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Air conditioning for photons [Invited]

J. B. Pendry

Open Access Open Access

Abstract

A simple time dependent system is solved analytically taking the form of a moving step-up or step-down in refractive index. We demonstrate that step motion, by a process of compression or expansion, adds or subtracts energy from radiation passing through, so heating or cooling the photons.

Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

1. Introduction

There is considerable interest in electromagnetic systems whose parameters change very rapidly with time. In these systems motion is only apparent generated by local changes in the refractive index without moving the medium itself. Moving gratings have been a favourite system to study where the refractive index profile moves with a constant velocity, $n({x - {c_g}t} )$ [1,2,3]. A review can be found in [4]. Most papers have been theoretical in nature due to the challenge of experimental realisation, especially at optical frequencies but recent work [5,6] shows promising progress in rapid time modulation of the refractive index.

The system studied in this paper consists of a step in refractive index whose profile moves with a constant velocity. Experiments on related systems have been around for some time in the context of the ‘push broom’ effect [7,8,9] where a step in refractive index is generated by a fast moving pump pulse which overtakes a probe pulse with a different frequency and travelling more slowly. The step is shown to sweep up the probe and crush it into a tighter pulse. In our system we simplify the problem by neglecting dispersion, impedance matching the system to eliminate back scattering, generating analytic solutions which we use to explore a wide range of parameters. Our justification for these approximations is that they produce an easily understood model which demonstrates the important physical features of these systems.

A step-up in refractive index sweeping through a field of thermal radiation uniformly crushes the wavelengths and in doing so heats the photons, which escape from the far side of the step with an enhanced temperature. We calculate the temperature increase. It can be considerable even for a modest step amplitude when the step velocity, ${c_g}$, approaches the velocity of light. Our model reveals that if somewhere within the step the local velocity of light is the same as that of the step, a singularity occurs into which light is captured never to escape. Here the temperature rises exponentially with time until cut-off when the step stops. The step-up is a furnace for photons.

In contrast a step-down does the opposite. As radiation sweeps through the step it is cooled because a step-down expands wavelengths of passing radiation. In this way it can create a propagating pool of electromagnetic silence, suppressing unwanted noise. It can also have a singular point similar to the singularity in the step-up where in contrast radiation is repelled from the step-down singularity rather than trapped. This gives rise to extreme cooling with temperature falling exponentially with the time for which the step has been in motion. A step-down is a refrigerator for photons.

2. Modelling a step

We take a simple model for the step,

$$\begin{array}{l} \varepsilon = {\varepsilon _1}{[{1 + \alpha \tanh \gamma ({gx - \Omega t} )} ]^{ - 1}},\;\;\mu = {\mu _1}{[{1 + \alpha \tanh \gamma ({gx - \Omega t} )} ]^{ - 1}}\\ n = \sqrt {{\varepsilon _1}{\mu _1}} {[{1 + \alpha \tanh \gamma ({gx - \Omega t} )} ]^{ - 1}} \end{array}$$

This model is chosen to be impedance matched so that there is no back scattering and solutions factor into forward and backward travelling waves. The backward waves travelling in a direction opposed to the step velocity move smoothly through the step and are not of interest here. This enables simple analytic solutions whilst retaining the essential features of a moving step. The following relationships hold for the fields,

$${H_y} ={\mp} \frac{1}{{{Z_m}}}{E_z},\;\;\;\;\;{B_y} ={\mp} {Z_m}{D_z},\;\;\;\;\;{Z_m} = \sqrt {{\mu / \varepsilon }} = \textrm{const}$$

Making a transformation to the step frame,

$$X = x - {c_g}t,\;\;\;\;\;t^{\prime} = t$$
enables us to write Maxwell’s equations in the form,
$$\frac{\partial }{{\partial X}}[{ \mp {c_l} + {c_g}} ]{D_z} ={+} \frac{\partial }{{\partial t^{\prime}}}{D_z},\;\;\;\;\;{c_g} = {\Omega / g},\;\;\;\;\;{c_l}(X )= {{{c_0}} / {\sqrt {\varepsilon \mu } }}$$
where ${c_g}$ is the step velocity and ${c_l}(X )$ the local velocity of light,
$${c_1} = {{{c_0}} / {\sqrt {{\varepsilon _1}{\mu _1}} }},\;\;\;{c_l} = {{{c_0}} / {\sqrt {\varepsilon \mu } }},\;\;\;{c_g} = {\Omega / g} = ({1 + \delta } ){c_1}$$

