12.1 Fundamentals of photonic crystals and band gaps

Photonic crystals are materials with periodic structures that control how light moves through them. They're like semiconductors, but for photons instead of electrons. These crystals can have patterns in one, two, or three dimensions, creating unique optical properties.

The key feature of photonic crystals is the photonic band gap, a range of frequencies where light can't pass through. This happens because of how waves interact with the crystal's structure. Understanding these gaps is crucial for designing optical devices and controlling light.

Photonic Crystal Fundamentals

Periodic Dielectric Structures

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• Photonic crystals consist of periodic dielectric structures that affect the motion of photons similar to how periodic potential in a semiconductor crystal affects electron motion
• These structures are composed of alternating regions of high and low dielectric constant materials arranged in a periodic fashion
• Periodicity can be in one dimension (1D), two dimensions (2D), or three dimensions (3D) depending on the design
• Examples of photonic crystal structures include multilayer films (1D), photonic crystal fibers (2D), and self-assembled colloidal crystals (3D)

Reciprocal Lattice and Brillouin Zone

• The reciprocal lattice is a Fourier transform of the spatial function describing the photonic crystal lattice
• It is used to analyze the wave properties of photonic crystals and determine the allowed modes of propagation
• The first Brillouin zone is a primitive cell in the reciprocal lattice that contains all the unique wave vectors ($k$-vectors) that characterize the propagation modes
• High symmetry points within the Brillouin zone (e.g., $\Gamma$, $X$, $M$) are used to describe the photonic band structure

Wave Propagation in Photonic Crystals

Bloch Waves and Dispersion Relation

• In photonic crystals, light propagates as Bloch waves, which are electromagnetic waves modulated by the periodic structure
• Bloch waves are characterized by a wave vector ($k$) and a periodic function ($u_k(r)$) that has the same periodicity as the photonic crystal lattice
• The dispersion relation $\omega(k)$ describes the relationship between the frequency ($\omega$) and the wave vector ($k$) of the Bloch waves
• It determines the allowed modes of propagation and the group velocity ($v_g = \frac{d\omega}{dk}$) of light in the photonic crystal

Bragg Reflection

• Bragg reflection occurs when the wavelength of light is comparable to the periodicity of the photonic crystal structure
• At certain wavelengths, the reflected waves from each dielectric interface interfere constructively, leading to strong reflection and the formation of a photonic band gap
• The condition for Bragg reflection is given by $m\lambda = 2nd\cos\theta$, where $m$ is an integer, $\lambda$ is the wavelength, $n$ is the refractive index, $d$ is the lattice spacing, and $\theta$ is the angle of incidence
• Examples of Bragg reflection can be found in multilayer dielectric mirrors and photonic crystal fibers designed for wavelength-selective filtering

Photonic Band Structure

Photonic Band Gap

• A photonic band gap is a range of frequencies where light propagation is prohibited in the photonic crystal
• It arises from the destructive interference of Bragg reflected waves, leading to the absence of allowed propagation modes
• The size and position of the band gap depend on the dielectric contrast, lattice geometry, and periodicity of the photonic crystal
• Photonic band gaps can be complete (omnidirectional) or partial (directional) depending on the crystal structure and the range of prohibited frequencies
• Applications of photonic band gaps include high-reflectivity mirrors, optical filters, and spontaneous emission control

Photonic Density of States

• The photonic density of states (DOS) describes the number of available optical states per unit frequency and volume in a photonic crystal
• It is determined by the photonic band structure and the dimensionality of the system
• In the presence of a photonic band gap, the DOS is significantly reduced or even vanishes for the corresponding frequency range
• The modification of the DOS in photonic crystals can lead to enhanced or suppressed spontaneous emission, as described by Fermi's golden rule
• Examples of DOS engineering include the use of photonic crystals for controlling the emission properties of quantum dots and enhancing the efficiency of solar cells