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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 2516–2527
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Intra-cavity stimulated emissions of photons in almost pure spin states without imposed nonreciprocity

Shu-Wei Chang  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 2516-2527 (2012)
http://dx.doi.org/10.1364/OE.20.002516


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Abstract

We propose a chiral Fabry-Perot cavity in which only the cavity modes in almost pure spin (circular polarization) states lase in the presence of gain. In absence of imposed nonreciprocal environments and time-reversal symmetry breaking of emitter states to favor the emission of circularly-polarized photons, only the resonance of modes with a specific spin orientation remains in the cavity. We demonstrate a prototype of the cavity using distributed Bragg reflectors and cholesteric liquid crystals. This reciprocal cavity may provide a method to control the angular momentum state of emitters based on stimulated emissions.

© 2012 OSA

1. Introduction

Nonvanishing angular momenta in quantum mechanical systems often come along with the time-reversal symmetry (T symmetry) breaking. For example, the Zeeman splitting of electron spin states requires a magnetic field which breaks the T symmetry. In optics, in addition to nonreciprocal responses such as the magneto-optic effect which distinguishes the two circular polarization (spin) states in a time-irreversible way, we gain another access to angular momenta of optical fields via reciprocal photonic structures. Such examples include quarter-wave plates which transform linearly polarized (LP) beams into circularly polarized (CP) ones, or cholesteric liquid crystals (CLCs) in the chiral phase that filter CP components of incident waves [1

1. H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta. Cryst. 4, 219–226 (1951). [CrossRef]

, 2

2. C. Elachi and C. Yeh, “Stop bands for optical wave propagation in cholesteric liquid crystals,” J. Opt. Soc. Am. 63, 840–842 (1973). [CrossRef]

]. Specific mask patterns can also transfer orbital angular momenta to optical beams [3

3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

9

9. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). [CrossRef]

]. In these schemes, the angular momentum difference between the incoming and outgoing beams is balanced by those of mechanical objects which may exhibit observable rotations [10

10. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936). [CrossRef]

, 11

11. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]

] if they are not too bulky.

The degree of circular polarization from CLC lasers, on the other hand, is more robust than that of the scheme based on external T-symmetry breaking because the circular polarization results from the less fragile photonic effect at room temperature. Despite the indirect spin assignment of emitted photons in the CLC case, chirality generally affects the spin and propagation (helicity) of photons. Experimentally, the photon emission from quantum dots coupling to the two CP radiation reservoirs through the planar chiral structure has been attempted [25

25. K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011). [CrossRef] [PubMed]

]. Our concern is whether with chiral cavities, the high purity of photon spin states can be achieved directly as photons are emitted. If this condition is achievable, we may control the angular-momentum states of emitters without imposed T-symmetry breaking such as an external magnetic field. Note that in reciprocal (or time-reversal invariant without loss or gain) cavities, spontaneous emissions (SPEs) cannot distinguish pairs of time-reversal states. However, the stimulated emissions (STEs) can do the job. This point will be addressed in section 2.

In this paper, we present a one-dimensional (1D) Fabry-Perot (FP) chiral cavity which exhibits significant intra-cavity CP photon emissions when lasing. Under certain criteria, the polarizations of the two lasing modes turn into the same spin state while the other spin state has no cavity resonance associated with it. As a result, only one circular polarization stands out in the presence of gain. We show that such a FP laser is realizable using isotropic distributed Bragg reflectors (DBRs) and CLCs, of which a few combinations had been considered in literature [26

26. Y. Matsuhisa, R. Ozaki, M. Ozaki, and K. Yoshino, “Single-mode lasing in one-dimensional periodic structure containing helical structure as a defect,” Jpn. J. Appl. Phys. Part 2 44, L629–L632 (2005). [CrossRef]

28

28. B. Park, M. Kim, S. W. Kim, and I. T. Kim, “Circularly polarized unidirectional lasing from a cholesteric liquid crystal layer on a 1-D photonic crystal substrate,” Opt. Express 17, 12323–12331 (2009). [CrossRef] [PubMed]

]. With an isotropic gain and randomly oriented dipole currents perpendicular to the propagation axis of the FP cavity, both of which do not favor particular CP emissions, we show that CP photon emissions are indeed due to chirality only.

In the remaining part of this paper, we first define reciprocal cavities and discuss the relation of spontaneous and stimulated emissions to T-symmetry breaking in these cavities (section 2). The features of FP cavities utilizing a unique type of chiral reflectors are then discussed, and we will show how to characterize the polarization state of the intra-cavity field with the 1D dyadic Green’s function and Stokes parameters (section 3). A prototype of the chiral FP cavity using CLCs and DBRs will be demonstrated, and the intra-cavity Stokes parameters are calculated to verify that the lasing mode is indeed significantly circularly-polarized inside the cavity (section 4). Although we begin with a reciprocal condition, we show that the feedback from CP stimulated emissions may turn the cavity nonreciprocal (section 5). A conclusion will be given at the end (section 6).

