Degenerate Perturbation Theory Describing the Mixing of Orbital Angular Momentum Modes in Fabry-Pérot Cavity Resonators

0.21 MB·English

by  David H. Foster

Degenerate Perturbation Theory Describing the Mixing of Orbital Angular
Momentum Modes in Fabry-P´erot Cavity Resonators
David H. Foster,1 Andrew K. Cook,2 and Jens U. N¨ockel2
1Deep Photonics Corporation, Corvallis, Oregon 97333, USA∗
2Department of Physics, University of Oregon, Eugene, Oregon 97403, USA
(Dated: January 2, 2009)
Wepresentananalyticperturbationtheorywhichextendstheparaxialapproximationforacom-
moncylindricallysymmetricstableopticalresonatorandincorporatesthedifferential,polarization-
dependent reflectivity of a Bragg mirror. The degeneracy of Laguerre-Gauss modes with distinct
orbital angular momentum (OAM) and polarization, but identical transverse order N, will become
9
observablyliftedatsufficientlysmallsizeandhighfinesse. Theresultingparaxialeigenmodespossess
0
two distinct OAMcomponents,thefractional composition subtly dependingon mirror structure.
0
2
PACSnumbers: 42.50.Tx,42.60.Da

n
a
J Polarization-dependent effects in three-dimensional

2 optical systems have in recent years received attention
under the aspect of orbital angular momentum (OAM)

] [1]. It has become important to understand the OAM
s
c interactionswithinterfaces[2,3],waveguides[4]andres-
i onators. One particular mechanism that warrants inves-
t
p tigationisthatofsmallcorrectionstotheparaxialtheory
o
of resonatorsgiving rise to what may be regardedas op-
.
s tical spin-orbit coupling [5, 6].
c
i
s In this Rapid Communication, we extend electromag-
y
netic resonatortheory [11] to provide a complete pertur-
h
bation analysis and numerical computations for an opti-
p
cal cavity which has an axis of rotational symmetry (ˆz),
[
but nevertheless does not conserve the component ℓ of

1 OAMalongthataxis. Thisphenomenonitselfisremark- Figure1: (a)Plano-concavemodelgeometry. Thetopmirror
v radius of curvature R is 100µm in the examples. (b) Wave
able because the model system, shown in Fig. 1(a), ap-
9 proaches the paraxial limit in which OAM conservation number splitting ∆k ≡ k2−k1 = δk2−δk1 from numerical
8 dataatL=25µm,λ≈400nm,indicatingtheavoidedcross-
2 might be taken for granted. Standard paraxial modes ing between a mixable pair of modes. (c) Numerical results
0 may be chosen to have well-defined ℓ because the po- (symbols) for the mode mixing angle α closely fit the (solid)
1. larization is transverse to zˆ and factors out of the wave curveα=arctan[(ǫp−ǫs)/∆ǫ−b]/2. Different symbols(col-
0 problem,leavingascalarHelmholtzequationwhichmaps ors) represent data series at L = 2.5 to 25µm (in steps of
9 to a quantum harmonic oscillator [7]. Labeling the re- 2.5µm)withkR≈780to1631. ∆ǫandbarefithereforeach
0 sulting transverse spectrum by ℓ Z and a radial node series.
: number p N , all modes with th∈e same transverse or-
v ∈ 0
i der N = 2p+ ℓ are degenerate in this set of approxi-
X mations. There|fo|re, going beyond this theory entails a heremaybeneededtodistinguishspectralanti-crossings
r degenerateperturbationtheory for whichwe constructa arising from passive cavity physics from those of strong
a coupling Hamiltonian V with the nominal Gaussian di- photon-electroncoupling.
vergenceangle,θ kw /2,asitssmallparameter. Here Themainresultofourworkisananalytical,quantita-
D 0
≡
k is the wave number and w is the waist radius. Since tiveexpressionforthe degreeofOAMmixinginadome-
0
V is typically not diagonalin ℓ, the solutionsmay be far shaped cavity with a Bragg mirror. The latter is crucial
from ℓ-eigenstates. This holds even for arbitrarily small to achieving high finesse in optical microcavities. Our
θ . Both polarization and ℓ mix in a paraxial spin-orbit results also shed new light on the recently-observedcou-
D
coupling [5, 6]. Our work, in part, yields a new way plingbetweenspatialandpolarizationdegreesoffreedom
of generating OAM and other non-uniformly polarized in broad-area vertical-cavity surface emitting lasers [9]:
light. Furthermore, in work involving cavity quantum polarization mixing was found to cause surprising spec-
electrodynamics, paraxial theory at the level developed tralcomplexityevenwithsimple(square)boundaries,ac-
companied by intricate polarization patterns in the far-
fieldemission. Theessentialphysicsisprovidedbythein-
teractionof differently polarizedplane-wavecomponents
∗Electronicaddress: [email protected] with the planar, high-reflectivity distributed Bragg re-
2
flector (DBR) on which both Ref. [9] and our system in bers of the pair Mm∓1σˆ by an amount
{ N ±1}
Fig. 1 (a) are based.
