## Modified Gouy phase in optical resonators with mixed boundary conditions, via the Born-Oppenheimer method

Optics Express, Vol. 15, Issue 9, pp. 5761-5774 (2007)

http://dx.doi.org/10.1364/OE.15.005761

Acrobat PDF (396 KB)

### Abstract

We investigate near-paraxial modes of high-finesse, planoconcave microresonators without using the paraxial approximation. The goal is to develop an analytical approach which is able to incorporate not only the spatial shape of the resonator boundaries, but also the dependence of reflectivities on angle of incidence. It is shown that this can be achieved using the Born-Oppenheimer method, augmented by a local Bessel wave approximation. We discuss how this approach extends standard paraxial theory. It is found that the Gouy phase of paraxial theory, which is determined purely by ray-optics, is no longer the sole parameter governing transverse mode splittings. The additional determining factor is the sensitivity with which boundary reflection phases depend on incident angle.

© 2007 Optical Society of America

## 1. Introduction

2. T. Klaassen, A. Hoogeboom, M. P.van Exter, and J. P. Woerdman, “Gouy phase of nonparaxial eigenmodes in a folded resonator,” J. Opt. Soc. Am. A, **21**, 1689–1692 (2004). [CrossRef]

4. D. H. Foster and J. U. Nöckel, “Bragg-induced orbital angular-momentum mixing in paraxial high-finesse cavities,” Opt. Lett. **29**, 2788–2790 (2004). [CrossRef] [PubMed]

6. H. Laabs and A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. **35**, 198–207 (1999). [CrossRef]

*θ*). Here,

*θ*is the angle of incidence of a ray with respect to the surface normal, and r(

*θ*) = exp(

*iφ*) with a real phase

*φ*=

*φ*(

*θ*) if there is no absorption or leakage at the mirror. It will be shown that even in the paraxial limit, the Fourier-relationship between position and angle makes it necessary to take the angular spread of Gaussian cavity modes into account when modeling the interaction with dielectric multilayer mirrors.

*Gouy phase*and transverse mode splittings in a paraxial dome-shaped resonator is modified when dielectric Bragg mirrors introduce phase shifts that depend on the angle of incidence. Gaussian-beam parameters such as the Gouy phase can be calculated purely based on ray optics[1], despite the fact that the transverse field pattern is typically not in the classical limit of large quantum numbers. A reflectivity that depends on angle of incidence is easily introduced in the ray picture, but not as easily incorporated into the paraxial theory, precisely because the transverse beam structure requires a wave description[3

3. G. W. Forbes, “Using rays better. IV. Theory for refraction and reflection,” J. Opt. Soc. Am. A **18**, 2557–2564 (2001). [CrossRef]

4. D. H. Foster and J. U. Nöckel, “Bragg-induced orbital angular-momentum mixing in paraxial high-finesse cavities,” Opt. Lett. **29**, 2788–2790 (2004). [CrossRef] [PubMed]

7. D. H. Foster and J. U. Nöckel, “Spatial and polarization structure in micro-domes: effects of a Bragg mirror,” in *Resonators and Beam Control VII* (A. V. Kudryashov and A. H. Paxton, eds.), 5333 of *Proceedings of SPIE*, 195–203 (2004). http://arxiv.org/abs/physics/0406131

8. D. H. Foster and J. U. Nöckel, “Methods for 3-D vector microcavity problems involving a planar dielectric mirror,” Opt. Commun. **234**, 351–383 (2004). [CrossRef]

9. M. Aziz, J. Pfeiffer, and P. Meissner, “Modal behaviour of passive, stable microcavities,” Phys. Stat. Sol. (a) **188**, 979–982 (2001). [CrossRef]

10. S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, “Spherical micromirrors from tem-plated self-assembly: Polarization rotation on the micron scale,” Appl. Phys. Lett. **83**, 767–769 (2003). [CrossRef]

11. G. V. Prakash, L. Besombes, T. Kelf, J. J. Baumberg, P. N. Bartlett, and M. Abdelsalem, “Tunable resonant optical microcavities by self-assembled templating,” Opt. Lett. **29**, 1500–1502 (2004). [CrossRef]

12. G. Cui, J. M. Hannigan, R. Loeckenhoff, F. M. Matinaga, M. G. Raymer, S. Bhongale, M. Holland, S. Mosor, S. Chatterjee, H. M. Gibbs, and G. Khitrova, “A hemispherical, high-solid-angle optical micro-cavity for cavity-qed studies,” Opt. Express **14**, 2289–2299 (2006). [CrossRef] [PubMed]

## 2. Adiabatic separation of variables

*z*axis is assumed to point perpendicular to the plane of the flat mirror at the base of the cavity. In the wave equation

*x,y*variation is of long wavelength, and use the envelope ansatz

*ψ*=

*η*(

*z;x,y*)χ(

*x ,y*) in the wave equation. In the lowest (adiabatic) approximation, we neglect the dependence of

*η*on

*x,y,*allowing us to separate off the equation for

*η*.

