Abstract

When a microscopic sphere is illuminated by a laser beam, interference fringes inside the sphere lead through the Kerr effect to a holographic grating. This grating is capable of phase conjugation, which is particularly strong when optical resonance takes place. According to numerical simulation, based on the Green function method, phase-conjugation reflectivity R=0.5 can be achieved in a single silica sphere of ∼9-μm diameter with realistic laser power. Such spheres can be aligned together to become a phase-conjugation mirror of large area and high reflectivity.

© 2004 Optical Society of America

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References

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  1. J. Feinberg, “Self-pumped, continuous-wave phase conjugator using internal reflection,” Opt. Lett. 7, 486–488 (1982).
    [CrossRef] [PubMed]
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    [CrossRef]
  3. G. Mie, “Beitrage zur optik truber speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  4. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).
  6. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  7. R. L. Sutherland, D. G. McLean, S. Kirkpatrick, Handbook of Nonlinear Optics (Marcel Dekker, New York, 2003).
  8. A. Yariv, Optical Electronics in Modern Communications (Oxford U. Press, New York, 1997).
  9. H.-B. Lin, J. D. Eversile, A. J. Campillo, “Vibrating orifice droplet generator for precision optical studies,” Rev. Sci. Instrum. 61, 1018–1023 (1989).
    [CrossRef]
  10. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature (London) 415, 621–623 (2002).
    [CrossRef]
  11. J.-Z. Zhang, R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989).
    [CrossRef]

2002 (1)

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature (London) 415, 621–623 (2002).
[CrossRef]

1999 (1)

M. Esselbach, G. Cedilnik, A. Kiessling, T. Baade, R. Kowarschik, V. Prokofiev, “Phase conjugation in fibre-like BTO crystals with applied electric ac field,” J. Opt. A Pure Appl. Opt. 1, 735–740 (1999).
[CrossRef]

1989 (2)

H.-B. Lin, J. D. Eversile, A. J. Campillo, “Vibrating orifice droplet generator for precision optical studies,” Rev. Sci. Instrum. 61, 1018–1023 (1989).
[CrossRef]

J.-Z. Zhang, R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989).
[CrossRef]

1982 (1)

1908 (1)

G. Mie, “Beitrage zur optik truber speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Baade, T.

M. Esselbach, G. Cedilnik, A. Kiessling, T. Baade, R. Kowarschik, V. Prokofiev, “Phase conjugation in fibre-like BTO crystals with applied electric ac field,” J. Opt. A Pure Appl. Opt. 1, 735–740 (1999).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).

Campillo, A. J.

H.-B. Lin, J. D. Eversile, A. J. Campillo, “Vibrating orifice droplet generator for precision optical studies,” Rev. Sci. Instrum. 61, 1018–1023 (1989).
[CrossRef]

Cedilnik, G.

M. Esselbach, G. Cedilnik, A. Kiessling, T. Baade, R. Kowarschik, V. Prokofiev, “Phase conjugation in fibre-like BTO crystals with applied electric ac field,” J. Opt. A Pure Appl. Opt. 1, 735–740 (1999).
[CrossRef]

Chang, R. K.

Esselbach, M.

M. Esselbach, G. Cedilnik, A. Kiessling, T. Baade, R. Kowarschik, V. Prokofiev, “Phase conjugation in fibre-like BTO crystals with applied electric ac field,” J. Opt. A Pure Appl. Opt. 1, 735–740 (1999).
[CrossRef]

Eversile, J. D.

H.-B. Lin, J. D. Eversile, A. J. Campillo, “Vibrating orifice droplet generator for precision optical studies,” Rev. Sci. Instrum. 61, 1018–1023 (1989).
[CrossRef]

Feinberg, J.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Kiessling, A.

M. Esselbach, G. Cedilnik, A. Kiessling, T. Baade, R. Kowarschik, V. Prokofiev, “Phase conjugation in fibre-like BTO crystals with applied electric ac field,” J. Opt. A Pure Appl. Opt. 1, 735–740 (1999).
[CrossRef]

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature (London) 415, 621–623 (2002).
[CrossRef]

Kirkpatrick, S.

R. L. Sutherland, D. G. McLean, S. Kirkpatrick, Handbook of Nonlinear Optics (Marcel Dekker, New York, 2003).

Kowarschik, R.

M. Esselbach, G. Cedilnik, A. Kiessling, T. Baade, R. Kowarschik, V. Prokofiev, “Phase conjugation in fibre-like BTO crystals with applied electric ac field,” J. Opt. A Pure Appl. Opt. 1, 735–740 (1999).
[CrossRef]

Lin, H.-B.

