Rotationally induced vortices in optical cavity modes

0.37 MB·English

by  Steven J. M. Habraken

Rotationally induced vortices in optical cavity modes
Steven J. M. Habraken∗ and Gerard Nienhuis
Leiden Institute of Physics, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands
We show that vortices appear in the modes of an astigmatic optical cavity when it is put into
rotation about its optical axis. We study the properties of these vortices and discuss numerical
results for a specific realization of such a set-up. Our method is exact up to first order in the
time-dependent paraxial approximation and involves bosonic ladder operators in the spirit of the
quantum-mechanicalharmonic oscillator.
9 I. INTRODUCTION amples of rotationally-induced vortices in material sys-
0 tems raise the question if and how rotation induces vor-
0 ticity in light fields. In this paper we address this topic
Rotation is a very natural source of vorticity: when a
2
by studying the optical properties of a two-mirror cav-
glass of water is stirred, a vortex appears at the center.
n ity that is put into rotation about its optical axis. This
Suchavortexisasingularityofthewatercurrentanditis
a set-up is schematically drawn in Fig 1. We expect an
ofthesametypeasthevortexthatappearsaboveasink.
J
effect of rotation only if the cavity lacks rotational sym-
When a glass of water is put into uniform rotation, for
5 metry. In analogy with the rotating elliptical potential
instance by placing it on a turntable, the water current

in which Bose-Einstein condensates can be trapped, we
] vanishes at the rotation axis but is not singular. It has
s beenknownforoverhalfacenturythatthisisdifferentin break the rotational symmetry by taking at least one of
c the mirrors cylindrical or astigmatic. In the absence of
i case of a superfluid [1]. Due to the zero viscosity of, and
t rotation, such a cavity has astigmatic Hermite-Gaussian
p the vortex quantization in, a superfluid, uniform rota-
modes [16], which have lines of zero intensity (line dis-
o tionsinducearegularpatternofequallychargedvortices.
locations) in the transverse plane. We show that rota-
. Vortices appear only if the rotation frequency exceeds a
s tion deforms the cavity modes into generalizedGaussian
c certain critical value and their number increases if the
i rotation frequency is further increased. Eventually rota- modes [17] and that the line dislocations are deformed
s
intoopticalvortices(pointsingularitiesinthe transverse
y tion may destabilize the superfluid. More recently, simi-
h larexperiments havebeen performedwith Bose-Einstein plane). We study the properties of these rotationally-
p induced optical vortices.
condensatesofdilutegassesbothbyopticallystirringthe
[ This paper is organized as follows. In the next sec-
condensate [2] and by trapping it in a rotating elliptical
tion we briefly review the propagation of optical fields
2 potential [3].
v During the past decades, optical vortices and their through time-dependent systems and focus on the spe-
5 cific case of a rotating astigmatic cavity. We discuss the
propagation have attracted a significant amount of at-
6 ladder-operatormethod that we apply to obtain explicit
tention [4, 5, 6, 7, 8]. An optical vortex is a singularity
2 expressionsoftherotatingcavitymodes[18]. Inthethird
4 of the phase of an optical beam and is characterized by sectionweusethe approachthatisdiscussedinRef. [19]
. itspositioninthetransverseplane,itstopologicalcharge
1 tocharacterizethedegreesoffreedomassociatedwiththe
and its morphology. The vortex chargeis determined by
1 astigmatism and vorticity of the rotating cavity modes.
the total phase change 2πq along a contour around the
8 Weapplytheanalogywiththequantum-mechanicalhar-
0 vortexcenterandmustbe integerforreasonsofcontinu-
monicoscillatortoderiveanalyticalexpressionsofthero-
: ity. Asopposedtothevorticesthatappearinsuperfluids
v tatingcavitymodes. Theseareusedtodiscusssomegen-
andBose-Einsteincondensates,opticalvorticescanbeel-
i eral properties of the vortices that appear in the modes.
X liptical; their morphology is characterizedby the partial
Inthefourthsectionweshowanddiscusssomenumerical
r derivatives of the beam profile close to the vortex center
a results for a specific realization of a uniformly rotating
and can be represented by a point on a sphere [9]. The
two-mirror cavity.
polar angle on the sphere determines the degree of el-
lipticity while the azimuth angle fixes the orientation in
thetransverseplane. Theinterplaybetweenastigmatism
II. PARAXIAL WAVE OPTICS BETWEEN
and the propagation of optical vortices may give rise to
ROTATING MIRRORS
veryrichbehaviour[10, 11,12]. The dynamicsofoptical
vortices in a laser cavity has also been studied [13].
