Subthreshold optical parametric oscillator with nonorthogonal polarization eigenmodes

polarization eigenmodes

Aiello, A.; Nienhuis, G.; Woerdman, J.P.

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Aiello, A., Nienhuis, G., & Woerdman, J. P. (2003). Subthreshold optical parametric oscillator

with nonorthogonal polarization eigenmodes. Physical Review A, 67, 043803.

doi:10.1103/PhysRevA.67.043803

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Subthreshold optical parametric oscillator with nonorthogonal polarization eigenmodes

A. Aiello, G. Nienhuis, and J. P. Woerdman

Huygens Laboratory, Leiden University, P.O. Box 9504, Leiden, The Netherlands

共Received 3 July 2002; revised manuscript received 27 November 2002; published 8 April 2003兲 We study the behavior of a type-II degenerate parametric amplifier in a cavity with nonorthogonal polariza-tion eigenmodes. The mode nonorthogonality is achieved by introducing circular birefringence and linear dichroism. We use a scattering matrix formalism to investigate the role of excess quantum noise in such a device. Since only two modes are involved we are able to derive an analytical expression for the twin-photon generation rate measured outside the cavity as a function of the degree of mode nonorthogonality. Contrary to recent claims we conclude that there is no evidence of excess quantum noise for a parametric amplifier working so far below threshold that spontaneous processes dominate. Using the same scattering matrix formalism we also investigate the output spectrum of the amplifier near the threshold of parametric oscillation. We find optical band structures very similar to those known for passive ring cavities. These optical band structures are studied as a function of mode nonorthogonality and mirror reflectivity.

DOI: 10.1103/PhysRevA.67.043803 PACS number共s兲: 42.50.Lc, 42.60.Da, 42.65.Yj I. INTRODUCTION

A linear amplifier is a device that takes an input signal and produces an output signal linearly related to the input signal. Under this definition fall frequency-conserving ampli-fiers, as laser ampliampli-fiers, and frequency-converting amplifi-ers, as parametric amplifiers. Quantum mechanics sets a lower limit on noise in linear amplifiers 关1兴 which corre-sponds, in a laser amplifier, to having ‘‘one noise photon’’ in the laser mode共Ref. 关2兴, p. 72兲 and, in a parametric ampli-fier, to having ‘‘one noise photon’’ in each of the input modes

关3兴. This limit is easily reached in small devices, particularly

in semiconductor lasers 关4兴. If the linear amplifier is part of an optical cavity the quantum limit on its performances is strongly affected by the optical characteristics of the cavity itself which offers the possibility to control and to manipu-late the quantum noise. This opens a wide range of possible studies which spans from cavity QED共see, e.g., 关5兴 and ref-erences therein兲 to the phenomenon of excess quantum noise

关6–13兴.

Recently there has been a large body of work pointing at the fact that the quantum noise may be enhanced by the so-called excess noise factor or Petermann K factor关6兴. From a physical point of view the K factor can be interpreted as if there are K noise photons in the lasing mode instead of the usual ‘‘one noise photon.’’ Semiclassically the noise en-hancement is due to nonorthogonality of the eigenmodes

关7,14兴. The existence of the Petermann K factor has been

experimentally verified in lasers with non-orthogonal eigen-modes, either longitudinal 关10兴, transverse 关11兴, or polariza-tion 关9,12兴 modes, showing that a noise enhancement really occurs. However, the physical origin of this enhancement is under debate; the two main points of view are that it stems from a cavity-enhanced single atom decay rate 关15–18兴 or from an amplification by the gain medium of the spontane-ously emitted photons 关13,19,20兴. If the single-atom decay rate were enhanced, excess noise would also be a valid con-cept共far兲 below the oscillations threshold of the device under consideration. In this case excess noise could be very useful; for instance, it has been claimed that it could lead to an

enhanced generation of twin photons in spontaneous para-metric down conversion 共SPDC兲, by placing a nonlinear crystal in an unstable cavity共which has nonorthogonal trans-verse eigenmodes兲 关16兴.

The most common experimental realization of mode non-orthogonality concerns the transverse modes of an unstable cavity. However, this case is intrinsically difficult to treat: one deals with an infinite manifold of transverse modes which cannot be truncated since there is no sharp distinction between system modes 共⫽cavity modes兲 and reservoir modes共⫽free space modes兲 关21兴. This unavoidable difficulty has motivated us to study the effect of excess noise on cavity-enhanced SPDC, for a case where one can construct an exactly solvable quantum theory of mode nonorthogonal-ity. This is possible for a cavity with nonorthogonal polar-ization eigenmodes 共instead of transverse eigenmodes兲 which has a nonlinear crystal inside.

In fact, SPDC constitutes a natural framework in which to study polarization excess noise in a quantum-mechanical context. Specifically, in a type-II SPDC process, two or-thogonally polarized photons are generated. Because of crys-tal anisotropy, for a fixed frequency only a restricted set of spatial directions is allowed to the emitted photons. In the degenerate case one can achieve a single allowed direction for a collinear emission 关22兴 thus, assuming perfect phase matching, single transverse mode operation can be realized. Although an optical cavity allows, in principle, several reso-nant longitudinal modes, the double resonance condition

共signal and idler兲 for SPDC restricts this number. It can be

shown 关23兴 that, because of crystal birefringence, for a type-II process the double resonance condition can only be satisfied at degenerate frequency so that the number of al-lowed longitudinal modes is reduced to one.

po-larization modes, both far below and near the threshold for parametric oscillation 关optical parametric oscillator 共OPO兲兴. We use and expand two existing theoretical models, one for the DPA and the other for the cavity, both of which have been experimentally verified. Our conclusion is that there is no enhancement in spontaneous parametric down conver-sion.

In the second part of this paper we discuss the behavior of the spectrum of a parametric oscillator working close to threshold. We first discuss the definition of spectral reso-nance within our scattering formalism, then we analyze the OPO spectrum for different cavity realizations. We find a quite unexpected behavior: the OPO spectrum exhibits band structures very similar to those known in passive ring cavi-ties. In fact, we find that because of the mode coupling in-duced by passive and active optical elements inside the cav-ity, four resonant peaks per free spectral range appear in the OPO spectrum.

The paper is organized as follows. In Sec. II we introduce a group-theoretical formalism for describing and analyzing the two-mode optical elements which are present in our model in terms of scattering matrices. In Sec. III such for-malism is applied to set up the cavity model. We also show explicitly the occurrence of the ‘‘geometrical’’ Petermann K factor in our cavity model. The results obtained in Sec. III are collected and analyzed in Sec. IV where the absence of a K-enhanced spontaneous down-conversion rate is proven. In Sec. V we exploit the scattering matrix formalism to inves-tigate the occurrence of band structures in the OPO spectrum in a cavity with nonorthogonal polarization eigenmodes. Fi-nally, we draw conclusions in Sec. VI.

II. TWO-MODE OPTICAL ELEMENTS AND GROUP THEORY

The optical devices we consider in this paper are com-posed of linear and lossless optical elements, and have two input ports 共say 1 and 2) and two corresponding output ports. When the elements are passive, no photons are created or destroyed, so that the number of photons entering the two input ports is equal to the number of photons leaving the two output ports. Such devices can be described by a unitary matrix belonging to the group U(2) 关25兴. Active optical de-vices can create and annihilate photons but when the differ-ence between the number of photons entering port 1 and that entering port 2 is conserved, the device can be described by a unitary matrix belonging to the group U(1,1) 关26,27兴. In this section we review briefly the matrix representation of lossless passive and active optical devices, characterizing them in terms of U(2) and U(1,1) group properties. We show how, introducing the so-called commutator matrix关28兴, the Schwinger model for angular momentum can be ex-tended to build the generators of U(1,1) group.

