ece 645 –optics and photonics lecture 02 –geometrical optics · lecture 02 –geometrical...
Post on 19-Jul-2020
13 Views
Preview:
TRANSCRIPT
1
ECE 645 – Optics and Photonics
Lecture 02 – Geometrical Optics
• HMY 645
• Lecture 02
• Spring Semester 2015
Stavros IezekielDepartment of Electrical and
Computer Engineering
University of Cyprus
iezekiel@ucy.ac.cy
THE RAY PICTURE OF LIGHT
2
3
A hierarchy of optics models
Geometrical optics
D >> λ
Wave optics
D ≈ λ
EM theory
D ≈ λ
Quantum electrodynamics (QED)
(Feynman)Increasing
“accuracy”
“Ease” of
use
• If λ → 0, wave optics → geometrical optics
4
• D ≈ λ
• Noticeable diffraction, so must use wave theory
• D >> λ
• Negligible diffraction; we can approximate behaviour using
rays to describe light (geometrical optics)
http://ngsir.netfirms.com/applets/diffraction/2X/Diffraction.htm
5
• The rays show us the direction that the light takes
6
Postulates of geometrical optics
1. Rays are perpendicular to the wavefronts. They show the direction the light
travels.
2. An optical medium is described by a refractive index n.
v
cn =
n = refractive index
c = speed of light in a vacuum (=2.998 × 108 ms-1)
v = speed of light in the optical medium
(1)
7
The optical path length (OPL) is the distance that light would travel in a vacuum during
the same time that it takes to travel a distance d in a medium with refractive index n.
c
OPL
v
dt ==
∫=B
A
dsn )(OPL r
In a non-uniform medium, the refractive index is a function of position:
ds
A
B
r = position vector = (x,y,z)
nddv
c== .OPL (2)
8
3. Fermat’s principle states that:
An optical ray travelling between points A and B will take a path so that the time
(or the OPL) between them is an extremum compared to adjacent paths. (An
extremum means the rate of change is zero.) For most cases, it is a minimum:
Rays of light travel along the path of least time
© M
IT ∫Γ
= dsn )(OPL r
Γ is chosen to minimise the
value of this integral compared
to other paths.
9
Example: Reflection from a mirror
A B
1 2 345
x
Mirror
x1 x2 x3 x4 x5
x
tAB(x)
tAB(x)MIN
x4
10
Example: Refraction in uniform media
n = n1
n = n2
A
B
θ1
θ2
h1
h2
x
w
variable
X
OPL =
AX n1 + XB n2
11
n = n1
n = n2
A
B
θ1
θ2
h1
h2
x
w
variable
X
[ ] 2
2
22
22
11
21
hxwc
n
xhc
n
c
nXBAXn
c
OPLtAB
+−+
+=
+=
=
22
1
1sinxh
x
+=θ
[ ] 2
2
22sin
hxw
xw
+−
−=θ
12
Solve for:
0=dx
dtAB
[ ] 2
2
2
2
22
1
1 )(
hxwc
nxw
xhc
xn
dx
dtAB
+−
−−
+=
[ ] 2
2
2
2
22
1
1 )(0
hxwc
nxw
xhc
xn
dx
dtAB
+−
−=
+⇒=
2211 sinsin θθ nn = Snell’s law (3)
13
Another example of the principle of least time: collect all the light that comes outof a point P, and collect it back at another point P'.
P P'
Optical Black Box
This is called focusing
14
How can this be possible? Make the paths take the same time.
In this picture, light near the top spends less time in the glass (which slows it), so:
tPQP
/ = tPRP
/ = tPP
/
This is called a focusing lens
P P'
R
Q
n1
n2
15
Geometrical Optics: The Paraxial Approximation
We define light rays as directions in space.
We will not worry about the phase.
Each optical system will have an axis, and all light rays will be assumed to
propagate at small angles to it. This is called the Paraxial Approximation.
axis
16
The Optical Axis
A mirror deflects the optic axis into a new direction.
This ring laser has an optic axis that scans out a rectangle.
Optical axis A ray propagating
through this system
We define all rays relative to the relevant optical axis.