2. Time-reversal symmetry breaking from photon emissions in reciprocal cavities

A reciprocal cavity is a resonance structure (assuming a relative permeability of unity) in which two current sources Js,1(r,ω) and Js,2(r,ω) and the corresponding electric fields E1(r,ω) and E2(r,ω) oscillating at a frequency ω satisfy the Lorentz reciprocity theorem in the regime of linear optics [29

29. H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Amsterdammer Akademie der Wetenschappen 4, 176 (1896).

, 30

30. J. A. KongElectromagnetic Wave Theory (EMW Publishing, 2008), last ed.

]:
ΩdrJs,1(r,ω)E2(r,ω)=ΩdrJs,2(r,ω)E1(r,ω),
(1)
where Ω is a region outside which Js,1(r,ω) and Js,2(r,ω) vanish. A nonreciprocal cavity is the one in which the integral identity in Eq. (1) does not hold. Violations of the theorem occur when the permeability tensor ε̿r(r,ω) represented in a real orthonormal basis set (for example , ŷ, and ) is nonsymmetric, namely ε¯¯r(r,ω)ε¯¯rT(r,ω), where the superscript “T” means matrix transpose. Thus, we classify the cavity as reciprocal or nonreciprocal based on whether ε̿r(r,ω) is symmetric [30

30. J. A. KongElectromagnetic Wave Theory (EMW Publishing, 2008), last ed.

, 31

31. S. W. Chang, “Full frequency-domain approach to reciprocal microlasers and nanolasers-perspective from Lorentz reciprocity,” Opt. Express 19, 21116–21134 (2011). [CrossRef] [PubMed]

]. For the nonlinear regime such as lasers, we generalize this criterion of symmetric matrices to the effective permittivity tensor dressed by nonlinearities [31

31. S. W. Chang, “Full frequency-domain approach to reciprocal microlasers and nanolasers-perspective from Lorentz reciprocity,” Opt. Express 19, 21116–21134 (2011). [CrossRef] [PubMed]

]. Note that the term (non)reciprocal here refers to the validity of the Lorenz reciprocity theorem rather than certain distinct behaviors of specifically-polarized waves which still satisfy Eq. (1) [23

23. J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005). [CrossRef] [PubMed]

].

In reciprocal cavities, spontaneous emissions cannot break the T symmetry of emitter states by distinguishing the occupation number of one of the time-reversal paired states from the other. To see that, let us compare the SPE rate Rsp,cv between states c and v (apart from a local correction factor due to dielectric constituents) [32

32. S. M. Barnett, B. Huttner, and R. Loudon, “Spontaneous emission in absorbing dielectric media,” Phys. Rev. Lett. 68, 3698–3701 (1992), (The dyadic Green’s function in that work has a sign difference from that used here). [CrossRef] [PubMed]

] with that Rsp,c̃ṽ between the time-reversal states and at the same location rs in a reciprocal cavity:
Rsp,cvRsp,c˜v˜=2μ0ωcv2h¯Im[μvc*G¯¯ee(rs,rs,ωcv)μvc]2μ0ωc˜v˜2h¯Im[μv˜c˜*G¯¯ee(rs,rs,ωc˜v˜)μv˜c˜]=2μ0ωcv2h¯Im[μvc*{G¯¯ee(rs,rs,ωcv)G¯¯eeT(rs,rs,ωc˜v˜)}μvc]=0,
(2)
where μo is the vacuum permittivity; is the Planck constant divided by 2π; μvc and ωcv (μṽc̃ and ωc̃ṽ) are the dipole moment and transition frequency between states c and v ( and ), respectively ( μv˜c˜=μvc* up to a constant phase; and ωcv = ωc̃ṽ in absence of T-symmetry breaking); G̿ee(r, r′,ω) is the dyadic Green’s function at frequency ω, which connects the current density at r′ to the optical field at r and exhibits the symmetry property G¯¯ee(r,r,ω)=G¯¯eeT(r,r,ω) (represented in a real orthonormal basis set) in a reciprocal environment; and we have used the fact that for a scalar X, X = XT. From Eq. (2), if the initial occupation probabilities of states c and v are identical to those of and , respectively, spontaneous emissions cannot make their occupations uneven at later times.