*iφ*). The height of the top mirror at

*x,y*is

*z*

_{M}(

*x,y*); e.g., for a spherical mirror as in Fig. 1(a) we have

*z*

_{M}(

*x,y*) =

*h*+ √

*R*

^{2}-

*x*

^{2}-

*y*

^{2}-

*R*, where

*R*is the radius of curvature and

*h*the maximum cavity length, as reached along the

*z*axis. In the paraxial approximation, this function is replaced by the quadratic expansion for small

*x,y,*

*z*

_{M}(

*x,y*) ≈

*h*-

*r*

^{2}/(2

*R*), where

*r*

^{2}=

*x*

^{2}+

*y*

^{2}. The solutions for

*η*are of the form

*η*(

*z;x,y*) =

*e*

^{-ikzz}+

*e*

^{iφ}

*e*

^{ikzz}At a given (

*x,y*), the condition fixing

*k*is then

_{z}*v*is the longitudinal quantum number.

*h̄*

^{2}≡ 2 and mass

*M*= 1:

*Born-Oppenheimer potential*. The shape of this potential

*V*

^{(ν)}

_{BO}depends on

*v*as defined in Eq. (1). The quantity

*K*

^{2}plays the role of the energy eigenvalue, except that

*K*as defined in Eq. (3) is a true constant only if the reflection phase

*φ*is a constant We will make this assumption

*φ*=

*const*in the following sections, and relax it again in Section 5.

## 3. Gaussian beams from the Born-Oppenheimer method

### 3.1. Transverse harmonic oscillator and mode waist

*V*

^{(ν)}

_{BO}in Eq. (4) is a harmonic potential. The main goal is to show how the Gouy phase and spreading angles of paraxial theory arise in our approach.

*axially symmetric*cavity, for which the preferred basis of solutions are the Laguerre-Gauss functions,

_{p}

^{∣ℓ∣}(

*u*) is the associated Laguerre polynomial. The integer ℓ is the

*orbital angular momentum*, and

*p*counts the number of radial nodes in the transverse wave field.

*W*

_{paraxial}= √2 (

*h*(

*R*-

*h*) /

*k*

^{2})

^{1/4}when

*h*≪

*R*. The eigenvalues, Eq. (3

3. G. W. Forbes, “Using rays better. IV. Theory for refraction and reflection,” J. Opt. Soc. Am. A **18**, 2557–2564 (2001). [CrossRef]

*N*= 2

*p*+ ∣

*ℓ*∣. For a given

*N*, the allowed angular momenta are

*ℓ*= -

*N*, -

*N*+ 2, …

*N*- 2,

*N*, with the highest value ∣

*ℓ*∣ =

*N*occurring when

*p*= 0. The radial wave equation for Ψ contains an

*effective radial potential*,

*V*

_{eff}(

*r*) =

*V*

^{(v)}

_{BO}(

*r*) +

*ℓ*

^{2}/

*r*

^{2}; it depends both on

*v*and on

*ℓ*, but not on the principal quantum number

*N*.

### 3.2. Gouy phase

*K*in Eq. (3) for the mode wave number

*k*in terms of the mode indices

*N*and

*v*:

*k*

^{2}that becomes quantized, not

*k*. In paraxial theory, on the other hand, one quantizes

*k*directly by imposing the round-trip single-valuedness of the field; the transverse excitation is viewed as giving rise to an additional phase shift in each round trip, called the

*Gouy phase*, which must be added on to

*k*. We now show how these two approaches are connected.

*K*

_{N;v}

^{2}to be much smaller than the longitudinal contribution

*k*

^{2}

_{z}(

*r*= 0), we get

*k*= 12.5μm and vary the cavity length from

*h*≈ 2μm to

*h*≈ 18μm. Comparing Eq. (11) with a Fabry-Pérot spectrum (plane-parallel mirrors), we see that the last term is the Gouy phase contribution to the wave number,

_{G}for a dome geometry is given by [1] cosθ

_{G}= √1-

*h*/

*R*, whence θ

_{G}→ √

*h*/

*R*for

*R*≫

*h*. This reduces to

*k*

_{G}as defined above when divided by

*h*to turn the angle θ

_{G}into a wavenumber.