Let us consider a pair of operators xˆ1, xˆ2 which satisfy

the following commutation rules:

关xˆi,xˆj兴⫽0, 关xˆi,xˆj †_{兴⬅共⌫兲}

i j 共i, j⫽1,2兲, 共1兲

where⌫ is a given diagonal 2⫻2 matrix. We arrange xˆ1and

xˆ2in a two-dimensional vector Xˆ 共and its adjoint Xˆ†) defined

as Xˆ⬅

冉

xˆ1 xˆ2

冊

, Xˆ†_⬅共xˆ 1 † _xˆ 2 †_{兲, 共2兲}

and define the inner product (⫺,⫺) between two vectors Xˆ and Yˆ as

共Xˆ,Yˆ 兲⫽xˆi †

yˆi 共i⫽1,2兲, 共3兲

where summation over repeated indices is understood. The three Pauli matrices together with the identity matrix form a basis in the vectorial space of 2⫻2 matrices; we write them as ␴0⫽

冉

1 0 0 1

冊

, ␴1⫽

冉

0 1 1 0

冊

, 共4兲 ␴2⫽

冉

0 ⫺i i 0

冊

, ␴3⫽

冉

1 0 0 ⫺1

冊

.

Using the Pauli matrices we can construct four Hermitian operators defined as

Sˆa⬅共Xˆ,␴aXˆ兲 共a⫽0, . . . ,3兲. 共5兲

These operators satisfy the following commutation rules:

关Sˆa,Sˆb兴⫽共Xˆ,␴abXˆ兲 共a,b⫽0, . . . ,3兲, 共6兲

where

␴ab⬅␴a⌫␴b⫺␴b⌫␴a 共a,b⫽0, . . . ,3兲. 共7兲

Because of completeness of the set of Pauli matrices, Eq.共4兲, we can always write, choosing adequately the constants fabc,

␴ab⫽i fabc␴c 共a,b,c⫽0, . . . ,3兲. 共8兲

Using Eqs.共8兲 we can then write Eq. 共6兲 as

关Sˆa,Sˆb兴⫽i fabcSˆc, 共9兲

which shows that the four operators Sˆasatisfy the same

com-mutation relations as the generators of a symmetry group. The numbers f_abc are called structure constants and com-pletely determine the group multiplication law关29兴. The op-erators Sˆ_a generate transformation of the vector operator Xˆ , in the form

exp共zSˆa兲Xˆexp共⫺zSˆa兲⫽exp共⫺z⌫␴a兲Xˆ, 共10兲

one notices that关Sˆ0,Sˆa兴⫽0 when ⌫⫽␴0, and that关Sˆ3,Sˆa兴

⫽0 when ⌫⫽␴3. These two cases are realized when one

chooses Xˆ⫽

冉

aˆ bˆ

冊

or X ˆ_⫽

冉

aˆ bˆ†

冊

, 共11兲

where aˆ, bˆ are independent harmonic oscillator operators which satisfy the boson commutation relations:

关aˆ,bˆ兴⫽0⫽关aˆ,bˆ†_兴,

共12兲 关aˆ,aˆ†_{兴⫽1⫽关bˆ,bˆ}†_兴.

Case 1:⌫⫽␴0. In this case the operators Sˆabelong to the

Lie algebra of the group U(2) and we recover the Schwinger representation of two modes,

Nˆa⫹Nˆb⫽Sˆ0⫽aˆ†aˆ⫹bˆ†bˆ, Jˆ_x⫽Sˆ1 2 ⫽ 1 2共aˆ †_bˆ⫹bˆ†_aˆ兲, 共13兲 Jˆ_y⫽Sˆ2 2 ⫽⫺ i 2共aˆ †_bˆ⫺bˆ†_aˆ兲, Jˆz⫽ Sˆ3 2 ⫽ 1 2共aˆ †_aˆ⫺bˆ†_bˆ兲.

The operators Jˆx,Jˆy,Jˆz obey the usual commutation rules of

angular momentum 关Jˆx,Jˆy兴⫽iJˆz, etc. The conserved

quan-tity associated with Sˆ0 is the total number of photons

repre-sented by the operator Nˆ_a⫹Nˆ_b which commutes with the three angular momentum operators Jˆx,Jˆy,Jˆz.

In order to see explicitly the connection between lossless passive optical devices and the elements of the group U(2) we denote with xˆ1 and xˆ2 the annihilation operators for the

field entering the two input ports and with yˆ1 and yˆ2 the

annihilation operators for the field leaving the two output ports. These four operators are connected by a scattering matrix M whose form is

冉

yˆ₁ yˆ2

冊

⫽

冉

M11 M12 M₂₁ M₂₂

冊

冉

xˆ₁ xˆ2

冊

. 共14兲

Conservation of probability in a scattering process demands that output operators satisfy the same commutation relations as the input operators. This requirement leads to the unitarity condition for M,

MM†⫽1. 共15兲

Here we write explicitly some scattering matrices and the associate transformations that will be used in the next sec-tion. The operators Jˆx and Jˆy generate two possible

scatter-ing matrices for a beam splitter and/or a rotator关30兴,

ei␣Jˆx

冉

aˆ bˆ

冊

e

⫺i␣Jˆx⫽

冉

cos共␣/2兲 ⫺i sin共␣/2兲

⫺i sin共␣/2兲 cos共␣/2兲

冊

冉

aˆ bˆ

冊

, 共16兲 ei␤Jˆy

冉

aˆ bˆ

冊

e⫺i␤Jˆy⫽

冉

cos共␤/2兲 ⫺sin共␤/2兲 sin共␤/2兲 cos共␤/2兲

冊冉

aˆ bˆ

冊

, 共17兲 while the scattering matrix accounting for free-field propaga-tion is generated by operator Jˆz,

ei␥Jˆz

冉

aˆ bˆ

冊

e ⫺i␥Jˆz⫽

冉

e⫺i␥/2 0 0 ei␥/2

冊

冉

aˆ bˆ

冊

. 共18兲 Case 2: ⌫⫽␴3. Using Eq. 共5兲 it is easy to see that the

operators Sˆa belong to the Lie algebra of the group U(1,1),

Nˆa⫺Nˆb⫺1⫽Sˆ3⫽aˆ†aˆ⫺bˆbˆ†, Kˆ_x⫽Sˆ1 2 ⫽ 1 2共aˆ †_bˆ†_{⫹bˆaˆ兲,} 共19兲 Kˆy⫽ Sˆ2 2 ⫽⫺ i 2共aˆ †_bˆ†_{⫺bˆaˆ兲,} Kˆz⫽ Sˆ0 2 ⫽ 1 2共aˆ †_aˆ⫹bˆbˆ†_兲.

In this case the difference in photon number is conserved, that is the operator Nˆa⫺Nˆbcommutes with Kˆx,Kˆy,Kˆzwhich

are generators of the group SU(1,1). The commutation rules for these operators are 关Kˆx,Kˆy兴⫽⫺iKˆz, 关Kˆy,Kˆz兴

⫽iKˆx, 关Kˆz,Kˆx兴⫽iKˆy. The scattering matrices generated

by the SU(1,1) operators follow by Eq.共10兲. They take the explicit form ei␣Kˆx

冉

aˆ bˆ†

冊

e⫺i␣Kˆx⫽

冉

cosh共␣/2兲 ⫺isinh共␣/2兲 i sinh共␣/2兲 cosh共␣/2兲

冊冉

aˆ bˆ†

冊

, 共20兲 ei␤Kˆy

冉

aˆ bˆ†

冊

e ⫺i␤Kˆy⫽

冉

cosh共␤/2兲 sinh共␤/2兲 sinh共␤/2兲 cosh共␤/2兲

冊冉

aˆ bˆ†

冊

, 共21兲 ei␥Kˆz

冉

aˆ bˆ†

冊

e ⫺i␥Kˆz⫽

冉

ei␥/2 0 0 e⫺i␥/2

冊冉

aˆ bˆ†

冊

. 共22兲

Using the last two equations we can construct the scattering matrix representing the nonlinear crystal, as shown in Ref. 关26兴.