3. Fabry-Perot cavities with a special chiral reflector

To construct FP cavities with a nearly perfect intra-cavity CP field, we incorporate the chirality into the reflector at one side of the cavity. Figure 1 shows the schematic diagram of the 1D FP cavity with a length h. The FP cavity is filled with an isotropic gain medium in the region z ∈ (z1, z2). Reflector 1 (2) characterized by the reflection matrix r̿1 (r̿2) in the LP basis ( and ŷ) is located at z1 (z2). These matrices connect the cartesian components of the incident wave to those of the reflected one. We set reflector 1 isotropic and adopt a special design for reflector 2. The reciprocity requires r̿2 (matrix elements indexed by x, y) to be symmetric [33

33. A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992). [CrossRef]

]:
r¯¯2=(r2,xx,r2,xyr2,yx,r2,yy)=(a,cc,b),
(4)
where a, b, and c are complex numbers. The reflection matrix r̿2,CP in the CP basis [matrix elements indexed by ± corresponding to the basis e^±=(x^±iy^)/2] is transformed from r̿2 with the change of basis:
r¯¯2,CP(r2,CP,++,r2,CP,+r2,CP,+,r2,CP,)=p¯¯r¯¯2p¯¯=(a+b2,ab2icab2+ic,a+b2),
(5a)
p¯¯=(1/2,1/2i/2,i/2),
(5b)
where p̿ is a unitary matrix describing the transformation between CP and LP representations.

Fig. 1 The schematic diagram of the 1D FP cavity. Reflector 1 is isotropic while reflector 2 is chiral with specific characteristics of reflections.

Fig. 2 The polarization states (a) v1,2 (case I), and (b) u1,2 (case II) when κ ≠ 0. The phase angle of κ is set to arg (ab). When κ = 0, the polarization states v1,2 become ê while u1,2 turn into ê+.

The polarization state of the field inside the FP cavity could be an almost pure spin state if the cavity modes carrying any of the two polarizations v1,2 (u1,2) are amplified. It does not matter which of v1,2 (u1,2) becomes the dominant polarization because they both resemble ê (ê+). To maintain the polarization eigenstates of reflector 2 throughout the FP cavity, isotropic reflector 1 is therefore adopted. In addition to these considerations, two issues still require clarifications. First, the two polarization eigenvectors v1,2 (u1,2) are linearly independent unless κ is strictly zero. Whenever κ ≠ 0, photons might have tended to be emitted in a combined polarization state of v1,2 (u1,2) which carries no spin angular momentum. This point can be resolved with the Stokes parameter S3(z,ω) inside the cavity, which is an indicator on the degree of circular polarization. Second, we need a prototype of reflector 2 to validate the feasibility of the concept in principle. Both issues will be addressed in section 4.

To see whether a nearly perfect CP field indeed exists inside the FP cavity with gain, we consider a SPE source Js(z,ω) which only has the in-plane (xy) components relevant to the transverse polarization state and satisfies the following ensemble average:
Js,α*(z,ω)Js,α(z,ω)=δα,αδ(zz)c,vDcv(z,ω)|jsp,vc(ω)|22,
(7)
where α,α = x,y; Dcv(z,ω) is the dipole density due to the radiative transition between states c and v, which contains the information of state filling and is only present inside the cavity; and jsp,vc(ω) is the amplitude of the SPE dipole current. For simplicity, only the randomness of the SPE source along the z direction [δ(zz′) in Eq. (7)] is taken into account, which is pertinent to the fundamental transverse mode of typical FP lasers. The Kronecker delta δαα indicates that the SPE source does not favor emissions in specific transverse polarization states. In the case of equal amounts of opposite-oriented but uncorrelated CP dipoles (no T-symmetry breaking), the form of the ensemble average remains identical to that in Eq. (7). Therefore, the polarization state of the intra-cavity lasing field is determined by the cavity alone. The optical field E(z,ω) is connected to the SPE source through a response integral:
E(z,ω)=z1z2dzG¯¯ee(z,z,ω)iωμ0Js(z,ω),
(8a)
where the dyadic Green’s function G̿ee(z, z,ω) is a 2-by-2 tensor now because only the in-plane components are considered. This Green’s function G̿ee(z, z,ω) satisfies the reflection boundary conditions at z1 and z2 and can be analytically expressed in terms of two dyadic response kernels G¯¯ee<(z,z,ω)[z(z1,z)] and G¯¯ee>(z,z,ω)[z(z,z2)] in different regions:
G¯¯ee(z,z,ω)=U(zz)G¯¯ee<(z,z,ω)+U(zz)G¯¯ee>(z,z,ω),
(8b)
G¯¯ee<(z,z,ω)=ηeikh2iωμ0[eik(zz1)r¯¯1+eik(zz1)I¯¯2][r¯¯2r¯¯1e2ikhI¯¯2]1×[eik(zz2)r¯¯2+eik(zz2)I¯¯2],
(8c)
G¯¯ee>(z,z,ω)=ηeikh2iωμ0[eik(zz2)r¯¯2+eik(zz2)I¯¯2][r¯¯1r¯¯2e2ikhI¯¯2]1×[eik(zz1)r¯¯1+eik(zz1)I¯¯2],
(8d)
where U(z) is the Heaviside step function [U(z) = 1 if z > 0; 0 otherwise]; η is the impedance of the gain medium; and k is the complex propagation constant incorporating the gain effect.