*monodromy matrix*describing the stable ray orbit on which the paraxial cavity mode is built [14, 15

15. H. E. Tureci, H. G. L. Schwefel, and A.Douglas Stone, “Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities,” Opt. Express **10**, 752–776 (2002) [PubMed]

2. T. Klaassen, A. Hoogeboom, M. P.van Exter, and J. P. Woerdman, “Gouy phase of nonparaxial eigenmodes in a folded resonator,” J. Opt. Soc. Am. A, **21**, 1689–1692 (2004). [CrossRef]

### 3.3. Range of validity

*h*≪

*R*. This is also the limit in which the transverse mode splittings of size

*k*

_{G}become small compared to the longitudinal free spectral range,

*π*/

*h*. The validity of the BO approximation relies precisely on this separation of frequency scales between neighboring longitudinal (

*v*) and transverse (

*N*) mode indices [13]: it insures the adiabaticity that prevents couplings between different

*v*. It is also worth noting here that the magnitude of

*v*does not enter the conditions of validity in our approach, and therefore we can describe modes that have either small or large values of the dimensionless wave number

*kh*, as is illustrated in Fig. 2.

*K*

^{2}

_{N;v}≪

*k*

^{2}

_{z}(

*r*= 0) which requires

*kh*≫

*hk*

_{G}= θ

_{G}; but this dependence on the absolute size of

*kh*is a limitation of the square root expansion leading to Eq. (11), and not of the the BO treatment. In Fig. 2, this condition is in fact valid for all data shown.

*ℓ*, and therefore it is straightforward to incorporate

*anharmonic*potentials into

*V*

_{BO}

^{(v)}(

*r*) without losing the effective-potential picture: in Fig. 3, the curves representing the effective radial potential will change shape, but the transverse field is still determined by a one-dimensional radial potential well with discrete bound states. This is one of the main advantages of the BO method in cavity mode calculations: the quadratic approximation loses the special status which it occupies in the paraxial approximation. In particular, for an axisymmetric cavity, the breaking of the degeneracies (8) is necessarily non-paraxial, but still falls within the adiabatic regime on which the BO approach relies.

*V*

_{BO}

^{(v)}(

*x,y*) ≠

*V*

_{BO}

^{(v)}(

*r*), we can still use Eq. (4), except that the transverse wave equation is not separable in cylinder coordinates. Aberrations such as astigmatism[17

17. S. J. M. Habraken and G. Nienhuis, “Modes of a twisted optical cavity,” Phys. Rev. A **75**, 033819 (2007). [CrossRef]

*other coordinate choices*(e.g., parabolic cylinder coordinates [16

16. J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E **62**, 8677–8699 (2000). [CrossRef]

*h*≈

*R*. Examples for BO approximations in different two-dimensional coordinate systems have been given in [19

19. O. Zaitsev, R. Narevich, and R. E. Prange, “Quasiclassical Born-Oppenheimer approximations,” Found. Phys. **31**, 7 (2001). [CrossRef]

## 4. The local plane wave approximation

### 4.1. Comparing Bessel-wave expansions

*q*and momenta

*p*in the Hamiltonian

*H*=

*p*

^{2}+

*q*

^{2}. In this notation, if

*ψ*(

*q*) is the position representation of an eigenstate, then

*ψ*(

*p*) is its momentum representation, which means

*ψ*(

*q*) is equal to its own

*Fourier transform*.

*R*(

_{p,ℓ,v}*r*) of Ψ

_{p,ℓ,v}(

*r,ו*) is formally identical to its own

*Fourier-Bessel transform*[18], which we call g

_{p,ℓ,v}(

*u*) This is defined to be the rotationally symmetrized form of the Fourier transform, expressed as the integral over the dimensionless radial coordinate

*ρ*=

*r*/

*w*:

_{v}*angular spectrum*of the paraxial Laguerre-Gauss beams; that is the symmetrized plane-wave decomposition of the field, in terms of the Bessel waves,

*J*(

_{ℓ}*kr*sinθ)exp{

*ikz*cosθ θ

*iℓϕ*} (in cylinder coordinates). We define the plane-wave spectrum g(θ) as the expansion coefficients in the integral decomposition

20. D. H. Foster, PhD thesis, http://hdl.handle.net/1794/3778 (University of Oregon, 2006).

### 4.2. Local angle of incidence from local wave number

*ϕ*dependence and then proceed as follows to eliminate

*z*: integrate both sides over

*z*, from the bottom of the resonator at

*z*= 0 to the top at

*z*=

*z*

_{M}(

*r*), with the factor exp(-

*ik*(

_{z}*r*)). This yields

*second*assumption motivated by paraxiality would be to set

*S*(

_{p,ℓ,v}*u*), i.e, g

_{p,ℓ,v}(θ) =

*C′*

*S*(θ).

_{p,ℓ,v}*a posteriori*. Equation (18) is important for what follows, and hence it is worth discussing its implications further. Accordingly, there is then a special angle θ(

*r*) for every radial position

*r*, given by

*local plane wave approximation*because Eq. (15) identifies θ(

*r*) as the angle of the Bessel wave that we would obtain if the cavity had the same height

*h*=

*z*

_{M}(

*r*) for all

*r*.

