Matrix Methods in Optics • For more complicated systems use Matrix methods & CAD tools • Both are based on Ray Tracing concepts • Solve the optical system by tracing may optical rays • In free space a ray has position and angle of direction y1 is radial distance from optical axis V1 is the angle (in radians) of the ray • Now assume you want to a Translation: find the position at a distance t further on • Then the basic Ray equations are in free space making the parallex assumption

tVyy 112 +=

12 VV =

Matrix Method: Translation Matrix • Can define a matrix method to obtain the result for any optical process • Consider a simple translation distance t • Then the Translation Matrix (or T matrix)

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

1

1

1

1

2

2

Vy

10t1

Vy

DCBA

Vy

• The reverse direction uses the inverse matrix

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

−−

=⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

1

1

2

2

2

21

1

1

1011

Vyt

Vy

ACBD

bcadVy

DCBA

Vy

General Matrix for Optical Devices • Optical surfaces however will change angle or location • Example a lens will keep same location but different angle • Reference for more lens matrices & operations A. Gerrard & J.M. Burch, “Introduction to Matrix Methods in Optics”, Dover 1994 • Matrix methods equal Ray Trace Programs for simple calculations

General Optical Matrix Operations • Place Matrix on the left for operation on the right • Can solve or calculate a single matrix for the system

[ ][ ][ ] ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

1

1

2

2

Vy

MMMVy

objectlensimage

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡ ′=⎥

⎦

⎤⎢⎣

⎡

1

1

2

2

101

1101

101

Vys

f

sVy

Solving for image with Optical Matrix Operations • For any lens system can create an equivalent matrix • Combine the lens (mirror) and spacing between them • Create a single matrix

[ ] [ ][ ] [ ] ⎥⎦

⎤⎢⎣

⎡==

DCBA

MMMM systemn 12L

• Now add the object and image distance translation matrices

[ ][ ][ ]objectlensimagess

ss MMMDCBA

=⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ ′=⎥

⎦

⎤⎢⎣

⎡10s1

DCBA

10s1

DCBA

ss

ss

( )⎥⎦

⎤⎢⎣

⎡+

+′++′+=⎥

⎦

⎤⎢⎣

⎡DCsC

DCssBAsCsADCBA

ss

ss

• Image distance s’ is found by solving for Bs=0 • Image magnification is

ss D

1m =

Example Solving for the Optical Matrix • Two lens system: solve for image position and size • Biconvex lens f1=8 cm located 24 cm from 3 cm tall object • Second lens biconcave f2= -12 cm located d=6 cm from first lens • Then the matrix solution is

⎥⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡10241

181

01

1061

1121

01

10X1

DCBA

ss

ss

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡2

81

241

23

121

61

10X1

DCBA

ss

ss

⎥⎦

⎤⎢⎣

⎡−−−−

=⎥⎦

⎤⎢⎣

⎡11042.0X12X1042.025

DCBA

ss

ss

• Solving for the image position:

cm12Xor0X12Bs ==−=

• Then the magnification is

1D1m

s

−==

• Thus the object is at 12 cm from 2nd lens, -3 cm high

Optical Matrix Equivalent Lens • For any lens system can create an equivalent matrix & lens • Combine all the matrices for the lens and spaces • The for the combined matrix where RP1 = first lens left vertex RP2 = last lens right most vertex n1=index of refraction before 1st lens n2=index of refraction after last lens

Example Combined Optical Matrix • Using Two lens system from before • Biconvex lens f1=8 cm • Second lens biconcave f2= -12 cm located 6 cm from f1 • Then the system matrix is

⎥⎦

⎤⎢⎣

⎡−

=⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡5110420

62501

81

01

1061

1121

01

...

DCBA

• Second focal length (relative to H2) is

cm..C

fs 76691042011

2 =−

−=−=

• Second focal point, relative to RP2 (second vertex)

cm...

CAfrP 4002

10420250

2 =−

−=−=

• Second principal point, relative to RP2 (second vertex)

cm..