A. Mode propagation in rotating cavity
In view of the recent interest in effects of rotating ele-
ments on optical beams [14] and the physical properties
of rotating mode patterns [15], the above-mentioned ex- The mathematical description of the propagation of
lightthroughopticalsystemssimplifiessignificantlyifthe
paraxialapproximationisapplied. Thisapproximationis
almost always justified in experimental set-ups with op-
∗Electronicaddress: [email protected] tical beams. In the present case of a rotating cavity, we
2
must account for effects that arise from the time depen-
dence ofthe mirrorsettings. Assuming that the rotation
frequencyΩismuchsmallerthantheopticalfrequencyω,
we use the generalization of the paraxial approximation
to the time-dependent case [20]. In this approximation,
the electric field is purely transverse. For a propagating
mode it can be written as
E(r,t)=Re E0ǫu(r,t)eikz−iωt , (1)
whereE0 istheamplitu(cid:8)de ofthe field,ǫis(cid:9)the transverse FIG.1: Schematicplotoftheset-upthatwestudyinthispa-
polarization, k is the wave number and ω = ck is the per: anastigmatic two-mirrorcavitythatisputintorotation
about its optical axis. The rotation frequency is denoted Ω
opticalfrequency with c the speed oflight. The complex
scalar profile u(r,t) characterizes the large-scale spatial and z0 indicates the transverse reference plane.
structure and slow temporal variations of the field. It
obeys the time-dependent paraxial wave equation
∇2ρ+2ik∂∂z + 2cik∂∂t u(ρ,z,t)=0, (2) widnehgsectrroeibUtˆehfse(zftr)iem=eepe-rixnopdp(cid:0)eap2igzkea∇ntido2ρe(cid:1)nnitosfpthaareappxariraoalpxawiagalavtbeoereaqcmuoarirtneiosapnosantndad--
(cid:18) (cid:19) tionaryframe. The operatorUˆ(z)has the significanceof
with ρ =(x,y) and 2 =∂2/∂x2+∂2/∂y2 is the trans-
∇ρ the propagator in the rotating frame. The rotation op-
verse Laplacian. If we omit the derivative with respect
erator arises from the Coriolis term in Eq. (4) and gives
to time, this equation reduces to the standard parax-
the propagating modes a twisted nature.
ial wave equation. This equation has the same form as
the Schr¨odinger equation for a free particle in 2D+1, z
playing the role of time. It describes the diffraction of a
freely propagating stationary paraxial beam. The time B. The modes of a rotating cavity
derivativeinEq. (2)accountsforthetimedependenceof
the profile and incorporates retardation between distant In a stationary cavity, modes are usually defined as
transverse planes. stationarysolutionsofthe time-dependentparaxialwave
In addition to diffraction, the propagationof the light equation(2)thatvanishonthe mirrorsurfaces[16]. The
inside a cavity is governed by the boundary condition transverse profiles of these modes are reproduced after
thattheelectricfieldmustvanishonthemirrorsurfaces. each round trip up to a Gouy phase factor exp( iχ),
−
In case of a rotating cavity this boundary condition is which determines the resonantwave numbers. From Eq.
explicitly time dependent. A natural way to eliminate (1)itfollowsthattheelectricfieldpicksupaphase2kL
−
this time dependence is by transforming to a co-rotating χ,whereListhemirrorseparation,aftereachroundtrip
frame, where it is sufficient to consider the behaviour of so that the resonance condition reads 2kL χ = 2πq
time-independentpropagatingmodesv(r). Thetransfor- with q Z. The necessary and sufficient co−ndition for
mationthat connects the profile in the rotatingframe to a statio∈nary cavity to have long-lived modes is that it
the profile in the stationary frame can be expressed as is geometrically stable, i.e. that the round-trip Gouy
u(ρ,z,t)=Uˆrot(Ωt)v(ρ,z), (3) phasesχ arerealso thatthe magnification|exp(−iχ)|is
equal to 1.
where Ω is the rotation frequency and Uˆrot(α) = Since a rotating cavity is time-dependent, we cannot
exp( iαLˆ )istheoperatorthatrotatesascalarfunction expect time-independent modes in this case. A natural
z
over−anangleαaboutthezaxiswithLˆ = i(x∂y y∂x) and continuous generalization of the mode criterion to
z
− − the rotating case is to require that the modes adopt the
the z-componentofthe orbitalangularmomentumoper-
ator. Substitution of the rotating mode (3) in the time- time-dependenceofthecavity,i.e. thattheyrotatealong
with the mirrors. These modes are time-independent in
dependent paraxial wave equation (2) yields the wave
equation for v(ρ,z) the co-rotating frame so that they obey Eq. (4) and
vanish on the mirror surfaces. Though the interplay be-
2+2ik ∂ + 2ΩkLˆ v(ρ,z)=0 (4) tweenrotationandstabilitygivesrisetosurprisinglyrich
∇ρ ∂z c z behaviour, geometrically stable rotating two-mirror cav-
(cid:18) (cid:19)
ities exist [21]. In a recent paper we have developed an
Intherotatingframe,theretardationtermisreplacedby
analytical-algebraic method to find explicit expressions
aCoriolisterm,whichisfamiliarfromparticlemechanics.