III. THE CAVITY MODEL

model of a degenerate parametric amplifier inside a Fabry-Pe´rot cavity is, in fact, an extension of the model of Gardiner et al.关31兴 to the case of a cavity with nonorthogonal polar-ization modes.

A. Scattering matrix for a cavity round-trip

We consider a cavity having one perfectly reflecting mir-ror at position x⫽⫺L, and a partially reflecting mirror at x

⫽0, as shown in Fig. 1.

We decompose the electric field inside the cavity into left

共subscript L) and right 共subscript R) propagating waves. In

degenerate type-II down-conversion two orthogonally polar-ized modes are excited at the same frequency ␻⫽⍀/2, where ⍀ is the frequency of the pump field. Let us denote with a and b these two field modes and assume that their polarization is parallel to the y and z axis, respectively. An-other mode f共also decomposed in fL and fR parts兲, is

intro-duced in order to assure the unitarity of the model; we call this mode the noise mode. We assume that mode f has the same polarization as mode a. The role of this noise mode will be soon made clear; for the moment we describe, as in Ref. 关8兴, the DPA cavity using a scattering matrix which is unitary only when it accounts both for field and noise modes. We shall see that nonorthogonality of the cavity modes natu-rally appears as a consequence of restricting the scattering matrix to the set of field modes a and b. However, truncating the scattering matrix to the field modes is not enough to achieve mode nonorthogonality; it is necessary to introduce a non-Hermitian coupling between them. In our model the mode nonorthogonality is achieved by inserting in the cavity a phase anisotropy due to circular birefringence共polarization rotator兲 and a loss anisotropy generated by linear dichroism

共polarization-dependent absorber兲, following the scheme

given in 关9兴. Another way to produce nonorthogonal

polar-ization modes is to use linear birefringence and linear dichro-ism at 45 ° as in Ref. 关12兴. Although these two alternative ways are implemented using physically different devices, both lead to basically the same expressions for the Peter-mann K factor, as we shall see in Sec. III B.

The canonical quantization scheme requires us to express the electromagnetic field inside the cavity in terms of a con-tinuous or discrete complete set of functions兵u_n其 共the eigen-modes of the cavity兲 and associating with them a correspond-ing set of field operators兵aˆ_n其. A serious problem arises when the set of cavity eigenmodes兵un其 is not orthogonal. In fact,

as shown in Refs. 关8,13兴, a set of nonorthogonal modes cannot be turned into a set of noncommuting operators. In order to avoid this problem our calculations are based on the orthogonal sets of operators 兵␣ˆin, fˆ␣_in其, 兵␣ˆout, fˆ␣_out其 (␣

⫽a,b) associated with a corresponding set of plane-wave

modes 关31兴. We assume that the input and output operators satisfy the usual共discrete兲 commutation relations

关aˆx,bˆx兴⫽0⫽关aˆx,bˆx † 兴, 关aˆx,aˆx †_{兴⫽1⫽关bˆ} x,bˆx †_{兴 共23兲} 共x⫽in, out兲,

and similarly for the noise operators.

The optical elements inside the cavity are as follows: an absorber modeled as a beam splitter acting only on mode a(y polarization兲, a crystal with nonlinear gain G, and a rotator which rotates the polarization axes by an angle␾along the x axis. The propagation of the modes over a cavity with length L is modeled by a delay line in front of the left mirror which introduces a phase shift␪⫽␻L/c. We assume that all optical elements are infinitesimally thin and that the operator phases at the position x⫽0 are equal to zero. The scattering matri-ces for the various optical elements inside the cavity are given below. On the output mirror the input annihilation erators belonging to the a mode are related to the input op-erators on the same mode, by the transformation

aˆout⫽T aˆ1R⫹Raˆin, 共24a兲

aˆ1L⫽Raˆ1R⫹T aˆin, 共24b兲

whereR⫽⫺

冑

R,T⫽i

冑

1⫺R, and 0⭐R⬍1. For the mode b the above relations hold if we make everywhere the substi-tution a→b. The effect of the rotator on left-traveling mode operators can be represented as关30兴

aˆ2L⫽cos␾aˆ1L⫹sin␾bˆ1L, 共25a兲

bˆ2L⫽⫺sin␾aˆ1L⫹cos␾bˆ1L. 共25b兲

The corresponding matrix for right-traveling modes is ob-tained substituting in the above formula 1↔2 and L→R. Note that we have chosen as a rotator, a device antisymmet-ric with respect to temporal inversion 关32兴 共e.g., a Faraday rotator兲; then the total rotation angle is doubled after a round trip. For completeness we note that in case of a device which

FIG. 1. Schematic representation of the degenerate-cavity para-metric amplifier. Modes a and b have orthogonal polarizations. The boxes indicated with␾, G, and ␪ represent the rotator, the nonlin-ear crystal, and the delay line, respectively. In the dotted box we show the absorber modeled as a beam splitter acting only on mode

a. For right-traveling modes we have put G⫽1 to indicate the

is symmetric with respect to temporal inversion 共e.g., a quartz crystal which displays optical activity兲, the light beam inside the cavity would retrieve its original polarization after one round trip. Polarization rotation can be also achieved using a half wave plate which introduces a ␲ phase differ-ence between fast and slow ( f and s) axes关33兴. This device has been used jointly with a linear dichroic element with its axes at⫾45 ° with respect to f and s in Refs. 关12,34兴. How-ever, we have preferred to use a Faraday rotator, jointly with a linear dichroic element with its axes parallel to the a and b polarization directions, since this configuration leads to a more clear separation between the phase anisotropy and the loss anisotropy inside the cavity.

The scattering matrix for the parametric crystal, in the nondepleted pump approximation关31,35兴, is given by

aˆ3L⫽Gaˆ2L⫹共G2⫺1兲1/2bˆ2L † , 共26a兲 bˆ_3L† ⫽共G2⫺1兲1/2aˆ2L⫹Gbˆ2L † , 共26b兲

where the real-valued gain G satisfies G⬎1. For the right-traveling modes the crystal is transparent due to the absence of phase matching and in this case the operator transforma-tions can be obtained from Eqs. 共26兲 after the substitutions 3↔2, L→R, and G⫽1. Since Eqs. 共26兲 preserve bosonic commutation rules it is not necessary, for a parametric am-plifier with a classical nondepleted pump, to add noise from an external bath 关1兴 to account for pump fluctuations. In our model only the down-converted field is confined by the cav-ity, not the pump field, therefore the cavity mode structure cannot affect the pump beam fluctuations. Incidentally, we note that when using this scattering matrix formalism, the difference between a linear and a nonlinear amplifier is rooted only in the choice of the operators which are coupled by the matrix, but not in the matrix itself, which is the same in both cases. In fact, in a linear amplifier the nondiagonal matrix elements couple a field annihilation operator with a noise creation operator, while in a nonlinear amplifier the coupling is between two different field modes, as in Eqs.

共26兲.