m
TypicalDBRscomprisedielectricmultilayersandhave ∆kLG ≡k(MNm+1σˆ−1)−k(MNm−1σˆ1)= 4kLR. (1)
a “form birefringence”: the reflectivities for TE (s-
polarized) and TM (p-polarized) plane waves, r = ThecalculationassumesadomeofverticallengthL,top
s(p)
|rs(p)|exp(iφs(p)), are unequal in phase at nonzero an- mirror radius of curvature R, and rs(p) = −1. We will
gles of incidence, θ. Previously [5, 10], we had found nowshowthatamorerealisticmodelforrs(p) leadstoac-
numerically that the reflectionphase difference φs(θD) tualresonatormodesΨN,m,1 andΨN,m,2 havingagiven
−
φ (θ ) = 0 is responsible for the polarization mixing. mbutformingarotationoftheLGbasispairbyamixing
p D
6
A consequence of great practical importance is that by angle α ( π/4,π/4):
∈ −
tuningcavityormirrorparameters,thecouplingofOAM
iomffpfoosredsebleyctMedaxmwoedlle’ss,egqiuvaintigonthsecmanwbeellr-idgeofirnoeudslℓy.tuArsnidede (cid:18)ΨΨNN,,mm,,12((αα))(cid:19)=(cid:18)csoinsαα −cosisnαα(cid:19)(cid:18)MMNmNm+−11σˆσˆ−11(cid:19). (2)
fromthe considerationofOAM,ourperturbationtheory
is fundamental to paraxial theory itself: we perform a Thestatesatα= π/4arehybrid modes,oneofwhichis
±
significant extension to and completion of previous work predominantly(thoughnotcompletely)composedofTM
byYuandLuk[11]whichderivedthelowestordercorrec- plane waves,the other being predominantly TE [13, 14].
tionstotheparaxialmodesoftwo-mirrorcavitieshaving This approximate polarization separation of the hybrid
perfect electrically conducting mirrors, r = r = 1. modes yields an intuitive picture of mode mixing; the
s p
Our approach, which allows a more general, dielec−tric separationactsasaleverarmbywhichtheBraggmirror,
planar mirror, requires the construction of a 2 2 per- having φs = φp, rotates the eigenmode basis away from
turbation matrix V and predicts very different×results the LG mo6des and toward the hybrid modes.
when r and r are different functions. The OAM-non-conservation described above is exhib-
s p
ited by the paraxial modes of two-mirror axisymmetric
TheworkofRef.[11]anditsprecursors(e.g.Ref.[12])
cavity resonators which 1) are of sufficiently small size
appearstobeforgotteninthecurrentliterature. Thiscan
with respect to wavelength, 2) have sufficiently narrow
be attributed to the difficulty of experimentally observ-
resonance widths (low loss), and 3) have at least one
ing the small spectral splittings caused by slightly non-
mirror for which φ = φ (commercial dielectric mirrors
s p
paraxial perturbations. Agreement with a degenerate 6
meet this requirement). The first two requirements are
perturbation theory for r = 1 was experimentally
s(p) − essentialinsplittingthedegeneraciesofhighorderGaus-
demonstrated using microwaves [12] where, due to com-
sian modes, and Eq. (1) allows us to estimate whether
parableresonatorsizeandwavelength,themode spacing
this is possible. To perform the degenerate perturbation
islargeenoughtoresolvethelifting oftheN+1-foldde-
theory in the presence of property 3), we first develop
generacies labeled by N. However, with recent progress
the perturbation Hamiltonian, V, in the two-mode basis
inminiaturization[8],comparablesizeparametersarebe- Mm∓1σˆ . V mustbesymmetric(V =V ),andfor
coming accessible to optical cavities similar to Fig. 1(a). { N ±1} 21 12
our purposesmay be takento be traceless(V = V ).