Since and Lˆ commute, the formal solution of the of these modes [18]. The method involves two pairs of
∇ρ z bosonic ladder operators in the spirit of the quantum-
paraxial wave equation in the rotating frame (4) can be
mechanicalharmonic oscillatorthat generate a complete
expressed as
andorthogonalsetofcavitymodes. Inagiventransverse
Ωz reference plane, which we take close to the first mirror
v(ρ,z)=Uˆf(z)Uˆrot − c v(ρ,0)≡Uˆ(z)v(ρ,0), (5) anddenotebyz =z0asisindicatedinFig. 1,theprofiles
(cid:18) (cid:19)
3
of the cavity modes can be expressed as from the ray vectors µ = (r ,t ) and transform accord-
i i i
ingly, the algebraic properties of the round-trip ray ma-
vnm(ρ,z0)= √n1!m! aˆ†1(z0) n aˆ†2(z0) mv00(ρ,z0), ttroixbe(8a)ppalriecaebslsee.ntFiarolmforthtehsee lpardodpeerrtoiepse,riattoforllaopwpsrotahcaht
(cid:16) (cid:17) (cid:16) (cid:17) (6) its eigenvalues are either realor pairwise complex conju-
where aˆ†1(z0) and aˆ†2(z0) are the two raising operators in gate phase factors [23]. The rotating cavity is geomet-
the reference plane. The fundamental mode v00(z0) is rically stable only in the latter case. In this case, the
fixed up to a phase factor by the requirement that act- corresponding pairwise complex conjugate eigenvectors
ingonitwiththecorrespondingloweringoperatorsmust µ1,2 = (r1,2,t1,2) and µ∗1,2 = (r1∗,2,t∗1,2) can be chosen
give zero, i.e. aˆ1(z0)v00(z0) = aˆ2(z0)v00(z0) = 0. The such that
ladder operators are linear combinations of the position
operators ρˆ=(x,y) and the conjugate momentum oper- ri(z0)tj(z0)−ri(z0)tj(z0)=0, (9)
ators kθˆ= i with k the wave number of the mode. and
ρ
− ∇
sTighneifiecxapneccetaotifonthveaaluveesra<gevt|ρrˆa|vnisvaenrdse<povs|iθˆt|ivoinhaanvde tthhee ri∗(z0)tj(z0)−ri(z0)t∗j(z0)=2iδij , (10)
average propagation direction of the beam. The posi- where the indices i and j take the values 1 and 2. The
tion and propagation direction operators obey canonical complex conjugate eigenvectors µ∗1,2 generate the rais-
commutation relations [ρˆ,kθˆ ]= iδ , where the indices ing operators aˆ† according to Eq. (7) and the special
i j ij 1,2
i and j run over the x and y and with δ the Kronecker properties of the eigenvectors (9, 10) guarantee that the
ij
delta function. In the reference plane z0 the lowering ladder operators obey bosonic commutation relations
operators can be expressed as
[ai(z0),aj(z0)]=0 and [aˆi(z0),aˆ†j(z0)]=δij , (11)
aˆi(z0)= k2 ti(z0)ρˆ−ri(z0)θˆ , (7) mwhoedreest(h6e)ifnodrmicesaicoamndpljetreuannodveorrt1hoannodrm2,alsosetthaint tthhee
r
(cid:16) (cid:17) reference plane.
wheretheindexirunsover1and2. Thecomplexvectors The four eigenvalues of the round-trip ray matrix (8)
ti(z0) and ri(z0) have two components and are chosen canbewrittenasexp(iχ1,2)andexp( iχ1,2),whereχ1,2
such that µi =(ri,ti) is an eigenvectorof the round-trip arethe round-tripGouy phase that a−repickedup by the
ray matrix in the co-rotating frame. In the present case lowering operators. They determine the resonant wave
ofanastigmaticopticalcavity,thisisareal4 4matrix, numbers for the (n,m) cavity mode according to
×
which can be expressed as
1 1
2kL χ1 n+ χ2 m+ =2πq , q Z. (12)
Mrt =M1 M(L) M2 M(L), (8) − 2 − 2 ∈
· · · (cid:18) (cid:19) (cid:18) (cid:19)
The ladder operators that generate the rotating cav-
where M(z) = Mf(z) Mrot( Ωz/c) is the ray matrix
· − ity modes in an arbitrary transverse plane can be con-
that corresponds to Uˆ(z) and describes free propagation
structed according to Eq. (7) and its hermitian conju-
intheco-rotatingframe,Mf(z)istheraymatrixthatde- gate by using that the ray vectors µ =(r ,t ) transform
i i i
scribes propagationin a stationaryframe andMrot(α) is according to
the ray matrix that describes a rotation over an angle α
itnhethmeirtrroarnss1vearnsedp2larensep.eTcthiveerlyayanmdaatrriecefusllMy1d,e2tderemscirnibede rtii((zz)) =M(z−z0) rtii((zz00)) (13)
(cid:18) (cid:19) (cid:18) (cid:19)
by their radii of curvature and orientation in the trans-
The properties given by Eqs. (9) and (10) are preserved
verse plane. These ray matrices are generalizations of
under this transformation so that the ladder operators
the standard 2 2 ray matrices, which can be found in
× obeythebosoniccommutationrelations(11)inalltrans-
any textbook of optics, to the astigmatic case. Explicit
verse planes. So far, we have considered only the mode
expressions of the 4 4 ray matrices are given in Ref.