The resonance effect of the FP cavity is also one of the causes to the potential CP field. When reflector 1 is isotropic (r̿1 = ΓI̿2, where Γ is the reflection coefficient), the dyadic Green’s function G̿ee(z, z,ω) is a matrix function of r̿2 only, as indicated by Eq. (8b) to (8d). Thus, the polarizations v1,2 (u1,2) are also the eigenvectors of G̿ee(z, z, ω). In particular, the dyadic Green’s function G̿ee(z, z, ω) contains a matrix [Γexp(2ikh) r̿2I̿2]−1, and the operation of G̿ee(z, z, ω) on v1,2 (take case I for example) leads to an eigenstate relation as follows:
G¯¯ee(z,z,ω)vn=gn(z,z,ω)ΓΛne2ikh1vn(n=1,2),
(12a)
gn(z,z,ω)={ηeikh2iωμ0[eik(zz1)Γ+eik(zz1)][eik(zz2)Λn+eik(zz2)]z(z1,z),ηeikh2iωμ0[eik(zz1)Γ+eik(zz1)][eik(zz2)Λn+eik(zz2)]z(z,z2),
(12b)
where gn(z, z,ω) is a regular scalar function. With the right frequency ω and gain effect, the magnitude of the denominator |ΓΛn exp(2ikh) – 1| in Eq. (12a) can be small, indicating huge responses for v1,2. This is just the round-trip oscillation condition of FP modes. Thus, if Js(z,ω) contains the components of v1,2, these components would be significantly induced on E(z,ω).

4. Construction of the chiral cavity

To implement the FP cavity, we use a quarter wavelength distributed Bragg reflector (λ/4 DBR) with a high reflectivity in its stop band as reflector 1 and the cascaded structure of a λ/4 DBR and CLC as reflector 2, as shown in Figs. 3(a) and 3(b), respectively. Thus, the reflector (cavity) is chiral. The DBR insertion in reflector 2 is critical. Although CLCs alone satisfy the matrix-element condition of |(ab)/2 ∓ ic| ≪ |ab| for r̿2, their characteristic of a ≈ – b leads to small magnitudes |Λ1,2| and makes the resonance condition ΓΛ1,2 exp(2ikh) ≈ 1 difficult. The additional insertion of DBR boosts |Λ1,2| (despite the smaller margin |ab| for the goal condition) and makes lasing easier.

Fig. 3 (a) Reflector 1 is a λ/4 DBR at the left side of the gain medium. (b) Reflector 2 is the cascade of a λ/4 DBR and CLC at the right side of the gain medium.

In the calculations, we set the DBR center wavelength to 850 nm. The refractive indices of the high- and low-index regions in both DBRs are assumed nondispersive with values 3.8 and 3.4, respectively. The thicknesses of the two regions are one quarter of the center wavelength divided by the corresponding refractive indices. Reflector 1 (2) has 15 (7) DBR pairs. For the CLC, the ordinary (extraordinary) refractive index no (ne) is 1.55 (1.65). We choose a chiral pitch of 531.25 nm to match the 850 nm wavelength due to the possibility of near-infrared lasing in CLCs [35

35. P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011). [CrossRef]

]. The number of pitches is 20. Outside reflectors 1 and 2 are two free spaces with refractive indices of 3.5 and 1.6, respectively. In the FP cavity, the complex refractive index of the gain medium is also frequency-independent with a value 3.5 – 0.0118i. The cavity length h is 486.3 nm.

The reflectivity spectrum of reflector 1 in Fig. 4(a) is calculated with the transfer-matrix method. The DBR has a high reflection (|Γ|2 > 99.3 %) near 850 nm. Similarly, we obtain the reflection matrix of r̿2 from propagation matrices of the CLC and DBR [36

36. D. W. Berreman, “Optics in smoothly varying anisotropic plannar structures: application to liquid-crystal twist cells,” J. Opt. Soc. Am. 63, 1374–1380 (1973). [CrossRef]

