Rami Arieli: "The Laser Adventure" Chapter 4 Section 1, page 9
Example 4.2 Calculating the Number of Longitudinal Modes in He-Ne laser

The length of the optical cavity in He-Ne laser is 30 [cm].

The emitted wavelength is 0.6328 [mm].

Calculate:

Solution to example 4.2:

Since He-Ne laser is a gas laser, the hidden information is the index of refraction of the active medium. It is quite accurate to approximate the index of refraction by 1.0.

    1. The equation for difference in frequency is the same as for the basic mode:
    (Delta n) = c/(2nL) = 3*108 [m/s]/(2*1.0*0.3 [m]) = 0.5*109 [Hz] = 0.5 [GHz]
    2. From the equation for the wavelength of the mth mode:
    lm = 2L/m
    m = 2L/ lm = 2*0.3 [m]/0.6328*10-6 [m ] = 0.948*106
      which means that the laser operate at a frequency which is almost a million times the basic frequency of the cavity.
    3. The laser frequency can be calculated in two ways:
a) By multiplying the mode number from section 2 by the basic mode frequency:
n = m*(Delta n) = (0.948*106)(0.5*109 [Hz]) = 4.74*1014 [Hz]
b) By direct calculation:
n = c/l = 3*108 [m/s]/0.6328*10-6 [m ] = 4.74*1014 [Hz]