× that propagates from left to right(from mirror 1 to mir-
[18].
ror 2 in Fig. 1). By using the ray matrix that de-
As opposed to the unitary propagator Uˆ(z) and the
scribes one of the mirrors (say mirror 2), one can con-
rotationoperatorUˆrot(α), whichactin the Hilbertspace structthe ladderoperatorsthatgeneratethe modesthat
of transverse modes, ray matrices describe real linear propagate in the opposite direction: (r←(z),t←(z)) =
i i
mtraanllsyfosrpmeaaktiinong,stihnisthpehtarsaensspvaecrseeisphaassyemsppalecceti(cρ,mθa).niFfoolrd- M(z0+L−z)M2M(z0+L−z)(ri→(z),t→i (z)). The ac-
tual field inside the cavity is a linear combination of the
and the real and linear transformations that preserve its
left and right propagating modes and can be expressed
mathematical structure form the real symplectic group
as
Sp(R,4). Symplectic groups and various physically rel-
evant aspects of symplectic geometry have been studied Enm(r,t)=Re −iE0ǫ vn→m(ρ,z)eikz−iωt−
indetail,seeforinstanceRef. [22]. Theladderoperators n
act in the mode space, but since they are constructed (cid:0) vn←m(ρ,z)e−ikz−iωt , (14)
(cid:1)o
4
t1,2 in two matrices, which are defined as
R(z)= i r1(z) and T(z)= t1(z) ,
− r2(z) t2(z)
(cid:18) (cid:19) (cid:18) (cid:19)
(15)
where the factor i is introduced for notational conve-
−
nience. Thespecialproperties(9,10)canbesummarized
as
RTT TTR=0 and R†T+T†R=2 (16)
−
As was mentioned already, the fundamental mode
FIG. 2: The Hermite-Laguerre sphere, which describes the
degrees of freedom associated with the nature of higher- v00(ρ,z) is fixed by the requirement that acting on it
with both lowering operators must give 0. The two first
order paraxial optical modes. Each point (φHL,θHL) corre-
order differential equations that are thus obtained have
spondstoacompleteandorthonormalsetofmodes;thepoles
(θHL = 0,π) correspond to Laguerre-Gaussian modes while the solution
points on the equator (θHL = π/2) correspond to Hermite-
gGivaeusrsiisaentmoogdeense.raInlitzeerdmGedaiuatsesiavnalumeosdoefs.theTpheolaarziamnugtlehθaHnL- v00(ρ,z)=sπdetkR(z)exp −kρTS(z)ρ/2 , (17)
gle φHL determines the orientation of the higher-order mode (cid:0) (cid:1)
patterns in thetransverse plane.
where S=R−1T is a 2 2 matrix. It makes sense to de-
composeitintoitsreal×andimaginarypartsS=Sr+iSi.
The matrices Sr and Si respectively characterize the
astigmatism of the Gaussian intensity and phase pat-
terns. From the properties of R and T (16) it can be
shown easily that S is symmetric and that its real part
Sr is positive definite so that the mode profile is square-
integrable. The profile in Eq. (17) has been normalized
properly. In order to characterize the other degrees of
freedom,itis convenienttorewritethe z-dependentlow-
ering operators (7) as
aˆ1(z) = kR(z) S(z)ρˆ+iθˆ . (18)
(cid:18)aˆ2(z)(cid:19) r2 (cid:16) (cid:17)
FIG. 3: A stationary cavity has inversion symmetry so that By using Eq. (16) we find that R†R = S−1 so that
it is equivalent to its mirror image (thedashed line indicates r
themirrorplane). Sincetherotationdirectionchangesunder R(z) can be written as σ(z)S(z)r−1/2, where σ is a uni-
inversion, this is not trueof a rotating cavity. tary 2 2 matrix. Notice that Sr is real, symmetric and
positiv×e definite so that Sr−1/2 is well-defined. By mak-
ing use of the operator identity eABe−A =B+[A,B]+
where the minus sign accounts for the fact that a mode 1[[A,[A,B]]+... and the canonical commutation rela-
2!
