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🔬Modern Optics

🔬modern optics review

8.1 Nonlinear susceptibility and wave equation

3 min readLast Updated on July 22, 2024

Nonlinear optics explores how intense light interacts with materials in unexpected ways. When powerful lasers hit certain substances, they create wild effects like turning red light into blue or making light waves mix together.

These phenomena happen because materials respond differently to super-bright light. Instead of just absorbing or reflecting light normally, they start doing funky things with the light waves, bending and twisting them into new forms.

Nonlinear Optical Phenomena

Nonlinear susceptibility in optics

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  • Nonlinear susceptibility quantifies nonlinear response of a medium to strong electric fields (lasers)
  • Denoted as χ(n)\chi^{(n)}, where nn represents the order of nonlinearity (χ(2)\chi^{(2)} for second-order, χ(3)\chi^{(3)} for third-order)
  • Relates induced polarization in the medium to applied electric field strength
  • Determines strength and nature of nonlinear optical effects enables phenomena like frequency mixing, harmonic generation, self-phase modulation
  • Allows manipulation of light-matter interactions beyond linear regime (superposition principle)

Derivation of nonlinear wave equation

  • Begin with Maxwell's equations for nonlinear medium:
    1. ×E=Bt\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} (Faraday's law)
    2. ×H=Dt\nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} (Ampère's law)
    3. D=ρ\nabla \cdot \mathbf{D} = \rho (Gauss's law)
    4. B=0\nabla \cdot \mathbf{B} = 0 (Gauss's law for magnetism)
  • Assume no free charges (ρ=0\rho = 0) and non-magnetic medium (B=μ0H\mathbf{B} = \mu_0 \mathbf{H}) for simplification
  • Express electric displacement field D\mathbf{D} in terms of electric field E\mathbf{E} and nonlinear polarization PNL\mathbf{P}_{NL}: D=ε0E+PNL\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}_{NL}
  • Substitute expressions for D\mathbf{D} and B\mathbf{B} into Maxwell's equations and simplify to obtain nonlinear wave equation: 2E1c22Et2=1ε0c22PNLt2\nabla^2 \mathbf{E} - \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = \frac{1}{\varepsilon_0 c^2} \frac{\partial^2 \mathbf{P}_{NL}}{\partial t^2}

Origins of nonlinear optical phenomena

  • Nonlinear optical phenomena arise from nonlinear response of material to intense electric fields (focused laser beams)
  • Under strong electric fields, induced polarization in medium becomes nonlinear deviates from linear proportionality
  • Nonlinear polarization acts as source term in nonlinear wave equation generates new frequency components and modifies wave propagation
  • Physical mechanisms contributing to nonlinear optical phenomena include:
    • Electronic polarization: distortion of electron cloud around atoms or molecules (Kerr effect)
    • Molecular orientation: alignment of polar molecules in strong electric fields (Pockels effect)
    • Electrostriction: change in material density due to applied electric field (stimulated Brillouin scattering)
    • Thermal effects: temperature changes induced by absorption of high-intensity light (thermal lensing)

Linear vs nonlinear optical responses

  • Linear optical response:
    • Occurs at low light intensities below nonlinear threshold
    • Induced polarization proportional to applied electric field follows superposition principle
    • Described by linear susceptibility χ(1)\chi^{(1)} governs linear optical properties
    • Examples: reflection, refraction, absorption, dispersion (prism)
  • Nonlinear optical response:
    • Occurs at high light intensities above nonlinear threshold (gigawatts per square centimeter)
    • Induced polarization has nonlinear dependence on applied electric field violates superposition principle
    • Described by higher-order nonlinear susceptibilities χ(2)\chi^{(2)}, χ(3)\chi^{(3)}, etc. enables novel optical phenomena
    • Examples: second-harmonic generation (frequency doubling), third-harmonic generation, sum-frequency generation (two photons in, one photon out), four-wave mixing (phase conjugation), self-phase modulation (spectral broadening)


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.