*k*≈

_{z}*k*to approximate

*const*, is recovered from this when

*w*→ ∞. Although this limit makes θ

_{v}_{p}go to zero, we can get finite θ(

*r*) by simultaneously choosing

*N*large. The term containing

*N*in Eq. (21) originates from the transverse mode splitting

*k*

_{G}(

*N*+ 1) of Eq. (11), which depends on the Gouy phase θ

_{G}.

*r*and θ. The largest allowed radius is the outer

*classical turning point*of the BO potential

*V*

_{BO}

^{(v)}(

*r*),

*root-mean-square width*,

*r*

_{rms}, of a two-dimensional harmonic oscillator in the

*N*-th excite state.

### 4.3. Semiclassical discussion

*r*, delimited by the

*classical turning points*,

*r*

_{min}and

*r*

_{max}, in the effective radial potential,

*V*

_{eff}(

*r*). The calculation of rmin and

*r*

_{max}amounts to reading off the intersections of the potential curves (solid lines) in Fig. 3 with the lines of

*K*

^{2}=

*K*

_{N}

^{2}=

*const*(the three horizontal lines in the graph). For the lowest orders

*N*of transverse excitation, we can expect the turning points to be washed out in the wave solutions, due to tunneling into the effective-potential barrier. This is illustrated by the transverse Laguerre-Gauss mode functions superimposed as dashed lines in Fig. 3. One can verify that in all cases the classical turning points give a reasonable estimate of the interval within which the respective mode functions have large amplitude, and that only exponential-type decay is observed in these functions when

*r*∉ [

*r*

_{min},

*r*

_{max}].

*r*

_{max}, of

*V*

_{eff}(

*r*) are less than or equal to

*r*

_{outer}. Specifically,

*r*

_{max}→

*r*

_{outer}for

*N*≫

*ℓ*. The points

*r*

_{max}are seen in Fig. 3 to coalesce at a single value that depends on

*N*, but not on

*ℓ*, when

*N*≫

*ℓ*; this is because the shapes of the effective potentials for different

*ℓ*approach each other on the large-

*r*side of the graph. The angle θ(

*r*) is largest when

*r*takes on its smallest value,

*r*

_{min}. Both

*r*

_{min}and

*r*

_{max}depend on the angular momentum of the mode, but

*r*

_{min}has a stronger

*ℓ*dependence due to the effect of the centrifugal barrier

*ℓ*

^{2}/

*r*

^{2}in

*V*

_{eff}(

*r*).

*ℓ*and principal transverse quantum number

*N*are given by

*ℓ*=

*N*= 0, we get

*r*

_{min}= 0 and

*r*

_{max}=

*w*, the conventional mode waist radius. Note also that we always have

_{v}*decrease*of θ with increasing

*r*. This is physically significant: e.g., for

*N*=

*ℓ*= 0 we find that there is a finite angle of incidence θ = θ

_{p}at

*r*= 0. In the BO approach, the smallest angle of incidence with respect to the normal occurs at large radial distance from the axis. This is what one also expects from the

*ray picture*.

*r*, cf. Eq. (16). The significant content of θ

_{max/min}is that they depend on the mode indices

*N*and

*ℓ*, whereas θ

_{p}is

*independent*of these transverse quantum numbers. For fixed value of the fundamental mode waist,

*w*, both

_{n}*r*

_{max}and θ

_{max}increase when the transverse order

*N*increases, in contrast to the reciprocal relation between

*w*and θ

_{n}_{p}.

## 5. Angle-dependent reflectivity

### 5.1. Including variable reflection phase shifts

*r*), Eq. (21). In a realistic cavity we want to allow the mirror reflectivity to depend both on position and on the angle of incidence. Here, we introduce a non-constant reflectivity only for the planar bottom mirror, and neglect absorption and leakage because we want to focus on the closed-resonator mode structure. If the planar interface is moreover translationally invariant, its reflectivity can be written as

*r*(θ) = exp(

*iφ*(θ)), i.e., it is a function of incident angle θ but not of position

*r*.

*need not be small*, but will be assumed to be a constant (the factor of two in front is introduced merely for convenience later). In fact, for multilayer mirrors, ε in general also depends on the wavelength; however, in what follows we shall neglect that dependence. It will be interesting for a complete analysis to allow either positive or negative sign for ε, even though in practice (see, e.g., [4

4. D. H. Foster and J. U. Nöckel, “Bragg-induced orbital angular-momentum mixing in paraxial high-finesse cavities,” Opt. Lett. **29**, 2788–2790 (2004). [CrossRef] [PubMed]

*meta-materials*with negative index of refraction, and in view of recent advances in scaling such materials to optical wavelengths[22

22. V. M. Shalaev,“Optical negative-index metamaterials,” Nature Photonics **1**, 41–48 (2007). [CrossRef]

^{2}≈ 1-

*k*

^{2}

_{z}/

*k*

^{2}, Eq. (1) becomes

*k*(

_{z}*r*) implicitly. Because the angle of incidence θ enters squared and not linearly, the square-root singularity of Eq. (21) plays no role here, and hence the classical turning points do not limit the interval of

*r*on which

*k*(

_{z}*r*) is defined. This is important to note because it means that Eq. (21) will allow us derive a modified BO potential for the radial wave equation, which is defined for

*r*∈ [0, ∞[. Quartic or higher powers of θ in the expansion for φ(θ) can be taken into account in straightforward generalization of this approach, but we will focus on the leading-order effects which directly modify the paraxial results.