changes sign when it is reflected by a mirror. In the rest tions [ρˆ,kθˆ]=iδ we find that
ij
of this paper, we focus on the mode profile v rather
nm
thanontheactualcavityfield(14),sincethisistheprofile iSiρˆ+iθˆ=e−ikρSiρ/2θˆeikρSiρ/2 (19)
thatwouldbemeasuredinanyrealisticexperiment. The
expressions of modes that we have given in this section so that the z dependent lowering operators can be ex-
hold in the co-rotating frame, the corresponding expres- pressed as −
sions of the rotating mode patterns u (ρ,z,t) in the
nm
stationary frame can be obtained by applying Eq. (3). aˆ1 = kσe−ikρSiρ/2 S1/2ρˆ+iS−1/2θˆ eikρSiρ/2 .
(cid:18)aˆ2(cid:19) r2 r r
(cid:16) (cid:17) (20)
III. LADDER OPERATORS AND VORTICES By introducing real scaled coordinates ρ′ = √kS1r/2ρ =
(x′,y′) that account for the astigmatism of the intensity
A. Analytical expressions of the modes pattern,theproductof k/2andthelinearcombination
ofρˆandθˆbetweenthebracketstakestheformofthelow-
p
In order to give more explicit expressions of the rotat- ering operators of a dimensionless quantum-mechanical
ing cavity modes (6), we combine the vectors r1,2 and harmonic oscillator in 2D+1. From right to left, both
5
the z-dependent lowering operators (20) and the corre- B. Vortices in higher order modes
sponding raising operators aˆ† first remove the curved
1,2
wave front, then modify the mode pattern and eventu- As is indicated in Fig. 3, a stationary two-mirror cav-
ally restore the wave front again. The unitary matrix σ ityhasinversionsymmetrysothatitmusthavethesame
describes the additional degrees of freedom, which char- modes as its mirror image. In the non-degenerate case,
acterize the nature of the higher-order modes. Overall thisimpliesthatthemodeprofilesclosetothemirrorsare
phase factors in the rows of σ are physically irrelevant realapartfromthe overallcurvedwavefronts[23]. Since
and without loss of generality we can fix its determinant this symmetry property holds for modes of all order, it
suchthat σ SU(2). Since σ is unitary, its rowsarenot impliesthatthehigher-ordermodesofastationarycavity
∈
independent and we can write closetothemirrorsmustbeHermite-Gaussian. Hermite-
Gaussian modes do have line dislocations (lines along
σ σ
σ = x y , (21) which the phase jumps by π) in the transverse plane,
σ∗ σ∗
(cid:18)− y x (cid:19) but do not have vortices (phase singularities).
with σ 2+ σ 2. IncompleteanalogywiththePoincar´e Rotation obviously breaks the inversion symmetry of
x y
| | | | a stationary cavity. As a result, the modes of a rotating
sphere for polarization vectors (or the Bloch sphere for
cavityhaveadditionalphasestructure. Thefundamental
spin-1/2 states) the degrees of freedom associated with
mode attains a twist; though the electric field vanishes
the complex vector (σ ,σ ) can be mapped onto the
x y
onthemirrorsurfaces,itswavefrontsdonotfitthelocal
so-called Hermite-Laguerre sphere [19]. As is indicated
curvature of the mirrors. From Eq. (17), it is clear that
in Fig. 2, every point on the sphere corresponds to
no vortices can appear in the fundamental mode. Vor-
two pairs of bosonic ladder operators that generate a
ticesappearaszerosofthepolynomialpartoftheprofile
complete set of higher-order modes. In this respect,
ofthehigherordermodes(22). Duetotherotationofthe
it is different from the Poincar´e and Bloch spheres, on
cavity, the line dislocations in the higher-order Hermite-
which every point corresponds to a single state. The
Gaussian modes are deformed into elliptical vortices.
