View
221
Download
1
Category
Preview:
Citation preview
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado
1: From Maxwell to Optics
• Maxwell’s equations– Constitutive relations
– Frequency domain
– The wave equation
• Geometrical optics– What is a ray
– Refraction and reflection
– Paraxial lenses
– Graphical ray tracing
• Fourier optics
• Application: spatial filtering
8
•Lecture 1–Outline
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 9
Maxwell’s equationsin differential form
t
BE
Jt
DH
0 B
ρD
Faraday’s law
Ampere’s law
Gauss’ laws
E Electric field [V/m]H Magnetic field [A/m]D Electric flux density [C/m2]B Magnetic flux density [Wb/m2]J Electric current density [A/m2] Electric charge density [C/m3] Curl [1/m] Divergence [1/m]
•Background–Maxell’s equations
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 10
Constitutive relationsInteraction with matter
tEε ε
tEεε
dE τ)(tεεD
Iε ε
f(t)ε
t
0
0
0
Dispersive & anisotropic
Anisotropic
Isotropic
H μdH τ)(tμμB cNonmagnetit
00
Permittivity of free space 8.854… 10-12 [F/m] Dielectric constant Permeability of free space 4 10-7 [H/m] Relative permeability Conductivity [/m]
E σJ
Ohm’s Law
•Lecture 1–Maxell’s equations
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 11
Monochromatic fieldsExpand all variables in temporal eigenfunction basis
deftf j t )(2
1)( Fourier Transform.
Note factor of 2which can be placed in different
locations.
)(EetE tjReMonochromatic fields E
transform like time-domain fields E for linear
operators
D
B
JDjH
BjE
0
Monochromatic Maxwell’s equations.
jdt
d Removes all time-derivates.
•Lecture 1–Maxell’s equations
dtetff j t )()(
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 12
Monochromatic constitutive relationsThe reason for using the monochromatic assumption
dE τ)(tεεDt
0 ED
)(0
Convolution Multiplication
0
)()( dtet tj Inverse Fourier Transform.Note that is now f() & not f(t).
If is not constant in , it causes “dispersion” of pulses.
dH τ)(tμμBt
0 HB
)(0
0
)()( dtet tj
•Lecture 1–Maxell’s equations
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 13
Wave equationEliminate all fields but E
E
D
Hj
BjE
002
02
0
Take curl of Faraday’s law
Magnetic constitutive
Electric constitutive
Ampere’s law
020 EkE
k Wave number of free space /c = 2 [1/m]c Speed of light in vacuum [m/s]001
Scalar simplification020
2 EkE
Monochromatic WE
•Lecture 1– Wave equation
EEE
2 Apply vector identity
0111 DDE In homogeneous,
charge-free space
Pedrotti3, Chapter 4
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado
Solutions of wave equationPlane wave
14
222
0
,,,
0
000
20
2222
020
222
0
20
2
nnc
fn
cnkkk
kkkkk
eEkkkk
eEtzyxE
EkE
zyx
zkykxktjzyx
zkykxktj
zyx
zyx
Scalar wave equation
Assumed solution
…is solution if
Vector wave equation has same solution but vector amplitude:
zkykxktj zyxeEtzyxE 0,,,
With the constraint (in lossless, isotropic media) that 00 kE
y
x
y
x
n Index of refraction n = sqrt(
y
x
2k
k
xxk 2yyk 2
Plug in
k
•Lecture 1– Wave equation
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado
More views of a plane wave
15
•Lecture 1– Wave equation
• Snapshot of the E and B fields versus space at one instant of time.• Note that E, B and k are perpendicular
BEk
• Equiphase fronts propagate forward at the speed of light.
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado
Solutions of wave equationSpherical wave
16
y
x 2220
222
,,,zyx
eEtzyxE
zyxktj
If you solve the scalar wave equation in spherical coordinates, you find the spherical wave solution:
n
ck
Ideal lenses turn portions of (infinite) plane waves into portions of spherical waves:
Note that complex valued, continuous field distributions (blue) can also be represented by straight lines that obey Snell’s laws (red). This is the foundation of geometrical optics.
•Lecture 1– Wave equation
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 17
Geometrical opticsApprox. solution of Maxwell’s equations
rSjkerrE
0EAssume slowly varying
amplitude E and phase S
zkykxkrSk zyx
0E.g. plane wave
22200 zyxkrSk
E.g. spherical wave
Contours of S(r) at multiples of 2
Sn(r)
S(r) Optical path length [m]
“Ray” = curve to S(r)
dsrnB
A
Ray approximation only retains information about phase and ignores amplitude. The approximation is invalid
anywhere amplitude changes rapidly.