### 5.2. Modified transverse wave equation

*k*

^{2}(

*r*), and the solution to quadratic order in

*r*can be written as

*γ*through

*k*=

_{z}*k*(

_{z}*r,k*), where the additional

*k*dependence in

*k*is not present for φ =

_{z}*const*. In other words, the eigenvalue

*k*, which is determined by the radial equation, appears in the longitudinal quantization. This is a new coupling that also makes the effective potential k-dependent:

*V*

_{BO}

^{(v)}(

*r,k*) ≈ Ω

^{2}

_{v,ε}(

*k*)

*r*

^{2}/2 with a

*k*-dependent oscillator frequency

*K*

^{2}

_{p,ℓ;v;ε}(

*k*) = √2Ω

_{v,ε}(

*k*) (

*N*+1) =

*K*

_{N;v;ε}

^{2}(

*k*), where we use the radial (

*p*) and azimuthal (

*ℓ*) quantum numbers as mode labels, and

*N*= 2

*p*+ ∣

*ℓ*∣. The solution to our problem consists in finding

*k*such that

*K*

_{N;v;ε}

^{2}(

*k*) =

*k*

^{2}-

*k*

^{2}(0), i.e.,

*k*

^{2}(0). Because Ω

^{2}

_{v,ε}(

*k*) and

*k*

^{2}(0) depend on

*k*

^{2}through γ, this is an implicit equation for

*k*

^{2}, but even without solving it we can see that the wavenumber

*k*will depend only on the principal quantum number

*N*, not on

*ℓ*or

*p*separately. Numerical solutions of Eq. (29) are shown in Fig. 4(a). They first of all confirm that despite this more complicated situation the BO separation of spectral scales is still valid. In fact, the global appearance of the spectrum is almost indistinguishable from the ε = 0 case, Fig. 2(a), except at small cavity lengths

*h*.

### 5.3. Results and discussion

*h*≪

*R*, are a domain where our approximations are especially good, so we will look at cavities of this type more closely, to illustrate the method we have introduced in the foregoing discussion. The angle-dependent reflectivity giving rise to Eq. (29) will not break the transverse-mode degeneracy of Eq. (8), and therefore we can label the multiplets of identical eigenvalues

*k*

_{p,ℓ,v;ε}=

*k*

_{N,v;ε}. For ε ≠ 0 the

*k*

^{2}

_{N,v;ε}for

*N*= 0, 1,2… are

*no longer strictly equidistant*, as they are for the standard harmonic oscillator. This is intriguing because we have therefore broken one spectral property of the harmonic oscillator (its equidistant spectrum) while strictly preserving another (its degeneracies). Quantitatively, however, the deviations from equidistant transverse mode spacing are negligible for most cavity parameters.

*Gouy phase*and boundary reflection phase. In Fig. 4(b), we inspect the level spacings on the scale of the transverse mode splitting, for a single longitudinal manifold. The most striking observation is that the spectral spacing of adjacent

*k*

_{N,v,ε}is

*no longer equal to the purely geometric quantity*

*k*

_{G}of Eq. (12), despite that fact that there is no splitting of the harmonic-oscillator degeneracies. In other words, the Gouy phase θ

_{G}cannot be read off directly from the mode splittings. The spectrum now shows a spacing between

*k*

_{N,v,ε}that is determined not only by

*k*

_{G}, but in addition by the angle-dependence of the mirror reflection phase (ε). Recall that a

*constant*reflection phase shift φ

_{0}would not affect the paraxial mode spacing, only cause a global spectral shift.

*k*-dependencies in the transverse eigenvalues

*K*

^{2}

_{N;n;ε}(

*k*) and the longitudinal wavenumber

*k*

_{z}

^{2}(0), entering in

*k*

^{2}=

*K*

^{2}

_{N;v;ε}+

*k*

_{z}

^{2}(0). Figure 5 shows the ε-dependence of (a) the oscillator frequency Ω

_{v,ε}entering

*K*

^{2}

_{N;v;ε}, and (b) the wave number

*k*

_{z}

^{2}(0). Increasing ε will

*reduce k*

^{2}

_{z}(0) but

*increase*Ω

_{v,ε}, but in Ω

_{v,ε}this is outweighed by the growth in cos

^{-1/2}〻 which in fact diverges for ε(α - ε) → (

*hk*/2)

