Nearly degenerate four-wave mixing applied to optical filters Joseph Nilsen and Amnon Yariv

Both authors are with California Institute of Technology, Pasadena, California 91125; J. Nilsen is in the Physics De­partment, and A. Yariv is in the Department of Electrical Engineering & Applied Physics. Received 21 October 1978. 0003-6935/79/020143-03$00.50/0. © 1979 Optical Society of America. Nearly degenerate four-wave mixing in a nondispersive

lossless medium1 has been shown capable of yielding a real­time optical bandpass filter. This Letter examines nonde-generate four-wave mixing in a resonantly enhanced medium. The two-level system, in many cases, has the advantage of providing much larger nonlinear coupling constants than a transparent medium. As a result, very modest pump inten­sities can be used to achieve a significant reflection coeffi­cient.2

The mixing involves two intense counterpropagating pump waves E1 and E2 of the same frequency ω and two weak counterpropagating waves E3 and E4. The geometry of Yariv and Pepper3 shown in Fig. 1 is used.

The fields are taken as plane waves:

with ri the distance along ki. We have

The two-level system is characterized by a dipole moment μ, an energy splitting hω0, and longitudinal and transverse relaxation times T1 and T2, respectively.4 The applied electric fields are polarized along the same direction, and the atoms are taken as stationary. The density matrix equations are solved to third order by perturbation theory.4 Only terms with resonant denominators are included in the solution.

Higher order effects such as the usual saturation effect in resonance absorption are not considered. The polarizations coupling the waves are

The coupling constants are given by

where δ = (ω - ω0)T2 is the normalized detuning of the pump fields from line center, v = (ω4 - ω)T2 is the normalized de­tuning of the signal field from the pump fields, α = T1/T2, A2

s = 4h2/T1T2μ2 is the line-center saturation intensity, a0 = μ2 N0T2k0/2h is the line-center small-signal-field attenuation coefficient with k0 the magnitude of the wavenumber at fre­quency ω0, and k = 2(ω4 - ω)/c.

Applying the standard methods of nonlinear optics,4 the wave equation (ε = μ = 1)

yields the set of coupled equations

when pump depletion is neglected, and the adiabatic ap­proximation

is used. The absorption of the pump waves is not included since it is geometry-dependent and unimportant in many cases.

For filter applications, the appropriate boundary conditions are A3(L) = 0, with A*

4(0) being the signal input to the system. The power reflection coefficient is defined as

Fig. 1. Geometry for nearly degenerate four-wave mixing (assuming nondepleting pump waves).

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Fig. 2. Reflection coefficient vs signal detuning v for several values of the pump detuning δ. All curves are normalized to unit reflectivity

in order to more clearly display the bandwidth of the system.

Fig. 3. Normalized bandwidth (B = ωT2) vs pump detuning δ.

Fig. 4. Reflectivity, for degenerate four-wave mixing, vs pump de­tuning δ.

144 APPLIED OPTICS / Vol. 18, No. 2 / 15 January 1979

The solution for R is

where

It should be noted that the reflected wave is frequency shifted from the incident wave. However, the frequency shift is very small for any significant reflection coefficient.

Using α0L = 1, T1 = T2, 2L/cT2 = 0.01, and A1,A2/A2S = 0.1,

the frequency dependence of the reflected (output) signal is studied. The values chosen for the parameters approximately model sodium vapor5 for pump fields below saturation, an atom density of 1011/cm3, and an interaction length of 1 cm.

Figure 2 shows the reflection coefficient, normalized to unity, plotted vs signal detuning v for several values of the pump detuning δ. The important feature to notice is the broadening of the curves as δ increases. As shown clearly in Fig. 3, the normalized bandwidth B = ωT2 ( ω is defined as the full width at half maximum) increases with the pump detuning δ. Figure 4 shows the dependence of the reflection coefficient on pump detuning for the degenerate case (v = 0). A recent paper by Abrams and Lind6,7 describes degenerate four-wave mixing in a two-level system in greater detail. The graphs indicate that operation near line-center is necessary to optimize the amplitude of the reflected wave. Under cer­tain situations, for example, α0L » 1,2 it would be advanta­geous to have the pump fields detuned from line-center.

This paper has demonstrated how nearly degenerate four-wave mixing in absorbing media can yield an active narrow bandwidth optical filter, which also has a large field-of-view (i.e., several steradians). The large field-of-view is a conse­quence of the fact that four-wave mixing does not depend on phase matching.3 The frequency response depends primarily on T2 for the values presented here. The bandwidth is di­rectly proportional to the inverse of the relaxation time T2 and monotonically increases with the pump detuning δ. When saturation effects are included, the bandwidth should become much narrower, and the peak reflectivity should become larger as the pump fields are increased.1,2,6 It will also be possible for the filter to yield an amplified output.1,2

A future publication will consider a Doppler-broadened two-level system and include saturation effects and pump depletion.

One of the authors (JN) thanks the University of California Lawrence Livermore Laboratory for the use of its computer facilities in developing the numerical results presented in the figures and the Fannie and John Hertz Foundation for sup­port of graduate studies. Joseph Nilsen is a Fannie and John Hertz Foundation Fellow.

References 1. D. M. Pepper and R. L. Abrams, "Narrow Optical Bandpass Filter

via Nearly Degenerate Four-Wave Mixing" (submitted to Opt. Lett.).

2. D. M. Bloom, P. F. Liao, and N. P. Economou, Opt. Lett. 2, 58 (1978).

3. A. Yariv and D. M. Pepper, Opt. Lett. 1, 16 (1977).

4. A. Yariv, Quantum Electronics (Wiley, New York, 1975), pp. 149-155, 418-421, 553-558.

5. P. F. Liao, J. E. Bjorkholm, and J. P. Gordon, Phys. Rev. Lett. 39, 15 (1977).

6. R. L. Abrams and R. C. Lind, Opt. Lett. 2, 94 (1978). 7. R. L. Abrams and R. C. Lind, Opt. Lett. 3, 205 (1978).

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