This equation can be solved by ‘the method of trajectories’. First we put (4) into standard format by defining,

$$\psi = [{{c_l} - {c_g}} ]{D_z}$$
to retrieve for forward travelling waves,
$$({{c_l} - {c_g}} )\frac{\partial }{{\partial X}}\psi + \frac{\partial }{{\partial t^{\prime}}}\psi = 0$$

The idea is find a trajectory, ${t_{traj}}(X )$, along which $\psi$ is constant. Solving this equation gives,

$${{t}_{traj}}\left( X \right)=+\int{\frac{dX}{\left( {{c}_{l}}-{{c}_{g}} \right)}}=+\int{\frac{dX}{{{c}_{1}}\left( \alpha \tanh \left( \gamma X \right)-\delta \right)}}$$

The trajectory can then be extrapolated back to the start time, ${t_0}({{X_0}} )$ where boundary conditions are imposed. After some straightforward integration we can define a function,

$$f(X )= [{\alpha \sinh ({\gamma X} )- \delta \cosh ({\gamma X} )} ]{e ^{ + \gamma \delta {\alpha ^{ - 1}}X}}$$
and find ${X_0}$ by solving,
$$f({{X_0}} )= f(X )\textrm{exp} ({ - {c_1}\gamma {\alpha^{ - 1}}({{\alpha^2} - {\delta^2}} )({t - {t_0}} )} )$$

Because we have impedance matched our system, the displacement field before the step is set in motion is given by,

$$D({{X_0},{t_0}} )= {E_0}\frac{1}{{{c_1}({1 + \alpha {\mathop{\rm sgn}} {\textrm{X}_0}} )}}{e ^{i\frac{{\omega {\textrm{X}_0}}}{{{c_1}({1 + \alpha {\mathop{\rm sgn}} {\textrm{X}_0}} )}}}}$$

Before motion is turned on energy flow is a constant everywhere and hence the electric field, ${E_0}$, is constant throughout, the phase being determined by the local refractive index. We make the approximation of a sharp step as far as the initial fields are concerned since ${X_0}$ will mainly be found far from the origin. Finally from (6),

$$D({X,t} )= D({{X_0},{t_0}} )\frac{{{c_l}({{X_0}} )- {c_g}}}{{{c_l}(X )- {c_g}}} = D({{X_0},{t_0}} )\frac{{\alpha \tanh ({\gamma {X_0}} )- \delta }}{{\alpha \tanh ({\gamma X} )- \delta }}$$

There are two types of step: a step up in refractive index, and a step down. Each has its own characteristics which we describe below.

3. Step up – an optical oven

Figure 1 shows a plot of the refractive index for sample up and down steps. If the step velocity, ${c_g}$, coincides with the local velocity of light somewhere within the step a singular point occurs [10]. For the step-up, light to the right of the point is slow and drifts towards the singularity, likewise light to the left is fast and also moves towards the point where energy accumulates. The longer the step continues in motion the more concentrated and more intense this energy becomes. Another process is at work: for a step-up, motion of the step pumps energy into the system further adding to the energy at the singular point. In contrast a step-down pushes light away from the singularity at the same time as subtracting energy. The step-down will be discussed in the next section. Energy is either supplied or taken up by the ‘engine’ driving the motion of the step. Although the equations contain a singularity, the fields, though distorted, are not singular provided that the step moves for only a finite time.

 figure: Fig. 1.

Fig. 1. Plot of the refractive index for up (left) and down (right) steps. High refractive index means that locally light travels slowly, conversely for low refractive index. The horizontal line shows the case where the step velocity, ${c_g},$ coincides with the local velocity of light somewhere within the step. The steps are defined by $\alpha ={\pm} 0.05,\textrm{ }\gamma = 2.0$.

