Abstract

Absorption-free Bragg reflector has been studied in ions doped in crystals. We propose a new scheme using Zeeman sublevels of atoms to construct an absorption-free Bragg reflector with practical laser power. Its spatial period of refractive index equals half of the wavelength of the incident standing-wave coupling light. The proposal is simulated in a helium atom scheme, and can be extended to alkali earth atoms.

© 2014 Optical Society of America

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  1. S. Wang, “Principles of distributed feedback and distributed Bragg-reflector lasers,” Quantum Electron.10(4), 413–427 (1974).
    [CrossRef]
  2. K. J. Vahala, “Optical microcavities,” Nature424, 839–846 (2003).
    [CrossRef] [PubMed]
  3. M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett.67, 1855–1858 (1991).
    [CrossRef] [PubMed]
  4. J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett.70, 1421–1424 (1993).
    [CrossRef] [PubMed]
  5. M. Fleischhauer, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich, and S. Y. Zhu, “Resonantly enhanced refractive index without absorption via atomic coherence,” Phys. Rev. A461468–1487 (1992).
    [CrossRef] [PubMed]
  6. D. D. Yavuz, “Refractive index enhancement in a far-off resonant atomic system,” Phys. Rev. Lett.95, 223601 (2005).
    [CrossRef] [PubMed]
  7. C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A84, 063835 (2011).
    [CrossRef]
  8. Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, “Refractive index enhancement with vanishing absorption in short, high-density vapor cells,” Phys. Rev. A85, 053810 (2012).
    [CrossRef]
  9. C. O’Brien and O. Kocharovskaya, “Resonant enhancement of refractive index in transition element doped crystals via coherent control of excited state absorption,” J. Mod. Opt.56, 1933–1940 (2009).
    [CrossRef]
  10. C. O’Brien and O. Kocharovskaya, “Optically controllable photonic structures with zero absorption,” Phys. Rev. Lett.107, 137401 (2011).
    [CrossRef]
  11. P. Anisimov and O. Kocharovskaya, “Decaying-dressed-state analysis of a coherently driven three-level Λsystem,” J. Mod. Opt.55, 3159–3171 (2008).
    [CrossRef]
  12. R. Newell, J. Sebby, and T. G. Walker, “Dense atom clouds in a holographic atom trap,” Opt. Lett.28, 1266–1268 (2003).
    [CrossRef] [PubMed]
  13. J. Sebby-Strabley, R. T. R. Newell, J. O. Day, E. Brekke, and T. G. Walker, “High-density mesoscopic atom clouds in a holographic atom trap,” Phys. Rev. A71, 021401(R) (2005).
    [CrossRef]
  14. W. L. Wiese and J. R. Fuhra, “Accurate atomic transition probabilities for hydrogen, helium, and lithium,” J. Phys. Chem. Ref. Data38, 565–715 (2009).
    [CrossRef]

2012 (1)

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, “Refractive index enhancement with vanishing absorption in short, high-density vapor cells,” Phys. Rev. A85, 053810 (2012).
[CrossRef]

2011 (2)

C. O’Brien and O. Kocharovskaya, “Optically controllable photonic structures with zero absorption,” Phys. Rev. Lett.107, 137401 (2011).
[CrossRef]

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A84, 063835 (2011).
[CrossRef]

2009 (2)

W. L. Wiese and J. R. Fuhra, “Accurate atomic transition probabilities for hydrogen, helium, and lithium,” J. Phys. Chem. Ref. Data38, 565–715 (2009).
[CrossRef]

C. O’Brien and O. Kocharovskaya, “Resonant enhancement of refractive index in transition element doped crystals via coherent control of excited state absorption,” J. Mod. Opt.56, 1933–1940 (2009).
[CrossRef]

2008 (1)

P. Anisimov and O. Kocharovskaya, “Decaying-dressed-state analysis of a coherently driven three-level Λsystem,” J. Mod. Opt.55, 3159–3171 (2008).
[CrossRef]

2005 (2)

J. Sebby-Strabley, R. T. R. Newell, J. O. Day, E. Brekke, and T. G. Walker, “High-density mesoscopic atom clouds in a holographic atom trap,” Phys. Rev. A71, 021401(R) (2005).
[CrossRef]

D. D. Yavuz, “Refractive index enhancement in a far-off resonant atomic system,” Phys. Rev. Lett.95, 223601 (2005).
[CrossRef] [PubMed]

2003 (2)

1993 (1)

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett.70, 1421–1424 (1993).
[CrossRef] [PubMed]

1992 (1)

M. Fleischhauer, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich, and S. Y. Zhu, “Resonantly enhanced refractive index without absorption via atomic coherence,” Phys. Rev. A461468–1487 (1992).
[CrossRef] [PubMed]

1991 (1)

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett.67, 1855–1858 (1991).
[CrossRef] [PubMed]

1974 (1)

S. Wang, “Principles of distributed feedback and distributed Bragg-reflector lasers,” Quantum Electron.10(4), 413–427 (1974).
[CrossRef]

Anisimov, P.

