www.iap.uni-jena.de Physical Optics Lecture 12: Gaussian beams 2018-06-27 Herbert Gross
www.iap.uni-jena.de
Physical Optics
Lecture 12: Gaussian beams
2018-06-27
Herbert Gross
Physical Optics: Content
2
No Date Subject Ref Detailed Content
1 11.04. Wave optics GComplex fields, wave equation, k-vectors, interference, light propagation,
interferometry
2 18.04. Diffraction GSlit, grating, diffraction integral, diffraction in optical systems, point spread
function, aberrations
3 25.04. Fourier optics GPlane wave expansion, resolution, image formation, transfer function,
phase imaging
4 02.05.Quality criteria and
resolutionG
Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point
resolution, criteria, contrast, axial resolution, CTF
5 09.05. Photon optics KEnergy, momentum, time-energy uncertainty, photon statistics,
fluorescence, Jablonski diagram, lifetime, quantum yield, FRET
6 16.05. Coherence KTemporal and spatial coherence, Young setup, propagation of coherence,
speckle, OCT-principle
7 23.05. Polarization GIntroduction, Jones formalism, Fresnel formulas, birefringence,
components
8 30.05. Laser KAtomic transitions, principle, resonators, modes, laser types, Q-switch,
pulses, power
9 06.06. Nonlinear optics KBasics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects,
CARS microscopy, 2 photon imaging
10 13.06. PSF engineering GApodization, superresolution, extended depth of focus, particle trapping,
confocal PSF
11 20.06. Scattering LIntroduction, surface scattering in systems, volume scattering models,
calculation schemes, tissue models, Mie Scattering
12 27.06. Gaussian beams G Basic description, propagation through optical systems, aberrations
13 04.07. Generalized beams GLaguerre-Gaussian beams, phase singularities, Bessel beams, Airy
beams, applications in superresolution microscopy
14 11.07. Miscellaneous G Coatings, diffractive optics, fibers
K = Kempe G = Gross L = Lu
Solution of the wave equation
Paraxial matrix calculus
Properties of Gaussian beams
Beam transformation
Mode matching
Generation of Gaussian beams in resonators
Ray equivalent auf gaussian beams
Generalized gaussian beams
Truncation of gaussian beams
Gaussian beams with aberrations
3
Contents
Helmholtz wave equation
Fast z-oscillation separated
Slowly varying envelope approximation
Paraxial approximation
paraxial wave equation
Conditions for scalar approximation:
1. Decoupling of field components,
wavelength small in comparison to free diameter
2. No large angles due to geometry,
Computation of field in large distances z
4
Solutions of the Wave Equation
02 EkE
0),,(
2
22
2
2
2
2
2
2
Ec
zyxn
z
E
y
E
x
E
ikzezyxEzyxE ),,(),,(
2
2
2 22
0
2
22E
zik
E
zE k
n x y
nn Eo
( , )
a
z
022
2
2
2
z
Eki
y
E
x
E
2
2
E
zk
E
z
5
Solutions of the Wave Equation
rapidly varying
envelope
slowly varying envelope
Slowly varying envelope approximation
Solution of the paraxial wave equation:
Gaussian beam
Interpretation:
- physical solution (diffraction included)
- only valid for small angles (paraxial)
