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🔬Modern Optics

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8.2 Second-order nonlinear effects: frequency doubling and parametric processes

3 min readLast Updated on July 22, 2024

Second-order nonlinear effects in optics manipulate light in fascinating ways. These phenomena, like second harmonic generation and optical parametric amplification, allow us to create new frequencies and amplify weak signals using special materials and precise conditions.

These effects have revolutionized laser technology and optical communications. By harnessing nonlinear processes, we can generate tunable light sources, create shorter wavelengths, and develop high-speed modulators for various applications in science and technology.

Second-order Nonlinear Effects

Process of second harmonic generation

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  • Nonlinear optical process doubles the frequency of an input light wave
    • Converts input wave with frequency ω\omega to output wave with frequency 2ω2\omega
    • Halves the wavelength of the output wave compared to the input wave (1064 nm to 532 nm)
  • Occurs in nonlinear optical materials with non-zero second-order susceptibility χ(2)\chi^{(2)}
    • Materials include KDP (potassium dihydrogen phosphate) and BBO (beta barium borate)
  • Nonlinear polarization P(2)P^{(2)} induced in the material proportional to the square of the input electric field EE: P(2)=ϵ0χ(2)E2P^{(2)} = \epsilon_0 \chi^{(2)} E^2
    • Nonlinear polarization acts as a source term for the second harmonic wave
  • Intensity of the second harmonic wave depends on the square of the input intensity and length of the nonlinear material
    • Doubling the input intensity quadruples the second harmonic intensity
    • Increasing the interaction length enhances the conversion efficiency

Phase-matching for nonlinear processes

  • Crucial for efficient second-order nonlinear processes (SHG, parametric amplification)
    • Ensures generated nonlinear waves maintain fixed phase relationship with input waves throughout the nonlinear material
  • Phase mismatch Δk\Delta k is the difference between the wave vectors of the interacting waves
    • For SHG: Δk=k22k1\Delta k = k_2 - 2k_1, where k1k_1 and k2k_2 are wave vectors of fundamental and second harmonic waves
  • Perfect phase-matching occurs when Δk=0\Delta k = 0
    • Leads to constructive interference and efficient nonlinear conversion
  • Methods to achieve phase-matching:
    1. Birefringent phase-matching: Utilizes difference in refractive indices for ordinary and extraordinary waves in birefringent crystals (BBO, LiNbO3)
    2. Quasi-phase-matching (QPM): Periodically modulates sign of nonlinear coefficient to compensate for phase mismatch
      • Achieved through periodic poling of ferroelectric materials (lithium niobate)

Optical parametric amplification and oscillation

  • Optical parametric amplification (OPA): Second-order nonlinear process amplifies weak input signal using strong pump wave
    • Pump photon with frequency ωp\omega_p splits into signal photon (ωs\omega_s) and idler photon (ωi\omega_i)
    • Energy conservation: ωp=ωs+ωi\omega_p = \omega_s + \omega_i
    • Phase-matching conditions must be satisfied for efficient OPA
  • Optical parametric oscillation (OPO): Extension of OPA incorporating feedback to generate tunable coherent light
    • Nonlinear crystal placed inside optical cavity resonant at signal or idler frequency (or both)
    • Above threshold pump power, OPO starts to oscillate, generating signal and idler waves
  • OPA and OPO generate tunable light sources in infrared and visible regions
    • Wavelength tuning achieved by adjusting phase-matching conditions (crystal angle, temperature)

Applications of second-order nonlinear effects

  • Frequency doubling (SHG) generates shorter wavelengths from available laser sources
    • Doubling 1064 nm Nd:YAG laser generates 532 nm green light
    • Applications: laser displays, microscopy, spectroscopy
  • Optical parametric oscillators (OPOs) employed as tunable coherent light sources
    • Applications: spectroscopy, remote sensing, quantum optics experiments
  • Electro-optic modulators (EOMs) utilize second-order nonlinear effects for high-speed light modulation
    • Pockels effect induces refractive index change proportional to applied electric field
    • Applications: telecommunications, optical switching, pulse picking
  • Nonlinear frequency conversion techniques (sum-frequency generation (SFG), difference-frequency generation (DFG)) generate light at specific wavelengths
    • Applications: quantum optics, spectroscopy, optical frequency metrology


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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.