17
The Ray Vector
A light ray can be defined by two co-ordinates
xin, θθθθin
xout, θθθθout
its position, x
its slope, θ
Optical axis
x
θ
These parameters define a ray vector,
which will change with distance and as
the ray propagates through optics.
x
θ
18
Ray Matrices
For many optical components, we can define 2 x 2 ray matrices.
An element’s effect on a ray is found by multiplying its ray vector.
Ray matrices
can describe
simple and complex
systems.
These matrices are often called ABCD Matrices.
in
in
x
θ
A B
C D
Optical system ↔ 2 x 2 Ray matrix
out
out
x
θ
19
Ray matrices as derivatives
out in
out inD
B x
C
x A
θ θ
=
out
in
θ
θ
∂
∂
out
in
x
x
∂
∂
out
inx
θ∂
∂
out
in
x
θ
∂
∂
angular
magnification
spatial
magnification
out ioutout
i n
i
n i
n nx xxx
xθ
θ= +
∂
∂
∂
∂
out in ioutut
nin i
onx
xθ
θ
θθ
θ= +
∂
∂
∂
∂
Since the displacements and angles are assumed
to be small, we can think in terms of partial
derivatives
(4)
We can write these
equations in matrix form.
20
For cascaded elements, we simply multiply ray matrices.
3 2 1 3 2 1
out in in
out in in
x x xO O O O O O
θ θ θ
= =
O1 O3O2
in
in
x
θ
out
out
x
θ
(5)
21
Ray matrix for free space or a medium
If xin and θin are the position and slope upon entering, let xout and θout be the
position and slope after propagating from z = 0 to z.
out in in
out in
x x z θ
θ θ
= +
=xin, θin
z = 0
xout θout
z 1
0 1
out in
out in
x xz
θ θ
=
Rewriting these expressions in
matrix notation:
1=
0 1space
zO
(6)
22
Ray Matrix for an Interface
At the interface, clearly:
xout = xin.
Now calculate θout
Snell's Law says: n1 sin(θin) = n2 sin(θout)
which becomes for small angles: n1 θin = n2 θout ⇒ θout = [n1 / n2] θin
θin
n1
θout
n2
xin xout
1 2
1 0
0 /interfaceO
n n
=
(7)
23
Ray matrix for a curved interface
At the interface, again:
xout = xin.
To calculate θout, we must
calculate θ1 and θ2.
If θs is the surface slope at
the height xin, then
θ1 = θin+ θs and θ2 = θout+ θs
If R is the surface radius of curvature, the surface z coordinate will be:
2 2 2 21
2
21
2
1 ( / ) 1 ( / )
( / )
in in in
in
z R R x R R x R R R x R
x R
= − − = − − ≈ − −
=/s in
in
dzx R
dxθ⇒ ≈ ≈
θin
n1
θout
n2
xin = xout
θ1θ2
θs
R
z
θs
z = 0
24
1 2 1 2
1 0
( / 1) / /curvedinterface
On n R n n
= −
1 2( / )( / ) /
out in in inn n x R x Rθ θ⇒ ≈ + −
1 2 1 2( / ) ( / 1) /out in inn n n n x Rθ θ⇒ ≈ + −
Now the output angle depends on the input position, too.
Snell's Law: n1 θ1 = n2 θ2
1 2( / ) ( / )
in in out inn x R n x Rθ θ⇒ + ≈ +
θ1 = θin+ xin / R and θ2 = θout+ xin / R
n1 n2
θ1θ2
(8)
25
A thin lens is just two curved interfaces.
1 2 1 2
1 0
( / 1) / /curvedinterface
On n R n n
= −
We neglect the glass in between (it is a very
thin lens!), and we take n1 = 1.
2 1 2 1
1 0 1 0
( 1) / (1/ 1) / 1/thin lens curved curved
interface interface
O O On R n n R n
= = − −
2 1 2 1
1 0 1 0
( 1) / (1/ 1) / (1/ ) ( 1) / (1 ) / 1n R n n R n n n R n R
= = − + − − + −
n=1
R1 R2
n≠1
n=1
2 1
1 0
( 1)(1/ 1/ ) 1n R R
= − −
1 0
1/ 1f
−
This can be written:
1 21/ ( 1)(1/ 1/ )f n R R= − − The Lens-Maker’s Formulawhere:
26
Ray matrix for a lens
The quantity, f, is the focal length of the lens. It is the most important
parameter of a lens. It can be positive or negative.