The scattering matrix representing the absorber, which in-troduces losses only for the mode a, is written as关36兴

aˆ4L⫽taˆ3L⫹r fˆLin, 共27a兲

bˆ4L⫽bˆ3L, 共27b兲

fˆLout⫽raˆ3L⫹t fˆLin, 共27c兲

where r⫽i

冑

1⫺t2 and the real parameter t(0⭐t⭐1) repre-sents the ratio between field amplitudes along y and z polar-ization directions. For right-traveling modes we obtain es-sentially the same equations by substituting 4↔3 and L

→R, that is we consider a device insensitive with respect to

the direction of the impinging light. We note that truncating the transformation equations 共27兲 to the field modes only leads to the following nonunitary transformation:

冉

aˆ4L bˆ_4L

冊

⫽

冉

t 0 0 1

冊

冉

aˆ3L bˆ_3L

冊

. 共28兲 Since as the absorber we have chosen a linear dichroic ele-ment with its axes parallel to the a and b polarization direc-tions, it introduces only anisotropic losses but no phase an-isotropy and therefore the matrix Eq. 共28兲 is diagonal.

The delay line with phase shift ␪ can be simply repre-sented as

aˆ5L⫽exp共i␪兲aˆ4L, 共29兲

aˆ4R⫽exp共i␪兲aˆ5R, 共30兲

where ␪⫽␻L/c. It allows us to evaluate the effects of the cavity length L. The same relations hold for mode b. Finally, on the left mirror the boundary condition requires

aˆ5R⫽⫺aˆ5L, 共31兲

and similarly for mode b.

Equations 共25兲–共31兲 can be straightforwardly solved to express right-traveling mode operators in terms of left-traveling mode operators,

冉

aˆ1R bˆ1R

冊

⫽G

冉

⫺␥⫹cos共2␾兲⫺␥⫺ ⫺␥⫹sin共2␾兲 ␥⫹sin共2␾兲 ⫺␥⫹cos共2␾兲⫹␥⫺

冊

⫻

冉

aˆ1L bˆ1L

冊

⫹共G2_⫺1兲1/2 ⫻

冉

␥⫺sin共2␾兲 ⫺␥⫺cos共2␾兲⫺␥⫹ ␥⫺cos共2␾兲⫺␥⫹ ␥⫺sin共2␾兲

冊

⫻

冉

aˆ1L † bˆ_1L†

冊

⫹

冉

fˆa fˆb

冊

, 共32兲

where␥_⫾⫽exp(2i␪)(t2⫾1)/2 and

fˆa⫽r共t fˆLin⫹ fˆRin兲cos␾, 共33兲

fˆb⫽⫺r共t fˆLin⫹ fˆRin兲sin␾. 共34兲

The effect of the noise on mode b appears as a consequence of introducing the rotator: fˆb⫽0 when ␾⫽0. At the same

time the noise disappears on both modes if t⫽1. This means that the full effect of the noise on the system becomes mani-fest only for␾⬎0 and t⬍1, that is when the cavity modes are nonorthogonal. Assuming that noise operators belonging to left- and right-traveling modes do commute,

关 fˆLin, fˆLin † _{兴⫽1⫽关 fˆ} Rin, fˆRin † _兴, 共35兲 关 fˆLin, fˆRin † _{兴⫽0⫽关 fˆ} Rin, fˆLin † _兴,

关 fˆa, fˆa †_{兴⫽共1⫺t}4_兲cos2_{␾, 共36a兲} 关 fˆb, fˆb †_{兴⫽共1⫺t}4_兲sin2_{␾, 共36b兲} 关 fˆa, fˆb †_{兴⫽⫺共1⫺t}4_{兲sin␾cos␾.} 共36c兲

This noise correlation disappears when the modes become orthogonal (␾⫽0, ␲/2, and/or t⫽1).

B. Nonorthogonal modes and the Petermann excess noise factor

Having found the relations between operators belonging to right-traveling and left-traveling modes after one round trip, we now show that our model effectively describes a cavity with nonorthogonal modes and can therefore show, in principle, excess quantum noise 关8,37兴. Although Eq. 共32兲 has been written in a quantum context, it is equally valid in a classical context if one substitutes for the various operators aˆ_1R, bˆ_1R, etc. the corresponding classical complex ampli-tudes A1R, B1R, etc. and disregards the noise operators fˆa

and fˆb. The remaining homogeneous equation describes the

round trip variation of a classical field inside the cavity. Fur-thermore, if one puts G⫽1 then the classical counterpart of Eq. 共32兲 reduces to

冉

A1R B1R

冊

⫽M

冉

A_B1L 1L

冊

, 共37兲 where M⬅⫺

冉

␥⫹ cos共2␾兲⫹␥_⫺ ␥_⫹sin共2␾兲 ⫺␥⫹sin共2␾兲 ␥⫹cos共2␾兲⫺␥⫺

冊

, 共38兲 which coincides, apart from a multiplicative factor, with the classical cold cavity round-trip matrix MRT. Now, following

Ref.关8兴, we find the eigenvalues and the eigenvectors of the matrix M and show that the latter ones form a nonorthogonal two-dimensional basis.

First we note that when ␾⫽0 or t⫽1, M_RT reduces to

␾⫽0⇒M⫽⫺

冉

t 2 ₀ 0 1

冊

共39兲 or t⫽1⇒M⫽⫺

冉

cos 2␾ sin 2␾ ⫺sin 2␾ cos 2␾

冊

. 共40兲 It is clear that in both these cases the eigenvectors are or-thogonal. In the general case the eigenvalues␭_⫾ are

␭⫾⫽ ⫺1 2 关共1⫹t 2_{兲cos 2␾⫾Z兴, 共41兲} where Z⬅关共1⫺t2兲2⫺共1⫹t2兲2sin22␾兴1/2. 共42兲

Depending on the values assumed for␾ and t we may have either Z real or purely imaginary. In the latter case it is con-venient to define

Z⬅i␨, ␨⬅关共1⫹t2兲2sin22␾⫺共1⫺t2兲2兴1/2, 共43兲 where␨ is real. The critical value of t for which Z becomes purely imaginary is given by

tc共␾兲⫽

冋

1⫺兩sin 2␾兩 1⫹兩sin 2␾兩

册

1/2

. 共44兲

For t⬍tc(␾) both eigenvalues are real and the cavity

eigen-modes are degenerate; this regime is usually referred to as the locked regime关9,38兴. Conversely, for t⬎tc(␾) the

eigen-values Eq.共41兲 acquire an imaginary part and the degeneracy between eigenmodes is removed 共unlocked regime兲. Let u_⫾ be the non-normalized eigenvectors corresponding to ␭_⫾, respectively,

u_⫾⫽

冉

共1⫺t

2_{兲cos 2␾⫾Z}

⫺共1⫹t2_{兲sin 2␾}

冊

. 共45兲

For arbitrary values of t and ␾ these eigenvectors are not orthogonal. This is shown in Fig. 2 where the angle ␤ be-tween u_⫹ and u_⫺ is plotted as a function of t for several values of␾. For t⫽1 we have ␤⫽␲/2共orthogonal modes兲 for all values of␾, while for the critical t⫽tc(␾) we see that

FIG. 2. 共a兲–共f兲 Angle␤ between the cavity eigenmodes u_⫾ ver-sus the absorber parameter t for different values of the rotator angle

␾. For t⫽1 the eigenmodes are always parallel (␤⫽␲/2)

irrespec-tive of the value of␾. For t⫽tc(␾) the eigenmodes become

␤⫽0 and the modes become parallel. In Fig. 2共c兲 the

‘‘geo-metrical’’ Petermann K factor for the cold cavity is plotted together with␤. As Siegman remarked years ago关7,14兴, the geometrical Petermann K factor, as given below, is an intrin-sic property of the cavity eigenmodes which has nothing to do with the gain medium inside the cavity. It can be calcu-lated using the well known recipe关9兴

