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