poles on the Hermite-Laguerre sphere correspond with
(σ ,σ ) = (1, i)/√2 and yield ladder operators that From the fact that σ (21) is a unitary matrix, it fol-
genxerayte Lague±rre-Gaussianmodes. Points on the equa- lowsthatthetworaisingoperatorsaˆ†1,2 generatevortices
tor correspond to (σ ,σ ) = (cosφ, sinφ) and give with opposite charge. As a result, the vortices that ap-
x y
rise to Hermite-Gaussian modes. Inter−mediate values of pear at the center of the v01 and v10 modes have equal
the polar angle 0 < θHL < π/2 correspond to general- but opposite topologicalcharges±1. Since both vortices
ized Gaussian modes [17]. The azimuth angle φHL fixes are spherical (canonical) in the scaled coordinates, their
the orientation of the higher-order mode patterns in the morphologiesare determined only by the astigmatismof
transverse plane. The rows of σ (21) correspond to an- the intensity pattern. In general, the raising operators
tipodal points on the sphere so that it is, strictly speak- split,displaceandintroducevorticesinthe cavitymodes
ing,sufficienttoconsideroneofthehemispheresonly. In sothatthevortexpatterninthehigher-ordermodes(22)
general,theseparationofthedegreesoffreedominterms can be very complicated. More explicit expressions have
of symmetric matrix S, which characterizes the astigma- been given only in limiting cases [10, 11].
tism, and the unitary matrix σ, which determines the The fundamental mode contains only even powers of
nature of the higher order modes, is local in the sense the position coordinates ρ = (x,y) and is even under
thatonlyholdsinasingletransverseplanez. Theevolu- inversion in the transverse plane: ρ ρ. The lad-
→ −
tionofσunderpropagationandopticalelementsdepends der operators are linear in the position and propagation
on the astigmatism and vice versa. direction operators and are obviously odd under this in-
UsingthedefiningidentityoftheHermitepolynomials version. Itfollowsthattherotatingcavitymodes(22)are
H (x)exp( x2/2)=(x ∂ )nexp( x2/2)andthebino- even or odd, depending on the parity of the total mode
minal expan−sion (a+b)n−=∂x n n−apbn−p for [a,b]=0 number n+m. Odd modes have a vortex at the center
p=0 p of the mode patterns, whereas even modes in general do
the higher order modes can be expressed as
P (cid:0) (cid:1) not.
v (ρ′,z)=e−ρ′2/2−iρ′Sr−1/2SiSr−1/2ρ′/2
nm
×
n m
np mq (σx∗′)p(σy′)q σy∗′ n−p(σx′)m−q× IV. EXAMPLES
p=0q=0(cid:18) (cid:19)(cid:18) (cid:19)
XX (cid:0) (cid:1)
Hk+l(x′)Hn+m−k−l(y′), (22) Inthissectionweillustratetherotationaleffectsonthe
structure of two-mirror cavity modes by further investi-
where ρ′ = (x′,y′) are the scaled coordinates that gatingaspecificexample. Wefocusonacavityconsisting
account for the astigmatism of the intensity pattern. of a stationary spherical and a rotating cylindrical mir-
The above expression holds in the co-rotating frame. ror,whichisthesimplestrealizationauniformly-rotating
The corresponding expressions for the rotating modes two-mirrorcavity. The radius of curvature of the spheri-
u (ρ′,z,t) in the stationary frame, can be obtained by calmirroristaken4L,withLthemirrorseparationwhile
nm
applying Eq. (3). the radius of curvature of the cylindrical mirror is cho-
6
dependent paraxial wave equation (2) that vanish on
bothmirrorsurfaces. Thetransverseintensityandphase
patterns close to the spherical mirror of the zeroth, first
and second order modes of the cavity are plotted in the
left column of Fig. 4. Though the astigmatism is dif-
u00 2 arg(u00) v00 2 arg(v00) ferent in other transverse planes (in particular close to
| | | |
the cylindricalmirror), the modes are Hermite-Gaussian
inalltransverseplanes. Theirphasestructureclearlyre-
flectsthefactthatthewavefrontsfitthecurvatureofthe
(spherical)mirror. Thephasejumpsof2π,whichappear
along circles in the transverse plane, are an artifact of
u01 2 arg(u01) v01 2 arg(v01) thedefinitionofthephaseofthe beamandarenotphys-
| | | | ical. The phase jumps of π, however, which arise from
the Hermite polynomials in Eq. (22) and appear along
horizontal and vertical lines in the transverse plane, are
physical and could be measured interferometrically.