1 K⫽1⫺ 兩共u⫹,u⫺兲兩2 共u⫹,u⫹兲共u⫺,u⫺兲, 共46兲 obtaining K_⬍⫽ 共1⫺t 2_兲2 共1⫺t2_兲2_⫺共1⫹t2_兲2_sin2_2␾, 共47兲 for t⬍tc(␾), and K_⬎⫽ 共1⫹t 2_兲2_sin2_2␾ 共1⫹t2_兲2_sin2_{2␾⫺共1⫺t}2_兲2, 共48兲

for t⬎tc(␾). Apart from notation these results agree with

earlier works 关9,12兴. In the limit of small rotator angle ␾

Ⰶ1 we have tc(␾)⯝1⫺2␾ which is very close to 1. If we

define the dissipative coupling ␶ as t⫽exp(⫺2␶)(␶⭓0) one simply notices that in the limit of small␾and␶, the behav-ior of K near the critical value t_cis given by

K_⬍⬇ 1 1⫺␾ 2 ␶2 , 共49a兲 K_⬎⬇ 1 1⫺ ␶ 2 ␾2 , 共49b兲

in agreement with Ref.关9兴.

IV. RESULTS AND DISCUSSION

In this section we calculate the SPDC rate of the sub-threshold OPO shown in Fig. 1 and study how it depends on the ‘‘nonorthogonality parameters’’ t and ␾. Equations 共24兲 together with Eqs. 共32兲 can be straightforwardly solved to express ‘‘out’’ operators in terms of ‘‘in’’ operators; this is done explicitly in the Appendix. The resulting expressions are very cumbersome and it is not useful to write them ex-plicitly. Their general form is

aˆout⫽

兺

␣⫽a,b共S1␣␣ ˆ_{in⫹S2␣␣}ˆ in †_⫹S 3␣fˆ␣⫹S4␣fˆ_␣†兲, 共50兲

and similarly for mode b, where Si␣ are complicated

func-tions of t, ␾, G, R, and ␻L/c. From the above results we calculate the average photon number emitted in modes a and b:

n ¯_␣⫽

_具

_␣ˆ

out † _␣ˆ

out

典

vac 共␣⫽a,b兲, 共51兲

where the subscript ‘‘vac’’ indicates that the quantum expec-tation value is calculated for the incoming vacuum field. When both the absorber and rotator are switched off

共orthogonal-mode case兲 we find n¯a⫽n¯b⬅n¯, where

n

¯_⫽共G2_⫺1兲

冋

1⫺R

1⫺2G

冑

R cos共2␻L/c兲⫹R

册

2

. 共52兲 This result is in agreement with Eq. 共16兲 in Ref. 关31兴. The term inside the square brackets, when calculated for G⫽1, coincides with the spontaneous emission modification factor F关39兴, but in our case it is quadratic because of nonlinearity

关40兴. At resonance (L⫽m␲c/␻, with m integer兲, a diver-gence appears for n¯ when G⫽(1⫹R)/(2

冑

R)⬎1, corre-sponding to the threshold of oscillation 关41兴. However, we are interested only in the subthreshold case where a privi-leged lasing mode is not selected. The average photon num-bers emitted on modes a and b, evaluated at resonance, in the general case␾⫽” 0 and t⫽”1, are shown in Fig. 3. The values of the nonlinear gain and the mirror reflectivity are G

⫽1.01 and R⫽0.2, respectively, corresponding to a

sub-threshold OPO. The behavior with respect to the variable␾ of n¯_a and n¯_b, is quite similar for t⬇1. When t→0, mode a is increasingly suppressed and n¯a→0. In the same limit

mode b does not disappear but is reduced by a factor ⬇3. We report in Fig. 4 the total average photon number N¯⬅n¯_b

⫹n¯a, evaluated at resonance, as a function of the absorber

transmission coefficient t and of the rotation angle␾ due to the rotator. The nonlinear gain G and the output mirror re-flectivity R have been chosen as G⫽1.01, R⫽0.2, so that subthreshold operation is achieved.

From Fig. 4 it is clear that the local maxima of N¯ , for the t variable, are located on the curve␾⫽0 which corresponds to a cavity with orthogonal modes. This curve constitutes the upper boundary of the gray band shown in Fig. 5. The other points in the gray band represent all possible values of N¯ , calculated with the same parameters as in Fig. 4, for cavities with nonorthogonal modes. All these points are below the curve corresponding to orthogonal modes; so we do not find any enhancement of the twin-photon rate under these condi-tions.

This may be compared with the behavior of the geometri-cal K factor, as given by Eqs.共47兲 and 共48兲. Figure 5 shows the behavior of this K factor with respect to N¯ , as a function of the absorber transmission t. Both K and N¯ are evaluated for ␾⫽␲/8; furthermore, N¯ is evaluated for G⫽1.01 and R⫽0.2. From a geometrical point of view, when t⫽tc the

cavity eigenmodes become parallel and the corresponding K factor diverges, as shown in Fig. 2共c兲. In Fig. 5 this resonant behavior of K, when t approaches tc, is evident, but at the

V. OPTICAL BAND STRUCTURE IN A PARAMETRIC OSCILLATOR

In the preceding section we have calculated the total av-erage photon number N¯⫽n¯a⫹n¯b of the subthreshold OPO

calculated at resonance, that is for ␻L/c⫽m␲, where m is an integer. In general the number N¯ varies as a function of the phase shift ␪⬅␻L/c which plays the role of a reduced length. It can be varied either by varying the length L of the cavity or by varying the pump frequency ⍀⫽2␻. Then we can regard the function N¯ (␪) 共calculated for fixed values of the other OPO parameters t, G, R, and ␾) as the cavity spectrum. In Fig. 6 we plot N¯ , calculated for G⫽1.01 and R⫽0.2 共below threshold OPO兲, versus the length␪ and the rotator angle␾, for several values of the absorber parameter t. The function N¯ (␪,␾) has, for␾⫽0 and all values of t, the expected periodic behavior共with period␲) which is charac-teristic of the spectrum of a Fabry-Pe´rot cavity. For decreas-ing t the height of the resonant peaks is lowered but their

shape and position are unchanged. For␾⫽␲/2 and all values of t we obtain the same spectrum as for␾⫽0 but shifted in the variable ␪ by an amount ␲/4. This happens because␾

⫽␲/2 simply corresponds, from a physical point of view, to an exchange of the role of the two orthogonal polarizations. For␾⬎0 and t⬎0 each resonant peak is split in two sepa-rate bands corresponding to cavity eigenmodes with y and z polarization. The degeneracy is removed because of the po-larization mode coupling induced by the rotator 关see Eqs. FIG. 3. 共a兲 Plot of the average number n¯aof photons emitted on

mode a for a subthreshold OPO at resonance as a function of the rotator angle ␾ and the absorber parameter t. The values of the other parameters are G⫽1.01, R⫽0.2. For t⫽0 and␾⫽0 the pho-tons in mode a are fully absorbed so that n¯a⫽0. 共b兲 Plot of the

average number n¯b of photons emitted in mode b under the same

conditions as in共a兲.

FIG. 4. Plot of the total average photon number N¯⬅n¯a⫹n¯bof

the subthreshold OPO, calculated at resonance, as a function of the absorber transmission t and of the rotator angle␾. The values of the other parameters are: G⫽1.01, R⫽0.2. For t⫽0 and ␾⫽0 the photons in mode a are fully absorbed and the residual value of N¯ is due to contribution of only mode b.

