## Electromagnetically-induced phase grating: A coupled-wave theory analysis |

Optics Express, Vol. 19, Issue 3, pp. 1936-1944 (2011)

http://dx.doi.org/10.1364/OE.19.001936

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### 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

## 1. Introduction

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]

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]

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

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

*ℓ*≈ 160

*z*

_{0}, where

*N*is the atomic density,

*μ*is the electric dipole moment, and

_{ac}*γ*is the natural linewidth of the atomic transition. For

_{c}*N*= 10

^{18}m

^{−3},

*μ*= 2.49 ×10

_{ac}^{−29}Cm and

*γ*= 2

_{c}*π*× 9.8 MHz, then

*ℓ*≈ 2 mm, which is three orders of magnitude larger than the grating period Λ ≈ 2

*μ*m for a probe wavelength

*λ*= 589 nm, suggesting the grating is thick.

*Q*= 2

*πλℓ*/Λ

^{2}, is greater than 10. In the atomic grating of Ref. [9

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

*Q*≈ 1850. Therefore, that grating, in fact, can be considered thick. It should show Bragg behavior and produce only one diffracted beam for Bragg angle incidence. Therefore, even higher diffraction efficiencies than originally predicted in Ref. [9

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

**35**, 977–979 (2010). [CrossRef] [PubMed]

## 2. Atomic model for an electromagnetically-induced phase grating

*N*four-level atom being driven by three cw lasers. Levels |

*c*〉 and |

*d*〉 are excited states that decay at rates

*γ*and

_{c}*γ*, respectively. Level |

_{d}*a*〉 is the ground state and |

*b*〉 is a metastable state with negligible decay rate (

*γ*≈ 0). These levels could be, for example, two hyperfine-split states of an alkaline atom. Levels |

_{b}*a*〉 and |

*c*〉 are connected by a probe beam with Rabi frequency Ω

*and wavelength*

_{p}*λ*, while the |

*b*〉 → |

*c*〉 transition is driven by a coupling beam (Rabi frequency Ω

*). Both coupling and probe beams are resonant with their respective transitions. The signal beam (Ω) is detuned from the |*

_{c}*b*〉 → |

*d*〉 transition by

*δ*=

*ω*–

_{bd}*ω*, where

*ω*is the atomic transition frequency, and

_{bd}*ω*is the signal optical frequency. We consider a homogeneously broadened medium.

*a*〉, |

*b*〉, and |

*c*〉. In the absence of the signal field, if

*γ*> Ω

_{c}*≫ Ω*

_{c}*, two indistinguishable pathways to probe absorption are created that interfere destructively. Probe absorption is canceled, and the atomic medium becomes transparent to the probe field [1*

_{p}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]

*b*〉 suffers an ac-Stark shift, which, because of the steep dispersion, leads to a large change in the index of refraction at the probe frequency [1

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]

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

*ω*is

_{p}*P*(

*ω*) =

_{p}*ɛ*

_{0}

*χ*(

*ω*)

_{p}*E*(

_{p}*ω*), where [9

_{p}**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]

*or*

_{c}*δ*. The nonlinear atomic susceptibility

*χ*is a function of two externally controllable parameters: the signal detuning

*δ*and the ratio of signal to coupling Rabi frequencies Ω/Ω

*.*

_{c}*δ*/

*γ*,

_{c}*R*= Ω/Ω

*, and Γ =*

_{c}*γ*/

_{d}*γ*. In the limit of large signal detunings (Δ ≫

_{c}*R*, Γ), Eqs. (1) become: where

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

*R*≪ 1, this excitation scheme is capable of yielding giant Kerr nonlinearities while keeping the linear susceptibilities identically zero for all fields. The XPM phase shift induced on the probe by the signal field is

*φ*= (

*πℓ*/

*λ*)Re[

*χ*], where

*ℓ*is the medium length. The scheme is limited only by the small nonlinear absorption given by

