to the right of the pump are equal. No such point exists when the pump is detuned to smaller wavenumbers, as the area on the left will always be larger than that on the right.

The result is the existence of two “special” modes which exactly conserve energy and momentum in the FWM process, and can therefore be used to produce narrow-band entangled photons with very large differences in wavelength. Entangled photon pairs from these modes can easily be filtered from the central comb modes with a broadband filter, and thus be used as an on-chip spectrally bright source of widely spaced narrowband entangled photons. If the quality factor of the resonator were increased to 106 (which is experimentally feasible [17

17. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4, 37–40 (2010). [CrossRef]

]), then the central comb region would be further suppressed (there would exist 10 times fewer modes in the central comb region), but the zero crossing in the angular group velocity dispersion in the wings of the spectra would still exist. As a chip-scale, CMOS compatible technology, it is anticipated that such a system could serve as a low cost source of spectrally bright entangled photons. Furthermore, on-chip filtering and processing of the entangled photons could lead to chip-scale quantum processing and networking.

4. Conclusion

In conclusion, we have derived an expression for the the quantum mechanical bi-photon wave-function and spectra resulting from four-wave mixing in an azimuthally symmetric optical resonator. In doing so, we have employed the angular wavenumber, angular group-velocity, and angular group velocity dispersion as opposed to their linear counterparts, yielding a more accurate and simple formulation. Furthermore, we have calculated the dispersion and spectra in two example systems, a thin silicon nitride disk in air and a silicon nitride ring clad in silicon dioxide. The physical origin of the qualitative features in each spectrum has been explained, and the existence of two “special” modes have been identified that conserve both energy and momentum far outside the bandwidth of the angular group velocity dispersion walk-off. It is anticipated that such a device may find use in the practical generation of widely separated narrow-band entangled photons.

5. Appendix 1: quantization of the field

Here we provide a derivation of Eq. (1), following the notation of Miller [31

31. D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University Press, 2008). [CrossRef]

]. Essential to deriving the proper quantum electric field operator in a Hamiltonian formalism is arriving at the correct expression for the total energy of the system, including energy in the material and fields. Therefore, we begin with a derivation of electromagnetic energy in non-uniform, dispersive systems. In order to gain intuition, we will do so in incremental steps, starting with vacuum modes (in a fictitious finite volume), and progress toward bound modes in non-uniform, dispersive media.

In the vacuum, calculating the total energy per mode in some volume is straightforward:
Uvac=12Vε0|Em(r)|2+μ0|Hm(r)|2=Vε0|Em(r)|2
(12)
In a nondispersive uniform dielectric, we need to take into account the dielectric constant ε, and calculate the energy as
Uε=12Vε0ε|Em(r)|2+μ0μ|Hm(r)|2=εUvac.
(13)
We immediately see that the effect of the dielectric is to increase the total energy of each mode by a factor of ε. Note that we have used the relation
12Vε0ε|Em(r)|2=12Vμ0μ|Hm(r)|2,
(14)
expressing the equal sharing of energy among the electric and magnetic fields for lossless, harmonic modes.

In a uniform medium with a frequency dependent dielectric constant, the total energy is
Uε(ω)=12Vε0d[ε(ω)ω]dω|ωm|Em(r)|2+μ0d[μ(ω)ω]dω|ωm|Hm(r)|2.
(15)

If we assume μ(ω) = 1, then we have
Uε(ω)=Vε0(ε(ω)+12ωdε(ω)dω|ωm)|Em(r)|2=Vε0(n(ω)2+n(ω)ωdn(ω)dω|ωm)|Em(r)|2=ε(ω)vpmvgmUvac.
(16)
where n(ω)=ε(ω), and vgm = c/[n(ω)+dn(ω)/dω] evaluated at ωm. We see from this that the effect of the material dispersion is to increase the energy by an additional factor of vpm/vgm. An expression similar to Eq. (16) was used by Milonni to quantize the electromagnetic field in uniform dispersive media [32

32. P. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Optic. 42, 1991–2004 (1995). [CrossRef]

]. The group velocity muliplier makes physical sense, since the material dispersion slows the energy propagation velocity by a factor of vpm/vgm, while maintaing the same power flow. The energy density must therefore increase by a factor of vpm/vgm. Even though the material dispersion is weak in our system, the intuition gained from calculating the dispersive energy can be used to quantify the effect of geometric dispersion on the total energy of the system.

