## Intra-cavity stimulated emissions of photons in almost pure spin states without imposed nonreciprocity |

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

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. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

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]

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]

*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.

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. 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]

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

**J**

_{s,1}(

**r**

*,*

*ω*) and

**J**

_{s,2}(

**r**

*,*

*ω*) and the corresponding electric fields

**E**

_{1}(

**r**

*,*

*ω*) and

**E**

_{2}(

**r**

*,*

*ω*) oscillating at a frequency

*ω*satisfy the Lorentz reciprocity theorem in the regime of linear optics [29, 30]: where Ω is a region outside which

**J**

_{s,1}(

**r**

*,*

*ω*) and

**J**

_{s,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

*ε*̿

_{r}(

**r**

*,*

*ω*) represented in a real orthonormal basis set (for example

*x̂*,

*ŷ*, and

*ẑ*) is nonsymmetric, namely

*ε*̿

_{r}(

**r**

*,*

*ω*) is symmetric [30, 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]

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]

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]

*R*

_{sp,}

*between states*

_{cv}*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]

*R*

_{sp}

*between the time-reversal states*

_{,c̃ṽ}*c̃*and

*ṽ*at the same location

**r**

_{s}in a reciprocal cavity:

*μ*

*is the vacuum permittivity;*

_{o}*h̄*is the Planck constant divided by 2

*π*;

*μ**and*

_{vc}*ω*

*(*

_{cv}

*μ**and*

_{ṽc̃}*ω*) are the dipole moment and transition frequency between states

_{c̃ṽ}*c*and

*v*(

*c̃*and

*ṽ*), respectively (

*ω*=

_{cv}*ω*in absence of T-symmetry breaking);

_{c̃ṽ}*G̿*

_{ee}(

**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

*X*,

*X*=

*X*

^{T}. From Eq. (2), if the initial occupation probabilities of states

*c*and

*v*are identical to those of

*c̃*and

*ṽ*, respectively, spontaneous emissions cannot make their occupations uneven at later times.

*R*

_{st}

*and*

_{,cv}*R*

_{st}

*triggered by the two transitions in the same cavity could be different. Let us look into the difference between the two transition rates:*

_{,c̃ṽ}*ε*

_{0}is the vacuum permittivity; ℛ

_{st}

*(*

_{,cv,c̃ṽ}**r**′,

**r**

_{s}) is an integrand proportional to the difference between two local transition-rate densities at

**r**′; “†” means hermitian conjugates of matrices; Δ

*ε*̿

_{r,a}(

**r**′

*,*

*ω*) is the variation of the permittivity tensor due to the presence of gain and is symmetric when represented in a real orthonormal basis set [

**E**

*(*

_{vc}**r**′) [

**E**

*(*

_{ṽc̃}**r**′)] is the field generated by

*μ**(*

_{vc}

*μ**): Further applying the symmetric form of the reciprocal dyadic Green’s function to Eq. (3b) and assuming an isotropic gain [Δ*

_{ṽc̃}*ε*̿

_{r,a}(

**r**′,

*ω*) =

*i*Δ

*ε*

_{r,a,I}(

**r**′

*,*

*ω*)

*I̿*

_{3}, where Δ

*ε*

_{r,a,I}(

**r**′,

*ω*) describes the isotropic gain; and

*I̿*

_{3}is the 3-by-3 identity matrix], we simplify ℛ

_{st}

_{,cv,c̃v}_{̃}(

**r**′,

**r**

_{s}) in Eq. (3b) as where “tr” means the trace sum. In Eq. (3d), the tensor product

_{st}

_{,cv,c̃v}_{̃}(

**r**′,

**r**

_{s}) always vanishes. On the other hand, in chiral (anisotropic) but reciprocal cavities, this product can be complex hermitian, leading to nonzero ℛ

_{st}

*(*

_{,cv,c̃ṽ}**r**′,

**r**

_{s}). In particular, the nonvanishing integrand ℛ

_{st}

_{,cv,c̃v}_{̃}(

**r**

_{s},

**r**

_{s}) at

**r**′ =

**r**

_{s}means that the transition rate densities of

*μ**and*

_{vc}

*μ**feedback by the corresponding dipole-induced fields (self-triggered STEs) may be different even with identical initial occupations of states*

_{ṽc̃}*c*and

*c̃*(

*v*and

*ṽ*). Physically, this phenomenon originates from the dipole-triggered lasing mode which carries a net angular momentum inside the cavity. In the next section, we will describe how to construct FP cavities of this type with a nearly perfect intra-cavity CP field when lasing.

