Optical Linewidth, Coherence Time, and Coherence Length by Line Shape

Good to see you again. In the previous lesson, we used a normalized spectral line shape to distribute a transition’s stimulated-emission strength across frequency. That was a property of the gain transition. We now turn to a related but distinct question: how long does the optical field itself retain a predictable phase relationship with a delayed copy of itself?

This is the final lesson of Module 1. You will connect an optical spectrum of specified shape and linewidth to its temporal coherence, then convert that time into a coherence length relevant to interferometers and path-length tolerances. The central message is simple, but the numerical prefactor depends on the line shape and on the definition of “coherence time.”

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Temporal coherence: interference with a delayed field

Consider a quasi-monochromatic complex optical field at one point in space,

$$E(t).$$

Temporal coherence asks whether the field now resembles the field a delay $\tau$ earlier. The standard normalized first-order field correlation is

$$\gamma (\tau )=\frac{⟨E^{\ast }(t)E(t+\tau )⟩}{⟨{\left|E(t)\right|}^2⟩}.$$

It has the properties

$$\gamma (0)=1,$$

and, for a finite-linewidth source,

$$\left|\gamma (\tau )\right|\to 0$$

as the delay becomes large. The magnitude $\left|\gamma (\tau )\right|$ is the quantity of practical interest: it describes how rapidly a fixed phase relationship is lost.

A Michelson interferometer makes this concrete. It combines a field with a delayed version of itself. If the two interfering beams have equal intensities and good spatial overlap, the fringe visibility is

$$\mathcal{V}(\tau )=\left|\gamma (\tau )\right|.$$

Thus, the familiar experimental statement “fringes disappear when the path difference becomes too large” means that the delayed field has become weakly correlated with the undelayed field.

It is important not to confuse:

• Temporal coherence, determined primarily by spectral linewidth and measured through a time delay or optical path difference.
• Spatial coherence, determined by correlations across different transverse points of a beam.

Laser beams often have high spatial coherence, but their temporal coherence can vary dramatically. A multimode diode laser, a narrow-linewidth single-frequency laser, and a broadband amplified-spontaneous-emission source may all be well collimated while having very different coherence lengths.

The following short video gives a useful interferometric picture of this distinction.

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Why linewidth determines coherence time

Let $p(\nu )$ be the source’s normalized optical power spectrum:

$${\int }_{-\infty }^{\infty }p(\nu )d\nu =1.$$

Unlike the $g_{\nu }(\nu )$ from the preceding lesson, which described a material transition, $p(\nu )$ here describes the spectrum of the radiation whose coherence is being assessed. They can have similar shapes, but they need not have the same width. For example, a gain medium may have a broad emission band while a laser cavity selects a much narrower lasing line.

For stationary light, the normalized correlation function and normalized spectrum form a Fourier-transform pair:

$$\gamma (\tau )={\int }_{-\infty }^{\infty }p(\nu )\exp \left(i2\pi \nu \tau \right)d\nu .$$

The carrier oscillation at the central frequency ${\nu }_0$ contributes a phase factor of the form

$$\exp \left(i2\pi {\nu }_0\tau \right).$$

It changes the location of bright and dark fringes, but not the envelope magnitude $\left|\gamma (\tau )\right|$. That envelope is set by the spectral width.

A broad spectrum contains frequency components that accumulate relative phase rapidly when one interferometer arm is delayed. Their interference terms then average out. A narrow spectrum has less differential phase accumulation, so visible interference persists over a longer delay.

The relationship is therefore an inverse one:

$${\tau }_{\mathrm{coh}}\sim \frac{1}{\Delta \nu }.$$

The symbol $\sim$ matters. A linewidth alone fixes the scale of the coherence time, but the exact coefficient depends on the spectral shape and on the convention used to compress a full correlation function into one number.

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A precise convention for coherence time

For this lesson, use the general integral convention

$$\boxed{{\tau }_{\mathrm{coh}}={\int }_{-\infty }^{\infty }{\left|\gamma (\tau )\right|}^{2}d\tau .}$$

This definition is particularly useful because it works for arbitrary line shapes. It measures the effective duration over which delayed copies of the field remain correlated.