With this in mind, we finally consider the case of a dispersive, non-uniform medium. In this case, the fields are truly normalizable in all space rather than an arbitrary volume. In contrast to the case of infinite media, the spatial function ε(r) can provide a potential well for bound modes, so that we may integrate over all space and obtain a finite result. The non-uniform dielectric constant has another important consequence: geometric dispersion, which, in addition to material dispersion, modifies the group velocity. We make the physical argument that, owing to conservation of energy in the volume, the relation Uε(r, ω) = [vpm/vgm]Uε still holds.

Note, however, that the group velocity is not simply c/[n(ω) + dn(ω)/dω] as in the case of uniform dielectrics, but is more generally vgm = ∇mωm, where ∇m is the gradient with respect to m. To find the group and phase velocities, one must solve Maxwell’s equations for the eigen-mode frequencies or field profiles and use either Eq. (3b) or Eq. (4). Also note that because ε(r) is no longer a constant in space, we cannot factor it from the energy density. This specifically precludes the possibility of normalizing the spatial function um(r) such that ∫V |um(r)|2 = 1.

To find the quantum mechanical electric field operator, we then postulate an electric field operator
E(r,t)=mpm(t)Dmum(r)
(17a)
B(r,t)=mqm(t)Dmcvm(r).
(17b)
where pm(t) and qm(t) are functions of time, Dm is a constant, and um(r) and vm(r) are spatial functions normalized such that.
Vε(r)um1(r)um2(r)=δm1,m2
(18a)
Vvm1(r)vm2(r)=δm1,m2.
(18b)

It is important to note that while these fields are normalized, the expression of orthogonality is an approximation based on the system being lossless. In reality, the field modes are not orthogonal when material dispersion is present (though geometric dispersion does not effect orthogonality). The reason for this can be most easily seen by writing down the mode solutions as an eigenvalue problem:
××E(r)=(ωc)2ε(r,ω)E(r).
(19)
The solutions to this generalized eigenvalue problem will only be orthogonal if both of the operators ∇ × ∇ × and ε(r, ω) are Hermitian. While ∇ × ∇ × is always Hermitian, ε(r, ω) is not Hermitian if it is frequency dependent. Owing to causality, the imaginary part of the dielectric constant must be nonzero:
Im[ε(ω)]=2ωπP0dωRe[ε(ω)]ω2ω2.
(20)
A complex valued dielectric function usually precludes its Hermiticity, and typically arises from frequency dependent loss. It is important to note, however, that geometric dispersion (e.g. boundaries) does not affect the Hermiticity of the operators, and hence the orthogonality of the modes, even though the effective index and phase velocity for each mode can vary greatly. For dielectric materials used to construct high quality optical resonators, the imaginary part of the dielectric constant is typically more than six (6) orders of magnitude smaller than the real part, making the approximation of orthogonality very good. In metallic resonators, however, the situation is quite different and great care should be taken when quantizing the field.

The fields must obey Maxwell’s equations, so by inserting the postulated fields into the Maxwell curl equations:
×Em(r,t)=Bm(r,t)t
(21a)
×Bm(r,t)=ε(r)c2Em(r,t)t
(21b)
we find that
×um(r)=ωmcvm(r)
(22a)
×vm(r)=ωmcε(r)um(r),
(22b)
if we require that
dqm(t)dt=ωmpm(t)
(23a)
dpm(t)dt=ωmqm(t).
(23b)
We write the classical Hamiltonian as the total energy of the system:
H=Uε(r,ω)=12vpmvgmVε(r)ε0|E(r,t)|2+1μ0|B(r,t)|2=12ε0mvpmvgmDm2[pm(t)2+qm(t)2],
(24)
where we have made use of the mode orthogonality and normalization. Note that this Hamilto-nian takes into account the energy in the fields and in the medium, which can be seen by noting the role of the vp/vg multiplier (this can also be accomplished by considering the total power flux rather than the total energy [33