FIG. 5. Dotted-dashed line: ‘‘geometrical’’ Petermann K factor, given by Eqs.共47兲 and 共48兲 for a cavity without crystal, calculated for␾⫽␲/8, as a function of the absorber transmission t. The value of K diverges for t→tc(␾⫽␲/8)⯝0.41. Dashed line: total average

photon number N¯ calculated at resonance and␾⫽␲/8. The values of the other parameters are G⫽1.01, R⫽0.2, corresponding to a subthreshold OPO. The gray band represents all possible values of N¯ for nonorthogonal modes. Note that N¯ is not enhanced for

共25兲兴. Finally, for t⫽0 there is an abrupt jump in the band

structure for ␾⫽␲/4 because mode a is totally suppressed and only a single linearly polarized mode can exist in the cavity. Actually this jump is not clearly visible in Fig. 6, but it becomes evident in Fig. 8.

The existence of optical band structures is well known for the case of a classical ring resonator, with passive polarization-optical elements 关32,42兴. In that case counter-propagating polarized waves are coupled by electro-optic modulators 共EOM兲, Faraday rotators, partial reflectors, etc., that are arranged in a ring configuration. The polarization-mode eigenfrequencies then display band structures as a function of a tuning parameter, e.g., the voltage across an EOM. A general method for determining the eigenfrequency band structure in a ring cavity containing various passive optical elements, has been developed in Refs. 关32,42兴. Opti-cal elements are represented by 4⫻4 matrices which couple two polarization degrees of freedom: x and y polarized waves, and two momentum degrees of freedom: clockwise

共cw兲 and counterclockwise 共ccw兲 waves. The spectrum of a

ring cavity is determined by solving the secular equation for eigenvalue unity,

det共MRT⫺1兲⫽0, 共53兲

where MRT is the matrix for one round trip along the

se-quence: MRT⫽Mn•••M2M1, and M1,M2, . . . are the

indi-vidual optical element matrices.

This approach is inherently classical because it neglects the coupling between the cavity modes and the world outside the cavity. Since our OPO is inherently a quantum system which, moreover, is based upon a Fabry-Pe´rot cavity instead of a ring cavity, we have to be careful before adopting the same method. Equation 共53兲 implicitly defines what is a spectral resonance for a classical ring cavity; we need an analogous definition in our quantum case. Input-output rela-tions for a field inside a cavity with nonorthogonal modes were already discussed from a very general point of view by Grangier and Poizat 关37兴; however, their analysis concerned only a cavity with a linear medium inside. In our case we shall find that the classical equation共53兲 remains valid in the quantum context but acquires a different meaning.

A. Resonance conditions: Quantum theory

can be generalized to an arbitrary linear amplifier 共in the sense of Caves关1兴兲 inside a cavity. A generalized one-output-mirror cavity is formed by a one-output-mirror M put in front of a bulk material B as shown in Fig. 7. Horizontal arrows represent field modes, that is modes of the electromagnetic both inside and outside the cavity. Vertical arrows represent noise modes, that is modes introduced to account for the loss chan-nels. We denote the set of left-traveling field modes by L and the set of right-traveling field modes by R and assume dim(L)⫽dim(R)⬅N. The set of annihilation opera-tors associated with the input and output field modes is denoted by ain⫽关(ain)1•••(ain)N(ain

†

)1•••(ain †

)N兴T and aout

⫽关(aout)1•••(aout)N(aout † ₎

1•••(aout † ₎

N兴T, respectively. The

set of annihilation operators associated with the input noise modes is denoted by f⫽关(F)1•••(F)N(F†)1•••(F†)N兴T.

All operators belonging to the input 共output兲 field modes commute with all operators 共and their corresponding ad-joints兲 belonging to the input 共output兲 noise modes. As shown with more details in the Appendix, if we indicate with

R, T, and M three 2N⫻2N matrices which represent the

reflectivity and transmittivity of the output mirror and the whole cavity, respectively, we find

aout⫽共R⫹TGT兲ain⫹T共1⫹GR兲f 共54a兲

⬅Sain⫹F, 共54b兲

where G⬅M(1⫺RM)⫺1. Equation共54a兲 can be straightfor-wardly interpreted in term of transmitted and reflected field amplitudes, exactly as in the classical Fabry-Pe´rot interfer-ometer theory. Looking at Eq.共54a兲 we see that the first term

Raincorresponds to the first reflected wave while the second

term (TGTain) is the product of the wave coupled into the

cavity which interacts with the optical elements represented by G and finally is coupled out of the cavity. In a similar manner we can interpret the noise term. Note that the solu-tion Eq.共54兲 exists only if

det共1⫺RM兲⫽” 0. 共55兲

Now we are ready to reexamine the definition of a spec-tral resonance. From the general equation共54兲 it is clear, by inspection, that all S-matrix elements have a common de-nominator D(␪) (␪⬅␻L/c) equal to D(␪)⫽det(1⫺RM). A natural definition of the resonant values ␪resis then given

by the complex zeros of D(␪)关43兴. From a physical point of view, since ␻ and L are real variables, we consider Re(␪res)

as the true resonant frequency. With this definition D„Re(␪res)…⫽”0 and our previous calculations apply. As an

example of this definition we show in Fig. 8 the frequency band structure corresponding to the spectra already shown in Fig. 6. Re(␪_res) is plotted versus the rotator angle ␾ for different values of t. When␾⫽0 the two modes correspond-ing to polarizations a and b are degenerate in frequency for all values of t. This degeneracy is removed by the rotator which induces a coupling between the two polarization modes. When ␾⫽␲/2 mod(2␲) the two modes exchange their role and the spectrum is simply shifted by ␲/2. For t

⫽0 the polarization mode a is completely suppressed and

the spectrum is again degenerate.

We now return to our discussion of the resonance condi-tion to notice that, when Im(␪res)⫽0, the determinant is zero

for real frequencies and our calculations break down. How-ever, the real solutions of the equation D(␪)⫽0 constitute a set of functions ␪i(R,G,␾,t) (i⫽1,2, . . . ), which fix the

boundary of the domain, in the space of the parameters R, G,

␾, and t, within which solutions of Eq.共54a兲 exist. In fact it is clear that, being ␪⬀␻⬀k, the solutions, in general com-plexes, of the equation D(␪)⫽0 are the analog of the circle of convergence of the geometrical series s(z) in the Fabry-Pe´rot transmission function. In the classical theory of the Fabry-Pe´rot interferometer a plane wave impinging on one of the mirrors of the interferometer is partially transmitted and partially reflected. The amplitude of both the transmitted and reflected wave is proportional to the sum of a geometri-cal series s(z)⫽1⫹z⫹z2⫹••• whose argument is z

⫽rei2(kL⫹␾) _{for normal incidence. This series can be}

summed only if 兩z兩⬍1. In our case 兩z兩⫽re⫺2kiL _{which is} less than 1 only if ki⬎ki

th_were

FIG. 7. Generalized one-output-mirror cavity. A mirror M is put in front of a bulk material B. The sets of annihilation field operators inside and outside the cavity are written as aR,aL and aout,ain,

respectively. Analogously F (G) represents the set of annihilation input共output兲 noise operators.

FIG. 8. Frequency band structures corresponding to the three-dimensional spectra shown in Fig. 6. We have plotted Re(␪res)

k_ith⫽⫺ 1 2Lln

冉

1

r

冊

. 共56兲

It is clear that the threshold condition corresponds to a value of z⫽x⫹iy which lies, in the complex plane (x,y), exactly on the radius of convergence of the geometrical series S(z). Analogously we identify the points lying on the boundary functions␪i(R,G,␾,t) with the set of the values of the

pa-rameters R, G,␾, and t for which oscillations start共threshold values兲 and therefore we write the threshold condition as Im(␪_res)⫽0. In the next section we analyze the distribution of these singular points in the plane (␪,␾) for different val-ues of R, G, and t. Finally it is interesting to note that the ‘‘quantum’’ equation D(␪)⫽det(1⫺RM)⫽0 does not con-tain any noise contribution and is, in fact, completely classi-cal and therefore fully equivalent to the ‘‘classiclassi-cal’’ equation

共53兲.