]. Due to the reciprocity in Eq. (5a) and (5b), the two diagonal matrix elements of r̿2,CP must be identical, reflecting the identical spectra of the square magnitudes for these two matrix elements in the upper graph of Fig. 4(b). In contrast, the square magnitude |r2,CP,+,−|2 is much larger than |r2,CP,−,+|2 [lower graph of Fig. 4(b)], which is the criterion for the resemblance between two polarizations u1,2 and ê+ (case II). From the two reflection matrices, we calculate the Stokes parameters inside the FP cavity in Fig. 5. The two peaks on the spectra correspond to the FP modes of u1 and u2 while the patterns in the z direction reflect the standing waves of the cavity modes. The magnitudes of 〈S1(z,ω)〉 and 〈S2(z,ω)〉 are smaller than those of 〈S0(z,ω)〉 and 〈S3(z,ω)〉 by about two orders of magnitude, indicating an almost perfect CP field. The ratio 〈S3(z,ω)〉/〈S0(z,ω)〉 is an indicator of the degree of circular polarization, and its maximum value at the main FP peak is about 99.8 % inside the cavity. Since we adopt an isotropic gain and a SPE source which does not favor particular CP emissions, an intrinsically reciprocal but chiral cavity is sufficient for the stimulated emission of photons in a nearly pure spin state. The transmission out of the isotropic reflector preserves the polarization state inside the cavity, and a nearly perfect CP radiation should be observable outside reflector 1.

Fig. 4 (a) The reflectivity of reflector 1 versus the wavelength. (b) The square magnitudes of matrix elements in r̿2 versus the wavelength. The square magnitudes |r2,CP,+,+|2 and |r2,CP,−,−|2 are equal (upper graph), while |r2,CP,+,−|2 is much larger than |r2,CP,−,+|2 near 850 nm (lower graph).
Fig. 5 (a)–(d) Spectra of the Stokes parameters 〈Sn(z,ω)〉 (n = 0 – 3). The magnitudes of 〈S1(z,ω)〉 and 〈S2(z,ω)〉 are much smaller than those of 〈S0(z,ω)〉 and 〈S3(z,ω)〉.

5. Effect of circularly-polarized stimulated emissions on reciprocity

If the gain medium consists of emitter systems which contain degenerate time-reversal paired states, the STEs in this chiral FP cavity can make the occupation numbers of the time-reversal pairs uneven. This effect may further change the appearance of the permittivity tensor. In particular, the dressed permittivity tensor represented in a real orthonormal basis set may become nonsymmetric, which breaks the reciprocity of the cavity.

To see the origin of this phenomenon, we use a simple model of double two-level systems which form a time-reversal pair for the gain medium. As shown in Fig. 6, the excited state c () is the spin angular momentum eigenstate |s,ms〉 = |1/2,1/2〉 (|1/2,–1/2〉) with the z angular momentum quantum number ms = 1/2 (−1/2) [the z axis here is identical to that of the chiral FP cavity]. The ground state v () is the angular momentum state |j,m〉 = |3/2,3/2〉 (|3/2,–3/2〉) with the z angular momentum quantum number m = 3/2 (−3/2) in the total angular momentum space j = 3/2. The dipole moment μvc = −μê (μṽc̃ = μê+) between states c and v ( and ) is circularly-polarized. Selection rules ensure that radiative transitions from c to v ( to ) are only accompanied by free-space photon emissions in the polarization state ê (z spin angular momentum quantum number ms,ph = −1), and those between states and result in photon emissions in the polarization state ê+ (ms,ph = 1). Such examples of the double two-level systems can be the fundamental heavy-hole excitons or the bandedge states of unstrained or compressively-strained [001] quantum wells [37

37. S. L. Chuang, Physics of Optoelectronic Devices (Wiley and Sons, 1995), 1st ed.

]. Let us approximate polarization eigenstates u1,2 of the two cavity modes in the chiral FP cavity as ê+ for simplicity. Since only photons in the polarization state ê+ experience cavity resonances and dominate in STEs, the population consumption in state is much faster than that of c. As a result, the occupation number f of is smaller than that fc of c, and the occupation f of is larger than the counterpart fv of v. This unbalance in state occupations leads to unequal occupation differences: fvfcff, which is the sign of T-symmetry breaking.

Fig. 6 The energy levels of double two-level systems forming a time-reversal pair. The dipole moment μvc between states c (|1/2,1/2〉) and v (|3/2,3/2〉) is −μê and that μṽc̃ between (|1/2,–1/2〉) and (|3/2,–3/2〉) is μê+. Due to the dominance of photons with a spin quantum number ms,ph = 1, only the STEs from to take place.