33. B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991). [CrossRef]

]). If we choose
Dm=vgmωmvpmε0,
(25)
then
dpm(t)dt=Hqm(t)
(26a)
dqm(t)dt=Hpm(t),
(26b)
satisfying Hamilton’s equations. Note that we could have chosen to set Dm to unity and instead normalized the spatial functions um(r), and vm(r) such that
Vε(r)um1(r)um2(r)=ωm12Uε(r,ωm1)δm1,m2
(27)
Vvm1(r)vm2(r)=ωm12Uε(r,ωm1)δm1,m2,
(28)
and such a formulation would still satisfy Hamilton’s equations, though it would bury the explicit dependence of the field operators on the phase and group velocities. Postulating a momentum operator m = −ih̄∂/∂qm(t), the quantum Hamiltonian can then be written
H^m=h¯ωm[a^ma^m+12],
(29)
where the creation and annihilation operators are
a^m12(ddξm+ξm)
(30a)
a^m12(ddξm+ξm),
(30b)
with ξmqm(t)/h¯. Note that the commutator [a^m1,a^m2]=δm1,m2 as expected. By inserting the expressions for Dm and p̂(t) into Eq. (17a), we find the quantized electric field operator for each mode to be
E^m(r,t)=ih¯ωmvgm2ε0vpm[a^m(t)a^m(t)]um(r).
(31)

6. Appendix 2: two-photon wavefunction

Here we provide the details of the derivation of Eq. (8), the two-photon wavefunction. The creation and annihilation operators â(t) in Eq. (1) are functions of time and the resonator is an open system in which photons enter and leave through ports. We may incorporate this fact phenomenologically by using an input/output formalism [28

28. D. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

]. It should be noted, however, that while incorporating the resonator loss via phenomenological decay rates allows a simple way to correctly calculate the signal and idler spectrum, it is not as rigorous as including loss via a direct coupling of the electric field operators to a reservoir of excitations. Physically, this is because the signal and idler photons from a generated pair may experience loss differently, (i.e. one may be absorbed or outcoupled while the other is not) affecting quantities such as coherence functions (e.g. g2).

The equations of motion for a ring with two ports and no reflections are
a˙=iΩaaκ0al=1,2κl+il=1,22κlsl+
(32a)
sl=sl++i2κla,
(32b)
where in this context |a|2 is the intracavity energy and |sl+|2 is the input power at port l, κ0 is the intrinsic linewidth of the resonator and κl is the coupling rate of the resonator to port l. Here, |sl|2 is the power flowing away from port l. Specializing to the case of launching power only into port (1) we can rewrite Eqs. (32a)–(32b) as
a˙=iΩaa(κ0+κ1+κ2)+i2κ1s1+
(33a)
sl=sl++i2κla.
(33b)
The steady state fields in the cavity and moving away from the cavity (in the absence of nonlinear frequency conversion) are then
ass=i2κ1iΩ+(κ0+κ1+κ2)s1+
(34a)
s1=s1++i2κ1ass
(34b)
s2=i2κ2ass.
(34c)
The Fourier components of âm(t) can thus be represented as
a^m(t)=12πa^m(ωm+Ω)i2κ1iΩ+(κ0+κ1+κ2)eiΩtdΩ.
(35)
The electric field creation operator is then
E^()(r,t)=mh¯mπfmε012πa^(ωm+Ω)i2κ1iΩ+(κ0+κ1+κ2)eiΩtdΩum(r)eimϕωmt.
(36)
Using first order perturbation theory, we may write the two-photon wavefunction as
|ψ=1ih¯dtHint|0=ε0ih¯tVχ(3)(r)|E^p(+)|2E^s()E^i()|0=ε0ih¯tVχ(3)(r)|Ep(r)|2a^pa^p[imsh¯πmsfmsε0a^ms(t)ums(r)][imih¯πmifmiε0a^mi(t)umi(r)]|0
(37)
where we have assumed a monochromatic pump with a classical amplitude. Explicitly integrating over the azimuthal coordinate ϕ to enforce the conservation of angular momentum, and making the approximation that the cavity linewidth is much less than the FSR (κtotfm), we find that only pair-wise products of modes m± occur, reducing the wavefunction to a single summation:
|ψ=2iπ2Δmtρ,zχ(3)(r)ρ|Ep(ρ,z)|2m+mfm+fma^pa^pa^ms(t)a^mi(t)um+(ρ,z)um(ρ,z)|0
(38)
where m± = mp ± Δm and we have used the identity
02πdϕei[2mp(mi+ms)]ϕ=2πδ2mp,ms+mi.
(39)
Finally, we explicitly integrate over time to enforce energy conservation:
|ψ=2iπ2Δmρ,zχ(3)(ρ,z)ρ|Ep(ρ,z)|2m+mfm+fma^pa^p×Ωsa^(ωm++Ωs)a^(ωm+Δ0Ωs)2κ1[iΩs+κtot][i(Δ0Ωs)+κtot]um+um|0
(40)
where we have used the identity
dtei[2ωmp(ωms+ωmi)]t=2πδ[2ωmp(ωms+ωmi)].
(41)