H.-B. Lin, J. D. Eversile, A. J. Campillo, “Vibrating orifice droplet generator for precision optical studies,” Rev. Sci. Instrum. 61, 1018–1023 (1989).
[CrossRef]

McLean, D. G.

R. L. Sutherland, D. G. McLean, S. Kirkpatrick, Handbook of Nonlinear Optics (Marcel Dekker, New York, 2003).

Mie, G.

G. Mie, “Beitrage zur optik truber speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Prokofiev, V.

M. Esselbach, G. Cedilnik, A. Kiessling, T. Baade, R. Kowarschik, V. Prokofiev, “Phase conjugation in fibre-like BTO crystals with applied electric ac field,” J. Opt. A Pure Appl. Opt. 1, 735–740 (1999).
[CrossRef]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature (London) 415, 621–623 (2002).
[CrossRef]

Sutherland, R. L.

R. L. Sutherland, D. G. McLean, S. Kirkpatrick, Handbook of Nonlinear Optics (Marcel Dekker, New York, 2003).

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature (London) 415, 621–623 (2002).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications (Oxford U. Press, New York, 1997).

Zhang, J.-Z.

Ann. Phys. (1)

G. Mie, “Beitrage zur optik truber speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

M. Esselbach, G. Cedilnik, A. Kiessling, T. Baade, R. Kowarschik, V. Prokofiev, “Phase conjugation in fibre-like BTO crystals with applied electric ac field,” J. Opt. A Pure Appl. Opt. 1, 735–740 (1999).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (London) (1)

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature (London) 415, 621–623 (2002).
[CrossRef]

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

H.-B. Lin, J. D. Eversile, A. J. Campillo, “Vibrating orifice droplet generator for precision optical studies,” Rev. Sci. Instrum. 61, 1018–1023 (1989).
[CrossRef]

Other (5)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

R. L. Sutherland, D. G. McLean, S. Kirkpatrick, Handbook of Nonlinear Optics (Marcel Dekker, New York, 2003).

A. Yariv, Optical Electronics in Modern Communications (Oxford U. Press, New York, 1997).

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Figures (11)

Fig. 1
Fig. 1

(a) Schematic of PC in a photorefractive crystal, where beam 3 branches twice to produce and mix with beams 1 and 2 (pumps) to generate beam 4 (phase conjugated). (b) In a microscopic sphere, beam groups 1 and 2 are bent symmetrically to form pumps. (c) In the same sphere, beam 3 (signal) mixes with the pumps to generate beam 4 (phase conjugated). The ray paths here represent Poynting vectors of plane waves rather than infinitely thin pencils of energy flux.

Fig. 2
Fig. 2

Microsphere in a laser beam marked as an arrow. The electric field of the incident radiation, Es, is linearly polarized in the x direction. The sphere has a radius a and is in the center of the coordinate system.

Fig. 3
Fig. 3

Linear gray-scale representation of |E|1/4 of the near Mie field, with n=1.5, ka=85, φ=0. The laser beam is from the left, in accord with the convention in Fig. 2.

Fig. 4
Fig. 4

Linear gray-scale representation of |EPC|1/4 when φ=0. The pump field is shown in Fig. 3.

Fig. 5
Fig. 5

Polar plot of the far field of |EPC| (maximum value normalized to 1): (a) φ=0, (b) φ=90°. The near field of EPC is shown in Fig. 4.

Fig. 6
Fig. 6

Linear gray-scale representation of the far field of |EPC|1/10 in two dimensions: r1, 120°<θ<240°, -15°<φ<15°. The near field of EPC is shown in Fig. 4.

Fig. 7
Fig. 7

Linear gray scale of |E|1/4 of the Mie field when the mode TM702 resonates. The pump beam is from the left.

Fig. 8
Fig. 8

Linear gray scale of |EPC|1/4. The pump field is shown in Fig. 7.

Fig. 9
Fig. 9

Polar distribution of the far field of |EPC| (maximum value normalized to 1): (a) in the φ=0 plane, (b) in the φ=90° plane. The pump field is shown in Fig. 7.

Fig. 10
Fig. 10

Linear gray-scale representation of the far field of |EPC|1/10 in two dimensions: r1, 120°<θ<240°, -15°<φ<15°. The near field of EPC is shown in Fig. 8.

Fig. 11
Fig. 11

Arrays of microspheres as a self-pumped PC mirror. Two such arrays are stacked together to improve phase-conjugated reflectivity.