- eigensolution (only scaling during propagation, no change of the gaussian shape)
- full orthogonal mode system
ikzzR
ikr
zw
r
zio eeeezw
wEzrE
)(2)()(
0
2
2
2
)(),(
Linear relation of ray transport
Simple case: free space
propagation
Advantages of matrix calculus:
1. simple calculation of component
combinations
2. Automatic correct signs of
properties
3. Easy to implement
General case:
paraxial segment with matrix
ABCD-matrix :
u
xM
u
x
DC
BA
u
x
'
'
z
x x'
ray
x'
u'
u
x
B
Matrix Formulation of Paraxial Optics
A B
C D
z
x x'
ray x'
u'u
x
Matrix Calculus
Paraxial raytrace transfer
Matrix formulation
Matrix formalism for finite angles
Paraxial raytrace refraction
Inserted
Matrix formulation
111 jjjj Udyy
1 jjjj Uyi in
nij
j
j
j''
1' jj UU
1 jj yy
1
'
''
j
j
j
j
j
jjj
j Un
ny
n
nnU
'' 1 jjjj iiUU
j
jj
j
j
U
yd
U
y
10
1
'
'1
j
j
j
j
j
jjj
j
j
U
y
n
n
n
nnU
y
'
'01
'
'
j
j
j
j
u
y
DC
BA
u
y
tan'tan
'
Linear transfer of spation coordinate x
and angle u
Matrix representation
Lateral magnification for u=0
Angle magnification of conjugated planes
Refractive power for u=0
Composition of systems
Determinant, only 3 variables
uDxCu
uBxAx
'
'
u
xM
u
x
DC
BA
u
x
'
'
mxxA /'
uuD /'
xuC /'
121 ... MMMMM kk
'det
n
nCBDAM
Matrix Formulation of Paraxial Optics
System inversion
Transition over distance L
Thin lens with focal length f
Dielectric plane interface
Afocal telescope
AC
BDM
1
10
1 LM
11
01
f
M
'0
01
n
nM
0
1L
M
Matrix Formulation of Paraxial Optics
Calculation of intersection length
Magnifications:
1. lateral
2. angle
3. axial, depth
Principal planes
Focal points
Matrix Formulation of Paraxial Optics
DsC
BsAs
'
DsC
BCADm
2'
DsC
BCAD
ds
ds
'sCA
BCADDsC
C
DBCADaH
C
AaH
1'
C
AaF '
C
DaF
2
2
)(
w
r
oeIrI
Gaussian Beams, Transverse Beam Profile
I(r) / I0
r / w
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-2 -1 0 1 2
0.135
0.0111.5
0.589
1.0
Transverse beam profile is gaussian
Beam radius w at 13.5% intensity
Radius r Diameter Amplitude A Intensity I Energy E P-truncation
0.58871 w 1.17741 w 0.7071 0.5000 0.5000 0.5000
w 2 w 0.3679 0.1353 0.8647 0.1353
1.0730 w 2.146 w 0.3162 0.1000 0.9000 0.1000
1.5 w 3 w 0.1054 0.0111 0.9889 0.0111
1.571 w w 0.0848 0.0072 0.9928 0.0072
2 w 4 w 0.0183 0.0003 0.9997 0.0003
Gaussian Beams: Parameters of the Profile
Typical parameters of a gaussian beam profile
2
2
2
1
2
2
2 1
2),(
o
To
z
zzw
r
o
To
e
z
zzw
PzrI
2
0 1)(
o
T
z
zzwzw
00000
zzw
o
o
Gaussian Beams, Definitions and Parameter
Paraxial TEM00 fundamental mode
Transverse intensity is gaussian
Axial isophotes are hyperbolic
Beam radius at 13.5% intensity
Only 2 independent beam parameter of the set:
1. waist radius wo
2. far field divergence angle o
3. Rayleigh range zo
4. Wavelength o
Relations
Expansion of the intensity distribution around the waist I(r,z)
Gaussian Beams
z
asymptotic
lines
x
hyperbolic
caustic curve
wo
w(z)
R(z)
o
zo
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
z / z
r / w
o
o
asymptotic
far field
waist
w(z)
o
intensity
13.5 %
Geometry of Gaussian Beams
-2
0
2-8
-6
-4
-2
0
2
4
6
8
0
0.5
1
z
intensity I
[a.u.]