From the density-matrix formalism [37

37. S. L. Chuang, Physics of Optoelectronic Devices (Wiley and Sons, 1995), 1st ed.

] and the assumption that the double two-level systems remain degenerate except for state occupations, the effective permittivity tensor ε̿r(ω) due to the feedback from the CP STEs can be written as
ε¯¯r(ω)=εr,bgdI¯¯3+Neh¯ε01ωcvωiγcv[μvcμvc(fvfc)+μv˜c˜μv˜c˜(fv˜fc˜)],
(13a)
where εr,bgd is the background permittivity; Ne is the volume density of emitters; and γcv is the dephasing parameter for the radiative transition between states c and v. If we substitute the expressions of dipole moments μvc = −μê and μṽc̃ = μê+ into Eq. (13a) and represent ε̿r(ω) in the basis set {x̂, ŷ,ẑ}, the matrix form of ε̿r(ω) becomes
ε¯¯r(ω)=εr,bgdI¯¯3+Neh¯ε0μ2ωcvωiγcv{[(fvfc)+(fv˜fc˜)]2(I¯¯2,02×101×2,0)+[(fvfc)(fv˜fc˜)]2(σ¯¯2,02×101×2,0)},
(13b)
where 02×1 (01×2) is the 2-by-1 (1-by-2) null matrix. If fvfcff, the antisymmetric Pauli matrix σ̿2 makes the permittivity tensor ε̿r(ω) nonsymmetric and turns the chiral FP cavity nonreciprocal. In fact, the matrix form of ε̿r(ω) is similar to the permittivity tensor in the presence of the magneto-optical effect, which explicitly breaks the reciprocity.

In Eq. (13b), the asymmetry of the permittivity tensor vanishes in the quasi equilibrium condition: fvfcff, which is often equivalent to fcf and fvf. Unless the occupations relax rapidly between angular momentum states forming time-reversal pairs, namely, fast angular momentum relaxation, the reciprocity is generally broken in this chiral FP cavity. In principle, the model in sections 3 and 4 should take this nonreciprocal effect into account. Nevertheless, the calculations based on reciprocal cavities and reasoning from Eq. (13a) and (13b) indicate that the reciprocity (in a generalized sense for lasers) may be unstable in the presence of both chirality and gain.

6. Conclusion

In summary, we have analyzed the criterion of the T-symmetry breaking of the emitter state in absence of imposed nonreciprocal environments, and proposed a prototype of reciprocal FP cavities which make the stimulated emission of photons directly in a nearly pure spin state possible. Due to the special chiral reflector, which is constructed with a λ/4 DBR and CLC, the potential lasing modes only have polarization states close to one of the circular polarizations. In this case, chirality in turn may lead to a nonreciprocal cavity via the CP stimulated emission.

Acknowledgments

This work is funded by the research project of Research Center for Applied Sciences, Academia Sinica, Taiwan, and support from National Science Council, Taiwan, under grant No. NSC100-2112-M-001-002-MY2.

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L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef] [PubMed]

8.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008). [CrossRef]

9.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). [CrossRef]

10.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936). [CrossRef]

11.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]

12.

H. Ando, T. Sogawa, and H. Gotoh, “Photon-spin controlled lasing oscillation in surface-emitting lasers,” Appl. Phys. Lett. 73, 566–568 (1998). [CrossRef]

13.

H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, “Circularly polarized lasing in a (110)-oriented quantum well vertical-cavity surface-emitting laser under optical spin injection,” Appl. Phys. Lett. 94, 131108 (2009). [CrossRef]

14.

M. Holub, J. Shin, S. Chakrabarti, and P. Bhattacharya, “Electrically injected spin-polarized vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 87, 091108 (2005). [CrossRef]

15.

M. Holub, J. Shin, D. Saha, and P. Bhattacharya, “Electrical spin injection and threshold reduction in a semiconductor laser,” Phys. Rev. Lett. 98, 146603 (2007). [CrossRef] [PubMed]

16.

M. Holub and P. Bhattacharya, “Spin-polarized light-emitting diodes and lasers,” J. Phys. D: Appl. Phys. 40, R179–R203 (2007). [CrossRef]

17.

V. I. Kopp, B. Fan, H. K. M. Vithana, and A. Z. Genack, “Low-threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals,” Opt. Lett. 23, 1707–1709 (1998). [CrossRef]

18.

B. Taheri, A. F. Munoz, P. Palffy-Muhoray, and R. Twieg, “Low threshold lasing in cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 358, 73–82 (2001). [CrossRef]

19.

J. Schmidtke, W. Stille, H. Finkelmann, and S. T. Kim, “Laser emission in a dye doped cholesteric polymer network,” Adv. Mater. 14, 746 (2002). [CrossRef]

20.

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4, 676–685 (2010). [CrossRef]

21.

V. I. Kopp and A. Z. Genack, “Twist defect in chiral photonic structures,” Phys. Rev. Lett. 89, 033901 (2002). [CrossRef] [PubMed]

22.

