Mechanical forces in electromagnetic systems are transmitted through “lines of force”, the electric and magnetic fields postulated by Michael Faraday and deployed by Maxwell in his equations. For example, a superconductor repels a magnet (Fig. 1) because it expels magnetic lines of force, compressing them and hence increasing their energy density. In this example, the lines are conserved and it is their compression rather than addition of new lines that is the key to understanding these quasi-static forces. Although this compression is a familiar concept at very low frequencies, it has so far played no role in explaining work done when amplifying electromagnetic waves. Here the familiar mechanisms are provided by gain media in the case of lasers or by parametric amplification. In these systems, no conservation of lines of force exists, and work is done on radiation by uniformly adding more lines. Hence little light is shed on the process by speaking in terms of lines of force. However, with new experimental possibilities emerging for systems whose parameters change on the same time scale as the frequency of the radiation both at optical [1,2] and terahertz frequencies [3–5] a different mechanism for amplification comes into play. Well known to computational studies [6,7], it has been little studied from an analytic point of view. Its modus operandi is the focus of this memorandum and constitutes the alternative mechanism for gain alluded to in our abstract by compressing lines of force.

 

Fig. 1. Superconductor repels a magnet by excluding its lines of force. As the superconductor approaches the magnet, the repulsive force stores energy by compressing the lines of force, which are conserved.

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We illustrate this new mechanism using a simple model: a Bragg grating synthetically moving with a uniform velocity, ${c_g} = \Omega /g $,

In Fig. 2, we show the three regimes supported by this model: Fig. 2A shows the dispersion, $\omega (k)$, of radiation in a uniform dielectric; Fig. 2B shows ${c_g} \ll {c_1}$, where the system is parity-time symmetric and Bloch waves are defined. Bandgaps in $\omega$ appear where bands intersect and within which the Bloch waves either grow or decay in space. In Fig. 2C, ${c_g} \gg {c_1}$ and Bloch waves are also defined, but the bandgaps are different. They are gaps in $k$ within which waves either grow or decay in time. In other words, in these gaps we find parametric amplification. For this mechanism to operate, it is essential to have bandgaps in $k$ where backscattering ensures conservation of momentum when a forward-traveling photon is generated and the reaction deposited in the grating. In the limit $g \to 0$, the model reduces to standard parametric amplification, where the entire system is modulated at twice the input frequency.

 

Fig. 2. Sketch of dispersion relation of light traveling through (A) a uniform dielectric; (B) a grating modulated as shown in Eq. (1), where ${c_g} \lt {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$; (C) a grating where ${c_g} \gt {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$; (D) the case of ${c_g} = {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$ where all the forward-traveling states are degenerate and therefore strongly coupled. Arrows show displacement of the bands by a space-time reciprocal lattice vector $({g,\Omega})$ with slope ${c_g}$.

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Figure 2D illustrates the third regime, the luminal one where ${c_g} \approx {c_1}$. Here Bloch waves are not defined, and computational studies on waves incident on a finite slab of grating show each period of the grating capturing a group of waves, which it compresses into pulses. The pulses grow ever more intense and narrow while locked to a specific point in the grating period until finally ejected from the far side of the slab as a supercontinuum. Their height, $h$, grows exponentially with length and with the strength of modulation and the pulse width scales as $1/\sqrt h$ so that there is overall amplification as well as local amplification. Their spacing is the period of the grating because they are trapped therein. Trapping occurs for a range of grating velocities close to the speed of light in the background medium [9]. This is illustrated in a schematic fashion in Fig. 3. This is not parametric amplification. In fact, we can turn off all parametric amplification by eliminating backscattering. This we do by ensuring that the grating is everywhere impedance-matched, ${\alpha _\varepsilon} = {\alpha _\mu}$. This closes all bandgaps and hence cuts off all parametric processes, but in the ${c_g} \approx {c_1}$ regime, amplification and compression remain.

 

Fig. 3. Schematic figure of the effect of a luminal grating, traveling to the right, on a plane wave incident from the left. The grating has little effect on waves incident from the right. Note the compression of lines of force shown schematically as gray lines.

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We stay with the impedance-matched model to ensure that no parametric processes intrude into our calculations, not because the new mechanism is specific to impedance matching. With no backscattering operating forward and backward, waves are independent, each obeying a first-order differential equation. The forward waves are of interest to us because they more nearly keep pace with the grating and they obey

In this memorandum, we have demonstrated that conventional descriptions of amplification fail to explain gain experienced in a luminal medium. A new mechanism, compression of lines of force, explains the processes at work with a highly intuitive picture. The model correctly predicts the relationship between amplification and pulse width. Further details can be found in [10].