B. Mode spectra of OPO

We start our analysis of the OPO optical band structure by considering what happens in a cavity with orthogonal modes (t⫽1) when the threshold condition Im(␪res)⫽0 is satisfied.

In Fig. 9 the frequency band structure of the OPO spectrum is plotted for increasing values of the gain G and fixed mirror reflectivity R⫽0.5. For G⫽1.01 共subthreshold OPO兲 we have Im(␪res)⫽” 0 and the spectrum is the same as Fig. 8共a兲.

For G⬎Gth(R)⬅(1⫹R)/(2

冑

R) each band is doubled and

shifted and the gap between two near bands is increasing along with G. In other words a degeneracy between two eigenmodes is removed when the OPO starts to oscillate. At first sight these band structures closely resemble the corre-sponding ones in passive ring cavities关see, e.g., Fig. 2共d兲 in

关32兴兴. In a ring cavity the doubling in the band structure

arises from the coupling between counterpropagating modes along the ring. In other words, in a ring cavity the four de-grees of freedom of the electromagnetic field that are respon-sible for the presence of four resonant peaks per free spectral range are two ‘‘polarization’’ degrees of freedom and two

‘‘momentum’’ degrees of freedom. Instead in our case the doubling is due to the coupling between annihilation and creation operators belonging to different polarization modes. In other words here we have two polarization degrees of freedom coupled in a linear way by passive optical devices, and the same two polarization degrees of freedom coupled in a nonlinear way by the crystal. To see this more clearly, the band structure for an OPO in a simple linear cavity (t⫽1 and␾⫽0) is shown in Fig. 10, where␪resis plotted versus G

for different values of R. When G approach the threshold value Gth(R), a bifurcation in the OPO spectrum appears.

This bifurcation should be, in principle, observable ex-perimentally. However, the well-known instability of a near threshold OPO 关41兴, which is perhaps connected with this bifurcation, could make its direct observation very difficult. However, a detailed analysis of the OPO instability and its connection with the spectrum bifurcation other than with self-phase-locking共see, e.g., 关44兴兲 goes beyond the scopes of the present work. We simply recall that in our calculation the crystal is considered infinitesimally thin so that the bifurca-tion cannot be explained as a refractive index-dependent propagation effect within the crystal. The true nature of this phenomenon lies in the nonlinear coupling due to the crystal between annihilation and creation operators belonging to dif-ferent polarization modes, as is made clear in Fig. 9. In order to understand this in detail, we rewrite the scattering matrix of a parametric amplifier as b1⫽cosh␥a1⫹sinh␥a2 †_{, 共57a兲} b2⫽cosh␥a2⫹sinh␥a1 † , 共57b兲

where G⫽cosh␥. Ou 关45兴 has shown that under the trans-formations

a_⫾⫽共a1e⫺i␦⫾a2ei␦兲/

冑

2, 共58a兲

b_⫾⫽共b1e⫺i␦⫾b2ei␦兲/

冑

2, 共58b兲

FIG. 9. Illustrating the doubling mechanism for an OPO in a cavity with orthogonal eigenmodes (t⫽1) and mirror reflectivity

R⫽0.5. For increasing values of G the gap between bands is also

increasing. Higher values of G are not considered here because our model is limited by the nondepleted pump approximation.

(᭙ ␦苸R) Eqs. 共57兲 decouple in the equations of two inde-pendent degenerate parametric amplifiers,

b_⫹⫽cosh␥a_⫹⫹sinh␥a_⫹† , 共59a兲 b_⫺⫽cosh␥a_⫺⫺sinh␥a_⫺† . 共59b兲 A degenerate parametric amplifier is, following the definition of Caves关1兴, a phase-sensitive amplifier, that is an amplifier which responds differently to the two quadrature phases of the field defined as

q共a兲⫽a⫹a †

冑

2 , 共60a兲 p共a兲⫽a⫺a †

冑

2i . 共60b兲

These operators are both Hermitian and thus, in principle, observable. From Eqs. 共59兲, 共60兲 it is easy to see that each quadrature phase is amplified with a different gain,

q_⫹共b兲⫽e␥q_⫹共a兲, p_⫹共b兲⫽e⫺␥p_⫹共a兲, 共61兲 q_⫺共b兲⫽e⫺␥q_⫺共a兲, p_⫺共b兲⫽e␥p_⫺共a兲,

共62兲

and we have four independent observable degrees of free-dom. In fact, from Eqs. 共61兲, 共62兲, we see that only two quadrature phases really exhibit different gain. Since the threshold condition Im(␪_res)⫽0 contains explicitly the gain,

each of these two quadrature phases reaches the threshold for a different set of values of the parameters R, G,␾, and t and two bands appear in the spectrum.

Now we consider the more general case of a cavity with nonorthogonal modes t⭐1 and ␾⫽” 0. In this case all four quadrature phases are coupled to each other and four bands appear. The frequency band structure of the OPO spectrum is shown in Fig. 11 where␪resis plotted versus the rotator angle

␾ for different values of t and fixed G⫽

冑

2 and R⫽0.5. The difference between these spectra and the ones usually ob-tained for lossless ring cavities is both in the shape of the bands and also in their disposal. In our case the bands are symmetric with respect to a vertical axis while in the passive-cavity case the symmetry is with respect to a hori-zontal axis. This is clearly illustrated in Fig. 12 where the two pictures differ for a rotation by a ␲/2 angle in the plane of the figure. This phenomenon is entirely due to the losses in our model represented by a nonunitary matrix. However, we stress the fact that this lack of unitarity only appears in the classical equation det(1⫺RM)⫽0 but not in the full quantum equation共54兲.

To illustrate this phenomenon we consider, for simplicity, a two-mode optical system which contains an absorbing el-ement whose matrix can be written as关46兴

A⫽

冉

e⫺␣ 0

0 e␣

冊

, 共63兲

where␣ is a real parameter. This matrix is trivially nonuni-tary. Let us now analytically continue the real parameter␣in

FIG. 11. Frequency band structures of an OPO in a cavity with nonorthogonal eigenmodes. The cavity ‘‘length’’␪⫽␻L/c is plot-ted versus the rotator angle ␾ for several values of the absorber parameter t. The values of the other parameters are G⫽

冑

2, R

⫽0.5.

the complex space via the transformation ␣→␣ei␩. After this transformation the matrix A becomes

A共␩兲⫽

冉

e

⫺␣ cos␩⫺i␣ sin␩ ₀

0 e␣ cos␩⫹i␣ sin␩

冊

, 共64兲 which is in general nonunitary for arbitrary values of the real parameter␩. When␩⫽0 we recover the original matrix A, while for␩⫽␲/2 we obtain

A共␲/2兲⫽

冉

e⫺i␣ 0

0 ei␣

冊

, 共65兲

which is unitary. In Fig. 12 we show the effects of the trans-formation␣→␣ei␩ _{on a eigenfrequency band structure: Fig.}

12共b兲 is obtained from Fig. 12共a兲 by writing the absorber parameter as t⫽exp(⫺␣) and by making the substitution␣

→i␣ in the boundary functions ␪i(R,G,␾,t⫽e⫺␣)

intro-duced in the preceding section. Looking at Eq.共10兲 it is clear that the trick ␣→␣ei␩ can be interpreted as a ‘‘Wick rota-tion’’关29兴 if one thinks of the parameter␣as proportional to a finite time interval.