P. Anisimov and O. Kocharovskaya, “Decaying-dressed-state analysis of a coherently driven three-level Λsystem,” J. Mod. Opt.55, 3159–3171 (2008).
[CrossRef]

Anisimov, P. M.

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A84, 063835 (2011).
[CrossRef]

Bowden, C. M.

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett.70, 1421–1424 (1993).
[CrossRef] [PubMed]

Brekke, E.

J. Sebby-Strabley, R. T. R. Newell, J. O. Day, E. Brekke, and T. G. Walker, “High-density mesoscopic atom clouds in a holographic atom trap,” Phys. Rev. A71, 021401(R) (2005).
[CrossRef]

Day, J. O.

J. Sebby-Strabley, R. T. R. Newell, J. O. Day, E. Brekke, and T. G. Walker, “High-density mesoscopic atom clouds in a holographic atom trap,” Phys. Rev. A71, 021401(R) (2005).
[CrossRef]

Dowling, J. P.

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett.70, 1421–1424 (1993).
[CrossRef] [PubMed]

Fleischhauer, M.

M. Fleischhauer, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich, and S. Y. Zhu, “Resonantly enhanced refractive index without absorption via atomic coherence,” Phys. Rev. A461468–1487 (1992).
[CrossRef] [PubMed]

Fuhra, J. R.

W. L. Wiese and J. R. Fuhra, “Accurate atomic transition probabilities for hydrogen, helium, and lithium,” J. Phys. Chem. Ref. Data38, 565–715 (2009).
[CrossRef]

Keitel, C. H.

M. Fleischhauer, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich, and S. Y. Zhu, “Resonantly enhanced refractive index without absorption via atomic coherence,” Phys. Rev. A461468–1487 (1992).
[CrossRef] [PubMed]

Kocharovskaya, O.

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A84, 063835 (2011).
[CrossRef]

C. O’Brien and O. Kocharovskaya, “Optically controllable photonic structures with zero absorption,” Phys. Rev. Lett.107, 137401 (2011).
[CrossRef]

C. O’Brien and O. Kocharovskaya, “Resonant enhancement of refractive index in transition element doped crystals via coherent control of excited state absorption,” J. Mod. Opt.56, 1933–1940 (2009).
[CrossRef]

P. Anisimov and O. Kocharovskaya, “Decaying-dressed-state analysis of a coherently driven three-level Λsystem,” J. Mod. Opt.55, 3159–3171 (2008).
[CrossRef]

Miles, J.

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, “Refractive index enhancement with vanishing absorption in short, high-density vapor cells,” Phys. Rev. A85, 053810 (2012).
[CrossRef]

Newell, R.

Newell, R. T. R.

J. Sebby-Strabley, R. T. R. Newell, J. O. Day, E. Brekke, and T. G. Walker, “High-density mesoscopic atom clouds in a holographic atom trap,” Phys. Rev. A71, 021401(R) (2005).
[CrossRef]

O’Brien, C.

C. O’Brien and O. Kocharovskaya, “Optically controllable photonic structures with zero absorption,” Phys. Rev. Lett.107, 137401 (2011).
[CrossRef]

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A84, 063835 (2011).
[CrossRef]

C. O’Brien and O. Kocharovskaya, “Resonant enhancement of refractive index in transition element doped crystals via coherent control of excited state absorption,” J. Mod. Opt.56, 1933–1940 (2009).
[CrossRef]

Proite, N. A.

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, “Refractive index enhancement with vanishing absorption in short, high-density vapor cells,” Phys. Rev. A85, 053810 (2012).
[CrossRef]

Rostovtsev, Y.

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A84, 063835 (2011).
[CrossRef]

Scully, M. O.

M. Fleischhauer, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich, and S. Y. Zhu, “Resonantly enhanced refractive index without absorption via atomic coherence,” Phys. Rev. A461468–1487 (1992).
[CrossRef] [PubMed]

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett.67, 1855–1858 (1991).
[CrossRef] [PubMed]

Sebby, J.

Sebby-Strabley, J.