*α*= (2

*π*/

*λ*)Im[

*χ*], and long medium lengths are then possible. Phase shifts of the order of

*π*with single photons in the signal field were proposed. But large nonlinearities at low light levels are also possible for short medium lengths if

*R*≳ 1 as was experimentally observed in cold Rb atoms in a magneto-optical trap under this same excitation scheme [15

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]

*x*direction. The signal Rabi frequency becomes: where

*π*/Λ is the spatial frequency of the standing wave modulation, which is controlled by the angle at which the two signal beams intersect. Throughout our analysis, we will consider Λ = 4

*λ*, without loss of generality. The signal-to-coupling Rabi frequency ratio

*R*also becomes modulated at this same spatial frequency. As a result, a spatial modulation is introduced to the atomic susceptibility:

*χ*=

*χ*(

*x*), creating a grating on which the probe beam can diffract.

## 3. Coupled wave analysis

*S*

_{0}(

*z*) and a first-order diffracted mode

*S*

_{1}(

*z*): where

*ρ⃗*

_{0,1}are the propagation vector of the zeroth (first) diffracted order, respectively. The two waves exchange energy as they propagate inside the atomic sample, and their complex amplitudes

*S*

_{0,1}vary along

*z*.

*x*and

*z*axes are in the plane of incidence and the

*y*-axis is perpendicular to the paper. The grating vector

**K**is oriented along the

*x*-axis. Although the signal standing wave is modulated at a spacial frequency equal to

*π*/Λ, the atomic susceptibility (and consequently, the grating) is modulated at twice that frequency as can be seen by making use of the trigonometric identity sin

^{2}

*u*= (1 – cos 2

*u*)/2 in Eq. (5). Therefore, the length of the grating vector is

*K*= 2

*π*/Λ. The probe beam enters the grating at an angle

*θ*, with respect to the

*z*axis, at or near Bragg incidence. The probe field is polarized perpendicularly to the plane of incidence, along the

*y*-axis. Figure 2b shows the propagation vectors

*ρ⃗*

_{0,1}, defined as and where

*β*= 2

*π*/

*λ*is the free-propagation wave number. Forced by the grating, the two vectors satisfy For incidence at the Bragg angle

*θ*, where the length of the two vectors will be equal to the free propagation constant:

_{B}*ρ*

_{0}=

*ρ*

_{1}=

*β*. For Λ = 4

*λ*, the Bragg angle is

*θ*≈ 7.2°.

_{B}*E*(

*x*,

*z*) satisfies the scalar wave equation The coupled wave equations for the

*S*

_{0}and

*S*

_{1}fields are obtained by substituting Eq. (6) into (11), combined with (9). We then compare terms with equal exponentials (

*e*

^{iρ⃗0·x⃗}and

*e*

^{iρ⃗1·x⃗}). Neglecting terms in

*ρ*⃗

_{0}+

*K*⃗,

*ρ*⃗

_{1}–

*K*⃗ and the other higher diffraction orders, after much algebra, we find where

*κ*=

*σ*/4

*z*

_{0},

*ψ*

_{1}= (

*α*

_{2}+ 3

*α*

_{4}/4)/2

*z*

_{0}and

*ψ*

_{2}= (

*α*

_{2}+

*α*

_{4})/4

*z*

_{0}.

*A*

_{0,1}and

*B*

_{0,1}are constants. Substituting Eqs. (15) and (16) into the coupled wave equations, we obtain the wave numbers with the plus sign corresponding to

*ξ*

_{0}and the minus sign to

*ξ*

_{1}.

*A*

_{0,1}and

*B*

_{0,1}, we need to specify the boundary conditions. We take the amplitude of the incident wave to be unity

*S*

_{0}(0) = 1 and that of the diffracted wave to be zero

*S*

_{1}(0) = 0 at

*z*= 0. From these conditions, we find At the Bragg angle,

*A*

_{1}= –

*B*

_{1}= −1/2.