^{2}, cf. Eq. (28). The meaning of Fig. 5 (a) is that increasing ε causes the transverse modes to be confined more tightly. In other words, the mode waist Ω

_{v,ε}given by Eq. (7) will expand for ε < 0 and shrink for ε > 0. In a usual paraxial resonator without angle-dependent reflection phase shifts, one would expect larger waist radius to come with smaller transverse mode splitting. However, in our case the

*opposite*happens: according to Fig. 5 (b), value of

*k*

_{z}

^{2}(0) increases for ε < 0, and it does so in a mode-dependent way, with larger slope for the higher-order transverse modes. As a result, the total spectral separation between the actual mode wave numbers

*k*

^{2}=

*K*

^{2}

_{N;v;ε}+

*k*

^{2}

_{z}(0) increases for ε < 0.

*k*

^{2}(0) and Ω

_{v,ε}both determine and are determined by

*k*, the limit ε (α - ε) → (

*hk*/2)

^{2}produces a fixed point to which all

*k*of a given longitudinal mode number

*v*(and hence phase α) flow. The curve

*k*(ε) near coalescence then approaches

*k*(ε) ≈ 2√ε(α - ε)/

*h*, which is independent of the transverse index

*N*.

*v*should be affected

*differently*by the angle-dependent reflectivity (measured by ε), given the fact that the spreading angle θ

_{p}of Eq. (9) is the

*same*for all modes with the same

*v*, assuming standard paraxial theory. In particular, θ

_{p}does not depend on the transverse mode index

*N*, but Fig. 4 (b) shows that modes with different

*N*have different slope in

*k*versus ε.

_{p}does not measure the details of the angular spectrum, and consequently introduce the angles θ

_{max}, θ

_{min}and θ

_{rms}. The root-mean-square angle is defined through the corresponding harmonic-oscillator length, Eq. (22), which depends on the quantum number

*N*: θ

_{rms}= √(

*N*+ 1)/2θ

_{p}. As was noted below Eq. (24), both

*r*

_{max}and θ

_{max}increase with

*N*, and this also holds for θ

_{rms}. Now assume that θ

_{rms}characterizes the plane-wave content of the mode, as it interacts with the mirrors. In paraxial theory, one obtains the modal wave number by quantizing the sum of the dynamical phase along the geometric path length plus the phases due to reflections (φ) and transverse excitation (the Gouy phase), cf. Eq. (11). When we introduce ε < 0, the reflection phase retardation φ(θ

_{rms}) is decreased without changing the other phase contributions. Writing Eq. (25) as φ(ε) = φ

_{0}+ 2εθ

_{rms}

^{2}, this amounts to a shortening of the “effective cavity length” which is more pronounced for larger

*N*because such modes have larger Ω

_{rms}. As a result, the quantized wave number is increased more rapidly for modes with larger

*N*(hence larger θ

_{rms}), as ε is made more negative. While this basic interpretation explains the trend in the left half of Fig. 4 (b), it is not designed to explain the dramatic coalescence in the vicinity of the square-root singularity for ε > 0.

## 6. Conclusion and outlook

23. G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A **48**, 656–665 (1993). [CrossRef] [PubMed]

24. O. Steuernagel, “Equivalence between focused paraxial beams and the quantum harmonic oscillator,” Am. J. Phys. **73**, 625–629 (2005). [CrossRef]

17. S. J. M. Habraken and G. Nienhuis, “Modes of a twisted optical cavity,” Phys. Rev. A **75**, 033819 (2007). [CrossRef]

25. F. Laeri, G. Angelow, and T. Tschudi, “Designing resonators with large mode volume and high mode discrimination,” Opt. Lett. **21**, 1324–1327 (1996). [CrossRef] [PubMed]

11. G. V. Prakash, L. Besombes, T. Kelf, J. J. Baumberg, P. N. Bartlett, and M. Abdelsalem, “Tunable resonant optical microcavities by self-assembled templating,” Opt. Lett. **29**, 1500–1502 (2004). [CrossRef]

10. S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, “Spherical micromirrors from tem-plated self-assembly: Polarization rotation on the micron scale,” Appl. Phys. Lett. **83**, 767–769 (2003). [CrossRef]

**n**(

*x,y*) of the curved mirror in order to define the angle of incidence entering the reflectivity. An additional extension of our work will be the inclusion of polarization-dependent reflectivities, in order to apply the BO method to the phenomenon of photonic spin-orbit coupling in paraxial dome resonators[4

**29**, 2788–2790 (2004). [CrossRef] [PubMed]

## Acknowledgements

## References and links

1. | A. E. Siegman, |

2. | T. Klaassen, A. Hoogeboom, M. P.van Exter, and J. P. Woerdman, “Gouy phase of nonparaxial eigenmodes in a folded resonator,” J. Opt. Soc. Am. A, |