J. Schmidtke, W. Stille, and H. Finkelmann, “Defect mode emission of a dye doped cholesteric polymer network,” Phys. Rev. Lett. 90, 083902 (2003). [CrossRef] [PubMed]

23.

J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater. 4, 383–387 (2005). [CrossRef] [PubMed]

24.

V. I. Kopp, Z. Q. Zhang, and A. Z. Genack, “Lasing in chiral photonic structures,” Prog. Quantum Electron. 27, 369–416 (2003). [CrossRef]

25.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106, 057402 (2011). [CrossRef] [PubMed]

26.

Y. Matsuhisa, R. Ozaki, M. Ozaki, and K. Yoshino, “Single-mode lasing in one-dimensional periodic structure containing helical structure as a defect,” Jpn. J. Appl. Phys. Part 2 44, L629–L632 (2005). [CrossRef]

27.

Y. Matsuhisa, Y. Huang, Y. Zhou, S. T. Wu, Y. Takao, A. Fujii, and M. Ozaki, “Cholesteric liquid crystal laser in a dielectric mirror cavity upon band-edge excitation,” Opt. Express 15, 616–622 (2007). [CrossRef] [PubMed]

28.

B. Park, M. Kim, S. W. Kim, and I. T. Kim, “Circularly polarized unidirectional lasing from a cholesteric liquid crystal layer on a 1-D photonic crystal substrate,” Opt. Express 17, 12323–12331 (2009). [CrossRef] [PubMed]

29.

H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Amsterdammer Akademie der Wetenschappen 4, 176 (1896).

30.

J. A. KongElectromagnetic Wave Theory (EMW Publishing, 2008), last ed.

31.

S. W. Chang, “Full frequency-domain approach to reciprocal microlasers and nanolasers-perspective from Lorentz reciprocity,” Opt. Express 19, 21116–21134 (2011). [CrossRef] [PubMed]

32.

S. M. Barnett, B. Huttner, and R. Loudon, “Spontaneous emission in absorbing dielectric media,” Phys. Rev. Lett. 68, 3698–3701 (1992), (The dyadic Green’s function in that work has a sign difference from that used here). [CrossRef] [PubMed]

33.

A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B 46, 3326–3336 (1992). [CrossRef]

34.

S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra (Prentice Hall, 1989), 2nd ed.

35.

P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE 8114, 81140T (2011). [CrossRef]

36.

D. W. Berreman, “Optics in smoothly varying anisotropic plannar structures: application to liquid-crystal twist cells,” J. Opt. Soc. Am. 63, 1374–1380 (1973). [CrossRef]

37.

S. L. Chuang, Physics of Optoelectronic Devices (Wiley and Sons, 1995), 1st ed.

OCIS Codes
(140.3430) Lasers and laser optics : Laser theory
(260.5430) Physical optics : Polarization
(160.1585) Materials : Chiral media

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: November 23, 2011
Revised Manuscript: January 11, 2012
Manuscript Accepted: January 11, 2012
Published: January 19, 2012

Citation
Shu-Wei Chang, "Intra-cavity stimulated emissions of photons in almost pure spin states without imposed nonreciprocity," Opt. Express 20, 2516-2527 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2516