## 3. Fabry-Perot cavities with a special chiral reflector

*h*. The FP cavity is filled with an isotropic gain medium in the region

*z*∈ (

*z*

_{1},

*z*

_{2}). Reflector 1 (2) characterized by the reflection matrix

*r̿*

_{1}(

*r̿*

_{2}) in the LP basis (

*x̂*and

*ŷ*) is located at

*z*

_{1}(

*z*

_{2}). 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

*r̿*

_{2}(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]

*a*,

*b*, and

*c*are complex numbers. The reflection matrix

*r̿*

_{2,CP}in the CP basis [matrix elements indexed by ± corresponding to the basis

*r̿*

_{2}with the change of basis: where

*p̿*is a unitary matrix describing the transformation between CP and LP representations.

**v**

_{1,2}(

**u**

_{1,2}) are amplified. It does not matter which of

**v**

_{1,2}(

**u**

_{1,2}) 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

**v**

_{1,2}(

**u**

_{1,2}) are linearly independent unless

*κ*is strictly zero. Whenever

*κ*≠ 0, photons might have tended to be emitted in a combined polarization state of

**v**

_{1,2}(

**u**

_{1,2}) which carries no spin angular momentum. This point can be resolved with the Stokes parameter

*S*

_{3}(

*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.

*r̿*

_{1}= Γ

*I̿*

_{2}, where Γ is the reflection coefficient), the dyadic Green’s function

*G̿*

_{ee}(

*z, z*′

*,*

*ω*) is a matrix function of

*r̿*

_{2}only, as indicated by Eq. (8b) to (8d). Thus, the polarizations

**v**

_{1,2}(

**u**

_{1,2}) are also the eigenvectors of

*G̿*

_{ee}(

*z, z*′

*,*

*ω*). In particular, the dyadic Green’s function

*G̿*

_{ee}(

*z, z*′

*,*

*ω*) contains a matrix [Γexp(2

*ikh*)

*r̿*

_{2}–

*I̿*

_{2}]

^{−1}, and the operation of

*G̿*

_{ee}(

*z, z*′

*,*

*ω*) on

**v**

_{1,2}(take case I for example) leads to an eigenstate relation as follows:

*g*(

_{n}*z, z*′

*,*

*ω*) is a regular scalar function. With the right frequency

*ω*and gain effect, the magnitude of the denominator |ΓΛ

*exp(2*

_{n}*ikh*) – 1| in Eq. (12a) can be small, indicating huge responses for

**v**

_{1,2}. This is just the round-trip oscillation condition of FP modes. Thus, if

**J**

*(*

_{s}*z,*

*ω*) contains the components of

**v**

_{1,2}, these components would be significantly induced on

**E**(

*z,*

*ω*).

## 4. Construction of the chiral cavity

*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 |(

*a*–

*b*)/2 ∓

*ic*| ≪ |

*a*–

*b*| for

*r̿*

_{2}, their characteristic of

*a*≈ –

*b*leads to small magnitudes |Λ

_{1,2}| and makes the resonance condition ΓΛ

_{1,2}exp(2

*ikh*) ≈ 1 difficult. The additional insertion of DBR boosts |Λ

_{1,2}| (despite the smaller margin |

*a*–

*b*| for the goal condition) and makes lasing easier.

*n*

_{o}(

*n*

_{e}) 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]

*i*. The cavity length

*h*is 486.3 nm.

^{2}> 99.3 %) near 850 nm. Similarly, we obtain the reflection matrix of

*r̿*

_{2}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]

*r̿*

_{2,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

*|r*

_{2,CP,+,−}|

^{2}is much larger than |

*r*

_{2,CP,−,+}|

^{2}[lower graph of Fig. 4(b)], which is the criterion for the resemblance between two polarizations

**u**

_{1,2}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

**u**

_{1}and

**u**

_{2}while the patterns in the

*z*direction reflect the standing waves of the cavity modes. The magnitudes of 〈

*S*

_{1}(

*z,*

*ω*)〉 and 〈

*S*

_{2}(

*z,*

*ω*)〉 are smaller than those of 〈

*S*

_{0}(

*z,*

*ω*)〉 and 〈

*S*

_{3}(

*z,*

*ω*)〉 by about two orders of magnitude, indicating an almost perfect CP field. The ratio 〈

*S*

_{3}(

*z,*

*ω*)〉/〈

*S*

_{0}(

*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.

