Mode spacing and gain coefficient: A Nd microchip laser has an index of refraction 1.78 and a cavity length of 3.20 mm. Mirrors of reflectivity 0.997 and 0.975 are mounted directly on the faces of the chip. The emission lineshape of the laser transition can be considered to be Lorentzian, with a linewidth of 4.80 THz. a) Calculate the frequency spacing between modes of the laser and the frequency width of the modes. (8 pts) b) Calculate the number of modes within the transition linewidth. (9 pts) c) Given the loss coefficient of 0.4 m-1 for the chip; calculate the gain coefficient at lasing threshold. (13 pts)
a) To calculate the frequency spacing between modes of the laser, we can use the formula:
Δν = c / (2L * n)
Where:
Δν is the frequency spacing between modes,
c is the speed of light in vacuum (3 × 10^8 m/s),
L is the cavity length (3.20 mm = 3.20 × 10^-3 m),
and n is the refractive index of the laser medium (1.78).
Substituting the given values into the formula, we can calculate Δν:
Δν = (3 × 10^8 m/s) / (2 * (3.20 × 10^-3 m) * 1.78)
Calculating this, we find:
Δν ≈ 26.7 GHz
The frequency spacing between modes is approximately 26.7 GHz.
To calculate the frequency width of the modes, we can use the formula for the full-width at half-maximum (FWHM) of a Lorentzian line:
FWHM = 2γ
Where FWHM is the frequency width of the modes and γ is the linewidth (4.80 THz = 4.80 × 10^12 Hz).
Substituting the given value into the formula, we can calculate the FWHM:
FWHM = 2 * (4.80 × 10^12 Hz)
Calculating this, we find:
FWHM ≈ 9.60 × 10^12 Hz
The frequency width of the modes is approximately 9.60 × 10^12 Hz.
b) To calculate the number of modes within the transition linewidth, we can use the formula:
N = FWHM / Δν
Where N is the number of modes and Δν and FWHM are the frequency spacing and width of the modes calculated above.
Substituting the values into the formula, we can calculate N:
N = (9.60 × 10^12 Hz) / (26.7 GHz)
Calculating this, we find:
N ≈ 359
There are approximately 359 modes within the transition linewidth.
c) To calculate the gain coefficient at lasing threshold, we can use the formula:
g = (1 - R1) * (1 - R2) / (2 * L)
Where g is the gain coefficient, R1 and R2 are the reflectivities of the two mirrors (0.997 and 0.975), and L is the cavity length (3.20 mm = 3.20 × 10^-3 m).
Substituting the given values into the formula, we can calculate g:
g = (1 - 0.997) * (1 - 0.975) / (2 * (3.20 × 10^-3 m))
Calculating this, we find:
g ≈ 124.8 m^-1
The gain coefficient at lasing threshold is approximately 124.8 m^-1.
To solve this problem, we will use the formula for mode spacing, mode width, and the number of modes within a linewidth, as well as the formula for gain coefficient.
a) Calculation of mode spacing and mode width:
The mode spacing (Δν) can be calculated using the formula:
Δν = c / (2L)
Where c is the speed of light and L is the cavity length.
Given:
Reflector 1 reflectivity (R1) = 0.997
Reflector 2 reflectivity (R2) = 0.975
Index of refraction (n) = 1.78
Cavity length (L) = 3.20 mm
Mode spacing (Δν) = c / (2L)
Convert the cavity length to meters.
L = 3.20 mm = 3.20 × 10^(-3) m
Substituting the values:
Δν = (3 × 10^8 m/s) / (2 × 3.20 × 10^(-3) m)
Calculate the value of Δν.
To find the mode width (Δν_width), we use the formula:
Δν_width = Δν / π(1 - √(R1 × R2))
Substitute the values to calculate Δν_width.
b) Calculation of the number of modes within the transition linewidth:
The number of modes (N_modes) within the transition linewidth (Δν_line) can be calculated using the formula:
N_modes = Δν_line / Δν
Given the linewidth (Δν_line) = 4.80 THz, convert this to Hz before substituting the values to calculate N_modes.
c) Calculation of the gain coefficient at lasing threshold:
The gain coefficient (α) at lasing threshold can be calculated using the formula:
α = (ln (1 / R1)) / (2L)
Substitute the values to calculate α.
This way, you can find the answers to all the parts of the problem.