•Lecture 1– Geometrical optics
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 18
Postulates of geometrical optics
• Rays are normal to equi-phase surfaces (wavefronts)
• The optical path length between any two wavefronts is equal
• The optical path length is stationary wrt the variables that specify it1
• Rays satisfy Snell’s laws of refraction and reflection
• The irradiance at any point is proportional to the ray density at that point
•Lecture 1– Geometrical optics
1 Pedrotti3, Section 2-2
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 19
Graphical ray tracingSolving Maxwell’s Eq. with a ruler
-t t’
Object
Image
1. A ray through the center of the lens is undeviated2. An incident ray parallel to the optic axis goes through the back focal point3. An incident ray through the front focal point emerges parallel to the optic axis.and occasionally useful4. Two rays that are parallel in front of the lens intersect at the back focal plane. 5. Corollary: two rays that intersect at the front focal plane emerge parallel.
-t t’
Object
Image
0
t
t
y
yM
y
y
•Lecture 1– Geometrical optics
Pedrotti3, Section 2-9
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 20
Graphical tracingNegative lenses
y
y’
-t
-t’
0
t
t
y
yM
-f
1. A ray through the center of the lens is undeviated2. An incident ray parallel to the optic axis appears to emerge from the front focal point3. An incident ray directed towards the back focal point emerges parallel to the optic axis.and occasionally useful4. Two rays that are parallel in front of the lens intersect at the back focal plane. 5. Corollary: two rays that intersect at the front focal plane emerge parallel.
Virtual image
•Lecture 1– Geometrical optics
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 21
Spatial frequencyBasis of Fourier optics
z
xinc
trans
inck
transk
xk
n
0
n0
incx
sin
0
nf transinc
xx
00
sinsin1
Spatial frequency in [1/m]
transincx
xx nfk
sin
2sin
222
00
Wave number in [1/m]
The electric field of a plane wave with wave-vector sampled on a line (the x axis) results in a sinusoidal field with spatial frequency
k
x
xxx
kxkf
1
2
2
22
ˆ
•Lecture 1– Fourier optics
Pedrotti3, Section 2-5, Snell’s Law
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 22
Lenses take Fourier transformsPhysical argument
E
sinx
x
x xF
xjxj
x
xx eeE
xEE
22
0
0
2
2cos
E
x
x
FF 0sin
xx
x
f
FxE
1
0
0F
xfx
fExE Fourier
•Lecture 1– Fourier optics
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado
Diffraction-limited spot size
23
F
dx 22
0
2,, FyxE
2,,0 zyE
F
dy 22
0
Lens
F
d d
y
z
F
d 2sin 1 3.83171
Neglecting diffraction, an infinitely-wide beam is Fourier transformed by a lens to an infinitely small focused spot. Finite beams are transformed to finite focused spots.
d
Fx
dF
xfx
0
0
1
d
F
d
Fx 00 22.1
2
83171.3
From circuit theory, we know that the Fourier tranform of a rect of width d is a sincfunction with its first null at f=1/d. Let’s use this to estimate the radius of the first null of a spot focused from a circular beam of diameter d through a lens of focal length F.
Use Fourier scale relationship from previous page
It turns out that the 2D Fourier transform of a circ or “top hat” function is a Bessel function – this strongly resembles a sync and is plotted above. The first null of this “Airy disk” focused spot field distribution is
So our estimate using a rectwas only off by 20%
•Lecture 1– Fourier optics
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado
Numerical aperture and F/#
24
2,, FyxE
The diffraction spot is the impulse response of the optical system. The image can thus be predicted by convolving the electric field distribution of the object with this point spread function.
•Lecture 1– Fourier optics
#22.1 0 Frspot
The resolution formula is sufficiently important that several quantities are defined to make it simpler. The F/# (pronounced “F number”) is the ratio of the focal length to the diameter of a lens.
This is convenient because the spot radius is ~ the (F/#) expressed in wavelengths.A similar and common quantity is the numerical aperture, which is the sin of the largest ray angle
NAr
F
dNA spot
061.02/
The radius of the first null is important because it defines the closest two points can be and just be resolved (the Rayleigh resolvability criterion).
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1.0
Image distance in units of spot radius
|E|2
Two incoherent point sources (e.g. stars) with their peaks on the first nulls of the adjacent point result in a small intermediate dip in intensity.
d
FF #
ECE 4606 Undergraduate Optics Lab
Robert R. McLeod, University of Colorado 25
Laser spatial filteringRay view
Pinhole
Collimate
Objective
fobj fcol
The incident collimated beam focuses to a point which passes through the pinhole, then expands until it hits the collimation lens, resulting in a magnified, collimated beam.
Rays that are not collimated, representing the noise on the beam, do not pass through the pinhole.
Pinhole
Collimate
Objective
fobj fcol
•Lecture 1– Laser spatial filtering
Recommended