J. Sebby-Strabley, R. T. R. Newell, J. O. Day, E. Brekke, and T. G. Walker, “High-density mesoscopic atom clouds in a holographic atom trap,” Phys. Rev. A71, 021401(R) (2005).
[CrossRef]

Sikes, D. E.

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, “Refractive index enhancement with vanishing absorption in short, high-density vapor cells,” Phys. Rev. A85, 053810 (2012).
[CrossRef]

Simmons, Z. J.

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, “Refractive index enhancement with vanishing absorption in short, high-density vapor cells,” Phys. Rev. A85, 053810 (2012).
[CrossRef]

Su, C.

M. Fleischhauer, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich, and S. Y. Zhu, “Resonantly enhanced refractive index without absorption via atomic coherence,” Phys. Rev. A461468–1487 (1992).
[CrossRef] [PubMed]

Ulrich, B. T.

M. Fleischhauer, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich, and S. Y. Zhu, “Resonantly enhanced refractive index without absorption via atomic coherence,” Phys. Rev. A461468–1487 (1992).
[CrossRef] [PubMed]

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature424, 839–846 (2003).
[CrossRef] [PubMed]

Walker, T. G.

J. Sebby-Strabley, R. T. R. Newell, J. O. Day, E. Brekke, and T. G. Walker, “High-density mesoscopic atom clouds in a holographic atom trap,” Phys. Rev. A71, 021401(R) (2005).
[CrossRef]

R. Newell, J. Sebby, and T. G. Walker, “Dense atom clouds in a holographic atom trap,” Opt. Lett.28, 1266–1268 (2003).
[CrossRef] [PubMed]

Wang, S.

S. Wang, “Principles of distributed feedback and distributed Bragg-reflector lasers,” Quantum Electron.10(4), 413–427 (1974).
[CrossRef]

Wiese, W. L.

W. L. Wiese and J. R. Fuhra, “Accurate atomic transition probabilities for hydrogen, helium, and lithium,” J. Phys. Chem. Ref. Data38, 565–715 (2009).
[CrossRef]

Yavuz, D. D.

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, “Refractive index enhancement with vanishing absorption in short, high-density vapor cells,” Phys. Rev. A85, 053810 (2012).
[CrossRef]

D. D. Yavuz, “Refractive index enhancement in a far-off resonant atomic system,” Phys. Rev. Lett.95, 223601 (2005).
[CrossRef] [PubMed]

Zhu, S. Y.

M. Fleischhauer, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich, and S. Y. Zhu, “Resonantly enhanced refractive index without absorption via atomic coherence,” Phys. Rev. A461468–1487 (1992).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

C. O’Brien and O. Kocharovskaya, “Resonant enhancement of refractive index in transition element doped crystals via coherent control of excited state absorption,” J. Mod. Opt.56, 1933–1940 (2009).
[CrossRef]

P. Anisimov and O. Kocharovskaya, “Decaying-dressed-state analysis of a coherently driven three-level Λsystem,” J. Mod. Opt.55, 3159–3171 (2008).
[CrossRef]

J. Phys. Chem. Ref. Data (1)

W. L. Wiese and J. R. Fuhra, “Accurate atomic transition probabilities for hydrogen, helium, and lithium,” J. Phys. Chem. Ref. Data38, 565–715 (2009).
[CrossRef]

Nature (1)

K. J. Vahala, “Optical microcavities,” Nature424, 839–846 (2003).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (4)

J. Sebby-Strabley, R. T. R. Newell, J. O. Day, E. Brekke, and T. G. Walker, “High-density mesoscopic atom clouds in a holographic atom trap,” Phys. Rev. A71, 021401(R) (2005).
[CrossRef]

M. Fleischhauer, C. H. Keitel, M. O. Scully, C. Su, B. T. Ulrich, and S. Y. Zhu, “Resonantly enhanced refractive index without absorption via atomic coherence,” Phys. Rev. A461468–1487 (1992).
[CrossRef] [PubMed]

C. O’Brien, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A84, 063835 (2011).
[CrossRef]

Z. J. Simmons, N. A. Proite, J. Miles, D. E. Sikes, and D. D. Yavuz, “Refractive index enhancement with vanishing absorption in short, high-density vapor cells,” Phys. Rev. A85, 053810 (2012).
[CrossRef]

Phys. Rev. Lett. (4)

C. O’Brien and O. Kocharovskaya, “Optically controllable photonic structures with zero absorption,” Phys. Rev. Lett.107, 137401 (2011).
[CrossRef]

D. D. Yavuz, “Refractive index enhancement in a far-off resonant atomic system,” Phys. Rev. Lett.95, 223601 (2005).
[CrossRef] [PubMed]

