The Area Theorem is the name given to a theoretical result that governs the coherent nonlinear transmission of short light pulses through materials that have an absorption resonance very near the frequency of the incident light, effectively two-level media. In 1967 McCall and Hahn identified a new parameter (called “Area” and denoted by θ) of optical pulses travelling in such media, and then predicted that θ obeys the simple equation [1]

where α is the linear optical attenuation coefficient for the material. They also reported [1,2] the first experiments testing the unusual predictions implied by the theorem (see also [3]). The two most striking consequences of the Area Theorem are: (i) pulses with special values of Area, namely all integer multiples of π, are predicted to maintain the same Area during propagation, and (ii) pulses with other values of Area must change in propagation until their Area reaches one of the special values. This property can be shown to be unstable for the odd multiples, but the even multiples enjoy the full immunity of the theorem.

The theorem is not empty. There exist soliton pulses that have area 2π and satisfy the Area Theorem and remain unchanged in shape and peak amplitude as they propagate through absorbing media. For example, the pulses of self-induced transparency are of this type. These special pulses have been extensively discussed [2–5] and will not concern us.

Here we give a new derivation of the Area Theorem, including pulse chirping, and obtain new results regarding pulse phase evolution. We expect that the method we use will have application in extensions of the theorem to situations that are currently under discussion, e.g., to matched pulses in three-level lambda and vee media [6–9].

In any near-resonant light-matter interaction the key parameter is the complex Rabi frequency, which is basically the light-matter interaction strength d ∙ E in frequency units. Let a plane-wave optical signal be defined in terms of its scalar slowly varying real envelope and phase: E(z,t) = 𝜀exp[-iϕ]exp[i(kz - ωt)] + c.c, and let the dipole matrix element of the near-resonant transition be denoted d (and taken to be real without loss of generality). Then the real Rabi frequency is defined to be: Ω ≡ (2d/ħ)𝜀. The definition of optical “Area” θ(z, t) is then:

Obviously θ is independent of t as soon as the integral covers the entire pulse.

We first recall the reduced Maxwell wave equation for the complex electric field of the propagating pulse. We follow convention and adopt the coordinate frame travelling at the speed of light in the medium. We denote the space and time variables in this frame by ζ and τ, respectively, and define them by ζ ≡ z and τ ≡ t - z/c. The source of the field is the complex polarization of the medium, which is determined by the off-diagonal element of the density matrix for the two-level medium. We use the u, v, w notation of Bloch [4] for the density matrix elements, and write the reduced Maxwell wave equation for the complex field as ∂(Ωe ^-iϕ)/∂ζ = i(μ/2)〈(u - iv)e ^-iϕ〉, or

On the right side, we have defined μ = 4π𝛮d ² ω)/ħc, where 𝛮 is the density of near-resonant atoms in the medium. The angular bracket signifies a sum over all interacting atoms at the position ζ. Each atom contributes differently, depending on exactly how near or far its individual transition frequency is from the carrier frequency of the propagating field. We assume a broad and continuous distribution of these frequencies. The inclusion of all atoms can be accomplished by an integration over the distribution of their detunings g(Δ).

In terms of u and v, the equation governing the off-diagonal density matrix element is

where w _Δ is the inversion of the two-level medium, and the suffix Δ is a reminder of its dependence on detuning. Note that the τ derivative of ϕ indicates that pulse chirping is accounted for, and the presence of the γ term signals the inclusion of homogeneous damping. The influence of the latter has only been sketched previously (see [4]), and chirping is included here for the first time.

For notational convenience we will define a new parameter a ≡ Δ - iγ as a complex detuning, and begin our new derivation in the manner of Lamb [5] by formally integrating the equation for u - iv:

where we have suppressed ζ dependences. The homogeneous part of the solution has been ignored on the grounds that there is no polarization of the medium in advance of the pulse (at τ = - ∞).

It is the detuning average of this solution that is needed in (3). This allows us to exploit the existence of widely separated time scales governing the different physical processes involved in the light-matter interaction, by using them to justify a quasi-Markovian assumption to simplify the time integration in (5). Our approach is analogous to a similar one commonly taken in scattering theory in defining complex self-energies. In scattering theory the “modes of free space” provide an effectively infinitely broad and continuous distribution of final state frequencies, which is similar to the broad detuning distribution g(Δ) here. In either case the associated inverse time scale is very short, providing an effectively Markovian (zero-memory) response.

That is, because of the broad bandwidth of g(Δ), in the integrand of (5) all terms are slowly varying compared to the very rapid exp{-ia(τ - τ′)}, and can be evaluated at τ′ = τ and removed from the τ′ integral. The integral of the phase factor itself is then trivial, giving a result that has a familiar singular limiting form:

The limit is realistic in the present case because g(Δ) has a halfwidth that can be assumed much wider than the homogeneous linewidth γ. In the time domain our various assumptions correspond to the double inequality $T_2^{*}$ ≪ τ_p ≪ T ₂, where these three times are the inhomogeneous lifetime (e.g., the inverse Doppler width), the pulse duration, and the homogeneous lifetime (i.e., 1/γ).

Now it is easy to separate the real and imaginary parts of the reduced wave equation. At the same time we integrate both sides of the real part over τ to get an equation for θ:

while the evolution of ϕ is given by

where we have reinserted the γ dependence to define the principal value integration more physically near to Δ = 0.

Because of the delta function in (7) only w ₀, the on-resonant inversion, and g(0), the detuning distribution at resonance, survive the bracket average. As McCall and Hahn first showed, w → -cosθ at resonance, so ∫ w ₀Ωdτ = - ∫ cosθdθ = -sinθ. Thus we recover the usual unchirped Area Theorem, given in (1), since one easily verifies that μπg(0) = α.

In (8) we easily obtain a new equation for the evolution of the pulse phase just by cancelling the factors of Ω, something that could not be done in the McCall-Hahn approach. We see explicitly that any τ dependence of ϕ (the chirping) that develops in the course of propagation must arise from τ dependence of the inversion. If the excitation by the pulse is weak, the medium will remain near its ground state where w = - 1 for all detunings. Then our result predicts no chirping, but still a non-zero ∂ϕ/∂ζ, corresponding exactly to the shift of phase already known from linear dispersion theory, which is the result that should be obtained for a ground-state medium.

In summary, we have presented a new derivation of the two-level Area Theorem for optical pulse propagation. Our derivation relies on the existence of a wide separation of physical time scales and reaches the same conclusion as McCall and Hahn regarding the Area of an unchirped pulse, but not regarding its phase. The method presented here may also be advantageous in applications to new versions of the Area Theorem appropriate to lambda and vee media, in which case two different pulses propagate together but satisfy a joint Area Theorem. These applications will be reported elsewhere.