*z*=

*ℓ*) of the atomic sample: This result is valid for an angle of incidence at or near the Bragg angle. Defining the diffraction efficiency as it is straightforward to substitute Eq. (19) into (20) to find

*η*. At Bragg incidence,

**35**, 977–979 (2010). [CrossRef] [PubMed]

*σ*. The effect of the amplitude grating is described by the second term, dependent on the nonlinear absorption through

*α*

_{2}and

*α*

_{4}. The total diffracted intensity is a simple addition of the intensities diffracted by the phase and the amplitude gratings. The exponential, which depends on the coefficients

*α*

_{2}and

*α*

_{4}, is an absorption term that limits the total diffraction efficiency, and it insures that Eq. (21) does not yield an efficiency larger than 1. However, as we will discuss in the next section, the phase grating contribution to diffraction is much larger than the amplitude grating contribution.

## 4. Numerical results and discussion

*λ*= 589 nm as an example of an atomic system suitable for implementing the EIG under consideration here. In this case,

*γ*=

_{d}*γ*= 2

_{c}*π*× 9.8 MHz,

*μ*= 2.49 × 10

_{ac}^{−29}Cm. We also take

*N*= 10

^{12}cm

^{−3}.

*ℓ*is given in units of the linear absorption length

*z*

_{0}. Bragg incidence is assumed, so

*η*is calculated from Eq. (21). The diffraction efficiency varies with the medium length as the signal-probe XPM phase shift changes. For a signal detuning Δ = 140, a peak diffraction efficiency of approximately 70% at

*ℓ*≈ 153

*z*

_{0}is predicted. This efficiency is significantly higher than the 30% efficiency reported in the Raman-Nath regime in Ref. [9

**35**, 977–979 (2010). [CrossRef] [PubMed]

*η*≈ 80%, at the expense of a longer medium. The small nonlinear absorption prevents the grating from reaching a 100% diffraction efficiency into the first order.

*ℓ*= 153

*z*

_{0}and Δ = 140, then

*σℓ*/8

*z*

_{0}cos

*θ*≈

*π*/2 and (

*α*

_{2}+

*α*

_{4})

*ℓ*/8

*z*

_{0}cos

*θ*≈ 0.12. As a result, the diffracted intensity from the phase component of the atomic grating is exp[−(

*α*

_{2}+ 3

*α*

_{4}/4)

*L*/2

*z*

_{0}cos

*θ*]sin

^{2}[

*σℓ*/8

*z*

_{0}cos

*θ*] ≈ 0.700, while that of the amplitude component is exp[−(

*α*

_{2}+ 3

*α*

_{4}/4)

*ℓ*/2

*z*

_{0}cos

*θ*]sinh

^{2}[(

*α*

_{2}+

*α*

_{4})

*ℓ*/8

*z*

_{0}cos

*θ*] ≈ 0.009. The small contribution of the latter is because, although the imaginary part of the atomic susceptibility is also spatially modulated, its amplitude is very small. The reason is that Im[

*χ*] ∝ 1/Δ

^{2}, while Re[

*χ*] ∝ 1/Δ. For a large enough signal detuning, the contribution of the amplitude modulation can be significantly reduced, while that of the phase modulation is kept relevant, particularly at large

*ℓ*. As we previously pointed out, this atomic grating is mostly a phase grating.