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett.67, 1855–1858 (1991).
[CrossRef] [PubMed]

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett.70, 1421–1424 (1993).
[CrossRef] [PubMed]

Quantum Electron. (1)

S. Wang, “Principles of distributed feedback and distributed Bragg-reflector lasers,” Quantum Electron.10(4), 413–427 (1974).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Level structure of transition 3S13P0,2. A coupling light couples all the transitions in 3S13P2 which satisfy Δm = 1, with detuning of Δc. A control light couples |3S1, m = 0〉 → |3P0, m = 0〉, with detuning of Δct. A probe light couples all transitions of 3S13P2 which satisfy Δm = 0 with detuning Δp.

Fig. 2
Fig. 2

Both of the Λ-systems can be regarded as a Ξ-system in the dressed picture. (a) The equivalent Ξ-system of the first Λ-system in the dressed state picture. |4′〉 (or |2′〉) is shifted by Ω c , 42 2 / Δ c (or Δ c Ω c , 42 2 / Δ c) compared to |4〉 in the bare state picture. |1′〉 is shifted by Ω ct 2 / Δ ct Ω c , 61 2 / Δ c by the coupling |1〉 → |8〉 and |1〉 → |6〉. (b) The equivalent Ξ-system of the second Λ-system in the dressed state picture. |6′〉 (or |3′〉) is shifted by Ω c , 61 2 / Δ c (or Δ c Ω c , 61 2 / Δ c Ω c , ct 2 / Δ ct) compared to |6〉 in the bare state picture. |3′〉 is shifted by Ω c , 73 2 / Δ c by the coupling |3〉 → |7〉 [11].

Fig. 3
Fig. 3

The effective Ξ-system. The probe light couples the two transitions of nearly the same frequency.

Fig. 4
Fig. 4

A possible experimental scheme of the Bragg reflector. The polarization of the signal and control light are set to be π and the polarization of the control light is set to be σ+ to ensure they couple the transitions stated above.

Fig. 5
Fig. 5

χ calculated by numerical calculation (blue) and by Eq. (1) (red). The real part (dispersion) and the imaginary part (absorption) are illustrated by solid and dashed curve, respectively. It can be seen that the absorption is very small and the dispersion is modulated periodically along the x-axis. |Im(χ)| < 0.006. x is normalized by 1/k, where k is the wavevector of the control light. Parameters are set as Ω c , 61 = 3 Ω c , 42, Ω c , 73 = 6 Ω c , 42, Δc = 100γ1, Ω ct = 2 γ 1 cos ( k x ) and Δc = 50γ1.

Fig. 6
Fig. 6

The frequency of the probe light is ωp = ω2′1′. (a) The relation between |ρ64/ρ41| and Δc. (b)The relation between |ρ64/ρ41| and Ωc,42.(c)The relation between |ρ84/ρ41| and Δct. (d)The relation between |ρ84/ρ41| and Ωct.

Equations (6)

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χ 1 = N | μ 41 | 2 ε 0 h ¯ [ ξ 1 ( ρ 1 ρ 2 ) ω 2 1 ω p + i γ 2 1 + ρ 1 ω 4 1 ω p + i γ 4 1 ] ,
χ 2 = N | μ 63 | 2 ε 0 h ¯ [ ξ 2 ( ρ 3 ρ 1 ) ω 1 3 ω p + i γ 1 3 + ρ 3 ω 6 3 ω p + i γ 6 3 ] ,
χ 3 = N | μ 5 2 | 2 ε 0 h ¯ ρ 2 ω 5 2 ω p + i γ 1 .
χ C = N ε 0 h ¯ ( | μ 52 | 2 ρ 2 + | μ 63 | 2 ρ 3 + | μ 41 | 2 ρ 1 Δ p ) ,
χ = N ε 0 h ¯ [ | μ 41 | 2 ξ 1 ( ρ 1 ρ 2 ) ω 2 1 ω p + i γ 2 1 + | μ 63 | 2 ξ 2 ( ρ 3 ρ 1 ) ω 1 3 ω p + i γ 1 3 ] + χ C .
ρ ˙ 41 = ( γ 1 + i Δ p ) ρ 41 i [ Ω p , 41 ( ρ 44 ρ 11 ) Ω c , 42 ρ 21 + Ω c . 61 ρ 64 + Ω ct ρ 48 ] ρ ˙ 63 = ( γ 1 + i Δ p ) ρ 63 i [ Ω p , 63 ( ρ 66 ρ 33 ) Ω c , 61 ρ 13 + Ω c , 73 ρ 67 ] .

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