laser problem solution

8/3/2019 Laser Problem Solution

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Problem

I would like you to find the mode size radius, w, in mm at mirror M1 and also to plot w(z) inside the

resonator (along its all length, z) in mm. Assume 1064 nm wavelength of laser light.

Ans.

This is equivalent to the optical circuit


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We can denote each optical element by corresponding ABCD transformation matrix

For propagation through , (1)

For thin lens having focal length , (2)

For mirrors, (3)

Considering the beam travelling from mirror to the effective ABCD matrix of the optical system

is given by

(4)

= (5)

Substituting values for , , and , we will

get ABCD matrix for half round trip as

= (6)

Then for one round trip, the total matrix can be obtained as

= = (7)

Here and hence the cavity is stable.

Considering gaussian beam propagation inside the resonator, we can define a complex q parameter

(8)

And the q parameter transform according to ABCD law, like

(9)

For one round trip, at mirror

Thus,

Solving, we will get


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, a pure imaginary number, because, to get a stable resonator the wave front curvature

should match that of the plane mirror at ends.

So taking radius of curvature, = infinity (for plane mirror) in equation (8) we will get

(10)

Taking =1064 nm, beam waist, at mirror

(11)

The width of beam just before entering the lens can be found using the formula

(12)

At a distance of from first plane mirror, the beam width is

. (13)

To find where the beam get focused at other end of first lens , we will consider the ABCD

matrix of optical system, which will include a propagation through , then through a lens of focal

length , and then a propagation through . i.e. at a distance of to the right of the lens , the beam

will be plane parallel. The corresponding ABCD matrix can be obtained as

(14)

Let is the complex beam parameter at first flat mirror ( , from

equation (10)), and let be the complex parameter at the waist in between the lenses,

then according to equation (9), and are related by

(15)

and , since input and output plane lies in the same medium.

will be purely imaginary, i.e. from equation (15)

(16)

Equating imaginary parts on either side of (15) give

, substituting value for


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(17)

Substituting values of A, B, C and D (from equation (14)) in the condition (16) and taking , ,

(18)

Substituting values for parameters will give

and (19.a)

(19.b)

Equation (9) can be used to find out the parameters at second lens position and at the

mirror . Since the cavity is stable, the parameter at will be purely imaginary. Looking at the

ABCD matrix of a thin lens we can easily find out that it wont change the width of the beam, it will

change the radius of curvature of the beam. The radius of beam at respective positions can be foundout from the imaginary part of parameter.

Thus the beam width at position of lens 2

. (20)

And the same at mirror is

. (21)

The variation of beam width inside the cavity will look like

0 500 1000 1500 2000 25000

0.5

1

1.5

2

2.5

3

Variation of beam width along the cavity

Length of cavity z(mm)

Widthofbeamw(z)(mm)

laser problem solution

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