.C

AHs 198710240

250112 −=

−−

=−

=

Matrix Method and Spread Sheets • Easy to use matrix method in Excel or matlab or maple • Use mmult array function in excel • Example [Space 1], [Lens 1] matrix multiply • Select array output cells (eg. [Matrix 1]) • In row 1, col 1 of output matrix location enter =mmult( • Select [Space 1] cells G5:H6 then comma • Select [Lens 1] cells I5:J6 (eg =mmult(G5:H6,I5:J6) ) • Then do control+shift+enter (very important) • Here is example from previous page

.

Gaussian Plane Waves • Plane waves have flat emag field in x,y • Tend to get distorted by diffraction into spherical plane waves and Gaussian Spherical Waves • E field intensity follows:

( )⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡ +−−=

R2yxKrtiexp

RU)t,R,y,x(u

220 ω

where ω= angular frequency = 2πf U0 = max value of E field R = radius from source t = time K= propagation vector in direction of motion r = unite radial vector from source x,y = plane positions perpendicular to R • As R increases wave becomes Gaussian in phase • R becomes the radius of curvature of the wave front • These are really TEM00 mode emissions from laser

Gaussian Beam • Assumes a Gaussian shaped beam

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −= 2

2

22

2

0 wr2exp

wP2

wr2expI)r(I

π

Where P = total power in the beam w = 1/e2 beam radius at point w(z)

Measurements of Spotsize • beam spot size is measured in 3 possible ways • 1/e radius of beam • 1/e2 radius = w(z) of the radiance (light intensity) most common laser specification value 13% of peak power point point where emag field down by 1/e • Full Width Half Maximum (FWHM) point where the laser power falls to half its initial value good for many interactions with materials • useful relationship

erFWHM 1665.1=

21177.1177.1e

rwFWHM ==

FWHMrwe

849.02

1 ==

Gaussian Beam Changes with Distance • The Gaussian beam radius of curvature with distance

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

220

zw1z)z(R

λπ

• Gaussian spot size with distance

21

2

20

0 wz1w)z(w

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

πλ

• Note: for lens systems lens diameter must be 3w0.= 99% of power • Note: some books define w0 as the full width rather than half width • As z becomes large relative to the beam asymptotically approaches

020

0)(wz

wzwzw

πλ

πλ

=⎟⎟⎠

⎞⎜⎜⎝

⎛≈

• Asymptotically light cone angle (in radians) approaches

( )0wZ

zwπλθ =≈

Rayleigh Range of Gaussian Beams • Spread in beam is small when width increases < 2 • Called the Rayleigh Range zR

λπ 2

0R

wz =

• Beam expands 2 for - zR to +zR from a focused spot • Can rewrite Gaussian formulas using zR

⎥⎦

⎤⎢⎣

⎡+= 2

2R

zz1z)z(R

21

2R

2

0 zz1w)z(w ⎥

⎦

⎤⎢⎣

⎡+=

• Again for z >> zR

R0 z

zw)z(w ≈

Beam Expanders • Telescope beam expands changes both spotsize and Rayleigh Range • For magnification m of side 2 relative side 1 then as before change of beam size is

0102 mww =

• Ralyeigh Range becomes

1R2

202

2R zmwz ==λ

π

• where the magnification is

1

2

ffm =

Example of Beam Divergence • eg HeNe 4 mW laser has 0.8 mm rated diameter. What is its zR, spotsize at 1 m, 100 m and the expansion angle • For HeNe wavelength λ = 632.8 nm • Rayleigh Range is

m794.010x328.6

)0004.0(wz 7

220

R === −

πλ

π

• At z = 1 metre

mm643.0m000643.0794.0110004.0

zz1w)z(w

21

2

221

2R

2

0 ==⎥⎦

⎤⎢⎣

⎡+=⎥

⎦

⎤⎢⎣

⎡+=

• At z = 100 m >> zR

Radians10x04.50004.0

10x328.6wz

)z(w 47

0

−−

===≈ππ

λθ

mm4.50m0504.0)10x04.5(100z)z(w 4 ===≈ −θ

• What if beam was run through a beam expander of m = 10

mmm.).(mww 4004000040100102 ====

Radiansx.x.mm

54

1004510

10045 −−

===θθ

mm04.5m00504.0)10x04.5(100z)z(w 5 ===≈ −θ

• Hence get a smaller beam at 100 m by creating a larger beam first

Focused Laser Spot • Lenses focus Gaussian Beam to a Waist • Modification of Lens formulas for Gaussian Beams • From S.A. Self "Focusing of Spherical Gaussian Beams" App. Optics, pg. 658. v. 22, 5, 1983 • Use the input beam waist distance as object distance s to primary principal point • Output beam waist position as image distance s'' to secondary principal point