Other definitions exist. One common alternative calls the delay at which $\left|\gamma (\tau )\right|$ falls to $1/e$ the coherence time. For a Lorentzian, that happens to equal the integral definition above; for a Gaussian, it does not. Therefore:

Never use a numerical linewidth-to-coherence-time formula without knowing both the assumed line shape and the convention for coherence time.

The two most useful cases are Lorentzian and Gaussian spectra.

Lorentzian spectral line

Let the optical power spectrum have Lorentzian FWHM $\Delta {\nu }_{\mathrm{L}}$:

$$p_{\mathrm{L}}(\nu )=\frac{1}{\pi }\frac{\Delta {\nu }_{\mathrm{L}}/2}{(\nu -{\nu }_0{)}^2+{\left(\Delta {\nu }_{\mathrm{L}}/2\right)}^2}.$$

Fourier transformation gives

$${\gamma }_{\mathrm{L}}(\tau )=\exp \left(i2\pi {\nu }_0\tau \right)\exp \left(-\pi \Delta {\nu }_{\mathrm{L}}\left|\tau \right|\right).$$

Hence the magnitude decays exponentially:

$$\left|{\gamma }_{\mathrm{L}}(\tau )\right|=\exp \left(-\pi \Delta {\nu }_{\mathrm{L}}\left|\tau \right|\right).$$

Substituting into the coherence-time definition gives

$$\boxed{{\tau }_{\mathrm{coh},\mathrm{L}}=\frac{1}{\pi \Delta {\nu }_{\mathrm{L}}}.}$$

This is the standard relation for a phase-diffusion-broadened single-mode laser. Random phase kicks, including the fundamental contribution associated with spontaneous emission into the lasing mode, naturally produce this exponential correlation decay and Lorentzian spectrum.

Gaussian spectral line

Suppose instead that the source spectrum is Gaussian with FWHM $\Delta {\nu }_{\mathrm{G}}$:

$$p_{\mathrm{G}}(\nu )=\frac{2\sqrt{\ln 2}}{\Delta {\nu }_{\mathrm{G}}\sqrt{\pi }}\exp \left[-4\ln 2{\left(\frac{\nu -{\nu }_0}{\Delta {\nu }_{\mathrm{G}}}\right)}^2\right].$$

The correlation function is also Gaussian:

$${\gamma }_{\mathrm{G}}(\tau )=\exp \left(i2\pi {\nu }_0\tau \right)\exp \left[-\frac{{\pi }^2\Delta {\nu }_{\mathrm{G}}^2{\tau }^2}{4\ln 2}\right].$$

Applying the same integral definition gives

$$\boxed{{\tau }_{\mathrm{coh},\mathrm{G}}=\frac{1}{\Delta {\nu }_{\mathrm{G}}}\sqrt{\frac{2\ln 2}{\pi }}\simeq \frac{0.664}{\Delta {\nu }_{\mathrm{G}}}.}$$

A Gaussian correlation envelope is typical when Gaussian spectral broadening dominates, as in an idealized Doppler-broadened source. Its coherence-time coefficient is about twice the Lorentzian coefficient for equal FWHM linewidth:

$$\frac{0.664}{1/\pi }\simeq \mathrm{2.09.}$$

This difference is not contradictory. The two profiles distribute the same FWHM bandwidth differently: a Lorentzian has long spectral wings and an exponential temporal decay, whereas a Gaussian falls rapidly in both domains.

| Spectral power profile | Correlation-envelope magnitude | Coherence time from $\int |\gamma {|}^{2}d\tau$ |
|---|---|---:|
| Lorentzian, FWHM $\Delta {\nu }_{\mathrm{L}}$ | $\exp (-\pi \Delta {\nu }_{\mathrm{L}}|\tau |)$ | $\frac{1}{\pi \Delta {\nu }_{\mathrm{L}}}$ |
| Gaussian, FWHM $\Delta {\nu }_{\mathrm{G}}$ | $\exp \left[-\frac{{\pi }^2\Delta {\nu }_{\mathrm{G}}^2{\tau }^2}{4\ln 2}\right]$ | $\frac{0.664}{\Delta {\nu }_{\mathrm{G}}}$ |

