laser optics-ii - iit bombayramesh/courses/me677/laser...me 677: laser material processing...

ME 677: Laser Material ProcessingInstructor: Ramesh Singh

Laser Optics-II

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Outline

• Absorption• Modes• Irradiance

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Reflectivity/Absorption

• Absorption coefficient will vary with the same effects as the reflectivity

• For opaque materials:– reflectivity = 1 - absorptivity

• For transparent materials:– reflectivity =1- (transmissivity + absorptivity)

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Reflectivity• In metals the radiation is predominantly absorbed by free

electrons in an “electron gas”• Free electrons are free to oscillate and reradiate without

disturbing the solid atomic structure• The reflectivity increases from visible to high wavelength• As a wavefront arrives at a surface all the free electrons in the

surface vibrate in phase generating an electric field 180° out of phase with the incoming beam

• The sum of this field will be a beam whose angle of reflection equals the angle of incidence

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Effect of Wavelength• Reflectivity is a function of the refractive index,

n, and the extinction coefficient, k

• At shorter wavelengths, the more energetic photons can be absorbed by a greater number of bound electrons– Reflectivity decreases and absorptivity increases

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Reflectivity of Metals

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Reflectivity of Non-metals

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Effect of Temperature

• Temperature increase results in increase in phonon population and phonon-electron energy exchanges– Reflectivity decreases– Absorption increases

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Surface Roughness

• Surface Roughness has a large effect on absorption due to:– The multiple reflections in the undulations– Also some "stimulated absorption" due to beam

interference with sideways reflected • If roughness is less than the beam wavelength, the

light will perceive the surface as flat

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Case Study-1045 steel

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Angle of Incidence• At certain angles the surface electrons may be constrained

from vibrating since to do so would involve leaving the surface. This they would be unable to do without absorbing the photon

• The electric vector is in the plane of incidence, the vibration of the electron is inclined to interfere with the surface and absorption is thus high

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Refraction

• On transmission the ray undergoes refraction described by Snell’s law:

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Refraction• Scattered intensity is a function of 1 /l4 Rayleigh

Scattering Law. • The normal form of a dispersion curve (refractive

index vs wavelength) is known as a Cauchy Equation:

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Beam Mode• Two spatial modes describe the beam

– Longitudinal – Transverse

• Essentially independent of each other– Transverse dimension in a resonator is normally considerably smaller

than the longitudinal• The standing wave condition will be amplified, i.e., there can be

only integer number of half wavelengths in the cavity, – D= q. λ/2 or qλ=2D– q is a large integer referring to the number of nodes in the longitudinal

standing, D is the cavity length (mirror separation), and λ is the wavelength.

• The longitudinal mode number is large in industrial lasers and is normally ignored on beam characteristics and performance.

• The transverse electromagnetic mode (TEM) is more important.

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Longitudinal Mode

• Longitudinal Mode (integral multiples of l/2)

l=2D, l=DD

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Longitudnal Mode• The frequency of the axial mode of the cavity,

! = #$2&

• If λ is replaced by '(, where c is the speed of light and ν is the frequency, the frequency separation between two adjacent nodes ∆ν between adjacent modes (∆q =1) is given by,

Δ! = '*+

• The axial modes of the laser cavity consists of a large number of frequencies spaced apart by '*+ as illustrated. However, the given mode can oscillate if the gain exceeds the losses at that frequency

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Spectral lines due to axial mode

(a)

(b)17

(a)

(b)

qq-3 q-2 q-1 q +1 q+2 q+3

q

Irrad

ianc

e

q

Irrad

ianc

e

Level representing losses


Transverse Mode• TEM describes the variation in beam intensity with position in

a plane perpendicular to the direction of beam propagation– It characterizes the intensity maxima in the beam

• The TEM is determined by: – The geometry of the cavity– Alignment and spacing of internal cavity optics– Gain distribution and propagation properties of the active medium– Presence of apertures in the resonator

