Abstract

We use a coupled-wave theory analysis to describe an atomic phase grating based on the giant Kerr nonlinearity of an atomic medium under electromagnetically induced transparency. An analytical expression is found for the diffraction efficiency of the grating. Efficiencies greater than 70% are predicted for incidence at the Bragg angle.

© 2011 Optical Society of America

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  1. M. Fleischhauer, “A. Imamoglu A and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
    [CrossRef]
  2. A. W. Brown, and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
    [CrossRef] [PubMed]
  3. M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
    [CrossRef] [PubMed]
  4. D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
    [CrossRef]
  5. H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
    [CrossRef]
  6. M. Mitsunaga, and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1990).
    [CrossRef]
  7. G. C. Cardoso, and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A 65, 033803 (2002).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. L. E. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010).
    [CrossRef] [PubMed]
  10. H. Schmidt, and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced tranparency,” Opt. Lett. 21, 1936–1938 (1996).
    [CrossRef] [PubMed]
  11. Z.-H. Xiao, S. G. Shin, and K. Kim, “An electromagnetically induced grating by microwave modulation,” J. Phys. B 43, 161004 (2010).
    [CrossRef]
  12. L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
    [CrossRef]
  13. Q1W. R. Klein, and B. D. Cook, “Unified approach to ultrasonic light diffraction,” Trans. Sonics Ultrason. SU14, 123 (1967).
  14. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  15. H. S. Kang, and Y. F. Zhu, “Observation of Large Kerr Nonlinearity at Low Light Intensities,” Phys. Rev. Lett. 91, 093601 (2003).
    [CrossRef] [PubMed]
  16. W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
    [CrossRef] [PubMed]
  17. S. Magkiriadou, D. Patterson, T. Nicolas, and J. M. Doyle, “Cold, Optically Dense Gases of Atomic Rubidium,” http://www.doylegroup.harvard.edu/wiki/index.php/Publications.

2010

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[CrossRef]

L. E. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010).
[CrossRef] [PubMed]

Z.-H. Xiao, S. G. Shin, and K. Kim, “An electromagnetically induced grating by microwave modulation,” J. Phys. B 43, 161004 (2010).
[CrossRef]

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[CrossRef]

2005

M. Fleischhauer, “A. Imamoglu A and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[CrossRef]

A. W. Brown, and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
[CrossRef] [PubMed]

2003

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[CrossRef] [PubMed]

H. S. Kang, and Y. F. Zhu, “Observation of Large Kerr Nonlinearity at Low Light Intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

2002

G. C. Cardoso, and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A 65, 033803 (2002).
[CrossRef]

1998

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[CrossRef]

1996

H. Schmidt, and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced tranparency,” Opt. Lett. 21, 1936–1938 (1996).
[CrossRef] [PubMed]

1993

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

1990

M. Mitsunaga, and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1990).
[CrossRef]

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1967

Q1W. R. Klein, and B. D. Cook, “Unified approach to ultrasonic light diffraction,” Trans. Sonics Ultrason. SU14, 123 (1967).

Bajcsy, M.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[CrossRef] [PubMed]

Brown, A. W.

A. W. Brown, and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
[CrossRef] [PubMed]

Cardoso, G. C.

G. C. Cardoso, and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A 65, 033803 (2002).
[CrossRef]

Cook, B. D.

Q1W. R. Klein, and B. D. Cook, “Unified approach to ultrasonic light diffraction,” Trans. Sonics Ultrason. SU14, 123 (1967).

Davis, K. B.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

de Araujo, L. E. E.

L. E. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010).
[CrossRef] [PubMed]

Duan, W.

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[CrossRef]

Felinto, D.

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer, “A. Imamoglu A and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[CrossRef]

Imamoglu, A.

H. Schmidt, and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced tranparency,” Opt. Lett. 21, 1936–1938 (1996).
[CrossRef] [PubMed]

Imoto, N.

M. Mitsunaga, and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1990).
[CrossRef]

Joffe, M. A.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

Kang, H. S.

H. S. Kang, and Y. F. Zhu, “Observation of Large Kerr Nonlinearity at Low Light Intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

Ketterle, W.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

Kim, K.

