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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