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TEM• Rectangular Modes

– X axis– Y axis

• Circular Modes TEMpl– Radial, p– Angular, l

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Number of zero fields along X axis (m)

Num

ber o

f zer

o fie

lds a

long

Y a

xis (n

)

Number of radial zero fields (m)Nu

mbe

r of a

ngul

ar z

ero

field

s (n)


Intensity Plots

a.TEM00; b. TEM 10; c. TEM01*

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Propagation…• Generic equation for propagation:

! ", $, %

= !'()'2 "

+(%) )(2 $

+(%)+

+(%) ./ 01231

41 ./5 67/(82'2() 9:;<= 77> ./5

6?1@A(7)

Equations for Hermite Gaussian beam in (x-y coordinates) at beam waist (z=0)

2 2

2

0

0

2 2( , )

_ _min _

_ tan _ __ tan _ _min _ _

x yw

m nx yE x y E H H ew w

whereE Electric field amplitudeE No al amplitudex x dis ce from axisy y dis ce from axisw No al beam radius

æ ö+-ç ÷ç ÷è ø

æ ö æ ö= ç ÷ ç ÷ç ÷ ç ÷

è ø è ø

=====

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Propagation…• Hn(x) is a Hermite polynomial

( ) ( ) 2 2

2 2

2

2

2

2

2

0

0

02

2

0

1

( ) 1

( , ) ._ ,

( )

( ) ( )

( )

nn x x

n n

x yw

rw

rw

dH x e edx

H x

E x y E eCircular coordinates

E r E e

I r E r

I r I e

-

æ ö+-ç ÷ç ÷è ø

æ ö-ç ÷ç ÷è ø


= -

=

=

=

µ

=

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Gaussian Beam

• Gaussian function goes out to infinity• Low powered lasers mimic the TEM00

• TEM00 beams can be focused to smallest spot as compared to any other distribution

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Beam Properties• The point where irradiance drops to 1/e2 of

the peak• The radius containing 1- 1/e2 power• A two dimensional plot, the x value of which

95% of the plot area is contained between x and –x.

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Gaussian Distribution

For different gaussian beams:

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2

2

2

2

2

0

0 2

9

0

0 2

( )2

Other beam,

( )9

rw

rw

I r I ePIw

I r I ePIw

p

p



=

=

=

=


Equations for a generic intensity

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Intensity I(x, y, z) propagating along z axis. The second moments in x and y axes at a given location, z, can be defined as !"# $ = ∬ "'"̅ ) * ",,,- .".,

∬ * ",,,- .".,

!,# $ = ∬ ,'/, ) * ",,,- .".,∬ * ",,,- .".,

The centroids, 0̅ and /1 are defined by, 0̅ = ∬ " * ",,,- .".,

∬ * ",,,- .".,

/1 = ∬ , * ",,,- .".,∬ * ",,,- .".,

The beam dimensions (widths) in x and y are given by,23" = 4 !"($)23, = 4 !,($)


Examples

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!(#′, &′) = )∗+, [1 − 012

32 ][1 −51262 ]

(1)!(#′, &′) = )∗+

, [1 − 01232 ][1 −

512

62 789:2;2] (2)

!(#′, &′) = 2∗+, [1 − &′2

02 ][1 −#′222 ] (1)

!(#′, &′) = 2∗+, [1 − &′2

02 ][1 −#′2

2231−&′ 202 4] (2)


Propagation of laser beam

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2 210

0 2( , ) tan2z z R z

w r z krE r z E Exp Exp i kz

w w z R-

æ öé ùæ ö æ ö= - - - +ç ÷ê úç ÷ ç ÷ç ÷è ø è øë ûè ø

For a monochromatic beam propagating in z the complex electric field amplitude

where E0 is the peak amplitude; w is beam waste radius; k = 2π/l ; ZR is the Rayleigh length; Rz is the radius of curvature of the wave front