x
z/zo
x/wo
I(x,z)
+ 4
- 4
+ 2
0
- 2- 4
- 8
+ 8
+ 4
00
1
Caustic of a Gaussian Beam
Intensity I(x,z)
Caustic of a Gaussian Beam
Parameter:
= 0.5 mm
wo = 0.3 / 0.6 / 0.9 mm
Normalization of intensity
Caustic of a Gaussian Beam
Parameter:
= 0.5 mm
wo = 0.3 / 0.6 / 0.9 mm
Normalization of power
Caustic of a Gaussian Beam
Parameter:
= 0.5 mm
wo = 0.3 / 0.6 / 0.9 mm
Normalization of intensity
Caustic of a Gaussian Beam
Parameter:
= 0.5 mm
wo = 0.3 / 0.6 / 0.9 mm
Normalization of power
21
Phase of a Gaussian Beam
Phase terms collected:
z
y
phase
plane wave Gouy phasewavefront bending
Ref: M. Dienerowitz
)(
)(2)(
0
2
2
2
)(),(
zzR
ikrkzi
zw
r
o eezw
wEzrE
)()(2
),(2
zzR
ikrkzzr
z
zzzR o
2
)( oz
zz arctan)(
2
2
0
0 1
''2)'(
''2)'(
)(
w
r
r
e
drrrI
drrrI
rE
2
02
2)( wIdrrrIP o
E(r) / E 0
r / w0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gaussian Beam, Energy Function
Encircled energy function
Total power in the beam
222
1
12
)(
2),0(
o
To
z
zzw
P
zw
PzrI
2
0
0
0
0
2
wwz o
o
I(z) / I0(z)
z / zo
-6 -4 -2 0 2 4 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gaussian Beam, Axial Intensity
Axial intensity: Lorentz function
Characteristic depth of the waist region:
Rayleigh range
T
oT
zz
zzzzR
2
)(
R / z0
z / zo
0 1 2 3 4 5 60
1
2
3
4
5
6
Gaussian Beams, Radius of Curvature
Gaussian beams have a spherical wavefront
Radius of curvature:
In the waist the phase is plane
The spherical wave is not concentric
to the waist center
In the distance of the Rayleigh range, the
curvature of the phase has ist maximum
Phase of a Gaussian beam:
Guoy phase
0
arctan)(z
zzz T
z
zo
-zo
-45°
-90°
+90°
+45°
Guoy Phase
f
zfz
zzz
TT
oTT
1
1
1112'
Transform of Gaussian Beams
Diffraction effects are taken into account
Geometrical prediction corrected in the waist region
No singulare focal point: waist with finite width
Focal shift: waist located towards the system, intra focal shift
Transform of paraxial beam
propagation
z'T / f
zT / f
-6 -5 -4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4
5
6
zo / f = 0.1
zo / f = 0.2
zo / f = 0.5
zo / f = 1
zo / f = 2
geometrical
limit zo / f = 0
Focussing Gaussbeam by a Lens
Two possibilities:
1. Start in waist
2. Start just before lens
waist position -z
Result
Comparison with lens
makers formula
(change of sign convention):
z
lens
focal length f'
z'
z
zo zo'
initial
plane
waist
final
plane
A BC D
1 0-1/f 1
Matrix :
z
z'
z
zo zo'
initial
plane
waist
final
planeA BC D
1 z0 1
Matrix :
1 0-1/f 1
waist
waist
lens
focal length f'
zf
zfz
zzfz
1
1
111
'
12
0
22
2
'ozzf
zfffz
nced
Wav
e
Optic
s
Focussing a Gaussbeam by a Lens
Setup:
Complex beam parameter
Transformation of Parameter
gives
geometrical model
z
focal length
f'
zT'zT
gaussian beam
waist
plane image plane
z
waist
plane
DCq
BAqq
'
0izzq
222
''o
oo
o
oo
zCDCz
DizzCBizzA
DizzC
BizzAizz
nced
Wav
e
Optic
s
-3 -2 -1 1 2 3 4
1
2
3
4
5
6
7
8
9
0
0.5
0.25
0.125
wo' / w
o
zT' / z'
o
1f_zo
=
22
'
oTo
o
zzf
f
w
w
0
max,2
1'z
ffz T
Transform of Gaussian Beams
Change of waist size / ‚magnification‘
The waist can not be located into an
arbitrary distance
largest possible distance:
-2
0
2-8
-6
-4
-2
0
2
4
6
8
0
0.5
1
z
intensity I
[a.u.]