*R*. An optimum ratio appears to exist around

*R*= 4.6 that maximizes the diffraction efficiency. A secondary ratio also exists that maximizes the efficiency, but at a much smaller value. A similar result was also observed in the Raman-Nath regime [9

**35**, 977–979 (2010). [CrossRef] [PubMed]

*η*[terms within braces in Eq. (21)], while the dashed blue line shows only the absorption component [exponential in Eq. (21)]. The diffraction efficiency plot in Figure 4a is obtained by multiplying the two curves in Fig. 4b. Without absorption, there are several values of

*R*for which

*σL*/8

*z*

_{0}cos

*θ*=

*mπ*/2 (where

*m*= 1, 3, 5, …) and that maximizes

*η*. The non-sinusoidal shape of the curve is due to the

*R*

^{2}dependence of

*σ*. But absorption of the probe beam (both zeroth and first order waves) also increases with

*R*. This is because if the signal field becomes too strong with respect to the coupling field, the atom is shifted out of the EIT condition, causing it to absorb the probe beam. At

*R*≈ 4.6, the signal beam is such that it optimizes the XMP phase shift to maximize diffraction without introducing a significant amount of absorption. But at

*R*≈ 8, when the second maximum in the diffraction efficiency occurs, absorption is severe, limiting the efficiency.

*|*

_{c}^{2}>

*γ*. In the limit that the ground state decoherence

_{c}γ_{b}*γ*≈ 0, the coupling field can be made arbitrarily weak. From the results shown above, Ω ≳ Ω

_{b}*. So the atomic phase grating can be created with very weak signal fields. In a real atomic sample, such as those in a magneto-optical trap,*

_{c}*γ*≈ 2

_{b}*π*× 1 KHz. Therefore, a grating created with a signal field with Rabi frequency Ω > 2

*π*× 455 KHz, will efficiently diffract a weak probe beam. This signal level is well below saturation level and even below the typical linewidth of lasers used in many EIT experiments, which would ultimately limit Ω.

*θ*=

*θ*–

_{B}*θ*from the Bragg angle. It is seen that the grating is very sensitive to the angle of incidence; with a deviation of less than 2 mrad from Bragg incidence, the efficiency drops to half its value at Bragg angle. From [14], the acceptance angle (full width at half maximum) of a thick grating can be estimated as 2Δ

*θ*≈ Λ/

*ℓ*. Because our grating thickness

*ℓ*is very much larger than its period Λ, its acceptance angle is very small. However, the angle between the two signal beams that form the atomic grating may be adjusted to increase the grating period, increasing the acceptance angle. The Bragg angle will decrease, but because cos

*θ*≈ 1, the diffraction efficiency [Eq.(21)] will not be affected.

_{B}## 5. Conclusion

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.

## Acknowledgments

## References and links

1. | M. Fleischhauer, A. Imamoglu A, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. |

2. | A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. |

3. | M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature |

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 |

5. | H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A |

6. | M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A |

7. | G. C. Cardoso and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A |

8. | J. W. Goodman, |

9. | L. E. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. |

10. | H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced tranparency,” Opt. Lett. |

11. | Z.-H. Xiao, S. G. Shin, and K. Kim, “An electromagnetically induced grating by microwave modulation,” J. Phys. B |

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 |

13. | W. R. Klein and B. D. Cook, “Unified approach to ultrasonic light diffraction,” Trans. Sonics Ultrason. |

14. | H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. |

15. | H. S. Kang and Y. F. Zhu, “Observation of Large Kerr Nonlinearity at Low Light Intensities,” Phys. Rev. Lett. |

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. |

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. |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(050.2770) Diffraction and gratings : Gratings

(270.1670) Quantum optics : Coherent optical effects

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: December 1, 2010

Revised Manuscript: January 12, 2011

Manuscript Accepted: January 12, 2011

Published: January 18, 2011

**Citation**

Silvania A. de Carvalho and Luis E. E. de Araujo, "Electromagnetically-induced phase grating: A coupled-wave theory analysis," Opt. Express **19**, 1936-1944 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-1936

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### References

- 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]
- M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638–641 (2003). [CrossRef] [PubMed]
- 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]
- 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]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
- 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]
- 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]
- Q1W. R. Klein, and B. D. Cook, “Unified approach to ultrasonic light diffraction,” Trans. Sonics Ultrason. SU14, 123 (1967).
- H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
- 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]
- 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.

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