The variation of beam radius in propagation

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2

0

20

1zR

R

zw wz

wz pl

æ ö= + ç ÷

è ø

=


Propagation of Laser Beams• A laser beam propagating in space (lower case for TEM00 and upper case

for real beams)– Beam waist or minimum diameter, d0 /D0

– Beam waist diameter, dz/Dz at a location z from the waist– Beam waist or minimum radius, w0 /W0

– Beam waist radius, wz/Wz at a location z from the waist– q/Q = Full-angle beam divergence– l = Wavelength of light

QBEAM WAIST DIAMTER, DZ

Q

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Propagation of Ideal Beam• For a TEM00 beam, the diameter dz for any distance z form the

waist is a hyperboloid2

0 20

2 2 2 20

0

00

1 2 1

2 1

41

4

_ can be approximated by

2 tan2( )

z

z

zd dd

d d z

dDivergence TEM

Divergence

d dz z

lp

qlq

pq

q -

æ ö= + ç ÷

è ø

= +

=

=

æ ö-= ç ÷-è ø

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- 3 - 2 - 1 1 2 3

- 3

- 2

- 1

1

2

3

d2d1

z2z1


Real Beam• Real beams can be defined in terms of TEM00

• It can be postulated a fictitious “embedded Gaussian beam” having a smaller dia d exists in the real beam; D=M.d , where M>1

qpl

pl

pl

pl

pl

2

0

2

2220

2

2

0

222

02

2

20

220

2

2

20

20

2

20

0

...4

...4

....41

....41

41

MDMzDD

DMzDD

DMzDD

DMz

MD

MD

dzdd

z

z

z

z

z

==Q

Q+=

÷÷ø

öççè

æ+=

÷÷

ø

ö

çç

è

æ÷÷ø

öççè

æ+=

÷÷ø

öççè

æ+=

÷÷ø

öççè

æ+=

\

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Focused Beam Calculations

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01

21

2 222

1

2 21 01

2 2 202 01

2

0

21

2 22 01

1 21

2

12 2 2 202 01 1

For ideal beam,

( )

( )4

1 1 11

4

4. .For real beam with ,

.

( )

( )

1 1 11

z f fz fd

z f

z dd d f f

MD

z f fz fDz f

zD D f f

pl

pl

lp

-= +

öæ- + ÷çç ÷

è ø

öæ öæ= - + ç ÷÷ç è øè ø

Q =

-= +

öæ- + ÷ç Qè ø

öæ= - +÷ç Qè ø


Final Calculation• Once D02 is calculated, Q2 could be found

• Depth of focus where focal spot size changes by ±5%.• Approximate solution for focused beam diameter if lens is

placed at z from the beam waist

If unfocused beam diameter at z, is Dz.

2

02

4. ..MDlp

Q =

22202

22 Q+= zDD z

2

024. . .

. z

f MDDl

p=

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Laser Optics Setup at IITB

Indian Patent Application No 442/MUM/2011 Filed on 17 February 2011

Method and device for generating laser beam of variable intensity distribution and variable spot size


Compact Focusing

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The magnifying lens can be selected for a given magnification(m) and length (l=u+v). ! = #

$% + ' = ()# +

)$ =

)*

Solving the above set of equations, the focal length of magnifying lens (a biconvex lens) can be determined in terms of length(l) and magnification(m),

+ = ,-,.) /


Aberrations

• Spherical Aberration• Thermal Distortion• Astigmatism• Damage

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Spherical Aberration

• There are two reasons why a lens will not focus to a theoretical point– Diffraction limited problem – Spherical lens is not a perfect shape.

• Most lenses are made with a spherical shape since this can be accurately manufactured economically

• The alignment of the beam is not so critical as with a perfect aspheric shape

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Thermal Distortion

• High power laser beams are absorbed by lenses/optics– Selection of right optics ZnSe with CO2– The power distribution in TEM00 causes more

severe gradients than Donut• Shape change of lens• Varies the refractive index, specially in ZnSe

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Astigmatism and Damage

• Due to optical misalignment• Damage

– Due to dirt accumulation and burning on lens surface

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Summary• Absorption• Beam Modes• Propagation• Focusing• Aberrations

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