Laser Beams And Resonators

29
Mode Matching of Lasers to External Resonators Mehmet Deveci Review of the theory of laser beams and resonators

Transcript of Laser Beams And Resonators

Mode Matching of Lasers to External

Resonators

Mehmet Deveci

Review of the theory of laser beams and resonators

Int r o duc t io n•Fabry-perot Interferometer as a Laser Resonator

•The modes in a optical structure

•Resonators with Spherical Mirrors

Par axial RaysNear the axis of an optical system

Ray Tr ans f e r Mat r ic e s

Wave Anal ys is o f Beams and Res o nato r sIs it plane wave?

The Scalar Wave Equation :

For light traveling in the z-direction :

Solving them gives :(Similar to time-dependent Schrödinger equation)

P(z) : complex phase shiftq(z) : complex beam parameter (Gaussian variation in the beam intensity)

Solution of above equation :

Pr o po g at io n Laws Fo r Fundamental Mo de

The general equation :

Two real beam parameters are introduced;

R and w

R: radius of the fieldw: measure of decrease of the field amplitude

Fundamental mode

Ampl itude dis t r ibut io n o f the f undamental beam

Distance at which 1/e times amplitude on the axisw: beam radius or spot size2w: beam diameter

Co nto ur o f a Gaus s ian Beam

•Minimum diameter at the beam waist

A distance z away from the waist

Expans io n o f the beam

and

Eq uat ing the r eal and imag inar y par t s o f :

we g et ;

waist 2w0 2 2 w√ 0

zR

Gaus s ian Beams

Hig her Or der Mo des

There are other solutions of

A solution for general wave equation :

Inserting above equation to general equation we get ;

Hermite Polynomial of order m

g: function of x and zh: function of y and z

Transverse mode numbers m and n

Hermite Polynomials

Phase shift

-4 -2 2 4

0.2

0.4

0.6

0.8

1

-4 -2 2 4

0.2

0.4

0.6

0.8

1

1.2

1.4

-4 -2 2 4

1

2

3

4

5

-4 -2 2 4

5

10

15

20

25

30

-4 -2 2 4

0.2

0.4

0.6

0.8

1

-4 -2 2 4

-1

-0.5

0.5

1

-4 -2 2 4

-2

-1

1

2

-4 -2 2 4

-4

-2

2

4

-4 -2 2 4

0.5

1

1.5

2

-4 -2 2 4

-10

-5

5

10

-4 -2 2 4

20

40

60

80

100

-4 -2 2 4

-750

-500

-250

250

500

750

Hig her -Or der Mo des - HG

Hn(x)

Hn(x) e -x /22

Hn(x) e -x /22

2

1 2 3 4

Beam Tr ans f o r mat io n by a Lens

•Focusing a Laser Beam

•Producing a beam of suitable diameter and phase front curvature

•Ideal Lens leaves unchanged

•However

a lens does change the parameters R(z) and w(z)

•What is the relationship between incoming and outgoing parameters?

If q ’ s ar e meas ur ed at dis t anc e d 1 and d 2

Beam Tr ans f o r mat io n by a Lens

Beam Tr ans f o r mat io n by a Lens

2

1 2 1 2

1

/ )

( / )

1/

1 /

(1 f

B d d d d f

C f

D d f

A d= + −= −= −

= −1

21

Aq Bq

Cq D

+=+

2 1 1 2 1 22

1 1

/ ( / )

/ ) (1 )

(1 )(f d d d d f

qf d f

d qq

+ + −=+ −

−−

Appl ic at io n

0 0q iz=

1

0 1

b

d

a dM

c=

=

01 01q

q dq d

+= = +

1 0

11

a bM

c df

= = −

'3 2q q d= +

' ' '013 2

1 01 1

q dqq q d d d

q q d

f f

+= + = + = ++− −

Seperating the real and imaginary part

'03 0

01

q dq d iz

q d

f

+= + =+−

' 00

0

( )q d fiz d

f q d

+− =− −

' '0 0 0( ) ( )( )iz d f f iz d iz d− = − − −

' 2 ' '0 0 0 0( )iz f df iz f d f z iz d dd+ = − + − +

' '0 0 0iz d iz d d d− = ⇒ =

2 202 0d fd z− + =' 2 '

0d f z dd df− + + =

the condition is, obviously, f >z0 .

Las er Res o nato r s

Self consistency requires q1=q2=q

Las er Res o nato r s

R is eq ual t o the r adius o f c ur vatur e o f t he mir r o r s

The w idt o f t he f undamental mo de is ;

Beam r adius w 0 in the c ent er o f t he r es o nato r , z=d/ 2

R1

z=z1 z=z2

z=0

w2w1

R2

q : number o f no desm and n: r ec t ang ul ar mo de number s

Re s ona nc e oc c ur s whe n t he pha s e s hif t f r on one mir r or t o ot he r is a mul t ipl e of π

the f r equency spacing between succes s ive l ongitudinal

r es onance:

Mo de Mat c hing

• Mo des o f Las er Res o nat o r s c an be c har ac t e r is ed by l ig ht beams• Thes e beams ar e o f t en inj ec t ed t o o ther o pt ic al s t r uc tur es w ith dif f e r ent s e t s o f beam par amet er s

• Thes e o pt ic al s t r uc tur es c an as s ume var io us phys ic al f o r ms

• To mat c h the mo des o f o ne s t r uc tur e t o tho s e o f ano ther we need t o t r ans f o r m a g ic en Gaus s ian beam

where

Fo r mul as f o r the c o nf o c al par amet e r and the l o c at io n o f beam wais t

The conf ocal par ameter b2 as a f unction of the l ens wais t

Co nc l us io n

It was nec e s s ar y f o r l eng th meas ur ement in met r o l o g y and c al ibr at io n t o c o nc ent r at e the dis c us s io n o f this wo r k o n the bas ic as pec t s o f l as e r beams and r e s o nato r s . A r eview o f t he theo r ie s f r o m 1 9 6 0 ’ s and o ur c o nt r ibut io n is do ne eac ho ther

Thank yo u Fo r Yo ur Int e r e s t