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.

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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. W. 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 (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]

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]

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

2005 (2)

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 (2)

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

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

2002 (1)

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 (1)

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

1996 (1)

1993 (1)

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 (1)

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

1969 (1)

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

1967 (1)

W. 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.

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.

W. 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.

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]

Goodman, J. W.

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

Imamoglu, A.

Imamoglu A, A.

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

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.

W. 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]

Marangos, J. P.

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

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.

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. (1)

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

J. Phys. B (2)

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]

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

Nature (1)

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. (3)

Phys. Rev. A (4)

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]

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]

Phys. Rev. Lett. (2)

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. (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]

Trans. Sonics Ultrason. (1)

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

Other (2)

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)

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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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