The nonunitary nature of the M matrix is also responsible for the lack of continuity in the band structure for values of t less then 1, as shown in Figs. 11共e兲 and 11共f兲. The breaking of the band structures and the appearance of ‘‘islands’’ is due to the fact that when t⬍1 one polarization mode 共mode a in the preceding sections兲 is increasingly suppressed because of the losses introduced by the absorber. Since in Fig. 11 the assigned value of the gain G corresponds to a near-threshold value Gth only for t⫽1, when t⬍1 the increasing losses

cause an increasing value of Gthand some eigenmodes

can-not start to oscillate. Particularly, for t⫽0, one mode is com-pletely suppressed and only two resonant peaks 共instead of four兲 per free spectral range are left. By explicit calculation it is easy to see that when using optical devices represented by a unitary matrix this phenomenon does not appear; this mode suppression can be achieved only using nonunitary optical devices.

VI. CONCLUSIONS

In the first part of this paper we have introduced and analyzed a model for an optical parametric oscillator in a cavity with nonorthogonal polarization modes. Our model comprises共and reduces to those as particular subcases兲 two theoretical models both of which have been experimentally verified. For the type-II degenerate parametric amplifier we use the model of Gardiner and Savage 关31兴 whose validity has recently been verified experimentally by Ou and Lu关23兴. For the cavity with two nonorthogonal polarization modes, where large polarization K factors have been demonstrated

关9,12兴, we adopt the model of van der Lee et al. 关9兴. By

using this model we have shown that there is no excess quan-tum noise enhancement in type-II SPDC. On the contrary, the use of a cavity with nonorthogonal 共instead of orthogo-nal兲 eigenmodes leads to a reduction of the twin photon gen-eration rate. Excess quantum noise must therefore be exclu-sively ascribed to amplification of spontaneously emitted

photons; the spontaneous emission process itself is not af-fected. Excess quantum noise becomes effective only very close to threshold when one of the cavity eigenmodes is ‘‘selected’’ as the oscillating mode which dominates over the other modes关47兴.

In the second part of this paper we have studied the eigen-frequency spectrum of the same OPO, but now working near threshold. In order to find the correct definition of a spectral resonance within our fully quantum treatment, we have de-rived the spectral dependence of this resonance from the OPO parameters by writing explicitly the scattering matrix for the whole cavity. Since a type-II parametric crystal couples annihilation operators belonging to a certain polar-ization mode with creation operators belonging to the or-thogonal polarization mode, we deal with a system which has four coupled degrees of freedom. Thus we have found that in the OPO spectrum four resonant peaks per free spec-tral range can exist. The ‘‘position’’␪⫽␻L/c of these peaks depends on the transmission t of the absorber and on the rotator angle␾which also fix the ‘‘degree of nonorthogonal-ity’’ of the cavity. Because of the ␾ dependence, different band structures, whose shapes depend on t, appear in the OPO spectrum. Since we are considering a degenerate para-metric amplifier with ␻⫽⍀/2, in order to experimentally detect the spectral band structures we can either scan the cavity length L or vary the pump frequency ⍀. These band structures closely resemble those found for a passive classi-cal ring cavity 关32兴.

ACKNOWLEDGMENTS

We acknowledge support from the EU under the IST-ATESIT contract. This project is also supported by FOM.

APPENDIX

In this appendix we derive explicitly Eq. 共54兲 utilized in Sec. V A. Let us consider the arrangement shown in Fig. 9. Horizontal arrows represent field modes, that is modes of the electromagnetic both inside and outside the cavity. Vertical arrows represent noise modes, that is modes introduced to account for the loss channels. We denote the set of left-traveling field modes byL and the set of right-traveling field modes by R and assume dim(L)⫽dim(R)⬅N. The set of annihilation operators associated with the input共output兲 field modes is denoted by a_in(a_out). The set of annihilation op-erators associated with the input共output兲 noise modes is de-noted byF (G). All operators belonging to the input 共output兲 field modes commute with all operators 共and their corre-sponding adjoints兲 belonging to the input 共output兲 noise modes. Finally we denote with aLand aRthe set of operators

belonging to the field modes inside the cavity. They satisfy the quite general linear relation

共aR兲␣⫽

兺

␤⫽1 N 关M␣␤共aL兲␤⫹L␣␤共aL †_兲 ␤兴⫹F␣, 共A1兲

in-side the cavity. Even if each of these elements is represented by a unitary operator, the requirement that the operators in the sets aR and aL obey the bosonic commutation relations

does not need to be satisfied since these operators are asso-ciated with intracavity modes 关48兴.

The mirror M generates a linear coupling between opera-tors belonging to the sets aL, aR, ain, aout which can be

represented as

共aout兲␣⫽

兺

␤⫽1

N

关T␣␤共aR兲␤⫹R␣␤共ain兲␤兴, 共A2a兲

共aL兲␣⫽

兺

␤⫽1

N

关T␣␤共ain兲␤⫹R␣␤共aR兲␤兴, 共A2b兲

(␣⫽1, . . . ,N). Is is easy to solve Eqs. 共A1兲 and 共A2兲 to-gether in order to express the operators a_outas linear combi-nations of the operators ain andF 共and their respective

ad-joints兲, as we have already done in Sec. I. However, in order to illustrate the nature of the solution that we have found and to show how the resonance condition can be imposed in a quantum theory, we solve again Eqs.共A1兲 and 共A2兲 introduc-ing a matrix notation. Let M stand for M_␣␤, L for L_␣␤, T for T_␣␤and R for R_␣␤. All these are N⫻N matrices. With aL,ain, etc., now we indicate the N-component vectors

a_L⫽关(a_L)₁(a_L)₂•••(a_L)_N兴T_{, a}

in⫽关(ain)1(ain)2•••(ain)N兴T,

etc., respectively and similarly for the corresponding adjoint operators. Using this notation we rewrite Eqs.共A1兲 as

冉

aR a_R†

冊

⫽

冉

M L L* M*

冊冉

aL a_L†

冊

⫹

冉

F F†

冊

, 共A3兲

and Eqs.共A2兲 as

冉

aout a_out†

冊

⫽

冉

T 0 0 T*

冊冉

aR a_R†

冊

⫹

冉

R 0 0 R*

冊冉

ain a_in†

冊

, 共A4a兲

冉

aL a_L†

冊

⫽

冉

T 0 0 T*

冊冉

ain a_in†

冊

⫹

冉

R 0 0 R*

冊冉

aR a_R†

冊

. 共A4b兲 This is only an intermediate step. We go ahead further intro-ducing the 2N-component vectors a_R⫽(a_R a_R†)T, f

⫽(F F†₎T_{, etc., and the 2N⫻2N matrices}

M⫽

冉

M L L* M*

冊

, 共A5兲 and T⫽

冉

T 0 0 T*

冊

, R⫽

冉

R 0 0 R*

冊

. 共A6兲

Now we are ready to rewrite Eqs. 共A3兲 and 共A4兲 as

aR⫽MaL⫹f, 共A7a兲

aout⫽TaR⫹Rain, 共A7b兲

aL⫽Tain⫹RaR. 共A7c兲

Inserting Eq. 共A7a兲 in Eq. 共A7c兲, solving for aL and using

this result in Eqs. 共A7a兲 and 共A7b兲 we finally find, for the operators belonging to the output field modes,

aout⫽共R⫹TGT兲ain⫹T共1⫹GR兲f 共A8a兲

⬅Sain⫹F, 共A8b兲

where G⬅M(1⫺RM)⫺1.

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