Department of Physics – 3 rd Class Laser Tutorial 1 December 3, 2012 Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected]1 1. In thermal equilibrium at T=300 K, the ratio of level populations N2/N1 for some particular pair of levels is given by 1/e. Calculate the frequency for this transition. In what region of the EM spectrum does this frequency fall? Homework 2. If levels 1 and 2 are separated by an energy E2 – E1 such that the corresponding transition frequency falls in the middle of the visible range, calculate the ratio of the populations of the two levels in thermal equilibrium at room temperature. Solution: At thermal equilibrium, the ratio N2/N1 is given as follows: The middle of the visible range is taken at ( ( ) ) 3. Consider a lower energy level situated 200 cm -1 from the ground state. There are no other energy levels nearby. Determine the fraction of the population found in this level compared to the ground state population at a temperature of 300 K. Solution: Boltzmann's constant is equal to The conversion from cm -1 to joules is given by: , where h is Planck's constant () and c is the speed of light in a vacuum (). Boltzmann's Law is used: By considering the energy of the ground state to be zero and calling 0 the ground state and 1 the lower energy level: ( ) ( ) After converting cm -1 to joules: Thus 38% of the population is in the lower energy level. 4. A helium-neon laser emitting at 633 nm makes a spot with a radius equal to 100 mm at 1/e 2 at a distance of 500 m from the laser. What is the radius of the beam at the waist (considering the waist and the laser are in the same plane)? Solution: The problem can be solved by using the formula that links the divergence of the beam and the waist size:
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Department of Physics – 3rd Class Laser Tutorial 1 December 3, 2012
1. In thermal equilibrium at T=300 K, the ratio of level populations N2/N1 for some
particular pair of levels is given by 1/e. Calculate the frequency for this transition. In
what region of the EM spectrum does this frequency fall? Homework
2. If levels 1 and 2 are separated by an energy E2 – E1 such that the corresponding
transition frequency falls in the middle of the visible range, calculate the ratio of the
populations of the two levels in thermal equilibrium at room temperature.
Solution:
At thermal equilibrium, the ratio N2/N1 is given as follows:
The middle of the visible range is taken at
(
( )
)
3. Consider a lower energy level situated 200 cm-1 from the ground state. There are no other energy levels nearby. Determine the fraction of the population found in this level compared to the ground state population at a temperature of 300 K.
Solution: Boltzmann's constant is equal to The conversion from cm-1 to joules is given by: , where h is Planck's constant ( ) and c is the speed of light in a vacuum ( ). Boltzmann's Law is used:
By considering the energy of the ground state to be zero and calling 0 the ground state and 1 the lower energy level:
(
) (
)
After converting cm-1 to joules:
Thus 38% of the population is in the lower energy level.
4. A helium-neon laser emitting at 633 nm makes a spot with a radius equal to 100 mm at 1/e2 at a distance of 500 m from the laser. What is the radius of the beam at the waist (considering the waist and the laser are in the same plane)?
Solution: The problem can be solved by using the formula that links the divergence of the beam and the waist size:
Department of Physics – 3rd Class Laser Tutorial 1 December 3, 2012
1. The brightness of the probably the brightest lamp so far available (PEK Labs type
107/109TM excited by 100 W of electrical power) is about 95 W/cm2 sr in its most
intense green line . Compare this brightness with that of a 1 – W argon
laser which can be assumed to be diffraction – limited.
Solution:
For diffraction limited beam, the divergence is given by:
[ ]
where and D are the wavelength and the diameter of the beam. The factor is a numerical coefficient of the order of unity whose value depends on the shape of the amplitude distribution. The brightness B of a given source of e.m. waves can be defined as the power P emitted per unit surface area ΔA per unit solid angle ΔΩ. Mathematically given as follows:
( )
( )
which is the maximum brightness that a beam of power P can have. Therefore, the brightness of the above 1 – W argon laser will be:
[ ]
For comparison purpose, the ratio
Or, the brightness
2. For laser light of wavelength =1.06*10-3 mm, D=3 mm, =1.1, calculate the beam divergence and compare it with convensional sources.
Solution: For diffraction limited beam, the divergence is given by:
[ ]
𝜃
R
ΔΩ
ΔA ΩTotal (sphere) = 4π
steradians
Ω Δ𝐴
𝑅 𝜋𝑟
𝑅 𝜋 𝜗𝑅
𝑅 𝜋𝜃
R
D
r
Department of Physics – 3rd Class Laser Tutorial 2 December 3, 2012
The single pass loss of the laser cavity is given by the following equation:
Where is called the logarithmic internal loss of the cavity. While
represent the logarithmic losses of the two cavity mirrors, given by:
Therefore, the single pass loss of the laser cavity will be:
The critical inversion or the threshold population inversion, Nc can be determined as follows:
[ ] [ ]
6. A 1 – mW He – Ne laser beam with a divergence of 0.5 mrad enters the eye. Find the irradiance on the retina if the focal length of the eye, from cornea to retina, is equal to f = 1.7 cm.
Given: P = 1 mW , = 0.5 mrad , f = focal length of eye focusing system (1.7 cm) Solution:
The solution of this problem requires three steps:
Calculate focal spot diameter. (Note: Laser propagation theory shows that, when a laser beam of divergence is focused by a lens of focal length f to a spot of diameter d, the spot diameter d is given by d = f .)
Calculate the area of the spot.
Calculate the retinal irradiance.
7. The beam of a YAG laser with a power of 50 W, 2 – mrad beam divergence and 6 – mm beam diameter is focused with a lens of focal length of 5 cm. Calculate:
1. The power density before the lens. 2. The beam diameter at the focal plan. 3. The power density at the focal plan.
Solution The power density is the laser power divided by the cross section of the beam:
Department of Physics – 3rd Class Laser Tutorial 2 December 3, 2012
50 W/ = 177 W/cm2 The beam diameter at the focal plane:
The cross section of the beam at the focal point:
The power density at the focal plane:
8. The diameter of a beam emitted from He – Ne laser is 1.2 mm, and its divergence angle is 1 mrad. A Kepler beam expander is used made of 2 positive lenses with focal lengths of 1 cm and 6 cm. Calculate: i. The beam diameter at the output of the beam expander. ii. The beam divergence angle.
Solution The relation between the beam diameters and the beam divergence angles is:
where f1 = Focal length [m] of the input lens – ocular, f2 = Focal length [m] of the output lens – objective, d1 = Diameter of the input beam [m] and d2 = Diameter of the output beam [m]. = Divergence angle (Rad) of the beam at the input to the beam expander while = Divergence angle (Rad) of the beam at the output to the beam expander. i. The beam diameter at the output of the beam expander is:
(
)
ii. The divergence angle at the output of the beam expander
(
) (
)
The beam expander caused a reduction of 6 times of the beam divergence (the ratio of the focal lengths of the lenses).
Department of Physics – 3rd Class Laser Tutorial 3 December 10, 2012