How much bandwidth is there in 1 micron of spectrum at a wavelength of 1 micron?

Concept

For light, frequency and wavelength are tied together by f = c/λ, where c is the speed of light in the medium. A window of spectral width Δλ around a wavelength λ is therefore also a window of frequencies. Differentiating f = c/λ gives |df/dλ| = c/λ2, so a narrow spectral window of width Δλ spans a frequency bandwidth given by Δf = c · Δλ / λ2.

So the bandwidth grows in proportion to the spectral width Δλ and falls with the square of the wavelength λ — it is not simply c/λ multiplied by nothing, and the λ2 in the denominator is what pushes an optical window into the terahertz range.

One caveat, stated up front: Δf = c · Δλ / λ2 is the local (differential) relation, exact in the limit Δλ ≪ λ. Here Δλ equals λ, so the figure it gives is the standard textbook estimate of the bandwidth of a spectral window at that wavelength, not an exact endpoint-to-endpoint span; the cross-check below shows that concrete endpoint choices in the vicinity of 1 micron land on the same order of magnitude.

Application

1. List the data: speed of light c = 3 × 108 m/s, spectral width Δλ = 1 micron = 1 × 10−6 m, centre wavelength λ = 1 micron = 1 × 10−6 m.

2. Write the relation: Δf = c · Δλ / λ2.

3. Square the wavelength: λ2 = (1 × 10−6 m)2 = 1 × 10−12 m2.

4. Multiply out the numerator: c · Δλ = (3 × 108 m/s)(1 × 10−6 m) = 3 × 102 m2/s.

5. Divide: Δf = (3 × 102) / (1 × 10−12) = 3 × 1014 Hz.

6. Convert to terahertz: 1 THz = 1012 Hz, so 3 × 1014 Hz = 300 THz.

Cross-check

Two independent checks confirm the result.

• Units: (m/s × m) / m2 = 1/s = Hz, so the expression really does return a frequency and not a length.

• Bracketing: pick concrete endpoints for a 1-micron-wide band in the vicinity of this wavelength and the order of magnitude does not move. Taking λ from 1 × 10−6 m to 2 × 10−6 m gives Δf = c/(1 × 10−6) − c/(2 × 10−6) = 1.5 × 1014 Hz; taking λ from 0.5 × 10−6 m to 1.5 × 10−6 m gives 4 × 1014 Hz. Both straddle the local estimate of 3 × 1014 Hz, so the terahertz scale is not an artefact of the approximation.

• Scale: 3 × 1014 Hz is 106 times larger than 3 × 108 Hz (300 MHz) and 105 times larger than 3 × 109 Hz (3 GHz), while 3 × 104 Hz (30 KHz) is smaller still by ten orders of magnitude; those lower magnitudes are radio and microwave carrier frequencies.

Result

Bandwidth = 3 × 1014 Hz = 300 THz.