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Interference is a characteristic phenomenon exhibited by waves. When two electromagnetic waves interact with their peaks and troughs coinciding, a resulting wave with enhanced amplitude is produced. This is known as constructive interference. In this case, the two waves interacting are in phase with each other.
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A close look at earthquakes provides evidence for the conditions appropriate for resonance, standing waves, and constructive and destructive interference. A building may vibrate for several seconds with a driving frequency matching the building's natural frequency of vibration; this produces a resonance that results in one building collapsing while the neighboring buildings do not. Often, buildings of a certain height are devastated, while other taller buildings remain intact. This...
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A household microwave and lasers are examples of standing electromagnetic waves in a cavity. When two conducting metal plates are placed parallel at the nodal planes, it creates a cavity where standing waves are formed. The cavity between the two planes is analogous to a stretched string held at the points x = 0 and x = L. Here, the distance 'L' between the two planes must be an integer multiple of half of the wavelength. The wavelengths that satisfy this condition are given by:
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Sometimes waves do not seem to move; rather, they just vibrate in place. Unmoving waves can be seen on the surface of a glass of milk kept in a refrigerator, which is one example of standing waves. Vibrations from the refrigerator motor create waves on the milk that oscillate up and down but do not seem to move across the surface. These waves are formed or created by the superposition of two or more identical moving waves in opposite directions. The waves move through each other, with their...
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Related Experiment Video

Updated: Mar 31, 2026

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
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Second and third harmonic waves excited by focused Gaussian beams.

Uri Levy, Yaron Silberberg

    Optics Express
    |October 20, 2015
    PubMed
    Summary

    Harmonic generation in nonlinear microscopy is not limited to the focal point. Optimal signal intensity occurs when the fundamental Gaussian beam is focused beyond the focal plane, with oscillations observed in thin materials.

    Area of Science:

    • Nonlinear Optics
    • Laser Physics
    • Materials Science

    Background:

    • Harmonic generation using tightly-focused Gaussian beams is crucial for nonlinear microscopy.
    • A common misconception is that nonlinear signals are generated solely within the focal region.
    • Harmonic wave intensity depends on excitation geometry and phase matching conditions over extended material regions.

    Purpose of the Study:

    • To analytically solve the amplitude integral for second and third harmonic waves.
    • To investigate the dependence of generated harmonic intensities on focal-plane position.
    • To analyze harmonic generation in thin materials and estimate wave-vector mismatch.

    Main Methods:

    • Analytical solution of the amplitude integral for second and third harmonic generation.

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  • Studying generated harmonic intensities as a function of focal-plane position.
  • Experimental reproduction of intensity oscillations in thin glass slabs.

    • Main Results:

    • Maximum harmonic intensity is achieved when the fundamental Gaussian beam is focused a few Rayleigh lengths beyond the front surface, particularly for positive wave-vector mismatch.
    • Strong intensity oscillations with material thickness are predicted by harmonic generation theory for thin samples.
    • Observed intensity oscillations of 517nm third-harmonic waves in glass slabs allowed estimation of the wave-vector mismatch.

    Conclusions:

    • Harmonic generation is influenced by regions beyond the focal plane, challenging the naive assumption of focal-plane dominance.
    • The focal position significantly impacts harmonic signal generation, with optimal focusing found beyond the focal plane.
    • Wave-vector mismatch in Soda-lime glass was estimated to be Δk(H) = -0.249 μm⁻¹ based on observed third-harmonic generation oscillations.