3. | G. W. Forbes, “Using rays better. IV. Theory for refraction and reflection,” J. Opt. Soc. Am. A |

4. | D. H. Foster and J. U. Nöckel, “Bragg-induced orbital angular-momentum mixing in paraxial high-finesse cavities,” Opt. Lett. |

5. | A. Fox and Y. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. |

6. | H. Laabs and A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. |

7. | D. H. Foster and J. U. Nöckel, “Spatial and polarization structure in micro-domes: effects of a Bragg mirror,” in |

8. | D. H. Foster and J. U. Nöckel, “Methods for 3-D vector microcavity problems involving a planar dielectric mirror,” Opt. Commun. |

9. | M. Aziz, J. Pfeiffer, and P. Meissner, “Modal behaviour of passive, stable microcavities,” Phys. Stat. Sol. (a) |

10. | S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, “Spherical micromirrors from tem-plated self-assembly: Polarization rotation on the micron scale,” Appl. Phys. Lett. |

11. | G. V. Prakash, L. Besombes, T. Kelf, J. J. Baumberg, P. N. Bartlett, and M. Abdelsalem, “Tunable resonant optical microcavities by self-assembled templating,” Opt. Lett. |

12. | G. Cui, J. M. Hannigan, R. Loeckenhoff, F. M. Matinaga, M. G. Raymer, S. Bhongale, M. Holland, S. Mosor, S. Chatterjee, H. M. Gibbs, and G. Khitrova, “A hemispherical, high-solid-angle optical micro-cavity for cavity-qed studies,” Opt. Express |

13. | A. Messiah, |

14. | F. Laeri and J. U. Nöckel, “Nanoporous compound materials for optical applications - Microlasers and microres-onators,” in: |

15. | H. E. Tureci, H. G. L. Schwefel, and A.Douglas Stone, “Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities,” Opt. Express |

16. | J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, “Mode structure and ray dynamics of a parabolic dome microcavity,” Phys. Rev. E |

17. | S. J. M. Habraken and G. Nienhuis, “Modes of a twisted optical cavity,” Phys. Rev. A |

18. | P. M. Morse and H. Feshbach, |

19. | O. Zaitsev, R. Narevich, and R. E. Prange, “Quasiclassical Born-Oppenheimer approximations,” Found. Phys. |

20. | D. H. Foster, PhD thesis, http://hdl.handle.net/1794/3778 (University of Oregon, 2006). |

21. | B. E. A. Saleh and M. C. Teich, |

22. | V. M. Shalaev,“Optical negative-index metamaterials,” Nature Photonics |

23. | G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A |

24. | O. Steuernagel, “Equivalence between focused paraxial beams and the quantum harmonic oscillator,” Am. J. Phys. |

25. | F. Laeri, G. Angelow, and T. Tschudi, “Designing resonators with large mode volume and high mode discrimination,” Opt. Lett. |

26. | M. Achtenhagen, A. Hardy, and E. Kapon, “Three-dimensional analysis of mode discrimination in vertical-cavity surface-emitting lasers,” Appl. Opt. |

27. | A. M. Sarangan and G. M Peake, “Enhancement of Lateral Mode Discrimination in Broad-Area VCSELs Using Curved Bragg Mirrors,” J. Lightwave Technol. |

28. | T. Gentsy, K. Becker, I. Fischer, W. Elsässer, C. Degen, P. Debernardi, and G. P. Bava, “Enhancement of Lateral Mode Discrimination in Broad-Area VCSELs Using Curved Bragg Mirrors,” Phys. Rev. Lett. |

**OCIS Codes**

(120.2230) Instrumentation, measurement, and metrology : Fabry-Perot

(140.4780) Lasers and laser optics : Optical resonators

(260.2110) Physical optics : Electromagnetic optics

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 3, 2007

Revised Manuscript: April 24, 2007

Manuscript Accepted: April 25, 2007

Published: April 26, 2007

**Citation**

Jens U. Nöckel, "Modified Gouy phase in optical resonators with mixed boundary conditions, via the Born-Oppenheimer method," Opt. Express **15**, 5761-5774 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5761