22 11
To write the splittings described in Ref. [11] in a form V then has eigenvectors~v = cosα ,~v = sinα−with
1 −sinα 2 cosα
we can use for the following discussion, let us first re-
(cid:0) (cid:1) (cid:0) (cid:1)
view the unperturbed vector-field basis spanning the N- α=(1/2)arctan(V12/V22), (3)
th transverse multiplet. These are the Laguerre-Gauss
(LG) modes, written in polar coordinates as a product and eigenvalues δk = (V2 + V2)1/2. As system
Mℓ (ρ,φ,z)σˆ of a scalar part Mℓ , having orbital an- parameters are vari1e(d2), the∓elem1e2nts o2f2V change and an
N s N
gular momentum ℓ, and a circular polarization vector anti-crossingof the hybrid modes emerges, cf. Fig. 1(b).
σˆ (xˆ+siyˆ)/√2, where s = 1 is the spin degree of All deviations from the paraxial limit must be con-
s
free≡dom. Expressionsfor the LG±modes and their Bessel sidered to lowest order in θ2, which for our cavity is
D
wavedecompositionsaregiveninRef.[13]. Perturbations θ2 = 2/[k L(R L)]. The physical derivation of V is
D −
thatpreserverotationalsymmetryaroundzˆwillonlycou- facilitated by noting that in the anti-crossing scenario,
ple those MNℓ (ρ,φ,z)σˆs for which the total angular mo- α = 0 is epquivalent to V12 = 0 and hence corresponds
mentum m=ℓ+s aroundthis axis is the same [13]. We to the assumptions underlying the known result Eq. (1).
henceforth consider N and m to be fixed, nonnegative Thus, Eq. (1) should be reproduced by our model at
parameters; we will briefly discuss later the allowance of α=0. In wave number units, we therefore set
m < 0. The non-conservation of OAM emerges here be-
causes cangenerally[17]take twovalues,corresponding 1 m L L θ2
to the basisstates MNm−1σˆ1 andMNm+1σˆ−1. No symme- V22 =−V11 = 2∆kLG = 16sR 1− R LD. (4)
try prevents these states from coupling, unless m=0 in (cid:18) (cid:19)
which case time-reversalinvariance applies.
Although Eq. (1) was derived for ideal-metal cavities,
Nevertheless, no such OAM coupling is found in our more general DBR boundary conditions do not af-
Ref.[11],whereperturbationsmerelysplitthewavenum- fect Eq. (4), because they shift all LG modes with the
3
same N equally. To explain this, consider the penetra-
tion depths δL of each of Mm∓1σˆ into the mir-
s(p) N ±1
ror layers. For a plane wave of s or p polarization with
incident angle θ at a DBR, δL = φ (k,θ)/(2k).
s(p) s(p)
In order to capture the relevant material properties of
the DBR, we neglect transmission and expand its reflec-
tion phase in the plane wave angle of incidence, θ, as
φ (k,θ) φ (k)+ǫ (k)θ2.
s(p) 0 s(p)
≈
Any vectorial mode Ψ can be decomposed into az-
imuthally symmetrized plane waves (Bessel waves) by
defining a“tilde”operator such that Ψ˜s(p)(θ) essentially
denotes the amplitudes of the TE(TM) plane waves of Figure 2: Mode-coupling width ∆ǫ from fit (symbols) com-
polarangleθ. TheparaxialΨ˜s(p)(θ)isnonzeroonlynear paredtoperturbativeprediction(solidline),plottedasafunc-
θ 0, with the averaged reflection phase of Ψ being tion of the cavity length L at fixed R = 100µm. Error bars
≈ were obtained from the fits of Fig. 1(c). Because the an-
alytic equation (10) is wavelength-independent, all data for
φ Ψ Ψ˜s(θ) 2φs(θ)+ Ψ˜p(θ) 2φp(θ) θdθ, (5) λ≈400nm and λ≈800nm fall onto thesame curve.
