Convert a laser spectral width between wavelength bandwidth (nm) and frequency bandwidth (GHz) using an exact end-point method and a quick small-bandwidth approximation. You can also estimate coherence time, coherence length in a medium, and quality factor (Q) from the same linewidth.
Advanced options
How to use our Laser Linewidth and Bandwidth Calculator
- Choose what you want to solve for: convert delta_lambda to delta_nu, or convert delta_nu to delta_lambda.
- Enter the center wavelength lambda0 (nm) from your laser spec (for example 1550 nm).
- If converting from wavelength to frequency, enter delta_lambda (nm). In Advanced options, confirm whether your value is a full width or a half-width.
- If converting from frequency to wavelength, enter delta_nu (GHz).
- Open Advanced options and set the coherence time convention if you want coherence time/length outputs.
- If you want coherence length in a material (like fiber), enter the refractive index n. Use n = 1.000 for vacuum/air (approx).
- Click Calculate.
- Read the exact and approximate conversions, then check the approximation error percent to see whether the small-bandwidth approximation is safe for your case.
Definitions
Wavelength (lambda): The distance between repeating points of a wave, like crest to crest. In this calculator, lambda0 is the center wavelength of the laser. [1]
Frequency (nu): How many wave cycles pass per second (Hz). Optical frequencies are often shown in THz.
Linewidth / bandwidth (delta): The width of the laser spectrum (how spread out it is). Here, delta_lambda is in nm and delta_nu is in Hz or GHz.
Refractive index (n): A number that tells how fast light travels in a material compared with vacuum; it affects coherence length in that medium. [2]
Quality factor (Q): A ratio that compares center frequency to linewidth: higher Q means a narrower line relative to its center.
Coherence time (tau_c): A time scale for how long the light stays phase-related; different books use different constant factors.
Coherence length (L_c): A distance scale related to coherence time; in a medium it is shorter when n is larger.
Methodology
Constants and unit conversions
Speed of light in vacuum:
c = 299,792,458 m/s (used for all conversions). [1]
Unit conversions used:
1 nm = 1e-9 m, 1 GHz = 1e9 Hz, 1 THz = 1e12 Hz.
Step 1: Center frequency from center wavelength
ν0 (Hz) = c / λ0 (m)
lambda0 is entered in nm and converted to meters before the math.
Step 2A: Exact conversion from wavelength width to frequency width
The calculator treats delta_lambda as a symmetric span around lambda0. If you select delta_lambda meaning = half-width, it first converts to full width:
delta_lambda_full = 2 * delta_lambda_input.
Δν_exact (Hz) = c/(λ0 - Δλ/2) - c/(λ0 + Δλ/2)
Inputs must satisfy λ0 > 0 and (λ0 - Δλ/2) > 0 (equivalently, Δλ < 2*λ0) so the end-point wavelengths stay positive.
Step 2B: Small-bandwidth approximation (useful for quick estimates)
Δν_approx (Hz) = (c / λ0^2) * Δλ
Δλ_approx (m) = (λ0^2 / c) * Δν
The approximation comes from differentiating ν = c/λ and is most accurate when delta_lambda is much smaller than lambda0.
Approximation error percent
Error (%) = 100 * (Δν_approx - Δν_exact) / Δν_exact
If Δν_exact is 0 or not finite (should not happen with valid inputs), the calculator shows N/A instead of dividing by zero. A note is shown when the absolute error is above 1% to encourage using the exact direction for better accuracy.
Q factor from linewidth
Q = ν0 / Δν
Δν is the frequency linewidth used in the active mode (computed exact when converting from delta_lambda, or taken from your entered delta_nu when converting from frequency). If the result is not finite (for example, extremely tiny delta_nu), the calculator shows a message like "Very large (check inputs)" instead of Infinity.
Coherence time and coherence length (optional)
You choose the coherence time convention. This changes only tau_c and L_c, not the wavelength-frequency conversion.
τ_c (s) = 1 / Δν
τ_c (s) = 1 / (2*π*Δν)
L_c (m) = (c / n) * τ_c
n is the refractive index of the medium (must be > 0). In a material (n > 1), the coherence length in that medium is shorter because the wave travels slower there. [2]