This Demonstration considers a pulse incident on an atomic frequency comb (AFC) [1] in some material, such as a rare-earth ion doped crystal with characteristics similar to that in [2]. The model here is simplified by neglecting any hyperfine splitting of levels.

The dielectric is modeled using the Clausius–Mossotti relation

,

where

is the number density of oscillators in the crystal,

is the polarizability of the dopant ions of class

and

is a frequency detuning from a typical value. The sum over

denotes different classes of oscillators in the crystal. In this case, although there is only one impurity ion, each ion's transition frequency depends on its local environment. This is called "inhomogeneous broadening." The continuum approximation is used to evaluate the sum,

,

where

denotes the density of states of the ground state and

represents the polarization of each ion in the crystal. Assuming that the density of states is a sum of Gaussians, as is the case in a crystal with inhomogeneous band structure:

.

The polarization around each ion is given by the Drude–Lorentz model

.

Since the dielectric function is expected to be close to 1, the Clausius–Mossotti relation can be approximated as:

.

Taking the

* *term in

allows evaluating

, which represents a single tooth in the AFC. For the

tooth in the comb we substitute

for

. The finesse of the AFC is defined as the ratio of free spectral range

to the full width at half maximum (FWHM) for each tooth. You can vary

, leaving

constant, to change the finesse of the comb. The FWHM is

. In [3], it was predicted that the number of photon echoes in the pulse train is linearly proportional to the finesse of the comb.

To model the pulse incident on the comb, assume the pulse is a Gaussian with a standard deviation much greater than the tooth spacing in the comb, but less than the entire width of the comb. The response of the pulse as a function of time is then given by:

where

denotes the inverse Fourier transform (

to

),

is the time,

denotes the initial pulse in the frequency domain,

* *is the wavevector of the undoped crystal,

, where

is the (real) index of refraction of the undoped crystal,

* *is the free-space wavelength and

* *is the propagation distance in the crystal.

You can see that the number of pulses increases as the finesse increases, although not necessarily in strict accordance with the linear rule suggested by Wiener and Leaird [3]. This Demonstration illustrates the phenomenon of photon echoes using classical electromagnetism. The model is similar in spirit to the quantum model of [1], although one qualitative difference is that [1] predicts that the second output pulse can be more intense than the first, but we observe the pulse intensities are always monotonically decreasing in time.

[1] M. Afzelius, C. Simon, H. de Riedmatten and N. Gisin, "Multimode Quantum Memory Based on Atomic Frequency Combs,"

*Physical Review A*,

**79**, 2009 052329.

doi:10.1103/PhysRevA.79.052329.

[2] A. N. Sharma, M. A. Ritter, K. H. Kagalwala, Z. H. Levine, E. J. Weissler, E. A. Goldschmidt and A. L. Migdall, "Effect of Hyperfine Structure on Atomic Frequency Combs in Pr:YSO."

arxiv.org/abs/2011.04086.

[3] A. M. Weiner and D. E. Leaird, "Generation of Terahertz-Rate Trains of Femtosecond Pulses by Phase-Only Filtering,"

*Optics Letters*,

**15**(1), 1990 pp. 51–53.

doi:10.1364/OL.15.000051.