Laser Physics - Assignment 3 Solutions | PHGN 480 | Assignments Engineering

PHGN 480: Laser Physics due 10 Oct 2008

Homework 3

1) Transition energies for iodine. In the simplest, ideal model, the vibrational

energies are given by

Evib =!

vib v+1

( )

, where

vib =

0.026eV. The rotational

energies are given by

Erot =!

rot J J +1

( )

, where

rot =

4.64x10-6 eV.

a. Calculate the number of rotational levels in between vibrational levels.

b. Make a plot of the distribution of population of rotational levels in the

ground and first two vibrational states vs energy at room temperature.

Here the 2J+1 degeneracy of the levels is important: the Boltzmann

occupancy factor is (2J+1) exp(E/kT). This gives the range of initially-

populated states for our absorption experiment.

2) ASE threshold for amplifiers. The maximum gain of an amplifier is limited by

depopulation by amplified spontaneous emission (see section 2.9.2). A cylindrical

rod of Nd:YAG with a diameter of 6.3mm (1/4 inch) and a length of 7.5cm is

pumped very hard with a flashlamp such that the pump energy is uniformly

distributed within the rod. Calculate the critical inversion for the onset of strong

ASE, assuming there is no feedback from the boundaries of the rod. (The end

faces are assumed to be perfectly anti-reflection coated.) Also calculate the

maximum energy that can be stored in the rod to avoid the ASE. Data for

Nd:YAG – peak cross-section for 1.064µm transition, σ = 2.8x10-19cm2,

refractive index of YAG, n = 1.82.

3) Gain saturation. In a high-power amplifier, the beam passes through the gain

medium once, and the beam size is adjusted to maximize the energy extracted

from the amplifier.

a. The saturated gain of a square pulse passing through an amplifier is given

by the Franz-Nodvick equation (1963):

Here, Fsat is the saturation fluence, Fin is the input fluence, and Fstor is the

stored fluence (total extractable energy density times the length of the gain

medium.) Show that in the limit of Fin/ Fsat<<1, the single-pass gain

reduces to the expected small-signal gain (independent of Fin).

b. Calculate an expression for the ratio of the extracted energy to the stored

energy. Plotting this versus input fluence for a few values of the small-

signal gain. Explain the trend you see, and determine a good estimate for

the value of Fin/ Fsat that allows for extraction of 50% of the stored energy.

c. The expression above assumes that electrons leave the lower lasing level

in a time much shorter than the pulse duration. How would you expect

your answer to change if the opposite were true?

Comment: This ratio for input fluence/saturation fluence is the major

determinant of how amplifiers are designed for different gain media. At one

G=F

sat

log 1+exp F

sat

&'1

&exp F

stor

sat

(

)

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PHGN 480: Laser Physics due 10 Oct 2008 Homework 3

Transition energies for iodine. In the simplest, ideal model, the vibrational

energies are given by Evib = !! vib ( v + 12 ), where !! vib =0.026eV. The rotational

energies are given by Erot = !! rot J ( J + 1 ), where !! rot =4.64x10-^6 eV.

a. Calculate the number of rotational levels in between vibrational levels. b. Make a plot of the distribution of population of rotational levels in the ground and first two vibrational states vs energy at room temperature. Here the 2J+1 degeneracy of the levels is important: the Boltzmann occupancy factor is (2J+1) exp(E/kT). This gives the range of initially- populated states for our absorption experiment.

ASE threshold for amplifiers. The maximum gain of an amplifier is limited by depopulation by amplified spontaneous emission (see section 2.9.2). A cylindrical rod of Nd:YAG with a diameter of 6.3mm (1/4 inch) and a length of 7.5cm is pumped very hard with a flashlamp such that the pump energy is uniformly distributed within the rod. Calculate the critical inversion for the onset of strong ASE, assuming there is no feedback from the boundaries of the rod. (The end faces are assumed to be perfectly anti-reflection coated.) Also calculate the maximum energy that can be stored in the rod to avoid the ASE. Data for Nd:YAG – peak cross-section for 1.064μm transition, σ = 2.8x10-^19 cm^2 , refractive index of YAG, n = 1.82.
Gain saturation. In a high-power amplifier, the beam passes through the gain medium once, and the beam size is adjusted to maximize the energy extracted from the amplifier. a. The saturated gain of a square pulse passing through an amplifier is given by the Franz-Nodvick equation (1963): Here, F sat is the saturation fluence, F in is the input fluence, and F stor is the stored fluence (total extractable energy density times the length of the gain medium.) Show that in the limit of F in/ F sat<<1, the single-pass gain reduces to the expected small-signal gain (independent of F in). b. Calculate an expression for the ratio of the extracted energy to the stored energy. Plotting this versus input fluence for a few values of the small- signal gain. Explain the trend you see, and determine a good estimate for the value of F in/ F sat that allows for extraction of 50% of the stored energy. c. The expression above assumes that electrons leave the lower lasing level in a time much shorter than the pulse duration. How would you expect your answer to change if the opposite were true? Comment: This ratio for input fluence/saturation fluence is the major determinant of how amplifiers are designed for different gain media. At one

G =

Fsat Fin log 1 + exp Fin Fsat

%&^

& exp^ Fstor Fsat

end, the saturation fluence for dyes is on the order of a few mJ/cm2, and at the other end, ytterbium amplifiers (e.g. Yb:KGW, Yb:glass) have a saturation fluence of around 10J/cm2. With low saturation fluence, the amplifier is easy to saturate, but there is a limited amount of energy that can be stored. With high saturation fluence, there is high energy storage, but it is difficult to extract the energy without damaging the laser rod.

Astigmatism correction in a beam expander. To expand a laser beam, it is often desirable to use curved mirrors instead of lenses, both to avoid extra dispersion and because the damage threshold for mirrors is higher than for lenses. The trouble is that tilting a curved mirror introduces astigmatism. A sketch of a system that avoids any intermediate focus is shown below. The mirrors must be tilted to get the beam through, and a tilted mirror is astigmatic (the focal lengths in the horizontal and vertical planes are different). If θ is the incident angle (measured from the local surface normal to the center of the beam), the horizontal focal length (in the plane of the diagram) of a tilted mirror will be shortened: fH = f 0 cos!, where the normal focal length is f 0. The vertical focal length is lengthened to fV = f 0 /cos!. Design a 5x beam expander that will take a collimated input (5mm diameter) and deliver a collimated output. The overall length should be no more than 300mm and there must be at least 25mm clearance between the center of the convex mirrors and the outgoing beam (distances x 2 ). As part of the design, determine a simple relationship between the ratios of the focal lengths f 1 and f 2 and the incident angles θ 1 and θ 2. You may assume that the incident angles are small, and that the mirror separation is close to the separation required for zero incident angle. You may use Mathematica if you need/want to: either with the lens imaging equation or with the ABCD matrices. Make a sketch of your final setup.

x 1 x 2

Laser Physics - Assignment 3 Solutions | PHGN 480, Assignments of Engineering

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energies are given by Evib = !! vib ( v + 12 ), where !! vib =0.026eV. The rotational

energies are given by Erot = !! rot J ( J + 1 ), where !! rot =4.64x10-^6 eV.

G =

%&^