1: from maxwell to optics - university of colorado...

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


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



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



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.


dtetff j t )()(



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




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



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




More views of a plane wave

15


• 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.



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.




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



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


1 Pedrotti3, Section 2-2



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


Pedrotti3, Section 2-9



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




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



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




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%




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.


#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 #



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



Laser spatial filteringFourier optics view

Convolved with Multiplied by

= =

REAL SPACE FOURIER SPACE

•Lecture 1– Laser spatial filtering

1: from maxwell to optics - university of colorado...

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