h i ≡
Z h(cid:12) (cid:12) (cid:12) (cid:12) i
with the norm(cid:12)alizati(cid:12)on (Ψ˜s 2(cid:12)+ Ψ˜p(cid:12) 2)θdθ = 1. Spe- where the width of the crossoverinterval is given by
| | | |
cializing to the LG modes with the abbreviation Λ
±1
|vMΛ˜erp±Nms1e∓(1θfiσ)ˆe|l±2d.1[,7T]chierenctuhalaiarlrsmtpohnoalitRac-rEoizqsac.ti(lil5oa)nt,owlreiantdhasttuhtroeeao|bΛ˜fos±tvh1ee(θet)xr|p2aann=≡s-- ∆ǫ(L/R,N,m)= m2 s((LN/R+)(11)2−−Lm/R2). (10)
sionforφ ,dependsonlyonthemodeorderN butnot Theformulasabovecompletethelowestorderdegener-
s(p)
on ℓ. This carries over to the average penetration depth ateperturbationtheoryforparaxialmodemixing. Inter-
hδLiΛ±1 = hφiΛ±1/(2k), and hence the effective cavity estingly,∆ǫisindependentofwavelength: forfixedcavity
lengthL+ δL is identicalforboth modesΛ ; this geometry and mode labels N, m, modes of different lon-
then impliehs eqiuΛa±l1spectral shifts, as claimed abo±ve1. gitudinalnodenumber(alongzˆ)willhavedifferentk but
identical anti-crossing behavior when α is plotted versus
To obtain the off-diagonal element V , we apply the
12
ǫ ǫ . This universalfunctional form provides a robust
samepenetration-depthargumenttothespecialcaseα= p s
−
way of tailoring any desired mixing angle α. Most im-
π/4 where Eq.(2) yields the hybridmodes. Their plane-
waveamplitudes, Ψ˜s(p) (θ) and Ψ˜s(p) (θ), can be writ- portantlyundertheaspectofOAMnon-conservation,we
N,m,1 N,m,2 can tune Eq. (2) to α=0.
ten purely in terms of the LG amplitudes Λ˜p±1(θ) using Once the cavity linewidth for the relevant paraxial
Eq. (2), and their splitting, ∆khybrid kN,m,2 kN,m,1, modes becomes less than the mode separation, ∆k
is given by ≡ − 2δk m/(4kLR), the two modes Ψ given b≡y
2 N,m,j
≥
Eqs. (2, 9, 10) are resolved at slightly different k (or
∆k = (k/L)(δL δL ) (6)
hybrid N,m,2 N,m,1 L). Any excitation of the cavity would generally have
− −
= ǫp−ǫs θ3Λ˜p (θ)Λ˜p (θ)dθ, (7) non-zero overlap with these modes, and thus compli-
L +1 −1 cated mode patterns [5, 13] can be generated by simple
Z excitation. The magnitude of α is zeroth order in θ2,
D
where k is the unperturbed wave number. Therefore, and excursions near the asymptotic values can be seen
we reach ∆khybrid ∆kLG if the form birefringence in Fig. 1(c). The magnitude of the relative frequency
| | ≫ | |
quantity, |ǫp−ǫs|, is made large. On the other hand, splitting, however, is O(θD4) as θD → 0. We note that
α π/4 implies V12 V22 , so that the eigenvalues of observed modes will not be restricted to m 0. Ax-
→ | |≫| | ≥
V in this limit are δk1(2) V12. Setting 2δk2 equal to ialsymmetrycreatesanexacttwo-folddegeneracyofthe
≈ ∓
Eq. (7) and performing the θ integral, one obtains vectorial LG basis modes under the transformation in
which m, ℓ, and s switch sign [10]. The presence of the
V =V = ǫp(k)−ǫs(k) (N +1)2 m2 θD2 . (8) exactdegeneracy[18]doesnot“washout”thegeneration
12 21 8 − L ofcomplicatedmodepatternsandismorefullydiscussed
p in Refs. [10, 13, 15].
Forsomesimpledielectricmirrors,ǫp ǫsmaybeswept WehavenumericallycalculatedmodesforR=100µm
−
acrosszeroby varyingthe cavitylengthacrossthe nomi-
and L = 2.5 to 25µm with a DBR comprising 36 pairs
nalLatwhichthemodepairofinteresthasunperturbed of quarter-wavedielectric layers A and B with refractive
kequaltothedesign(center)wavenumberofthemirror,
indices n = 3.52 and n = 3.00, with layer B at the
A B
kd. The mixing angle α can then be written as topsurface. Typicalvaluesofǫ werearound 3,with
s(p)
−
d(ǫ ǫ )/d(k k ) 2.8µm. Data were analyzed
p s d
α=(1/2)arctan [ǫ (k) ǫ (k)]/∆ǫ , (9) − − ≈ −
p s for modes close to λ 2π/k 400 and 800 nm. We
− d d
≡ ≈
(cid:8) (cid:9)
4
consideredmodepairswithN =2andm=1,thelowest order N family provides N +1 nearly degenerate energy
values for which OAM mixing can occur. levels corresponding to modes having different vectorial
The numerical data for the mixing angle are fit ex- spatial patterns of the electric field. This spectral and
tremely well by Eq. (9) if we allow for an offset b in the spatialstructurecombinedwithquantumdotsatthepla-
argument of the arctan, as done in Fig. 1(c). The com- nar mirror may possess quantum logic capability.