x
Gaussian Beam Propagation
Paraxial transform of
a beam
Intensity I(x,z)2
)(2
2 )(
2),(
zw
r
ezw
PzrI
Propagation of Gaussian Beams
Paraxial transport through
ABCD-system
Astigmatic beam transfer
thin lensesx
y
z
Gauss Beam in Quadratic Grin-Medium
I(z)
I(r,z)
I(r)
z
r
Transform of Gaussian Beam
R
w
w'
R'
starting
plane
receiving
plane
paraxial
segment
A B
C D
incoming
Gaussian beam
z
outgoing
Gaussian beam
Transfer of a Gaussian beam by a paraxial ABCD system
w wB
wA
B
R'
2
2 2
R
AB
R
B
w
AB
RC
D
RD
B
w
'
2
2
2
2
2
given input
beam required output
beam
f
z
f1 2
z zL1L2
ww'
00
zTT '
2
00
2
'
2
0000
2
00'00'2
00
2/1 ''''2'2'
1zzzzzzzzzzzzz
zzf TTTTTT
Gaussian Beam Mode Matching
Given beam at before and after the system: search for transfer optic
In general two solutions
Location of lens at the intersection points of the two hyperbolic curves
0'
ww o
0' o
f
z
2
Lz
w0
f1
T
z'T
w'0
z'TT'
Beam Transformation by a Telescope
Afocal telescope of the Kepler or Galilean type
Application: adaptation of beam width
Beam radius enlarged: divergence reduced and vice versa
Kepler system:
Internal waist, can make problems with energy density
Gaussian beam with minimal diameter in a given distance L
Waist size
located in the half distance
Maximum value of beam radius
Mode Matching
wmax
w0
L
f
2
Lwo
Lw max
37
Laser Resonator Types
Ref: B. Böhme
Description of curved mirrors as combination of thin lens and plane mirror
RR1 2
A B
C D
f
f 1
2
A B
C D
221
1
22121
1
11222112
1G
B
GG
BG
R
BD
RR
B
R
A
R
DC
BR
BA
dc
baLMLmm
T
Equivalent Plan-plan-Resonator
Resonator Mode Calculation after Fox-Li
From the physical point of view the field inside a stable resonator is given as the
eigensolution of the electromagnetic field, that is reproduced for one round trip
through the resonator
Typically a system of eigensolutions is found by this boundary value problem, t
these are he modes of the resonator
The transverse limitations of the field due to a stop governs the modes
The eigenvalues g determine the losses of the modes
, , ,( , ) ( , , ', ') ( ', ') dx'dy'n m n m n mE x y K x y x y E x y
RR1 2
A BC D
En,m(x,y)
round tripstop
Condition of stability for a reproduced wave
Mirrors are phase surfaces
g1-g2 diagram of stability
yellow region delivers stable
operation
10 21 gg
A Do o
21
g
-3 -2
1
-1
2
3
g1
-3
-2
1
-1
2 3
2
stable regimes
symmetric
confocal
plane-plane
symmetric
concentric
Stability of a Gaussian Resonator
To
o
To zz
z
zzwzw
)(
Gaussian Beams: Asymptotic Limiting Case
Limiting case for large distances from waist z >> zo
Linear geometrical propagation
Geometrical aperture angle corresponds to divergence angle
Geometrical prediction are approximately valid
Equivalent description of the gaussian beams by two paraxial rays possible:
1. Waist ray corresponds to chief ray
2. divergence ray corresponds to marginal ray
Therefore description of propagation by identical ABCDS matrices
x
z
waist ray
divergence rayo
wo
z
beam hyperbola
General case:
Two rays with heights y and ybar and anglers u and ubar
Corresponds to marginal (MR) and chief (CR) ray
Rays must fulfill the Lagrange condition
x
zy ray MRy ray CR
o
beam hyperbola
o
oo
uyuyL
Gaussian Beam Ray Equivalent
Astigmatic Gaussian beams:
different values for wx, wy and Rx, Ry in both cross sections
Simple case of decoupled sections without twist
The intensity profile is
elliptical with changing axes
)()(
2
)()()(
2
)(
22
2
2
22
2
),,(zR
y
zR
xy
zR
xi
zw
y
zw
xy
zw
x
o
yxyxyxyx eeEzyxE
x / y
z
y-z cross
section
x-z cross
section
woxwoy
zTy
zTx
T
y
x
y
x
y
x
y
x
y
x
Astigmatic Gaussian Beams
Beams with twist:
ellipse of intensity
ellipse of curvature
with different phase
Beam rotates during
propagation in the azimuth
)()()( 2
2
22
2
intzw
y
zw
xy
zw
xE
yxyx
