Fringe visibility (fringe contrast)

Fringe visibility, also called fringe contrast, is the number that says how deep the modulation of an interference pattern is:

$$V = \frac{I_\text{max} - I_\text{min}}{I_\text{max} + I_\text{min}}.$$

It runs from 0, a uniform field with no fringes, to 1, fringes whose dark bands reach zero intensity. A well-aligned interferometer with a single-frequency laser and equal arm powers gives values above 0.95; a white-light source shows fringes only within a few micrometers of zero path difference; and light from two independent sources gives $V = 0$ at any path difference.

Intensity ratio, coherence and polarization

Three factors set the visibility of two-beam fringes, and they multiply. For beams of intensity $I_1$ and $I_2$ with complex degree of coherence $\gamma_{12}$ and parallel polarizations,

$$V = \frac{2\sqrt{I_1 I_2}}{I_1 + I_2}\,|\gamma_{12}|.$$

The first factor depends only on the intensity ratio $r = I_2/I_1$ and equals $2\sqrt{r}/(1+r)$. It is 1 for equal beams, 0.80 for 1:4, 0.57 for 1:10 and 0.198 for 1:100. The dependence is weak: a beam carrying only 7.2% of the power of the other still gives $V = 0.5$, which is why a stray reflection of a few percent produces clearly visible fringes. With a 1:4 ratio and $|\gamma_{12}| = 0.5$, the two factors give $V = 0.40$.

The second factor, $|\gamma_{12}|$, contains the coherence of the source. Its dependence on the delay between the beams is the temporal coherence; its dependence on the separation of the two points from which the light is taken is the spatial coherence. Measured with equal beams, the visibility equals $|\gamma_{12}|$ directly, which is how coherence is measured in practice.

The third factor is polarization. Only parallel field components interfere, so two linearly polarized beams at an angle $\theta$ to each other give, for equal intensities, $V = \cos\theta$: 0.98 at 10°, 0.87 at 30°, 0.71 at 45° and zero for orthogonal polarizations. In fiber interferometers a slow drift of the polarization state in one arm appears as a slow fading of the fringes.

Visibility and path difference

For a source with a Lorentzian line of full width $\Delta\nu$, the visibility of equal-beam fringes falls with delay $\tau$ as

$$|\gamma(\tau)| = \exp(-\pi\,\Delta\nu\,|\tau|).$$

A laser with a 1 MHz linewidth has $V = 0.50$ at a path difference of 66 m in air and 0.35 at 100 m; at the coherence length $c/(\pi\Delta\nu) = 95$ m it has fallen to $1/e$. A multimode laser behaves differently: its visibility falls and recovers periodically with path difference, at a period of twice the cavity length, because the longitudinal modes beat against each other. The coherence length calculator plots visibility against path difference for several line shapes.

Measuring it

Visibility is read from a recording of the fringes, either across a camera image or from a photodiode while one arm is scanned. Two practical points matter. The minimum must be measured above the detector's dark level and any background light, both of which add to $I_\text{min}$ and $I_\text{max}$ equally and lower $V$. And the detector must resolve the fringes: a pixel or aperture that spans a large fraction of a fringe period averages over it and reports a visibility below the true one.

Scanning the delay in a Michelson interferometer and recording $V$ against path difference gives the modulus of the coherence function; the Fourier transform of the full interferogram gives the spectrum. Recording the full fringe signal rather than only its envelope is the principle of Fourier transform spectroscopy, used in FTIR spectroscopy. In Young's double-slit experiment, recording visibility against slit separation measures spatial coherence: for sunlight at 550 nm the fringes vanish at a separation of about 72 µm.

Visibility and extinction ratio

In a Mach-Zehnder interferometer used as a modulator or switch, the same imperfections appear as a finite extinction ratio, $I_\text{max}/I_\text{min} = (1+V)/(1-V)$. A visibility of 0.9 corresponds to 12.8 dB, 0.99 to 23.0 dB and 0.999 to 33.0 dB, so the 20–30 dB extinction of a telecom modulator requires the arm powers to match within about 25% (23 dB) to 12% (30 dB) and the polarizations to within a few degrees, with everything else ideal.

Common questions

Why is the fringe visibility below 1 with a laser?

Usually because the arm powers are unequal, the polarizations are not parallel, the wavefronts are tilted or curved relative to each other across the detector, or background light is present. Coherence limits the visibility only when the path difference approaches the coherence length.

Is fringe visibility the same as fringe contrast?

Yes. The two terms are used interchangeably for the same quantity, sometimes also called the Michelson contrast.

How much intensity imbalance can an interferometer tolerate?

A good deal for most purposes: a 1:2 ratio still gives $V = 0.94$. The imbalance matters most where high extinction is required, as in modulators and balanced detection.

References: M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 7 and 10; L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995); J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015); E. Hecht, Optics, 5th ed. (Pearson, 2017).