1 0=
-1/ 1lens
Of
If f > 0, the lens deflects
rays toward the axis.
f > 0
If f < 0, the lens deflects
rays away from the axis.
f < 0
1 21/ ( 1)(1/ 1/ )f n R R= − −
R1 > 0
R2 < 0
R1 < 0
R2 > 0
(9)
GEOMETRICAL OPTICS ANALYSIS OF
OPTICAL FIBRES
27 28
Optical Fibres: Basic Structure
• A circular core of refractive index n1 is surrounded by cladding with a slightly lower value of
refractive index (n2 < n1).
• Light is confined to the core of the fibre by total internal reflection – TIR at the core-cladding
interface.
CORE CLADDING BUFFER COATING
Not to scale!
29
• Snell’s law: 2211 sinsin φφ nn =
Refraction occurs when: φ1 < φC
Note: n2 < n1
n1
n2Normalline Refracted
ray
Reflected ray
Incident ray
Material boundary
θ1
θ2
θ1
φ1
φ2
2211 coscos θθ nn =
(10)
(11)
30
• At the critical angle, φ 1 = φ C :
211 sin nn =φ
Note: n2 < n1
n1
n2
Incident
ray
Material
boundary
φ1 = φC
φ2 = π/2
( )12
1sin nnC
−=φ
Ray propagates
along the material
boundary
(12)
31
2211 sinsin φφ nn =
• What happens when φ1 > φ C ?
1
2
12 sinsin φφ
n
n=⇒
Now, sin φ2 ≤ 1, hence:
1sin 1
2
1 ≤φn
n
1
21sin
n
n≤∴ φ
• Since n2 < n1, this implies that although the maximum value (“mathematically”)
for sin φ1 is unity, in this case it is limited to: 0 ≤ sin φ1 ≤ n2/n1.
32
• Beyond the critical angle, φ 1 > φ C :
Note: n2 < n1
n1
n2
Incident
ray
Material
boundary
φ1
φ2 = π/2
φ1
TIR is the physical principle upon which optical waveguides, and specifically optical
fibres, are based.
Total internal
reflection (TIR)
Totally
internally
reflected
ray
33
θ1 = θC
n1
n2
θ1
θ1 > θCθ1 < θC
n1
n2θ2
θ1
n1
n2
θ1
θ2=900
Summary
Total internal reflection (TIR) is the mechanism with which light propagates in
optical fibres.
It also has other applications (e.g. prisms).
Refraction Critical angle Total internal reflection
34
Does not propagate; φ < critical angle at core-cladding interface
Propagates through repeated TIR at core-cladding interface;θ0 denotes the acceptance angle.
Propagation in an ideal step-index fibre
Airn0
Cladding
Core
Cladding
n1
n2
n2
n1 > n2 > n0
θ0
θφ
Reflected ray
Refracted
ray
35
Numerical aperture (NA) in a step-index fibre
What is the maximum acceptance angle θ0?
Apply Snell’s law:
θθ sinsin 100 nn =21 sin nn C =φφC
θ
θ0
Cladding n2
Core n1
Air n0
2πφθ =+ C
36
2
2
2
1
2
1
21
2
1
1
1
100
1
sin1
cos
)2(sin
sinsin
nn
n
nn
n
n
n
nn
C
C
C
−=
−=
−=
=
−=
=
φ
φ
φπ
θθ
This equation defines the numerical aperture (NA) of a step-index fibre:
21 sin nn C =φφC
θ
θ0
Cladding n2
Core n1
2πφθ =+ C
2
2
2
100 sinNA nnn −== θ (13)
37
Basic types of optical fibre
• Single-mode step-index (monomode)
• Multimode step-index
• Multimode graded index
• Dispersion shifted and dispersion flattened fibre
38
Refractive index profile Fibre cross sections &
ray paths
Typical
dimensions
125 μm cladding
8 - 12 μm
core
n
n1n2
Step-index single mode
125 - 400 μm cladding
50 - 200 μm
coren
n1n2
Step-index multimode
125 – 140 μm cladding
50 - 100 μm
coren
n1n2
Graded index multimode
39
Single-mode fibres
• advantage: low dispersion
• disadvantages:
1) small numerical aperture (NA)
2) high tolerance connectors required
3) difficult to couple LED light into them
• used with laser diodes and for relatively long links.