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Figure 2 shows a series of calculations of $|{D(X )} |$ for a step-up defined by $\alpha ={-} 0.05,\textrm{ }\gamma = 2.0$ running for a time $t = 40$. The top left panel calculates for several step velocities starting from a stationary step up to a velocity just below the slowest light in the step. The picture is one of minimal disturbance to the light as it passes through the step as we should expect for this very weak change in refractive index. Matters begin to change as the step velocity increases, then it becomes evident that light passing over the step is being amplified. The leading edge of the red curve marks light that passed over the step as it was set in motion. This point is clarified in Fig. 3 where the step has been in motion for a much longer time. Four calculations are made and here we are interested in the left hand figure, where the black curve corresponds to ${c_g} = 0.925{c_1}$ and shows the step increasing the amplitude of the light by a factor of $4.75$ for as long as it remains in motion. At this step velocity we are outside the singular region and light escapes from the far side of the step. The height of the step can be calculated from (12) in the limit $t \to \infty$ since in this regime $X \to + \infty ,\textrm{ }{X_0} \to - \infty$, or vice versa, so that,

$$\left|{\frac{{{D_{step}}}}{{{D_0}}}} \right|= \left|{\frac{{D({ - \infty } )}}{{{D_0}}}} \right|\left|{\frac{{\alpha \tanh ({ - \infty } )- \delta }}{{\alpha \tanh ({ + \infty } )- \delta }}} \right|= 4.75$$

 figure: Fig. 2.

Fig. 2. Evolution of the electric field as a function of $X$ for a rising step, $\gamma = 2.0\textrm{, }\alpha ={-} 0.05\textrm{, }t = 40$ for several different step velocities. Note the change of vertical scale.

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 figure: Fig. 3.

Fig. 3. Evolution of the electric field as a function of X, $\gamma = 2.0\textrm{, }t = 200$ for a rising step, $\alpha ={-} 0.05$, and for a falling step, $\alpha ={+} 0.05$. for two step velocities: one above, the other below the light velocity, showing how the step adds energy to or subtracts energy from the field.

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Moving now to the top right panel of Fig. 2, calculations here are for velocities where the light is trapped at a singular point as illustrated in Fig. 1. Singular points occur in the luminal regime where the step velocity is in the range,

$$({1 - |\alpha |} )< {c_g} < ({1 + |\alpha |} )$$

In this calculation the step has been moving for a time $t = 40$ and substantial amplification of the light is seen. Total energy grows exponentially with time and the width of the peak decreases exponentially with time, much the same as observed in our previous work on a moving grating. After the step ceases to move, the pulse continues to propagate though no longer increasing in amplitude.

In a previous paper [11] it was shown that in impedance matched systems lines of force and phase are conserved so that energy can only be added by compressing the lines of force: locally fields are scaled in time and space by the compression factor as is the local frequency of the light. This enables a connection between the well known effect that physically compressing a container of radiation in thermal equilibrium maintains thermal equilibrium but raises the temperature. Hence if radiation captured by the step-up was thermal in nature it would be heated by the compressive effect of the step. The compression factor is the same factor as determines $|{D(X )} |$. Outside the luminal region where the singularity does not operate there is a finite amount of heating of radiation as it passes over the step which can be seen in Fig. 3. This may be substantial if ${c_g}$ is close to but not within the luminal regime.

When ${c_g}$ lies within the luminal regime light enters the step but does not re-emerge. Instead it is captured at the singularity where it is continuously heated. For a step up and ${c_g} = {c_1}$ there is an analytic expression for the compression factor. We calculate the local increase in temperature created by a step-up moving for a given time to be,

$$T(X )= {T_0}\frac{{\cosh ({\gamma X} ){e^{ - \alpha \gamma {c_1}t}}}}{{\sqrt {1 + {{\sinh }^2}({\gamma X} ){e^{ - 2\alpha \gamma {c_1}t}}} }},\textrm{ }T({X = 0} )= {T_0}{e^{ - \alpha \gamma {c_1}t}},\textrm{ }{c_g} = {c_1}$$
bearing in mind that for a step up $\alpha < 0$. The centre of the compression could become very hot indeed for quite modest values of $\alpha$. Indeed the top right panel of Fig. 2 shows the effect on $|{D(X )} |$ which as we have stated is compressed by the same factor as the temperature.

Two important messages emerge from this result: the first is that in this regime, the so called ‘luminal regime’, where a singularity exists, it makes no sense to speak of a continuously moving step because several quantities diverge. Restricting movement to a finite time is essential to remove any singularities in the fields and give physically meaningful results. The second is that an infinitely sharp step also produces a divergence even when moving for only a finite time, pumping infinite amounts of energy into an infinitesimally narrow peak. The latter would be hidden within the step. This has consequences for approaches which try to solve the problem by matching fields either side of a sudden step.