If the cylindricalmirroris put into rotation, the mode
structure changes significantly. The modes of a rotating
u10 2 arg(u10) v10 2 arg(v10) cavity are defined as co-rotating solutions of the time-
| | | |
dependent paraxial wave equation (2), or equivalently
solutions of Eq. (4), that vanish on the mirror sur-
faces. In the co-rotating frame, propagating modes at-
tain a twist due to retardation, which is reflected by the
Coriolis term in Eq. (4). The effect of rotation on the
intensity and phase patterns of the zeroth, first and sec-
u11 2 arg(u11) v11 2 arg(v11) ondordercavitymodes is illustratedinthe rightcolumn
| | | |
of Fig. 4, for which the rotation frequency is equal to
Ω0/10. The mode patterns are clearly affected by ro-
tation. The intensity patterns become more similar to
Laguerre-Gaussian modes and have obviously the struc-
ture of generalizedGaussianmodes. Thoughthe electric
u02 2 arg(u02) v02 2 arg(v02) field vanishes on the mirror surfaces, the wave front no
| | | | longer fit their local curvature. This is most apparent in
case of the fundamental mode v00, where the curves of
constantphaseclosetothesphericalmirrorhavebecome
elliptical. Thenon-parallelorientationoftheellipticalin-
tensity and phase patterns of the fundamental mode v00
reflects its twisted nature. The higher-order modes also
u20 2 arg(u20) v20 2 arg(v20) attain a twist. Moreover, vortices appear in their phase
| | | | patterns. Theresultsconfirmthatthevorticesinthev01
and v10 have opposite chargesand that only modes with
FIG.4: False-colorintensityandphasepatternsofthemodes oddn+mhaveavortexatthecenter. Inthemodeswith
ofacavityconsistingofasphericalandastationary(left)and equal mode numbers n and m, the total vortex chargeis
rotating(right)cylindricalmirror. Theradiusofcurvatureof equal to zero so that the curves of constant phase that
thesphericalmirrorisequalto4L,withLthemirrorsepara- contain all vortices are closed.
tion. Theradiusofcurvatureofthecylindricalmirrorisequal
Due to a combined symmetry that survives rotation,
to 2L. The plots show the mode patterns in the co-rotating
the intensity patterns of the rotating modes are aligned
frameclose tothesphericalmirror andthecylindrical mirror
along the axes of the cylindrical mirror while the phase
isflatintheverticaldirection. Intherightplots,therotation
frequency is equal to Ω0/10 with Ω0 =cπ/L. patterns are not [18]. As a result the azimuth angle
onthe Hermite-Laguerresphere φHL, which specifies the
orientation of the higher-order mode patterns, does not
depend on the rotation frequency. This is not true for
sen 2L. This cavity is geometrically stable for rotation
the polar angle θHL, which specifies whether the modes
frequencies up to Ω0/6, where Ω0 = cπ/L, with c the are Hermite-Gaussian, Laguerre-Gaussianor generalized
speed of light, is frequency corresponding to the cavity
Gaussian modes and is a measure of their vorticity. Its
round-trip time [21].
dependence onthe rotationfrequency Ω is shownin Fig.
In the absence of rotation, the modes of the cavity 5. In general, the characterization of the cavity modes
u are defined as the stationary solutions of the time- in terms of a vector on the Hermite-Laguerre sphere is
nm
7
andvanishonthemirrorsurfaces. Thismodecriterionis
a continuous generalization of the requirement that the
HL
modes of a stationary cavity are stationary solutions of
e
gl Eq. (2) that vanish on the mirror surfaces. The rotat-
n
ar a 3 /4 ingcavitymodesarestationarysolutionsinaco-rotating
ol frame where mode propagation is twisted due to the fi-
P
nite speed of light. Due to this twist, rotation deforms
the astigmatic Hermite-Gaussian modes of a stationary
/2 cavityintogeneralizedGaussianmodes. Thelinedisloca-
0.00 0.05 0.10 0.15 0.20
tions in the Hermite-Gaussian modes are deformed into
Scaled rotation frequency elliptical vortices. In a recent paper [21], we have shown
that rotation can destabilize a two-mirror cavity that is
FIG.5: Dependenceontherotation frequencyΩofthepolar stableintheabsenceofrotation. Whensuchatransition
angle θHL on the Hermite-Laguerre sphere for a cavity con- is approached, the cavity modes lose their confinement
sisting of a spherical and a rotating cylindrical mirror. The in one transverse direction so that the elliptical vortices
characterization of the modes in terms of (φHL,θHL) is local are stretched to become line dislocations again. This is
andtheplotspecifiesthenatureofthehigher-ordermodesin illustrated in Fig. 5.
thetransverse plane close to thespherical mirror.
Aninterestingbutopenquestionishowrotationwould
affect the optical properties of a geometrically unstable
cavity, especially when close to the rotationally induced
local and the plot in Fig. 5 characterizes the higher-
transition from an unstable to a stable geometry. A ge-
order cavity modes close the the spherical mirror. The
ometrically unstable cavity is intrinsically lossy [16] and
plot confirms that, due to the fact that rotation breaks
thepropagationoflightinsideanunstablecavityisdom-
the inversionsymmetry ofa stationarycavity,the cavity
inated by diffraction at the sharp edges of the mirrors
modesarecontinuouslydeformedfromHermite-Gaussian
[24]. Both mathematically and physically this system is
modes in the stationary case into generalized Gaussian
fundamentally different from the stable cavities that we
modesintherotatingcase. Fornottoolargevaluesofthe
havestudied in this paper and it is not possible to apply
rotation frequency, the rotationally induced vorticity is
or generalize our method to such a system.
