Tool · Laser characterization

Coherence Length Calculator

Over what path difference will a source still give interference fringes? From the centre wavelength and the linewidth, in frequency or wavelength, and the shape of the spectrum, the calculator gives the coherence time and length, the fringe visibility at a chosen path difference, the path difference where visibility halves, and for a Gaussian spectrum the OCT axial resolution. Background: coherence length, temporal coherence, linewidth, interferometer, and optical coherence tomography.

Source

Centre wavelength (nm)

Linewidth, full width at half maximum

Linewidth unit

Line shape

Interferometer

Group index of the medium (1 for air)

Path difference between the arms (m)

A free-running semiconductor or fiber laser has a close to Lorentzian line; broadband sources such as LEDs and superluminescent diodes are closer to Gaussian. Changing the unit keeps the same linewidth. The path difference is the physical length difference in the medium; for fiber, enter its group index, about 1.468.

Presets

The laser and LED linewidths are the representative values tabulated in the site’s coherence length entry; the OCT and filtered white-light cases are illustrative. A datasheet value for a real source replaces them.

Readouts

Fringe visibility against path difference

LorentzianGaussianrectangular

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. Coherence times and lengths for the three line shapes, the bandwidth conversion, fringe visibilities and the OCT axial resolution are compared with values worked out by hand and with the figures in the site’s coherence length entry; the coherence time of each shape is also integrated numerically from its correlation function, and the number parser is checked on decimal and thousands commas.

Check	Expected	Computed	Tolerance

The coherence-time factors for the three line shapes follow Saleh and Teich, Fundamentals of Photonics, ch. 12. The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

A spectrum of full width at half maximum $\Delta\nu$ has a complex degree of temporal coherence $g(\tau)$ , the normalized Fourier transform of its power spectrum. The coherence time is defined as

\tau_c = \int_{-\infty}^{\infty} |g(\tau)|^2\, d\tau = \frac{1}{\pi\Delta\nu},\quad \frac{\sqrt{2\ln 2/\pi}}{\Delta\nu},\quad \frac{1}{\Delta\nu}

for Lorentzian, Gaussian and rectangular spectra, and the coherence length in a medium of group index $n$ is $\ell_c = c\tau_c/n$ . A bandwidth in wavelength converts as $\Delta\nu = c\,\Delta\lambda/\lambda_0^2$ . In a two-beam interferometer with equal powers in the arms, the fringe visibility at a path difference $\Delta L$ is $|g(n\Delta L/c)|$ : for a Lorentzian line $e^{-\pi\Delta\nu\tau}$ , for a Gaussian $e^{-(\pi\Delta\nu\tau)^2/4\ln 2}$ , and for a rectangular spectrum $|\mathrm{sinc}(\Delta\nu\tau)|$ . Many tables quote the simpler $c/\Delta\nu$ , which for a Lorentzian line is $\pi$ times longer; the calculator shows both. The OCT axial resolution for a Gaussian spectrum is $(2\ln 2/\pi)\,\lambda_0^2/\Delta\lambda$ .

Worked example

A DFB laser at 1550 nm with a 1 MHz Lorentzian linewidth has a coherence time of 318.3 ns and a coherence length of 95.43 m in air, against 299.8 m from $c/\Delta\nu$ . In an interferometer whose arms differ by 10 m of fiber with group index 1.4682, the fringe visibility is 0.8574. An OCT source at 840 nm with a 50 nm Gaussian spectrum gives an axial resolution of 6.227 µm in air.

References: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007), ch. 12. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), ch. 4. W. Drexler and J. G. Fujimoto, eds., Optical Coherence Tomography, 2nd ed. (Springer, 2015).