)()()( 2
2
22
2
zR
y
zR
xy
zR
xE
yxyx
phas
22
2
11
2
2tan
yx
xy
Int
ww
w
22
2
11
2
2tan
yx
xy
Phas
RR
R
isophotes of
intensity
isophase
surfaces
z
z
Beams with Twist
a/wo = 1
0 1 2 3 4 5 610
-12
10-10
10-8
10-6
10-4
10-2
100
a/wo = 2
a/wo = 3
a/wo = 4
x
Log |A|
Truncated Gaussian Beams
Untruncated gaussian beam: theoretical infinite extension
Real world: diameter D = 2a = 3w with 1% energy loss acceptable
Truncation: diffraction ripple occur, depending on ratio x = a / wo
Gaussian beam with width w
Stop with radius a
Truncation/transmission for a chaning lateral offset v in the range
v / w = 0.05 ...2.5
a
w
t
w
v
dtw
vtIete
wT
0
20
22
2
44 2
2
2
2
Gaussian Beam: Truncation by Offset
T
a/w
v/w = 0.05
v/w = 2.5
Transmission of a gaussian beam
with width w by a ring stop
inner radius b
outer radius a
Optimal beam width for maximal
transmission
2 2
2 2a a
w wT e e
w / aopt
E / E
1a
b
ln
12 awopt
Gaussian Bema: Truncation by Ring Stop
ri
Ii wi
circular
stop
gaussian
profil
z
a
b
Transmission of a gaussian beam by a ring stop with
for stop ratios = 0.05 ...2.5
Gaussian Beam: Truncation by Circular Ring Stop
Transmission of an elliptical gaussian beam with aspect ratio
by a circular stop withg radius a
Change of ratio a/w:
a / w = 6 , 5 , 4.5 , 4 , 3.5
3.0 , 2.5 , 2.0 , 1.75
1.5 , 1.25 , 1.0 , 0.9
0.8 , 0.7 , 0.6 , 0.5
0.4 , 0.3 , 0.2 , 0.1
Elliptical Gaussian Beam: Truncation by Stop
x
y
w
w
Transmission of a gaussian beam by a
centered slit of width b
Change of ratio a/w:
a / w = 2 , 1.5 , 1.4 , 1.3 , 1.2 , 1.1 , 1.0 0.95 ,
0.90 , 0.85 , 0.80 , 0.75 , 0.70 , 0.65 , 0.60 ,
0.55 , 0.50 , 0.45 , 0.40 , 0.35 , 0.30 ,
0.25 , 0.20 , 0.15 , 0.10 , 0.05
Gaussian Beam: Truncation by a Slit
x
y
bslit
w
gaussian
beam2
Slit
bT erf
w
Focussed Gaussian beam with spherical aberration
Asymmetry intra - extra focal
depending on sign of spherical aberration
Gaussian profile perturbed
Gaussian Beam with Spherical Aberration
c9 = -0.25
c9 = 0.25
c9 = 0
Gaussian beam with spherical aberration
The intensity becomes asymmetrical
Position and height of the peak are changing
0
11
2
1
0
24
1
2
2)( drree
zi
EzE
rRzw
rSik
w
r
Gaussian Bemas with Spherical Aberration
unperturbed
kS1 = 0.05 0.07 0.085 -0.07
I(0,z)
z
I(0,z)
z
I(0,z)
z
I(0,z)
z
Focussed Gaussian beam with spherical aberration
Focal length : f = 200 mm , beam radius: w = 3 mm,
Spherical aberration: Zernike coefficient c9 = 0.10 (normalization at r = 1.5 w, PV is 2c9)
Colored plot: logarithmic, cross sections linear at z = 180, 200, 210 mm
Gaussian Beam with Spherical Aberration
Focussed Gaussian beam with spherical aberration
Focal length : f = 200 mm , beam radius: w = 3 mm,
Spherical aberration: Zernike coefficient c9 = 0.25 (normalization at r = 1.5 w, PV is 2c9)
Colored plot: logarithmic, cross sections linear at z = 180, 200, 210 mm
Gaussian Beam with Spherical Aberration
Focussed Gaussian beam with spherical aberration
Focal length : f = 200 mm , beam radius: w = 3 mm,
Spherical aberration: Zernike coefficient c9 = 0.50 (normalization at r = 1.5 w, PV is 2c9)
Colored plot: logarithmic, cross sections linear at z = 160, 180, 200, 210 mm
Gaussian Beam with Spherical Aberration
Focussed Gaussian beam with spherical aberration
Focal length : f = 200 mm , beam radius: w = 3 mm,
Spherical aberration: Zernike coefficient c9 = 1.0 (normalization at r = 1.5 w, PV is 2c9)
Colored plot: logarithmic, cross sections linear at z = 160,180, 200, 210 mm
Gaussian Beam with Spherical Aberration
Astigmatism
c = 0.3
20 21 22 23 24 25 26 27 28 29 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
I(z) I(y,z)I(x,z)
Coma
c = 0.3
20 21 22 23 24 25 26 27 28 29 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
I(z) I(y,z)I(x,z)
Gaussian Beam with Coma and Astigmatism