125 μm cladding
8 - 12 μm
core
n
n1n2
Step-index single mode
n1n2
40
Multimode step-index fibres
• advantages:
1) large NA
2) low tolerance connectors
3) easy to couple LED light into them
• disadvantage: intermodal dispersion due to
different group velocities of the many modes
present
• used with LEDs and for “short” links, e.g.
LANs (local area networks)
125 - 400 μm cladding
50 - 200 μm
coren
n1n2
Step-index multimode
n1n2
41
Graded index fibres
• Multimode step index fibres cause dispersion because high order modes travel further than low
order modes in multimode fibre (causing broadening of the transmitted light pulse)
• This is minimised by using graded index fibres, where the refractive index n decreases smoothly
with increasing radius
125 – 140 μm cladding
50 - 100 μm
coren
n1n2
Graded index multimode
42
Dispersion
• If attenuation was the only source of signal degradation, it would not be too bad, due to
the existence of optical amplifiers:
• Unfortunately, real fibres also have dispersion ....
1 00
1 00
FIBRE
Input bit stream Output bit stream:
attenuated
due to optical losses in fibre
1 00
Original bit stream
OPTICAL
AMPLIFIER
43
p (t)p(t - τ)
pIN (t)
t
pOUT (t)
tτ
tτ
tτ
Attenuation onlyReduction in pulse energy
Attenuation & dispersionReduction in pulse energy
• Pulse spreading
Fibre
No change in
pulse shape
44
• We can (normally) consider the fibre to be a linear system, with an impulse
response as shown:
tt
h(t)
σ
δ (t)
pin(t)
pin(t)pout(t)
pout(t)
pin(t) = δ (t), hence pout(t) = h(t)
t
t = mean arrival time
σ = rms pulse spread
45
• Consider the output pulse:
σ
t
pout(t)
t
dttpE out∫∞
∞−
= )(
• Energy content
• E = area under pulse
dttptE
t out∫∞
∞−
= )(1
• Mean time of pulse arrival
FWHM = ∆τ
( )
22
22
)(1
)(1
tdttptE
dttpttE
out
out
−=
−=
∫
∫∞
∞−
∞
∞−
σ
• σ is root mean square spread of
pulse around mean arrival time
• It gives a measure of the dispersion
• An alternative measure is the
full width at half maximum (FWHM)
46
• If a pulse with an rms pulse width of σ1 is applied to a fibre, then the output
pulse spread will be given by:
h(t)
tm2
σ2
pout(t)
tm1
σ1
pin(t)
σ
22
1
2
2 σσσ +=
47
Dispersion leads to pulse spreading and overlapping:
48
49
Physical Causes of Dispersion
There are two major types of dispersion in optical fibres:
• Intermodal
- only occurs in multimode (MM) fibres, not in
single mode (SM)
- dominant source of dispersion for MM fibres
• Intramodal
- occurs in both SM & MM fibres
- dominates in single-mode (SM) fibres
- consists of:
• Material dispersion
• Waveguide dispersion
50
Intermodal Dispersion
• Light is transmitted along a multimode optical fibre by several modes (ray
paths).
• The distances travelled by the various paths are different, and hence the
transit times through the fibre also differ.
• A pulse of light, even if it is monochromatic, will have a spread of delays
and the received pulse will have a wider FWHM.