Z.-H. Xiao, S. G. Shin, and K. Kim, “An electromagnetically induced grating by microwave modulation,” J. Phys. B 43, 161004 (2010).
[CrossRef]

Klein, W. R.

Q1W. R. Klein, and B. D. Cook, “Unified approach to ultrasonic light diffraction,” Trans. Sonics Ultrason. SU14, 123 (1967).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Lezama, A.

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[CrossRef]

Li, Y.-Q.

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[CrossRef]

Ling, H. Y.

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[CrossRef]

Lukin, M. D.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[CrossRef] [PubMed]

Martin, A.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

Mitsunaga, M.

M. Mitsunaga, and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1990).
[CrossRef]

Moretti, D.

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[CrossRef]

Pritchard, D. E.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

Schmidt, H.

H. Schmidt, and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced tranparency,” Opt. Lett. 21, 1936–1938 (1996).
[CrossRef] [PubMed]

Shin, S. G.

Z.-H. Xiao, S. G. Shin, and K. Kim, “An electromagnetically induced grating by microwave modulation,” J. Phys. B 43, 161004 (2010).
[CrossRef]

Tabosa, J. W. R.

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[CrossRef]

G. C. Cardoso, and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A 65, 033803 (2002).
[CrossRef]

Xiao, M.

A. W. Brown, and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
[CrossRef] [PubMed]

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[CrossRef]

Xiao, Z.-H.

Z.-H. Xiao, S. G. Shin, and K. Kim, “An electromagnetically induced grating by microwave modulation,” J. Phys. B 43, 161004 (2010).
[CrossRef]

Yelin, S. F.

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[CrossRef]

Zhao, L.

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[CrossRef]

Zhu, Y. F.

H. S. Kang, and Y. F. Zhu, “Observation of Large Kerr Nonlinearity at Low Light Intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

Zibrov, A. S.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[CrossRef] [PubMed]

Bell Syst. Tech. J.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

J. Phys. B

Z.-H. Xiao, S. G. Shin, and K. Kim, “An electromagnetically induced grating by microwave modulation,” J. Phys. B 43, 161004 (2010).
[CrossRef]

D. Moretti, D. Felinto, J. W. R. Tabosa, and A. Lezama, “Dynamics of a stored Zeeman coherence grating in an external magnetic field,” J. Phys. B 43, 115502 (2010).
[CrossRef]

Nature

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003).
[CrossRef] [PubMed]

Opt. Lett.

A. W. Brown, and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30, 699–701 (2005).
[CrossRef] [PubMed]

L. E. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35, 977–979 (2010).
[CrossRef] [PubMed]

H. Schmidt, and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced tranparency,” Opt. Lett. 21, 1936–1938 (1996).
[CrossRef] [PubMed]

Phys. Rev. A

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57, 1338–1344 (1998).
[CrossRef]

M. Mitsunaga, and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59, 4773–4776 (1990).
[CrossRef]

G. C. Cardoso, and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A 65, 033803 (2002).
[CrossRef]

L. Zhao, W. Duan, and S. F. Yelin, “All-optical beam control with high speed using image-induced blazed gratings in coherent media,” Phys. Rev. A 82, 013809 (2010).
[CrossRef]

Phys. Rev. Lett.

H. S. Kang, and Y. F. Zhu, “Observation of Large Kerr Nonlinearity at Low Light Intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[CrossRef] [PubMed]

Rev. Mod. Phys.

M. Fleischhauer, “A. Imamoglu A and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[CrossRef]

Trans. Sonics Ultrason.

Q1W. R. Klein, and B. D. Cook, “Unified approach to ultrasonic light diffraction,” Trans. Sonics Ultrason. SU14, 123 (1967).

Other

S. Magkiriadou, D. Patterson, T. Nicolas, and J. M. Doyle, “Cold, Optically Dense Gases of Atomic Rubidium,” http://www.doylegroup.harvard.edu/wiki/index.php/Publications.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (5)

Fig. 1
Fig. 1

(a) The atomic model: An open four-level atom interacting with three laser beams: probe (Ωp), coupling (Ωc) and signal (Ω). (b) Sketch of the probe- and signal-beam spatial configuration with respect to the atomic sample showing the zeroth and first diffraction orders. The coupling beam (not shown) is parallel to the incident probe beam.