Gaussain Beam Lens Formulas • Normal lens formula in regular and dimensionless form

1

fs1

fs1or

f1

s1

s1

=⎟⎠

⎞⎜⎝

⎛ ′+

⎟⎠

⎞⎜⎝

⎛=

′+

• This formula applies to both input and output objects • Gaussian beam lens formula for input beams includes Rayleigh Range effect

f1

s1

fszs

12R

=′

+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+

• in dimensionless form

1

fs1

1fsfz

fs

12

R

=⎟⎠

⎞⎜⎝

⎛ ′+

⎟⎠

⎞⎜⎝

⎛−

⎟⎠

⎞⎜⎝

⎛

+⎟⎠

⎞⎜⎝

⎛

• in far field as zR goes to 0 (ie spot small compared to lens) this reduces to geometric optics equations

Gaussain Beam Lens Behavior • Plot shows 3 regions of interest for positive thin lens • Real object and real image • Real object and virtual image • Virtual object and real image

Main Difference of Gaussian Beam Optics • For Gaussian Beams there is a maximum and minimum image distance • Maximum image not at s = f instead at

Rzfs +=

• There is a common point in Gaussian beam expression at

1fs

fs

=′′

=

For positive lens when incident beam waist at front focus then emerging beam waist at back focus • No minimum object-image separation for Gaussian • Lens f appears to decrease as zR/f increases from zero i.e. Gaussian focal shift

Magnification and Output Beams • Calculate zR and w0, s and s'' for each lens • Magnification of beam

21

2R

20

0

fz

fs1

1wwm

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠

⎞⎜⎝

⎛+⎥

⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−

=′′

=

• Again the Rayleigh range changes with output

RR zmz 2=′′

• The Gaussian Beam lens formula is not symmetric From the output beam side

( )f1

fszs

1s1

2R

=

⎥⎦

⎤⎢⎣

⎡−′′′′

+′′+

Special Solution to Gaussian Beam • Two cases of particular interests Input Waist at First Principal Surface • s = 0 condition, image distance and waist become

2

Rzf1

fs

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=′′

21

2

R

0

zf1

wf

w

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

=′′ πλ

Input Waist at First Focal Point • s = f condition, image distance and waist become

fs =′′

0wfw

πλ

=′′

Gaussian Spots and Cavity Stability • In laser cavities waist position is controlled by mirrors • Recall the cavity g factors for cavity stability

ii r

L1g −=

• Waist of cavity is given by

( )[ ][ ]

41

22121

21212

1

0 gg2gggg1ggLw

⎭⎬⎫

⎩⎨⎧

−+−

⎟⎠⎞

⎜⎝⎛=

πλ

where i=1=back mirror, i=2= front

Gaussian Waist within a Cavity • Waist location relative to output mirror for cavity length L is

( )2121

212 gg2gg

Lg1gz−+

−=

[ ] [ ]4

1

212

12

1

2

41

211

22

1

1 11 ⎭⎬⎫

⎩⎨⎧

−⎟⎠⎞

⎜⎝⎛=

⎭⎬⎫

⎩⎨⎧

−⎟⎠⎞

⎜⎝⎛=

ggggLw

ggggLw

πλ

πλ

• If g1 = g2 = g=0 (i.e. r = L) waist becomes

5.0z2

Lw 2

21

0 =⎟⎠⎞

⎜⎝⎛=

πλ

• If g1 =0, g2 = 1 (curved back, plane front) waist is located z2 = 0 at the output mirror (common case for HeNe and many gas lasers) • If g1 = g2 = 1 (i.e. plane mirrors) there is no waist