Next we turn to the last panel for Fig. 2. Here the step is moving faster than the light so that in the step frame light appears to move backwards. Just as in the subluminal case finite amount of energy is imparted to the light as is passes the step. This is further evidenced by Fig. 3: the red curve in the left panel almost mirrors the subluminal case. The other curve in Fig. 2 corresponds to a step moving much faster than the light velocity. Light quickly passes the step which has much less time to exert its influence, and hence there is much less amplification.

4. Step down – an optical refrigerator

In some ways the step down is more intriguing than the step up. In this case the step extracts energy from the light. This is illustrated in the right of Fig. 3 for the sub-luminal (black) and super luminal (red) cases when ${c_g}$ is well clear of any singularities. The step-down makes a ‘hole’ in the amplitude of light and this effect is most pronounce when ${c_g}$ is almost in the singular regime. Away from this regime the effect is small. The depth of the hole can be calculated from Eq. (13) but now inserting positive values for $\alpha$.

$$\left|{\frac{{{D_{step}}}}{{{D_0}}}} \right|= \left|{\frac{{D({ - \infty } )}}{{{D_0}}}} \right|\left|{\frac{{\alpha \tanh ({ - \infty } )- \delta }}{{\alpha \tanh ({ + \infty } )- \delta }}} \right|= 0.21$$

When ${c_g}$ enters the luminal regime shown in Fig. 4 the singularity exerts itself and the situation changes dramatically. A deep hole is created around the singularity, ever expanding in size but with the singularity remaining at its centre. After the step ceases to move, the hole continues to propagate, a silent zone cleansed of all noisy signals whose length is determined by how long the step down was in motion,

$${X_{hole}} = 2t|\alpha |{c_1}$$

 figure: Fig. 4.

Fig. 4. Evolution of the electric field as a function of X, $\gamma = 2.0\textrm{, }t = 200$, for a falling step, $\alpha ={+} 0.05$, for a step velocity of ${c_g} = 1.025{c_1}$, which is in the luminal regime where a singular point exists within the step. Field intensities within the ‘hole’ tend to zero exponentially with t.

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For example we could imagine a region of refractive index expanding into vacuum giving the required step down, and ejecting into the vacuum a silent zone.

If the step-down encountered thermal radiation it would act in a converse manner to the step-up, rapidly expanding and hence cooling the radiation rather than heating it. Outside the luminal regime radiation is cooled by a finite amount however long the step runs for.

The temperature at the singularity would be given by Eqn. (14) but with a positive value of $\alpha .$ For example if $\alpha ={+} 0.05,\textrm{ }t = 40$ then the temperature would be reduced by a factor of $0.017$. Increasing the time to $t = 200$ reduces the temperature by a factor of $2 \times {10^{ - 9}}$.

This novel manner of refrigerator could also be implemented for vibrational heat. Phonons obey similar equations to electromagnetic fields and would also be cooled in the same manner given the correct time dependent elastic properties.

5. Conclusions

A simple model of a smooth step in refractive index in uniform motion reveals several novel features of time dependent electromagnetic systems. In essence a moving step is a compressor or decompressor of radiation. Like its companion system, the moving grating, there is a phase transition in the manner of compression. This occurs when there is a point within the step at which the local velocity of light coincides with that of the grating: light enters this singularity never to leave and finite compression is replaced by exponential growth of compression in the trapped light in the case of a step-up and by exponentially increasing stretching for a step down.

Funding

Gordon and Betty Moore Foundation.

Acknowledgments

J.B.P. acknowledges funding from the Gordon and Betty Moore Foundation, productive conversations with Yu Luo, Simon Horsley and Steve Barnett.

Disclosures

The author declares no conflicts of interest.

Data availability

There are no data beyond those presented in the text.

References

1. E. S. Cassedy and A. A. Oliner, “Dispersion relations in time-space periodic media: Part I—Stable interactions,” Proc. IEEE 51(10), 1342–1359 (1963). [CrossRef]  

2. E. S. Cassedy, “Dispersion relations in time-space periodic media part II—Unstable interactions,” Proc. IEEE 55(7), 1154–1168 (1967). [CrossRef]  

3. Y. Sharabi, A. Dikopoltsev, E. Lustig, et al., “Spatiotemporal photonic crystals,” Optica 9(6), 585–592 (2022). [CrossRef]  

4. E. Galiffi, R. Tirole, S. Yin, et al., “Photonics of time-varying media,” Adv. Phot. 4(01), 014002 (2022). [CrossRef]  

5. Eran Lustig, Ohad Segal, Soham Saha, et al., “Time-refraction optics with single cycle modulation,” Nanophotonics 12(12), 2221–2230 (2023). [CrossRef]  

