Rayleigh length

In optics and especially laser science, the Rayleigh length or Rayleigh range, z R {displaystyle z_{mathrm {R} }} , is the distance along the propagation direction of a beam from the waist to the place where the area of the cross section is doubled. A related parameter is the confocal parameter, b, which is twice the Rayleigh length. The Rayleigh length is particularly important when beams are modeled as Gaussian beams. In optics and especially laser science, the Rayleigh length or Rayleigh range, z R {displaystyle z_{mathrm {R} }} , is the distance along the propagation direction of a beam from the waist to the place where the area of the cross section is doubled. A related parameter is the confocal parameter, b, which is twice the Rayleigh length. The Rayleigh length is particularly important when beams are modeled as Gaussian beams. For a Gaussian beam propagating in free space along the z ^ {displaystyle {hat {z}}} axis with wave number k = 2 π / λ {displaystyle k=2pi /lambda } , the Rayleigh length is given by where λ {displaystyle lambda } is the wavelength (the vacuum wavelength divided by n {displaystyle n} , the index of refraction) and w 0 {displaystyle w_{0}} is the beam waist, the radial size of the beam at its narrowest point. This equation and those that follow assume that the waist is not extraordinarily small; w 0 ≥ 2 λ / π {displaystyle w_{0}geq 2lambda /pi } . The radius of the beam at a distance z {displaystyle z} from the waist is The minimum value of w ( z ) {displaystyle w(z)} occurs at w ( 0 ) = w 0 {displaystyle w(0)=w_{0}} , by definition. At distance z R {displaystyle z_{mathrm {R} }} from the beam waist, the beam radius is increased by a factor 2 {displaystyle {sqrt {2}}} and the cross sectional area by 2. The total angular spread of a Gaussian beam in radians is related to the Rayleigh length by The diameter of the beam at its waist (focus spot size) is given by These equations are valid within the limits of the paraxial approximation. For beams with much larger divergence the Gaussian beam model is no longer accurate and a physical optics analysis is required.