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### References

- A. E. Siegman, Lasers (University Science Books, 1986).
- T. Klaassen, A. Hoogeboom, M. P. van Exter, and J. P. Woerdman, "Gouy phase of nonparaxial eigenmodes in a folded resonator," J. Opt. Soc. Am. A 21, 1689-1692 (2004). [CrossRef]
- G. W. Forbes, "Using rays better. IV. Theory for refraction and reflection," J. Opt. Soc. Am. A 18, 2557-2564 (2001). [CrossRef]
- D. H. Foster and J. U. Nöckel, "Bragg-induced orbital angular-momentum mixing in paraxial high-finesse cavities," Opt. Lett. 29, 2788-2790 (2004). [CrossRef] [PubMed]
- A. Fox and Y. Li, "Resonant modes in a maser interferometer," Bell Syst. Tech. J. 40, 453-488 (1969).
- H. Laabs and A. T. Friberg, "Nonparaxial eigenmodes of stable resonators," IEEE J. Quantum Electron. 35, 198-207 (1999). [CrossRef]
- D. H. Foster and J. U. Nöckel, "Spatial and polarization structure in micro-domes: effects of a Bragg mirror," in Resonators and Beam Control VII (A. V. Kudryashov and A. H. Paxton, eds.), Proc. SPIE 5333, 195-203 (2004). http://arxiv.org/abs/physics/0406131
- D. H. Foster and J. U. Nöckel, "Methods for 3-D vector microcavity problems involving a planar dielectric mirror," Opt. Commun. 234, 351-383 (2004). [CrossRef]
- M. Aziz, J. Pfeiffer, and P. Meissner, "Modal behaviour of passive, stable microcavities," Phys. Stat. Solidi A 188, 979-982 (2001). [CrossRef]
- S. Coyle, G. V. Prakash, J. J. Baumberg, M. Abdelsalem, and P. N. Bartlett, "Spherical micromirrors from templated self-assembly: Polarization rotation on the micron scale," Appl. Phys. Lett. 83, 767-769 (2003). [CrossRef]
- G. V. Prakash, L. Besombes, T. Kelf, J. J. Baumberg, P. N. Bartlett, and M. Abdelsalem, "Tunable resonant optical microcavities by self-assembled templating," Opt. Lett. 29, 1500-1502 (2004). [CrossRef]
- G. Cui, J. M. Hannigan, R. Loeckenhoff, F. M. Matinaga, M. G. Raymer, S. Bhongale, M. Holland, S. Mosor, S. Chatterjee, H. M. Gibbs, and G. Khitrova, "A hemispherical, high-solid-angle optical micro-cavity for cavity-qed studies," Opt. Express 14, 2289-2299 (2006). [CrossRef] [PubMed]
- A. Messiah, Quantum Mechanics (North Holland, John Wiley & Sons, 1966) Vol. 2.
- F. Laeri and J. U. Nöckel, "Nanoporous compound materials for optical applications - Microlasers and microresonators," in Handbook of Advanced Electronic and Photonic Materials, H. S. Nalwa, ed., 6, 103-151 (Academic Press, 2001).
- H. E. Tureci, H. G. L. Schwefel, and A. Douglas Stone, "Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities," Opt. Express 10, 752-776 (2002) [PubMed]
- J. U. Nöckel, G. Bourdon, E. L. Ru, R. Adams, I. Robert, J.-M. Moison, and I. Abram, "Mode structure and ray dynamics of a parabolic dome microcavity," Phys. Rev. E 62, 8677-8699 (2000). [CrossRef]
- S. J. M. Habraken and G. Nienhuis, "Modes of a twisted optical cavity," Phys. Rev. A 75, 033819 (2007). [CrossRef]
- P. M. Morse and H. Feshbach, Methods of Theoretical Physics, (Feshbach Publishing, LLC, 1981) Vol. 2.
- O. Zaitsev, R. Narevich, and R. E. Prange, "Quasiclassical Born-Oppenheimer approximations," Found. Phys. 31, 7 (2001). [CrossRef]
- D. H. Foster, PhD thesis, http://hdl.handle.net/1794/3778 (University of Oregon, 2006).
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc, 1991). [CrossRef]
- V. M. Shalaev,"Optical negative-index metamaterials," Nature Photonics 1, 41-48 (2007). [CrossRef]
- G. Nienhuis and L. Allen, "Paraxial wave optics and harmonic oscillators," Phys. Rev. A 48, 656-665 (1993). [CrossRef] [PubMed]
- O. Steuernagel, "Equivalence between focused paraxial beams and the quantum harmonic oscillator," Am. J. Phys. 73, 625-629 (2005). [CrossRef]
- F. Laeri, G. Angelow, and T. Tschudi, "Designing resonators with large mode volume and high mode discrimination," Opt. Lett. 21, 1324-1327 (1996). [CrossRef] [PubMed]
- M. Achtenhagen, A. Hardy, and E. Kapon, "Three-dimensional analysis of mode discrimination in vertical-cavity surface-emitting lasers," Appl. Opt. 44, 2832-2838 (2005). [CrossRef] [PubMed]
- A. M. Sarangan and G. M Peake, "Enhancement of lateral mode discrimination in broad-area VCSELs using curved Bragg mirrors," J. Lightwave Technol. 22, 543-549 (2004). [CrossRef]
- T. Gentsy, K. Becker, I. Fischer, W. Elsässer C. Degen, P. Debernardi, and G. P. Bava, "Enhancement of lateral mode discrimination in broad-area VCSELs using curved Bragg mirrors," Phys. Rev. Lett. 94, 233901 (2005).

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