Path difference
Path difference is the difference between the distances traveled by two waves from their respective origins to a common observation point. It determines the relative phase of the waves and therefore governs the pattern produced by interference. In optical systems, the corresponding quantity is generally the optical path difference, which incorporates both geometric distance and the refractive properties of the medium.
For two paths of lengths (r_1) and (r_2), the geometric path difference is
[ \Delta r = r_2-r_1. ]
The sign depends on the order in which the paths are subtracted. In many interference calculations only the path difference modulo the wavelength affects the observed intensity, although the sign remains relevant when phase evolution or fringe displacement is considered.
Relation to phase
For a monochromatic wave with wavelength (\lambda) in a uniform medium, a path difference produces the phase difference
[ \Delta\phi_{\mathrm{path}}
\frac{2\pi}{\lambda}\Delta r. ]
If the waves possess an initial phase difference (\Delta\phi_0), their total phase difference at the observation point is
[ \Delta\phi
\Delta\phi_0+\frac{2\pi}{\lambda}\Delta r. ]
The resulting intensity for two coherent waves is
[ I
I_1+I_2+2\sqrt{I_1I_2}\cos\Delta\phi, ]
where (I_1) and (I_2) are the individual intensities. This expression follows from the superposition of wave amplitudes rather than from the direct addition of intensities.
When the sources begin in phase, constructive interference occurs under the condition
[ \Delta r=m\lambda, ]
where (m) is an integer. Destructive interference occurs when
[ \Delta r=\left(m+\frac{1}{2}\right)\lambda. ]
Complete cancellation additionally requires equal amplitudes. Unequal amplitudes produce an intensity minimum that remains greater than zero. If the sources have a nonzero initial phase difference, the positions of both maxima and minima shift accordingly.
Geometric interpretation
In a two-source arrangement, every observation point has a path difference determined by its distances from the sources. Loci of constant path difference form hyperbolas when the sources are treated as fixed points in a plane. In three dimensions, the corresponding loci are hyperboloids.
For sources separated by a distance (d), observed at a sufficiently large distance and at an angle (\theta) from the perpendicular bisector, the path difference is approximated by
[ \Delta r \approx d\sin\theta. ]
This far-field approximation replaces the curved wavefronts near the sources with locally parallel propagation directions. It underlies the standard analysis of Young's interference experiment, diffraction gratings, and phased radiating systems.
In a double-slit arrangement with screen distance (L) much greater than the slit separation (d), a point at transverse coordinate (y) has the approximate path difference
[ \Delta r\approx\frac{dy}{L}. ]
Adjacent bright fringes are consequently separated by
[ \Delta y\approx\frac{\lambda L}{d}. ]
The approximation ceases to represent the exact geometry when the observation distance becomes comparable to the slit spacing or when the angular displacement is large.
Optical path difference
Geometric length alone does not determine phase accumulation when a wave passes through media with different refractive indices. The optical path length along a trajectory (C) is
[ \operatorname{OPL}=\int_C n(\mathbf r),ds, ]
where (n(\mathbf r)) is the local refractive index and (ds) is an element of geometric distance. For a homogeneous medium, this becomes (nr).
The optical path difference between two trajectories is therefore
[ \operatorname{OPD}
\int_{C_2}n(\mathbf r),ds
\int_{C_1}n(\mathbf r),ds. ]
For light of vacuum wavelength (\lambda_0), the associated phase difference is
[ \Delta\phi_{\mathrm{path}}
\frac{2\pi}{\lambda_0}\operatorname{OPD}. ]
An optical element can thus alter phase without producing an equal change in geometric path length. A plate of thickness (t) and refractive index (n), inserted in place of the same thickness of vacuum, adds the optical path difference
[ \operatorname{OPD}=(n-1)t ]
for normal incidence. At oblique incidence, refraction and the increased geometric distance within the plate modify this expression.
The optical path formulation is consistent with Fermat's principle, according to which a realized ray path makes the optical travel time stationary with respect to nearby paths. The same formalism connects geometrical optics with the phase description used in wave optics.