Acknowledgments

We thank P. Davids, P. Kumar, and P. Milonni, F. Intravaia, and R. Behunin for discussions relating to this work. We acknowledge funding by DARPA/DSO ZENO-Based Optoelectronics program. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

References and links

1.

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett. 78, 3221–3224 (1997). [CrossRef]

2.

T. C. Ralph and G. J. Pryde, “Optical quantum computation,” in Prog. Optics,vol. 54, E. Wolf, ed. (Elsevier Science2010), pp. 209–269. [CrossRef]

3.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature 423, 731–734 (2003). [CrossRef] [PubMed]

4.

A. Haase, N. Piro, J. Eschner, and M. W. Mitchell, “Tunable narrowband entangled photon pair source for resonant single-photon single-atom interaction,” Opt. Lett. 34, 55–57 (2009). [CrossRef]

5.

K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys. 11, 013049 (2009). [CrossRef]

6.

Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. 83, 2556–2559 (1999). [CrossRef]

7.

S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, P. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express 17, 16558–16570 (2009). [CrossRef] [PubMed]

8.

L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous four-wave mixing in microring resonators,” Opt. Lett. 35, 3006–3008 (2010). [CrossRef] [PubMed]

9.

J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a Silicon-on-Insulator micro-resonator,” Opt. Express 19, 1470–1483 (2011). [CrossRef] [PubMed]

10.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983 (2002). [CrossRef]

11.

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388–12393 (2006). [CrossRef] [PubMed]

12.

H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. I. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007). [CrossRef]

13.

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053601 (2005). [CrossRef] [PubMed]

14.

H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-i. Itabashi, “Generation of polarization entangledphoton pairs using silicon wirewaveguide,” Opt. Express 16, 5721–5727 (2008). [CrossRef] [PubMed]

15.

K.-i. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-i. Itabashi, “Generation of high-purity entangled photon pairs using silicon wire waveguide,” Opt. Express 16, 20368–20373 (2008). [CrossRef] [PubMed]

16.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007). [CrossRef]

17.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4, 37–40 (2010). [CrossRef]

18.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010). [CrossRef]

19.

M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun 1, 29 (2010). [CrossRef] [PubMed]

20.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based optical frequency combs,” Science 332, 555 –559 (2011). [CrossRef] [PubMed]

21.

F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

22.

A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100GHz repetition rates,” Opt. Lett. 37, 875–877 (2012). [CrossRef] [PubMed]

23.

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. 25, 554–556 (2000). [CrossRef]

24.

A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16, 4881–4887 (2008). [CrossRef] [PubMed]

25.

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010). [CrossRef]

26.

Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. 104, 103902 (2010). [CrossRef] [PubMed]

27.

M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun. 282, 3518–3523 (2009). [CrossRef]

28.

D. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).

29.

T. Bååk, “Silicon oxynitride; a material for GRIN optics,” Appl. Optics 21, 1069–1072 (1982). [CrossRef]

30.

I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1208 (1965). [CrossRef]

31.

D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University Press, 2008). [CrossRef]

32.

P. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Optic. 42, 1991–2004 (1995). [CrossRef]

33.

B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177–182 (1991). [CrossRef]

OCIS Codes
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(230.3990) Optical devices : Micro-optical devices
(230.5750) Optical devices : Resonators
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Nonlinear Optics

History
Original Manuscript: July 24, 2012
Revised Manuscript: August 17, 2012
Manuscript Accepted: August 17, 2012
Published: September 11, 2012

Citation
Ryan M. Camacho, "Entangled photon generation using four-wave mixing in azimuthally symmetric microresonators," Opt. Express 20, 21977-21991 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-21977