n1
n2
21
3
nO
n
Multimode step
index fibre. Ray
paths are different
so that rays arrive at
different times
input
t
+
+
=
t
t
output
• Consider a simple example, in which input light is launched into a fibre
and then propagates along three modes:
52
• Consider worst case scenario for step-index MM fibre:
A B
n2
n1
longest
pathshortest
path
φc
Intermodal dispersion in step-index multimode fibres
53
φc
LS
LL
sin φc = n2/n1 {Snell’s law for critical angle}
• Also, LS = LL sin φc = LL n2/n1
• Both rays have same velocity: v = c/n1
• τS = LS /v = LSn1/c
• τL = LL n1/c = LS . (n12/cn2)
• Hence δτ = τL - τS = (LSn1/c).{n1/n2 - 1}
• Consider time delay between longest and shortest paths through the fibre:
• So the pulse spread per unit length for intermodal dispersion in a
multimode fibre is:
−=
2
211
n
nn
c
n
L
τδ(14)
54
n1
n2
21
3
nO
n1
2
1
3
n
n2
OO' O''
n2
23
• Intermodal dispersion can be minimised by using graded index fibre:
Multimode step
index fibre. Ray
paths are different
so that rays arrive at
different times
Graded index fibre.
Ray paths are
different
but so are the
velocities
along the paths so
that all rays arrive
at approximately
the same time
55
GEOMETRICAL OPTICS ANALYSIS OF
LASER RESONATORS
56
57
Laser – Light Amplification by Stimulated Emission
• Three “ingredients” are needed for a laser:
1. Medium with optical gain
2. Pump (optical or electrical)3. Optical resonator
58
Types of Optical Resonator
Ring resonator
Planar mirror resonator
59
Spherical mirror resonator
zOptical
axis
R1 R2
d
Radius of curvature
60
Stability of optical resonators
Two flat mirrors, the flat-flat
laser cavity, is difficult to align
and maintain aligned.
Two concave curved mirrors, the
usually stable laser cavity, is
generally easy to align and
maintain aligned.
Two convex mirrors, the unstable
laser cavity, is impossible to
align!
Optical gain medium
http://webphysics.davidson.edu/applets/optics4/default.html
61
We will now examine the stability of a spherical mirror resonator
Define one round-trip as: going through the cavity twice, and being reflected
twice (once by the left mirror, once by the right mirror)
z
0
1
2
x0
x1
x2
d
62
• Unit cell
• Equivalent to one round trip in the
optical cavity
• Continually repeated, infinitely
We can model multiple reflections in the previous optical cavity with an
equivalent system of lenses spaced equally apart:
d
mm - 1 m + 1
z
63
z
d
R1 R2
z
f1 f2
CAVITY
MODEL OF CAVITY FOR
HALF A ROUND-TRIP
− 11
01
1f
− 11
01
2f
− 12
01
1R
− 12
01
2R
By inspection:
11
21
Rf=
22
21
Rf=
64
z
d
R1 R2
d
z
f1 f2 f1 f2 f1
CAVITY
MODEL OF CAVITY
A
A
B
B C
C
D
D
E
E
65
DC
BA
DC
BA
DC
BA
0
0
θ
x
DC
BA
DC
BA
1
1
θ
x
m
mx
θ
+
+
1
1
m
mx
θ
1 2 m-1 m+1m
=
0
0
θθ
x
DC
BAxm
m
m
A cascade of identical optical components
=
+
+
m
m
m
m x
DC
BAx
θθ 1
1
(22)
(23)
66
mmm
mmm
DCx
BAxx
θθ
θ
+=
+=
+
+
1
1
B
Axx
B
Axx
mmm
mmm
121
1
+++
+
−=⇒
−=
θ
θ
mmmm DCx
B
Axxθ+=
− ++ 12
B
ADxDxCx
B
Axx mmm
mm −+=
− +++ 112
67
( ) ( ) mmm xBCADxDAx −−+= ++ 12
mmm xOxOx ][Det ][ Trace 12 −= ++
DAO +=][ Trace
BCADO −=][Det
(24)
In many cases, the determinant of the ABCD matrix is one.
(25)
68
If we know the initial offset from the optical axis (x0) and the initial angle (θ0),
we can use the iterative equation (24) to find where the ray of light is at fixed points
(solid dots):
z
x
0 1 2 3 4……….
z
x
0 1 2 3 4……….
