Negative Refraction, Metamaterials and Diffraction: What Light Can Do
A free light and metamaterials lab: see negative refraction and the Veselago flat lens with rays, the perfect lens and its loss limit, a live Maxwell simulation of a material with negative permittivity and permeability, a diffraction calculator for slits and gratings, and ten live demos of reflection, refraction, dispersion, interference, polarization, slow light and cloaking.
In glass, a ray bends toward the normal but stays on the other side of it. In a material with a negative refractive index the ray bends to the same side of the normal it came from. Snell's law still holds, n₁ sin θ₁ = n₂ sin θ₂, with a negative n₂, so θ₂ comes out negative. Such materials do not occur in nature; they are built as metamaterials (tiny wires and split rings, smaller than the wavelength) and were first made for microwaves in 2000-2001.
The perfect lens (Pendry, 2000)
An ordinary lens can never show details finer than about half a wavelength: the fine detail is carried by evanescent waves, which die away within a wavelength of the object. Pendry showed that a slab with ε = μ = −1 amplifies evanescent waves by exactly the amount they decay, so in principle every detail reaches the image. The plot is the exact transfer function of the slab (one plane-wave component at a time): 1 means that detail is copied perfectly. Real metamaterials have loss, and loss is what limits the resolution.
A live solution of Maxwell's equations on a grid (2D, electric field pointing out of the screen). A plain "n = −1" cannot be simulated directly: a negative index only makes sense at one frequency, so the material is modelled the honest way, as a Drude medium for both ε and μ, tuned so that ε = μ = −1 at the source frequency. Watch the wavefronts inside the slab move toward the source (the phase runs backwards) while the energy still flows away from it.
What is modelled, and what is not
- Modelled: Maxwell's equations in 2D (Ez, Hx, Hy, Yee grid, 20 cells per wavelength). The slab has Drude currents in both the electric and the magnetic equation: ε(ω) = 1 − ωp²/(ω(ω + iγ)), and the same form for μ. That is causal and stable, and it gives ε = μ = −1 only near the source frequency (it is positive again above ωp).
- Not modelled: the wires and rings a real metamaterial is made of (it is treated as a smooth medium), 3D, other polarisations. The grid shifts ε and μ a little from their target values, and the corners of the slab start a surface wave (a surface plasmon, a real effect at ε = −1) that the loss slider calms down.
- The foci (dashed marks) are where ray optics puts the two images of an n = −1 slab: inside it, and as far behind it as the source is in front. Switch to Brightness to see them as two bright spots.
Light that passes through narrow openings spreads out (diffraction) and the waves from different openings add up or cancel (interference). Far from the slits (the Fraunhofer limit) the brightness at angle θ is a single-slit envelope times an interference pattern: \(I(\theta) = I_0\,\mathrm{sinc}^2\beta\,\left(\dfrac{\sin N\alpha}{N\sin\alpha}\right)^2\) with \(\beta = \pi a\sin\theta/\lambda\) and \(\alpha = \pi d\sin\theta/\lambda\).
Want to watch the waves themselves go through slits? The Light in a Medium Simulator has single slit, double slit and grating scenes that solve the wave equation live.
Ten things light does, each one live: change the sliders and the picture follows the physics.
Negative refraction, metamaterials and what light can do
The refractive index n says how much a material slows light down and how much it bends a ray. For every natural transparent material n is bigger than 1. In 1968 Victor Veselago asked what would happen if both the permittivity ε and the permeability μ of a material were negative at the same time. The answer is a material with n = −√(εμ), a negative refractive index: rays bend the "wrong" way, the phase of a wave moves backward while its energy moves forward, and a flat slab acts as a lens. Nobody could test it until John Pendry showed how to make negative ε (thin wires) and negative μ (split-ring resonators) from structures smaller than the wavelength, and David Smith's group measured negative refraction of microwaves in 2001. Such engineered structures are called metamaterials.
Things to try
- One surface: slide the index from +1.5 down through 0 to −1.5 and watch the refracted ray swing across the normal.
- Flat lens: with n = −1 every ray from the source meets again inside the slab and again behind it, at the same points whatever the angle. Try n = −2: the images move and blur (aberration), because only n = −1 is perfect.
- Wave simulation: run the flat lens, wait a few periods and switch to Brightness to see the two bright images. Then pick "Only ε negative": a single negative quantity just reflects the light, like a metal mirror.
- Perfect lens: make the loss tiny and the slab thin and the transfer function stays near 1 far beyond kx = k₀, the classical limit.
- Diffraction: load the double slit with missing orders, then increase N to see the fringes sharpen into grating lines.
- Cloaking: in "What light can do", rays bend around the hidden core and come out exactly where they would have gone, so the core is invisible (for one colour, in ray optics).
Things people ask
- Is negative refraction real? Yes, it has been measured many times for microwaves and, with more loss, for infrared and visible light. Photonic crystals can also refract negatively without a negative index.
- Does it break causality or let anything go faster than light? No. The energy (group velocity and Poynting vector) still moves away from the source; only the wavefronts (phase) run backward. A negative index also needs dispersion: it cannot be negative at every frequency, which is why the simulation uses the Drude model.
- Why is there no perfect lens on the market? Loss. The amplified evanescent waves are very sensitive to any absorption, and making ε and μ both −1 at the same frequency with low loss is hard, especially for visible light. Near-field "superlenses" using silver (ε ≈ −1 only, for one polarisation) have imaged details of about λ/6.
- Is the invisibility cloak real? Transformation-optics cloaks have been demonstrated for microwaves (2006) and as carpet cloaks for light, but only for a narrow band of colours and in a limited range of directions.
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