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References

  1. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum state transfer and entanglement distribution among distant nodes in a quantum network,” Phys. Rev. Lett.78, 3221–3224 (1997). [CrossRef]
  2. T. C. Ralph and G. J. Pryde, “Optical quantum computation,” in Prog. Optics,vol. 54, E. Wolf, ed. (Elsevier Science2010), pp. 209–269. [CrossRef]
  3. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles,” Nature423, 731–734 (2003). [CrossRef] [PubMed]
  4. A. Haase, N. Piro, J. Eschner, and M. W. Mitchell, “Tunable narrowband entangled photon pair source for resonant single-photon single-atom interaction,” Opt. Lett.34, 55–57 (2009). [CrossRef]
  5. K. Akiba, K. Kashiwagi, M. Arikawa, and M. Kozuma, “Storage and retrieval of nonclassical photon pairs and conditional single photons generated by the parametric down-conversion process,” New J. Phys.11, 013049 (2009). [CrossRef]
  6. Z. Y. Ou and Y. J. Lu, “Cavity enhanced spontaneous parametric Down-Conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett.83, 2556–2559 (1999). [CrossRef]
  7. S. Clemmen, K. P. Huy, W. Bogaerts, R. G. Baets, P. Emplit, and S. Massar, “Continuous wave photon pair generation in silicon-on-insulator waveguides and ring resonators,” Opt. Express17, 16558–16570 (2009). [CrossRef] [PubMed]
  8. L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous four-wave mixing in microring resonators,” Opt. Lett.35, 3006–3008 (2010). [CrossRef] [PubMed]
  9. J. Chen, Z. H. Levine, J. Fan, and A. L. Migdall, “Frequency-bin entangled comb of photon pairs from a Silicon-on-Insulator micro-resonator,” Opt. Express19, 1470–1483 (2011). [CrossRef] [PubMed]
  10. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett.14, 983 (2002). [CrossRef]
  11. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express14, 12388–12393 (2006). [CrossRef] [PubMed]
  12. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. I. Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett.91, 201108 (2007). [CrossRef]
  13. X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-Fiber source of Polarization-Entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett.94, 053601 (2005). [CrossRef] [PubMed]
  14. H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-i. Itabashi, “Generation of polarization entangledphoton pairs using silicon wirewaveguide,” Opt. Express16, 5721–5727 (2008). [CrossRef] [PubMed]
  15. K.-i. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-i. Itabashi, “Generation of high-purity entangled photon pairs using silicon wire waveguide,” Opt. Express16, 20368–20373 (2008). [CrossRef] [PubMed]
  16. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1217 (2007). [CrossRef]
  17. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics4, 37–40 (2010). [CrossRef]
  18. L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics4, 41–45 (2010). [CrossRef]
  19. M. Ferrera, Y. Park, L. Razzari, B. E. Little, S. T. Chu, R. Morandotti, D. J. Moss, and J. Azaa, “On-chip CMOS-compatible all-optical integrator,” Nat. Commun1, 29 (2010). [CrossRef] [PubMed]
  20. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based optical frequency combs,” Science332, 555 –559 (2011). [CrossRef] [PubMed]
  21. F. Ferdous, H. Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics5, 770–776 (2011). [CrossRef]
  22. A. R. Johnson, Y. Okawachi, J. S. Levy, J. Cardenas, K. Saha, M. Lipson, and A. L. Gaeta, “Chip-based frequency combs with sub-100GHz repetition rates,” Opt. Lett.37, 875–877 (2012). [CrossRef] [PubMed]
  23. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett.25, 554–556 (2000). [CrossRef]
  24. A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express16, 4881–4887 (2008). [CrossRef] [PubMed]
  25. Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A82, 033801 (2010). [CrossRef]
  26. Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett.104, 103902 (2010). [CrossRef] [PubMed]
  27. M. Scholz, L. Koch, and O. Benson, “Analytical treatment of spectral properties and signal/idler intensity correlations for a double-resonant optical parametric oscillator far below threshold,” Opt. Commun.282, 3518–3523 (2009). [CrossRef]
  28. D. Walls and G. J. Milburn, Quantum Optics (Springer, 2007).
  29. T. Bååk, “Silicon oxynitride; a material for GRIN optics,” Appl. Optics21, 1069–1072 (1982). [CrossRef]
  30. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am.55, 1205–1208 (1965). [CrossRef]
  31. D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University Press, 2008). [CrossRef]
  32. P. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Optic.42, 1991–2004 (1995). [CrossRef]
  33. B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett.16, 177–182 (1991). [CrossRef]

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