Unstable: ray eventually leaves system
Stable and periodic: ray never leaves system
Unit cell
69
DC
BA
DC
BA
DC
BA
0
0
θ
x
DC
BA
DC
BA
1
1
θ
x
m
mx
θ
+
+
1
1
m
mx
θ
1 2 m-1 m+1m
d
mm - 1 m + 1
z
For our model of the spherical mirror cavity, we have:
Need to
calculate
ABCD matrix
for this unit
cell
70
+−−+−−
−−
=
−−
−−=
−
−=
+
+
21
2
122121
2
2
2
2211
211
1
21
11
21
11
1
11
1
10
1
11
01
10
1
11
01
ff
d
f
d
f
d
ff
d
ff
f
dd
f
d
x
f
d
f
d
f
d
f
d
xd
f
d
f
x
m
m
m
m
m
m
θ
θθ(26)
71
+−−+−−
−−
=
21
2
122121
2
2
2
21
11
21
ff
d
f
d
f
d
ff
d
ff
f
dd
f
d
DC
BA
To calculate determinant, consider equation (26) on previous slide:
(27)
−
−=
10
1
11
01
10
1
11
01
21
d
f
d
fDC
BA
Det = 1 Det = 1 Det = 1 Det = 1
1][Det =−=∴ BCADO
72
( ) mmm xxDAx −+= ++ 12
So the iterative equation that we use to calculate the ray path now becomes
simpler:
Let:
( )DAb +=2
02 12 =+−∴ ++ mmm xbxx
(28)
(29)
Assume that the solution to (29) has the following form:
θjm
m exx 0= (30)
If we substitute (30) into (29) we get:
( ) 0122
0 =+− θθθ jjjmbeeex (31)
73
0122 =+−∴ θθ jj
bee
(32)21 bjbe
j −±=θ
Solve this quadratic in ejθ to get:
b
21 bj −
1
θje
So the general solution to (29) if b is real is:
θθ jmjm
m exexx−+= *
00
We can also write this using trigonometric functions, for example:
)sin(max αθ += mxxm
(33)
(34)
74
this analysis will be OK provided:
Since:2
1 bjbej −±=θ
012 ≥− b
1≤⇒ b
From the definition in (28),
12
≤+ DA
(35)
(36)
Condition for stability
75
12
1 ≤+
≤−DA
In other words:
If we add 1 to this equation and then divide by two, we get:
14
20 ≤
++≤
DA
Condition for stability
(37)
This version of the stability condition is easier to use for our equivalent lens system.
From (27):
21
2
12
21
ff
d
f
d
f
dD +−−=
2
1f
dA −=
76
−
−=
+−−=
+−−=
++
21
21
2
21
21
2
12
21
21
4221
224
4
1
4
2
f
d
f
d
ff
d
f
d
f
d
ff
d
f
d
f
dDA
Let:
−=
i
if
dg
21
(38)
(39)
77
• The condition for stability can now be written as:
10 21 ≤≤ gg (40)
−=
1
12
1f
dg
−=
2
22
1f
dg
78
−=
1
12
1f
dg
−=
2
22
1f
dg
1 2 3 4
-4 -3 -2 -1
-1
-2
-3
-4
1
2
3
4
STABLE REGION
421
dff ==
∞±== 21 ff
221
dff ==
79
ii Rf
21=
To relate our equivalent lens model with the actual optical cavity, remember:
−=
−=∴
ii
iR
d
f
dg 1
21
80
−=
1
1 1R
dg
−=
2
2 1R
dg
1 2 3 4
-4 -3 -2 -1
-1
-2
-3
-4
1
2
3
4
STABLE REGION
221
dRR ==
∞±== 21 RR
dRR == 21
∞=
=
2
1
R
dR
dR
R
=
∞=
2
1
htt
p:/
/co
rd.o
rg/c
m/l
eo
t/co
urs
e0
1_
mo
d0
7/m
od
01
-07
fra
me
.htm
http://perg.phys.ksu.edu/vqm/laserweb/java/cavity/cavstabe.htm
81
Closing Remarks
• Ray optics (geometrical optics) is a useful tool, especially when combined with
ABCD matrices
• It can be used for both analysis and design of optical systems
• Here, we have used it to analyse ray propagation in optical fibres and to look at
the stability of laser resonators
• However, the ray optics approach has limitations:
• It depends on the optical component size being bigger than the wavelength
• It is not useful for looking at phenomena such as diffraction
• It tells us nothing about the spectral content (i.e. wavelength) of the light
or other important properties like the state of polarisation.
top related