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From coherence time to coherence length

The conventional temporal coherence length is the vacuum distance light travels during the coherence time:

$$\boxed{L_{\mathrm{coh}}=c{\tau }_{\mathrm{coh}}.}$$

For the two line shapes above,

$$\boxed{L_{\mathrm{coh},\mathrm{L}}=\frac{c}{\pi \Delta {\nu }_{\mathrm{L}}}}$$

and

$$\boxed{L_{\mathrm{coh},\mathrm{G}}=\frac{0.664c}{\Delta {\nu }_{\mathrm{G}}}.}$$

These lengths characterize an allowable optical path difference, not the physical length of a beam or the transverse diameter over which the beam is spatially coherent.

If propagation occurs in a uniform dispersive medium, a physical excess length $\Delta L$ produces a delay approximately given by

$$\Delta \tau =\frac{n_g\Delta L}{c},$$

where $n_g$ is the group index. Therefore, the corresponding physical propagation distance in that medium is

$$\Delta L_{\mathrm{coh}}=\frac{c{\tau }_{\mathrm{coh}}}{n_g}.$$

For most elementary interferometer calculations in air or vacuum, use $L_{\mathrm{coh}}=c{\tau }_{\mathrm{coh}}$. In fiber or a dispersive optical material, distinguish vacuum-equivalent coherence length from the physical distance associated with the same delay.

The image illustrates a central experimental fact: the rapidly oscillating fringes are governed mainly by the optical carrier wavelength, while the slowly varying fringe envelope is governed by temporal coherence. Narrowing the spectrum broadens this envelope.

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Converting a wavelength linewidth

Spectrometers often report a wavelength FWHM $\Delta \lambda$, while coherence formulas are usually expressed in frequency bandwidth. Since

$$\nu =\frac{c}{\lambda },$$

a small linewidth relative to its central wavelength satisfies

$$\boxed{\Delta \nu \simeq \frac{c\Delta \lambda }{{\lambda }_0^2}.}$$

The approximation requires

$$\Delta \lambda \ll {\lambda }_0.$$

Substitution gives convenient narrowband expressions:

$$\boxed{L_{\mathrm{coh},\mathrm{L}}\simeq \frac{{\lambda }_0^2}{\pi \Delta \lambda }}$$

for a Lorentzian spectrum, and

$$\boxed{L_{\mathrm{coh},\mathrm{G}}\simeq 0.664\frac{{\lambda }_0^2}{\Delta \lambda }}$$

for a Gaussian spectrum.

For broadband sources, do not assume that a spectrum is equally Gaussian or symmetric in frequency and wavelength. The nonlinear transformation between $\lambda$ and $\nu$ changes the profile. In that situation, transform the full measured spectrum before calculating $\gamma (\tau )$.

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Worked comparisons

A narrow-linewidth laser

A single-frequency laser has a Lorentzian optical linewidth

$$\Delta {\nu }_{\mathrm{L}}=100\mathrm{kHz}.$$

Its coherence time is

$${\tau }_{\mathrm{coh},\mathrm{L}}=\frac{1}{\pi (1.00\times 10^5\mathrm{Hz})}=3.18\mu \mathrm{s}.$$

The corresponding vacuum coherence length is

$$L_{\mathrm{coh},\mathrm{L}}=(3.00\times 10^8\mathrm{m}{\mathrm{s}}^{-1})(3.18\times 10^{-6}\mathrm{s}),$$

so

$$\boxed{L_{\mathrm{coh},\mathrm{L}}\simeq 955\mathrm{m}.}$$

Thus, even a linewidth that is nonzero by spectroscopic standards can support interference over hundreds of metres of path mismatch.