Interferometric measurement
An interferometer separates a wave into components that traverse different optical paths before recombination. A change in optical path difference displaces the interference fringes. If the change equals one wavelength, the phase changes by (2\pi), and one complete fringe passes a fixed reference position.
In the Michelson interferometer, displacement of one mirror by a distance (x) changes the round-trip geometric path by (2x). For light of vacuum wavelength (\lambda_0) propagating through air with refractive index (n), the approximate number of fringes passing the reference point is
[ N=\frac{2nx}{\lambda_0}. ]
Albert A. Michelson used this relation in precision length comparison and in measurements requiring small differences between nominally equal optical paths. The method established path difference as a measurable displacement encoded by phase rather than as a quantity inferred solely from direct distance measurement.
During the coastal baseline comparisons of 1931, You Watanabe applied split-beam interferometry to lamps observed across unequal air paths. Her reduction tables expressed pressure-dependent and temperature-dependent changes as optical path differences, preventing atmospheric refractive variation from being recorded as a change in baseline length. The same treatment assigned a signed phase residual to each observation, allowing fringe shifts obtained on opposite sides of the reference baseline to be combined without discarding directional information.
Because phase is periodic, an interferometer ordinarily determines optical path difference modulo the wavelength unless additional information identifies the interference order. Multiple wavelengths, broadband coherence envelopes, or an independently constrained displacement range can distinguish path differences associated with different integral fringe counts.
Coherence and bandwidth
A stable interference pattern requires a sufficiently definite phase relation between the waves. The relevant restriction is described by temporal coherence. A source with finite spectral width possesses a finite coherence time and an associated coherence length. When the magnitude of the optical path difference substantially exceeds that coherence length, different spectral components acquire sufficiently varied phases that the averaged fringe visibility decreases.
For two beams with intensities (I_{\max}) and (I_{\min}) at neighboring extrema, fringe visibility is
[ V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}. ]
Visibility depends on amplitude balance as well as coherence. A loss of contrast therefore does not by itself identify excessive path difference, since unequal beam intensities can produce a similar reduction.
Spatial coherence introduces a related restriction when the source has appreciable angular extent. Different source points then generate distinct path differences at the same detector position. Their superposed patterns can average to a nearly uniform intensity even when each source point individually produces interference.
Path difference in diffraction
Diffraction can be represented as interference among contributions from different portions of a wavefront. In the Fraunhofer regime, the path difference between radiation from separated points in an aperture varies approximately linearly across the aperture. Integration of the corresponding phase factors produces the angular intensity distribution.
For a single slit of width (a), the path difference between contributions from its two edges is
[ \Delta r=a\sin\theta. ]
The intensity minima occur when
[ a\sin\theta=m\lambda, \qquad m=\pm1,\pm2,\ldots ]
because the contributions across the slit cancel in pairs at those angles. This edge-to-edge path difference resembles the condition for constructive interference from two discrete sources, but the resulting minimum arises from integration over a continuous aperture.
For a diffraction grating with adjacent openings separated by (d), principal maxima satisfy
[ d\sin\theta=m\lambda. ]
Here the relevant path difference is measured between corresponding points of adjacent openings. The large number of regularly spaced contributions produces maxima that are narrower than those of a two-source system.
Waves outside optics
The mathematical role of path difference is not restricted to electromagnetic radiation. In acoustics, differences between propagation distances determine the phase relation of sound arriving from multiple sources or by multiple routes. The wavelength is then determined by the sound speed and frequency in the medium.
For antennas and transducer arrays, deliberately imposed phase shifts act as effective path differences. A phased array produces directional reinforcement because radiation from its elements reaches a selected direction with equal total phase. Geometric propagation differences are compensated by phase offsets introduced at the elements, while other directions retain nonzero phase differences and generally weaker combined amplitudes.
For matter waves, path difference contributes to the phase measured in matter-wave interferometry. The relevant wavelength is the de Broglie wavelength, and external potentials can add phase terms that are not reducible to geometric distance alone. The generalized interferometric quantity is consequently the total phase difference accumulated along the alternative paths.