Tables (2)

Tables Icon

Table 1 PC Parameters for Microspheres

Tables Icon

Table 2 Pump Power Coupled into Resonant Modes

Equations (42)

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E=EtwithinthesphereEs+Eroutsidethesphere,
Es=E0l=1il2l+1l(l+1) [Mo1l1(r)-iNe1l1(r)],
Er=E0l=1il2l+1l(l+1) [rlTEMo1l1(r)-irlTMNe1l1(r)],
Et=E0l=1il2l+1l(l+1) [tlTEMo1l3(r)-itlTMNe1l3(r)],
Mσml1,3(r)=×rˆfσml1,3(r),
Nσml1,3(r)=1nk ××rˆfσml1,3(r),
feml1(r)=ψl(nLkr)Plm(cos θ)cos(mφ),
nL=1whenr>anwhenr<a,
H=-inL0/μ0E0Nσml1,3 when E=E0Mσml1,3,
H=-inL0/μ0E0Mσml1,3 when E=E0Nσml1,3,
rˆ × Et=rˆ×(Es+Er),
rˆ × Ht=rˆ×(Hs+Hr),
rlTE=-ψl(nka)ψl(ka)+nψl(nka)ψl(ka)ψl(nka)ζl(1)(ka)-nψl(nka)ζl(1)(ka),
rlTM=-nψl(nka)ψl(ka)+ψl(nka)ψl(ka)nψl(nka)ζl(1)(ka)-ψl(nka)ζl(1)(ka),
tlTE=inψl(nka)ζl(1)(ka)-nψl(nka)ζl(1)(ka),
tlTM=innψl(nka)ζl(1)(ka)-ψl(nka)ζl(1)(ka).
nNL=1whenr>an+n2|E|2whenr<a.
[2+(nNLk)2](E+EPC)=0
[2+(nk)2]EPC=-2k2nn2|E|2E,
EPC=EsPC+ErPCwithinthesphereEtPCoutsidethesphere,
EsPC(r)=-2k2n0n2G(r, r)E(r)|E(r)|2dr,
G(r, r)=nk4πil=12l+1l(l+1)m=0lσm(l-m)!(l+m)!×σ[Mσml3(r)Mσml1(r)+Nσml3(r)Nσml1(r)]
EsPC(a)=in2E03m=1,3lm2l+1l(l+1)(l-m)!(l+m)!×[SmlTEMoml3(a)+SmlTMNeml3(a)],
SmlTE=n02k32πE03Mσml1(r)Et(r)|Et(r)|2dr,
SmlTM=n02k32πE03Nσml1(r)Et(r)|Et(r)|2dr;
ErPC(r)=in2E03m=1,3lm2l+1l(l+1)(l-m)!(l+m)!×[RmlTEMoml1(r)+RmlTMNeml1(r)],
RmlTE=-ζl(1)(nka)ζl(1)(ka)+nζl(1)(nka)ζl(1)(ka)ψl(nka)ζl(1)(ka)-nψl(nka)ζl(1)(ka) SmlTE,
RmlTM=-nζl(1)(nka)ζl(1)(ka)+ζl(1)(nka)ζl(1)(ka)nψl(nka)ζl(1)(ka)-ψl(nka)ζl(1)(ka) SmlTM.
EtPC(r)=in2E03m=1,3lm2l+1l(l+1)(l-m)!(l+m)!×[TmlTEMoml3(r)+TmlTMNeml3(r)],
TmlTE=iψl(nka)ζl(1)(ka)-nψl(nka)ζl(1)(ka) SmlTE,
TmlTM=inψl(nka)ζl(1)(ka)-ψl(nka)ζl(1)(ka) SmlTM.
ζl(1)(kr)(-i)l+1exp(ikr),
S=R(EPC*×HPC),
PPC=(n2E03)20μ02πa2(ka)2×lmm=1,32l+1l(l+1)(l-m)!(l+m)!×(|TmlTE|2+|TmlTM|2)
P=E020μ02πa2(ka)2l=1(2l+1)×(|rmlTE|2+|rmlTM|2)
PPC/P=(κL)2,
PPC/P=tan2|κ0L|(κ0L)2
κ0=ω2μ00 χ(3)E02,
κκ0=lmm=1,32l+1l(l+1)(l-m)!(l+m)! (|TmlTE|2+|TmlTM|2)1/2×2nka32l=1(2l+1)(|rmlTE|2+|rmlTM|2)-1/2.
Q=ωWP=43 n2kaEtE02
κ=ω2μ001/2χ(3)E023Q4n2ka
κκ0=3Q4n2ka,

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