A broadband Gaussian source

Consider a source centered at

$${\lambda }_0=633\mathrm{nm}$$

with a Gaussian wavelength width

$$\Delta \lambda =10\mathrm{nm}.$$

Because the fractional bandwidth is modest, first convert its width:

$$\Delta {\nu }_{\mathrm{G}}\simeq \frac{(3.00\times 10^8\mathrm{m}{\mathrm{s}}^{-1})(10\times 10^{-9}\mathrm{m})}{(633\times 10^{-9}\mathrm{m}{)}^2}.$$

This yields

$$\Delta {\nu }_{\mathrm{G}}\simeq 7.49\times 10^{12}\mathrm{Hz}.$$

The Gaussian coherence time is

$${\tau }_{\mathrm{coh},\mathrm{G}}=\frac{0.664}{7.49\times 10^{12}\mathrm{Hz}}\simeq 88.7\mathrm{fs}.$$

Therefore,

$$\boxed{L_{\mathrm{coh},\mathrm{G}}\simeq 26.6\mu \mathrm{m}.}$$

The contrast with the narrow-line laser is stark: a broad source can still display interference, but only when the interferometer’s optical path difference is matched to within a few tens of micrometres. That short coherence length is precisely what makes broad-bandwidth sources useful for depth-selective interferometry.

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A reliable calculation procedure

When a problem specifies a linewidth and line shape, use this sequence:

1. Identify the relevant spectrum. Use the optical emission or laser spectrum, not automatically the atomic or gain-transition linewidth.

2. State the line shape and width convention. Confirm that the supplied $\Delta \nu$ is an intensity or power-spectrum FWHM.

3. Convert wavelength width to frequency width when justified. For narrow relative bandwidth, use

   $$\Delta \nu \simeq \frac{c\Delta \lambda }{{\lambda }_0^2}.$$

4. Use the line-shape-specific coherence relation. In this lesson’s convention:

   $${\tau }_{\mathrm{coh},\mathrm{L}}=\frac{1}{\pi \Delta {\nu }_{\mathrm{L}}},$$ $${\tau }_{\mathrm{coh},\mathrm{G}}=\frac{0.664}{\Delta {\nu }_{\mathrm{G}}}.$$

5. Convert time to length. Use

   $$L_{\mathrm{coh}}=c{\tau }_{\mathrm{coh}}$$

   for a vacuum-equivalent optical path difference, or account for $n_g$ when a physical distance in a medium is requested.

A quick dimensional check is valuable: linewidth in hertz has units of ${\mathrm{s}}^{-1}$, so its inverse is a time; multiplying by $c$ produces a length.

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Takeaways

Temporal coherence quantifies how strongly an optical field remains correlated with a delayed copy of itself. In an interferometer, the magnitude of the field correlation determines fringe visibility.

The spectrum and correlation function are Fourier-transform partners. Consequently, a narrower optical linewidth means a longer coherence time and coherence length, but the exact numerical coefficient depends on the spectral line shape.

For the integral definition

$${\tau }_{\mathrm{coh}}={\int }_{-\infty }^{\infty }|\gamma (\tau ){|}^{2}d\tau ,$$

the principal results are

$$\boxed{{\tau }_{\mathrm{coh},\mathrm{L}}=\frac{1}{\pi \Delta {\nu }_{\mathrm{L}}},L_{\mathrm{coh},\mathrm{L}}=\frac{c}{\pi \Delta {\nu }_{\mathrm{L}}}}$$

for a Lorentzian spectrum, and

$$\boxed{{\tau }_{\mathrm{coh},\mathrm{G}}=\frac{0.664}{\Delta {\nu }_{\mathrm{G}}},L_{\mathrm{coh},\mathrm{G}}=\frac{0.664c}{\Delta {\nu }_{\mathrm{G}}}}$$

for a Gaussian spectrum.

This completes Module 1’s foundation: radiative transitions, broadening, gain cross sections, and coherence. Next, Module 2 begins with population dynamics and the key limitation of a continuously pumped ideal two-level medium: it cannot sustain a population inversion.