proportionalto the rotation frequency. For larger values
Asopposedtoanormalfluid,aspatiallyconfinedopti-
something surprising happens: at some point, the vor-
cal cavity mode attains vorticity when it is put into uni-
ticitystartstodecreasewithincreasingrotationfrequen-
formrotation. Inthisrespect,ithassomesimilaritywith
ciesandeventuallythemodesbecome Hermite-Gaussian
superfluidsandBose-Einsteincondensates. Comparedto
again. This is due to the fact that this cavity is desta-
those systems, however, optical beams have many more
bilized by rotationata rotationfrequency Ω0/6. At this
degrees of freedom to cope with rotation. The phase
point, the modes lose their confinement in one of the
of Bose-Einstein condensates and superfluids is fixed by
transverse directions (in this case in the vertical direc-
the broken gauge invariance. The only way in which
tion, i.e. the direction in which the cylindrical mirror is
such systems can attain orbital angular momentum, is
flat) so that the elliptical vortices are stretched to be-
bylocallycreatingcylindricallysymmetricvorticeswhile
comeline dislocationsagain. Thoughthevorticityinthe
keeping their phase fixedelsewhere. As a result, vortices
modesdisappearsifthistransitionisapproached,theor-
appear only when the rotation frequency exceeds a cer-
bital angular momentum diverges at this point due to
tainthresholdandthenumberofvorticesincreaseswhen
the diverging astigmatism [21]. In specific cases, rota-
the rotation frequency is further increased. For optical
tion can also stabilize a cavity that is unstable in the
cavitymodes,ontheotherhand,boththeastigmatismof
absenceofrotation. Insuchcasesweexpecttheopposite
thephaseandintensitypatternsandthepropertiesofthe
behaviour. Due to the rotation of the cavity, we retrieve
opticalvorticesareaffectedbyrotation. Bothcontribute
Hermite-Gaussianmodesatthepointwherethecavityis
totheorbitalangularmomentum[19],whichtypicallyin-
stabilized,whilemodedeformationandinducedvorticity
creases with increasing values of the rotation frequency
appear for even larger values of the rotation frequency.
even though the number of vortices does in general not
increase.
Fortypicalvaluesofthemirrorseparation,therotation
V. CONCLUSION AND OUTLOOK
frequency corresponding to the cavity round-trip time is
in the MHz-range. Although mechanical rotation fre-
In this paper, we have applied a ladder operator quencies in this range are not experimentally feasible,
methodtostudythevorticesthatappearinthemodesof there are several possible routes towards experimentally
an astigmatic two-mirrorcavity when it is put into rota- realizingaset-upthatcapturestheessentialfeaturesofa
tionabouttheopticalaxis. Themodesofarotatingcav- rotatingastigmatictwo-mirrorcavity. Mechanicalvibra-
ity are defined as solutions ofthe time-dependent parax- tionfrequenciesintheMHz-rangecanbeachievedbyus-
ial wave equation (2) that rotate along with the cavity ing a piezoelectric actuator. Developing a similar device
8
to simulate rotations at MHz-frequencies is challenging job. Instead, one could use a ring resonator with four,
but, at least in principle, not impossible. Another pos- pairwise identical, mirrors in which the light passes an
sible way of realizing a set-up that is similar to the one imagerotatoraftereachmirror. Althoughthesettingsof
that is discussed in this paper, is by applying a rotating the optical elements in this set-up are time-independent
astigmaticmodepattern,whichcanbe constructedfrom it does capture the optical properties of a rotating two-
its stationary Doppler-shifted frequency components, to mirror cavity. A difficulty with this set-up is that the
optically induce a rotating refractive-index pattern in a clockwise and counterclockwise propagating modes are
material with a Kerr-nonlinearity. Such a pattern could frequency degenerate. In a passive set-up one of the two
be used in transmission to realize a two-mirror cavity modes can be selected by injecting light in one direction
with a rotating lens. The optical properties of such a only. Inanactiverealizationofsuchaset-up,the degen-
cavitycouldbeobservedbyusingalaseratanotherwave eracycanbeliftedbyapplyingthepolarizationdegreeof
length. Besides the dynamical realizations of the set-up, freedom.
one could try to mimic the twisted mode propagation
in the rotating frame by using image rotators. Since it It is a pleasure to thank Michiel J. A. de Dood and
is essential that back and forth propagating modes are Eric R. Eliel for fruitful discussions regarding possible
rotated in the same direction, a two-mirror cavity with experimental realizations of the set-up that is discussed
an image rotator between the mirrors does not do the in this paper.
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