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References

  1. H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta. Cryst.4, 219–226 (1951). [CrossRef]
  2. C. Elachi and C. Yeh, “Stop bands for optical wave propagation in cholesteric liquid crystals,” J. Opt. Soc. Am.63, 840–842 (1973). [CrossRef]
  3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45, 8185–8189 (1992). [CrossRef] [PubMed]
  4. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett.17, 221–223 (1992). [CrossRef] [PubMed]
  5. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.112, 321–327 (1994). [CrossRef]
  6. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Optical helices and spiral interference fringes,” Opt. Lett.22, 52–54 (1997). [CrossRef] [PubMed]
  7. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett.96, 163905 (2006). [CrossRef] [PubMed]
  8. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev.2, 299–313 (2008). [CrossRef]
  9. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon.3, 161–204 (2011). [CrossRef]
  10. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev.50, 115–125 (1936). [CrossRef]
  11. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett.75, 826–829 (1995). [CrossRef] [PubMed]
  12. H. Ando, T. Sogawa, and H. Gotoh, “Photon-spin controlled lasing oscillation in surface-emitting lasers,” Appl. Phys. Lett.73, 566–568 (1998). [CrossRef]
  13. H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, “Circularly polarized lasing in a (110)-oriented quantum well vertical-cavity surface-emitting laser under optical spin injection,” Appl. Phys. Lett.94, 131108 (2009). [CrossRef]
  14. M. Holub, J. Shin, S. Chakrabarti, and P. Bhattacharya, “Electrically injected spin-polarized vertical-cavity surface-emitting lasers,” Appl. Phys. Lett.87, 091108 (2005). [CrossRef]
  15. M. Holub, J. Shin, D. Saha, and P. Bhattacharya, “Electrical spin injection and threshold reduction in a semiconductor laser,” Phys. Rev. Lett.98, 146603 (2007). [CrossRef] [PubMed]
  16. M. Holub and P. Bhattacharya, “Spin-polarized light-emitting diodes and lasers,” J. Phys. D: Appl. Phys.40, R179–R203 (2007). [CrossRef]
  17. V. I. Kopp, B. Fan, H. K. M. Vithana, and A. Z. Genack, “Low-threshold lasing at the edge of a photonic stop band in cholesteric liquid crystals,” Opt. Lett.23, 1707–1709 (1998). [CrossRef]
  18. B. Taheri, A. F. Munoz, P. Palffy-Muhoray, and R. Twieg, “Low threshold lasing in cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst.358, 73–82 (2001). [CrossRef]
  19. J. Schmidtke, W. Stille, H. Finkelmann, and S. T. Kim, “Laser emission in a dye doped cholesteric polymer network,” Adv. Mater.14, 746 (2002). [CrossRef]
  20. H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics4, 676–685 (2010). [CrossRef]
  21. V. I. Kopp and A. Z. Genack, “Twist defect in chiral photonic structures,” Phys. Rev. Lett.89, 033901 (2002). [CrossRef] [PubMed]
  22. J. Schmidtke, W. Stille, and H. Finkelmann, “Defect mode emission of a dye doped cholesteric polymer network,” Phys. Rev. Lett.90, 083902 (2003). [CrossRef] [PubMed]
  23. J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nat. Mater.4, 383–387 (2005). [CrossRef] [PubMed]
  24. V. I. Kopp, Z. Q. Zhang, and A. Z. Genack, “Lasing in chiral photonic structures,” Prog. Quantum Electron.27, 369–416 (2003). [CrossRef]
  25. K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett.106, 057402 (2011). [CrossRef] [PubMed]
  26. Y. Matsuhisa, R. Ozaki, M. Ozaki, and K. Yoshino, “Single-mode lasing in one-dimensional periodic structure containing helical structure as a defect,” Jpn. J. Appl. Phys. Part 244, L629–L632 (2005). [CrossRef]
  27. Y. Matsuhisa, Y. Huang, Y. Zhou, S. T. Wu, Y. Takao, A. Fujii, and M. Ozaki, “Cholesteric liquid crystal laser in a dielectric mirror cavity upon band-edge excitation,” Opt. Express15, 616–622 (2007). [CrossRef] [PubMed]
  28. B. Park, M. Kim, S. W. Kim, and I. T. Kim, “Circularly polarized unidirectional lasing from a cholesteric liquid crystal layer on a 1-D photonic crystal substrate,” Opt. Express17, 12323–12331 (2009). [CrossRef] [PubMed]
  29. H. A. Lorentz, “The theorem of poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Amsterdammer Akademie der Wetenschappen4, 176 (1896).
  30. J. A. KongElectromagnetic Wave Theory (EMW Publishing, 2008), last ed.
  31. S. W. Chang, “Full frequency-domain approach to reciprocal microlasers and nanolasers-perspective from Lorentz reciprocity,” Opt. Express19, 21116–21134 (2011). [CrossRef] [PubMed]
  32. S. M. Barnett, B. Huttner, and R. Loudon, “Spontaneous emission in absorbing dielectric media,” Phys. Rev. Lett.68, 3698–3701 (1992), (The dyadic Green’s function in that work has a sign difference from that used here). [CrossRef] [PubMed]
  33. A. L. Shelankov and G. E. Pikus, “Reciprocity in reflection and transmission of light,” Phys. Rev. B46, 3326–3336 (1992). [CrossRef]
  34. S. H. Friedberg, A. J. Insel, and L. E. Spence, Linear Algebra (Prentice Hall, 1989), 2nd ed.
  35. P. J. W. Hands, C. A. Dobson, S. M. Morris, M. M. Qasim, D. J. Gardiner, T. D. Wilkinson, and H. J. Coles, “Wavelength-tuneable liquid crystal lasers from the visible to the near-infrared,” Proc. SPIE8114, 81140T (2011). [CrossRef]
  36. D. W. Berreman, “Optics in smoothly varying anisotropic plannar structures: application to liquid-crystal twist cells,” J. Opt. Soc. Am.63, 1374–1380 (1973). [CrossRef]
  37. S. L. Chuang, Physics of Optoelectronic Devices (Wiley and Sons, 1995), 1st ed.

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