6. R. Tirole, S. Vezzoli, E. Galiffi, et al., “Double-slit time diffraction at optical frequencies,” Nat. Phys. 19(7), 999–1002 (2023). [CrossRef]  

7. C. Martin de Sterke, “Optical push broom,” Opt. Lett. 17(13), 914 (1992). [CrossRef]  

8. N.G.R. Broderick, D. Taverner, D.J. Richardson, et al., “Optical pulse compression in fibre Bragg gratings,” Phys. Rev. Lett. 79(23), 4566–4569 (1997). [CrossRef]  

9. Mahmoud A. Gaafar, Alexander Yu. Petrov, and Manfred Eich, “Front-induced transitions,” Nat. Photonics 13(11), 737–748 (2019). [CrossRef]  

10. P.A. Huidobro, M.G. Silveirinha, E. Galiffi, et al., “Homogenization Theory of Space-Time Metamaterials,” Phys. Rev. Appl. 16(1), 014044 (2021). [CrossRef]  

11. JB Pendry, E Galiffi, and PA Huidobro, “Gain in time-dependent media—a new mechanism,” J. Opt. Soc. Am. B 38(11), 3360–3366 (2021). [CrossRef]  

Data availability

There are no data beyond those presented in the text.

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Figures (4)
Fig. 1.
Fig. 1. Plot of the refractive index for up (left) and down (right) steps. High refractive index means that locally light travels slowly, conversely for low refractive index. The horizontal line shows the case where the step velocity, ${c_g},$ coincides with the local velocity of light somewhere within the step. The steps are defined by $\alpha ={\pm} 0.05,\textrm{ }\gamma = 2.0$.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Evolution of the electric field as a function of $X$ for a rising step, $\gamma = 2.0\textrm{, }\alpha ={-} 0.05\textrm{, }t = 40$ for several different step velocities. Note the change of vertical scale.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Evolution of the electric field as a function of X, $\gamma = 2.0\textrm{, }t = 200$ for a rising step, $\alpha ={-} 0.05$, and for a falling step, $\alpha ={+} 0.05$. for two step velocities: one above, the other below the light velocity, showing how the step adds energy to or subtracts energy from the field.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Evolution of the electric field as a function of X, $\gamma = 2.0\textrm{, }t = 200$, for a falling step, $\alpha ={+} 0.05$, for a step velocity of ${c_g} = 1.025{c_1}$, which is in the luminal regime where a singular point exists within the step. Field intensities within the ‘hole’ tend to zero exponentially with t.

View in Article | Download Full Size | PDF

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

ε=ε1[1+αtanh⁡γ(gx−Ωt)]−1,μ=μ1[1+αtanh⁡γ(gx−Ωt)]−1n=ε1μ1[1+αtanh⁡γ(gx−Ωt)]−1
Hy=∓1ZmEz,By=∓ZmDz,Zm=μ/ε=const
X=x−cgt,t′=t
∂∂X[∓cl+cg]Dz=+∂∂t′Dz,cg=Ω/g,cl(X)=c0/εμ
c1=c0/ε1μ1,cl=c0/εμ,cg=Ω/g=(1+δ)c1
ψ=[cl−cg]Dz
(cl−cg)∂∂Xψ+∂∂t′ψ=0
ttraj(X)=+∫dX(cl−cg)=+∫dXc1(αtanh⁡(γX)−δ)
f(X)=[αsinh⁡(γX)−δcosh⁡(γX)]e+γδα−1X
f(X0)=f(X)exp(−c1γα−1(α2−δ2)(t−t0))
D(X0,t0)=E01c1(1+αsgnX0)eiωX0c1(1+αsgnX0)
D(X,t)=D(X0,t0)cl(X0)−cgcl(X)−cg=D(X0,t0)αtanh⁡(γX0)−δαtanh⁡(γX)−δ
|DstepD0|=|D(−∞)D0||αtanh⁡(−∞)−δαtanh⁡(+∞)−δ|=4.75
(1−|α|)<cg<(1+|α|)
T(X)=T0cosh⁡(γX)e−αγc1t1+sinh2(γX)e−2αγc1t, T(X=0)=T0e−αγc1t, cg=c1
|DstepD0|=|D(−∞)D0||αtanh⁡(−∞)−δαtanh⁡(+∞)−δ|=0.21
Xhole=2t|α|c1
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