Fig. 2
Fig. 2

(a) Illustration of the thick atomic grating showing the probe’s angle of incidence θ, the grating vector K⃗, the grating period Λ, and the grating thickness . (b) Vector diagram showing the relation between the zeroth- and first-order propagation vectors and the grating vector.

Fig. 3
Fig. 3

The diffraction efficiency η as a function of medium length , in units of z0. A signal detuning Δ = 140 corresponds to approximately 2π × 1.4 GHz in Na. The signal-to-coupling Rabi frequency ratio was set to R = 4.6.

Fig. 4
Fig. 4

(a) The first-order diffraction efficiency η as a function of the ratio of signal-to-coupling Rabi frequencies R for = 135z0, Δ = 140γc. (b) Absorption (dashed blue line) and modulation (solid red line) components of η as a function of R.

Fig. 5
Fig. 5

Diffraction efficiency η as a function of angular deviation Δθ from Bragg incidence. Here, R = 4.6, Δ = 140 and = 153z0.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

Re [ χ ] = ( 2 N μ a c 2 / h ¯ ɛ 0 ) 2 ( Ω / Ω c ) 2 δ 4 δ 2 + [ γ d + γ c ( Ω / Ω c ) 2 ] 2 , Im [ χ ] = ( 2 N μ a c 2 / h ¯ ɛ 0 ) γ d ( Ω / Ω c ) 2 + γ c ( Ω / Ω c ) 4 4 δ 2 + [ γ d + γ c ( Ω / Ω c ) 2 ] 2 .
Re [ χ ] = A R 2 2 Δ , Im [ χ ] = A Γ R 2 + R 4 4 Δ 2 ,
Ω ( x ) = Ω sin ( π x / Λ ) ,
Re [ χ ] = A σ sin 2 ( π x / Λ ) ,
Im [ χ ] = A α 2 sin 2 ( π x / Λ ) + A α 4 sin 4 ( π x / Λ ) ,
E ( x , z ) = S 0 ( z ) e i ρ 0 x + S 1 ( z ) e i ρ 1 x ,
ρ 0 = ( ρ 0 x 0 ρ 0 z ) = β ( sin θ 0 cos θ )
ρ 1 = ( ρ 1 x 0 ρ 1 z ) = β ( sin θ K / β 0 cos θ ) ,
ρ 1 = ρ 0 K .
sin θ B = K / 2 β = λ / 2 Λ ,
2 E + β 2 ( 1 + χ ) E = 0 .
S 0 + 2 i ρ 0 z S 0 + ( 2 κ β + i β ψ 1 ) S 0 = ( κ β + i β ψ 2 ) S 1 , S 1 + 2 i ρ 1 z S 1 + [ 2 κ β + i β ψ 1 + ( β 2 ρ 1 2 ) ] S 1 = ( κ β + i β ψ 2 ) S 0 ,
( cos θ ) S 0 + α A S 0 = i κ A S 1 ,
( cos θ ) S 1 + ( α A i ϑ ) S 1 = i κ A S 0 .
S 0 ( z ) = A 0 exp ( ξ 0 z ) + β 0 exp ( ξ 1 z ) ,
S 1 ( z ) = A 1 exp ( ξ 0 z ) + B 1 exp ( ξ 1 z ) ,
ξ 0 , 1 = ( i ϑ 2 α A ) ± i ϑ 2 + 4 κ A 2 2 cos θ ,
A 1 = B 1 = κ A ϑ 2 + 4 κ A 2 .
S 1 ( ) = 2 i κ A ϑ 2 + 4 κ A 2 e i ϑ / 2 cos θ e α A / cos θ sin [ ϑ 2 + 4 κ A 2 ( / 2 cos θ ) ] .
η = S 1 S 1 * ,
η = exp [ ( α 2 + 3 α 4 / 4 ) / 2 z 0 cos θ ] { sin 2 [ σ / 8 z 0 cos θ ] + sinh 2 [ ( α 2 + α 4 ) / 8 z 0 cos θ ] } .

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