## 5. Effect of circularly-polarized stimulated emissions on reciprocity

*c*(

*c̃*) is the spin angular momentum eigenstate |

*s,m*〉 = |1/2,1/2〉 (|1/2,–1/2〉) with the

_{s}*z*angular momentum quantum number

*m*= 1/2 (−1/2) [the

_{s}*z*axis here is identical to that of the chiral FP cavity]. The ground state

*v*(

*ṽ*) 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

*μ**= −*

_{vc}*μ*

*ê*

_{−}(

*μ**=*

_{ṽc̃}*μ*

*ê*

_{+}) between states

*c*and

*v*(

*c̃*and

*ṽ*) is circularly-polarized. Selection rules ensure that radiative transitions from

*c*to

*v*(

*c̃*to

*ṽ*) are only accompanied by free-space photon emissions in the polarization state

*ê*

_{−}(

*z*spin angular momentum quantum number

*m*

_{s,}_{ph}= −1), and those between states

*c̃*and

*ṽ*result in photon emissions in the polarization state

*ê*

_{+}(

*m*

_{s,}_{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]. Let us approximate polarization eigenstates

**u**

_{1,2}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

*c̃*is much faster than that of

*c*. As a result, the occupation number

*f*of

_{c̃}*c̃*is smaller than that

*f*of

_{c}*c*, and the occupation

*f*of

_{ṽ}*ṽ*is larger than the counterpart

*f*of

_{v}*v*. This unbalance in state occupations leads to unequal occupation differences:

*f*–

_{v}*f*≠

_{c}*f*–

_{ṽ}*f*, which is the sign of T-symmetry breaking.

_{c̃}*ε*̿

_{r}(

*ω*) due to the feedback from the CP STEs can be written as where

*ε*

_{r,bgd}is the background permittivity;

*N*

_{e}is the volume density of emitters; and

*γ*is the dephasing parameter for the radiative transition between states

_{cv}*c*and

*v*. If we substitute the expressions of dipole moments

*μ**= −*

_{vc}*μ*

*ê*

_{−}and

*μ**=*

_{ṽc̃}*μ*

*ê*

_{+}into Eq. (13a) and represent

*ε*̿

_{r}(

*ω*) in the basis set {

*x̂, ŷ,ẑ*}, the matrix form of

*ε*̿

_{r}(

*ω*) becomes

**0**

_{2×1}(

**0**

_{1×2}) is the 2-by-1 (1-by-2) null matrix. If

*f*–

_{v}*f*≠

_{c}*f*–

_{ṽ}*f*, the antisymmetric Pauli matrix

_{c̃}*σ*̿

_{2}makes the permittivity tensor

*ε*̿

_{r}(

*ω*) nonsymmetric and turns the chiral FP cavity nonreciprocal. In fact, the matrix form of

*ε*̿

_{r}(

*ω*) is similar to the permittivity tensor in the presence of the magneto-optical effect, which explicitly breaks the reciprocity.

*f*–

_{v}*f*≈

_{c}*f*–

_{ṽ}*f*, which is often equivalent to

_{c̃}*f*≈

_{c}*f*and

_{c̃}*f*≈

_{v}*f*. 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.

_{ṽ}## 6. Conclusion

*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

## References and links

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2. | C. Elachi and C. Yeh, “Stop bands for optical wave propagation in cholesteric liquid crystals,” J. Opt. Soc. Am. |

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 |

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30. | J. A. Kong |

31. | S. W. Chang, “Full frequency-domain approach to reciprocal
microlasers and nanolasers-perspective from Lorentz
reciprocity,” Opt. Express |

32. | S. M. Barnett, B. Huttner, and R. Loudon, “Spontaneous emission in absorbing dielectric media,” Phys. Rev. Lett. |

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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 |

36. | D. W. Berreman, “Optics in smoothly varying anisotropic plannar structures: application to liquid-crystal twist cells,” J. Opt. Soc. Am. |

37. | S. L. Chuang, |

**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

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