Parameters of a Gaussian Laser Beam. A 1-mW He-Ne laser produces a Gaussian beam

Question

Parameters of a Gaussian Laser Beam.

A 1-mW He-Ne laser produces a Gaussian

beam of wavelength A = 633 nm and a spot size 2Wo = 0.1 mm.

a) Determine the angular divergence of the beam, its depth of focus, and its diameter at

z = 3.5 x 10^5 km ( approximately the distance to the moon).

(b) What is th e radius of curvature of the wavefront at z = 0, z = Zo, and z = 2Zo,?

(c) What is the optical intensity (in W/cm2) at the beam center (z = 0, p = 0) and at the

axial point z = Zo? Compare this with the intensity at z = Zo of a 100-W spherical

wave produced by a small isotropically emitting light source located at z = 0.

Explanation / Answer

A Gaussian beam has a beam waist and a Rayleigh range. The beam waist is 0.1mm/2. The depth of focus is two times the Rayleigh Range

Depth of focus = 2*Pi*(0.05mm)^2/633nm = 2.48 cm (this is two Rayleigh ranges, 2 z subR, where R is for Rayleigh, one on each side of the focal point - one half may be inside the laser). The spot size is SQRT(2) times longer at each end of the focal region.

The diameter of the beam is

2 omega = 2 omega zero * SQRT(1 + z/ zR) = 0.1mm*SQRT(1 + 3 x 10^5 km/0.0124m) = 0.1mm*SQRT(1+3 x 10^8/0.0124)

2 omega = 0.1mm*SQRT(24178818956) = 15,549 mm = 15.55 m

b) Radius of curvature

R(z) = z( 1 + (z/zR)^2)

we substitute in the various distances z, we know zR is 1.24 cm

z = 0, radius of curvature = 0

z = z0, this must be your notation for Rayleigh range, so 1.24cm(1 +(z0/z0)^2) = 1.24cm(2) = 2.48 cm

z = 2 Z0, so 2.48cm(1 +4) = 12.4 cm

C) the spot size grows in length by SQRT(2) and in width by SQRT(2) going from center to Zo. We square these to find area, so intensity is half as much.

If we had a point source the area would increase by 4*Pi*distance squared.

Laser

1 mW, 0.1mm diameter = 0.01cm diameter, area is Pi*(0.005cm)^2 = 7.85 x 10 ^-9 cm^2

intensity is 1 mW/7.85 x 10 ^-9 cm^2 = 127,000 W/cm^2 at the focal point

= 63,600 W/cm^2 at Zo (area twice as much, so intensity half as much)

Light bulb

1.24 cm radius surface area of sphere of radius 1.24cm = 4*pi*(1.24cm)^2 = 19.34 cm^2

Intensity 100W/19.34 cm^2 = 5.2 W/cm^2

So, the 1 mW laser is 63,600/5.2 or 12,000 times more intense.

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