parison between numerical fit and Eq. (10) is shown in Inconclusion,boththemixingangleandthefrequency
Fig. 2. The offset b has median 0.32 for our data and splitting for OAM-mixed paraxial resonator modes in
−
empirically behaves as C/[kL(1 L/R)], where C is a an axisymmetric cavity have been analytically derived
−
slowly-varying function of k and k . Taking the limit here in a degenerate perturbation theory which includes
d
θ 0 such that L/R is bounded away from 0 and 1 the form birefringence of a practical mirror. The phe-
D
→
implies that b = O(θ2). This next-highest-order cor- nomenon is similar in principle to the polarization cou-
| | D
rection to our perturbation theory will be discussed in a plingobservedinRefs.[9,16]: differencesinthepenetra-
subsequent publication [15]. tion depths for TE and TM plane wavesmodify the vec-
To spectrally resolve the transverse mode splitting torialresonatormodes. Forourcasehowever,eigenmode
along the entire mixing curve, the cavity must obey coupling persists from the non-paraxial regime to the
2kR[1 (R R )1/2]<m,whereR arethepowerreflec- deeplyparaxialregime. Byexaminingthelatter,wehave
1 2 1(2)
−
tivities of the two mirrors. Microwave experiments may shownherethatthecouplingcaninfactbeturnedonand
bethemostdirectapproach. Alternatively,paraxialspin- off via the material parameters ǫ . The underlying in-
s(p)
orbit coupling may be realized in microcavityresonators adequacyof a scalar paraxialtreatment is washedout in
for quantum-informationapplications,cf. Ref. [8], where macroscopic cavities, but must be regarded as a funda-
small size and high finesse are required. Such cavities mentallimitationinhigh-finessemicrocavities,wherethe
couldgenerateOAMorhybridbeamsatlightlevelsfrom correct starting point for any paraxial formulation must
single-photon-on-demandto that of a macroscopic laser. be the mode basis of Eq. (2), which is heterogeneous in
Inparticular,onecouldutilizethefinestructureandvar- orbital angular momentum.
ied spatial patterns of the split modes. This has partic- This work is supported in part by National Science
ularpotentialforquantuminformationapplications: the Foundation Grant No. ECS-0239332.
[1] L. Allen, et al., in Progress in Optics, edited by E. Wolf 23, 218 (1975).
(Elsevier, Amsterdam, 1999), vol. 39, pp.291–372. [13] D. H. Foster, Ph.D. thesis, University of Oregon (2006),
[2] O. Hosten and P. Kwiat, Science 319, 787 (2008). URL http://hdl.handle.net/1794/3778.
[3] R. Loudon,Phys.Rev.A 68, 013806 (2003). [14] D. H. Foster, A. K. Cook, and J. U. N¨ockel, Opt. Lett.
[4] A. V.Dooghin, et al., Phys.Rev.A 45, 8204 (1992). 32, 1764 (2007).
[5] D.H.FosterandJ.U.N¨ockel,Opt.Lett.29,2788(2004). [15] J. U. N¨ockel, D. H.Foster, and A. K. Cook, (to be pub-
[6] K.Y.Bliokh and D.Y.Frolov,Opt.Commun. 250,321 lished).
(2005). [16] L. Fratta, et al., Phys. Rev.A 64, 031803(R) (2001).
[7] J. U.N¨ockel, Opt.Expr.15, 5761 (2007). [17] Specifically,whenN ≥2and0<|m|<N+1, thereare
[8] G. Cui, et al., Opt.Expr. 14, 2289 (2006). two mixable basis states and Eq. (2) applies.
[9] I. V. Babushkin, et al., Phys. Rev. Lett. 100, 213901 [18] Thisresultsinadegenerate(superimposable)SU(2)sec-
(2008). torwhichexternallymultipliesthenon-degenerate(mix-
[10] D. H. Foster and J. U. N¨ockel, Opt. Commun. 234, 351 able) SU(2) sector we have considered in Eq. (2). The
(2004). location, or“generalized polarization”, of each observed
[11] P. K. Yu and K. Luk, IEEE Trans. Microwave Theory mode within the exactly degenerate sector is excitation
Tech. 32, 641 (1984). dependent,while thecavityfixesthelocation (α) within
[12] C. W. Erickson, IEEE Trans. Microwave Theory Tech. themixable sector.