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arXiv:1805.08725

Frequency crystal

R. N. Shakhmuratov

Published May 21, 2018• 0 citations
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Abstract

Pulse filtering through a medium with infinite periodic structure in transmission spectrum is analyzed. Two types of filters are considered. The first, named harmonic frequency crystal (HFC), is the filter whose widths of transmitting and absorbing windows are equal. The second, named anharmonic frequency crystal (AHFC), has narrow absorption peaks separated by wide transmission windows. AHFC of moderate optical thickness demonstrates properties quite similar to those of high finesse atomic frequency comb (AFC) with limited number of the absorption peaks, which produce a few pulses from a short input pulse. On the contrary, HFC transforms the input pulse into a train of pulses whose maximum amplitudes follow a wide bell-shaped envelope. HFC allows to find an exact universal analytical solution, which describes transformation of a broadband pulse into a train of short pulses, slow light propagation for a pulse whose spectral width fits one of the transparency windows of the crystal, and absorption typical for a single line absorber if the pulse spectrum falls into one of the absorption peaks. Potential applications of HFC are discussed.

DOI
10.48550/arxiv.1805.08725

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Pulse filtering through a medium with infinite periodic structure in transmission spectrum is analyzed. Two types of filters are considered. The first, named harmonic frequency crystal (HFC), is the filter whose widths of transmitting and absorbing windows are equal. The second, named anharmonic frequency crystal (AHFC), has narrow absorption peaks separated by wide transmission windows. AHFC of moderate optical thickness demonstrates properties quite similar to those of high finesse atomic frequency comb (AFC) with limited number of the absorption peaks, which produce a few pulses from a short input pulse. On the contrary, HFC transforms the input pulse into a train of pulses whose maximum amplitudes follow a wide bell-shaped envelope. HFC allows to find an exact universal analytical solution, which describes transformation of a broadband pulse into a train of short pulses, slow light propagation for a pulse whose spectral width fits one of the transparency windows of the crystal, and absorption typical for a single line absorber if the pulse spectrum falls into one of the absorption peaks. Potential applications of HFC are discussed.
Filtering of weak pulses through a medium with infinite periodic structure in the absorption and/or transmission spectrum is analyzed. Two types of filters are considered. The first, named harmonic frequency crystal (HFC), is the filter whose periodic structure is harmonic and described by the sine or cosine function of frequency with the fundamental period. The second, named anharmonic frequency crystal (AHFC), also has a periodic structure with one fundamental period but it is described by the function containing many harmonics of the fundamental period. AHFC demonstrates properties quite similar to those of a high-finesse atomic frequency comb (AFC) with a limited number of absorption peaks separated by transparency windows. Filtering of the pulse through AHFC produces a prompt pulse accompanied by a few delayed pulses. Time spacing TTT of the delayed pulses is inversely proportional to the comb period. On the contrary, HFC transforms the input pulse into a train of many delayed pulses, generated at times nTnTnT, where nnn is an integer. Maximum amplitudes of the delayed pulses follow a wide bell-shaped envelope. Both HFC and AHFC are extensions of AFC, which was proposed to implement quantum memory protocols for single photons. In this paper, it is proposed to create time-bin qubits with the help of a short pulse filtering through HFC. HFC is also proposed to implement tomography of these quantum states. Another promising application of HFC is the optical detection of ultrasound in biological tissues for ultrasound-modulated optical tomography. Since HFC allows one to generate pulses with longer delay with respect to the excitation pulse, one could obtain higher discrimination between weak acoustically generated sidebands and the carrier.
A simple analytical solution for the propagation of a weak Gaussian pulse in a dense absorptive medium with electromagnetically induced transparency is found. This solution is applied to the analysis of three regimes: (1) and (2) the pulse spectrum is narrower than the transparency window [which is narrow (1) or wide (2) with respect to the width of the absorption line] and (3) the pulse spectrum is broader than the transparency window. It is shown that the pulse maintains its area in all three regimes and maintains its Gaussian shape but narrows in spectrum in regime 1. In regime 2, the pulse begins to distort after a certain distance. In regime 3, the pulse is split into two parts. One part is an adiabatic part with a spectrum defined by the effective width of the transparency window for a thick medium and the other is an oscillating nonadiabatic part of short duration. The adiabatic part propagates slowly and the nonadiabatic part propagates with a velocity close to the speed of light. Thus in regime 3, the medium acts as a time-frequency filter, separating the narrow and wide spectrum components of the pulse in time at the output of the absorber.
This report shows that the electromagnetically-induced transparency (EIT) window can be used as a time-to-frequency filter for the probe pulse whose spectrum is much broader than the width of the EIT window. This study demonstrates that the narrow spectral region of the pulse, coincident with the EIT window, develops in a long Gaussian pulse with slow group velocity. An analytical theory is developed describing the propagation of a small amplitude pulse through an optically dense absorptive medium with a narrow EIT window.
The resonant filtering method transforming the frequency-modulated radiation field into a train of short pulses is proposed to be applied in the optical domain. Effective frequency modulation can be achieved by using an electro-optic modulator. Due to frequency modulation, a narrow-spectrum cw radiation field is seen by the resonant filter as a comb of equidistant spectral components separated by the modulation frequency. Tuning a narrow-bandwidth filter in resonance with the nthn\mathrm{th}nth spectral component of the comb transforms the radiation field into bunches of pulses, with nnn pulses in each bunch. The transformation is explained by the interference of the coherently scattered resonant component of the field with the whole comb. Constructive interference results in the formation of pulses, while destructive interference is seen as dark windows between pulses. It is indicated that the optimal thickness of the resonant filter is several orders of magnitude smaller than the necessary thickness of the dispersive filters used before in the optical domain to produce short pulses from the frequency-modulated field.
We have analyzed the transmission properties of pulses through one-dimensional periodic structures in order to systematically explore the best conditions to achieve the maximum delay with the minimum possible distortion. In the absence of absorption and no layer variation, the transmission coefficient tN{t}_{N}tN​ can be well approximated by a sum of Lorentzian resonances. The ratio between their width ${\ensuremath{\Gamma}}_{\mathrm{r}}$ and their separation $\ensuremath{\Delta}{\ensuremath{\omega}}_{\mathrm{r}}$ is a crucial parameter to characterize the distortion of the transmitted pulse. For typical values of the parameters used in telecommunications and high index of refraction contrasts n2/n1{n}_{2}/{n}_{1}n2​/n1​, the distortion of the transmitted pulse is unacceptably large for frequencies near the edge of the transmission window. We estimate fractional delays achievables in terms of the central frequency used and the pulse bandwidth.
The propagation of broadband pulse in a medium with electromagnetically induced transparency (EIT) was studied. The results show that if the carrier frequency of pulse is exactly tuned to the center of the transparency window, and the pulse spectrum is broader than transparency window, the pulse is temporally broadened and the delay effect is significant. If the carrier frequency of pulse the is exactly tuned to the region of absorption, and the pulse spectrum is wide with respect to the width of absorption line, the pulse begins to distort after a certain distance and is split into two parts at last. The peak value depends not only on the frequency and intensity of the coupling field, but also on the length of the sample. Under appropriate conditions, strong and time domain narrowed pulse can be obtained in the output light.