•Design of ideal imaging syst ems with geometrical …ecee.colorado.edu/.../lecturenotes/FoundationsOfGeometricalOptics.pdf · that all three of the laws of geometric optics can

ECE 5616 OE System Design

Robert McLeod 18

Geometrical opticsDesign of imaging systems

“Geometrical optics is either very simple, or else it is very complicated”

Richard P. Feynman

What’s it good for?1. Where is the image?2. How large is it?3. How bright is it?4. What is the image quality?1

“Useful when size of aperture is > 100 ”

Why?

1P. Mouroulis & J. Macdonald, Geometrical Optics and Optical Design, Oxford, 1997

•Design of ideal imaging systems with geometrical optics


Robert McLeod 19

The ray equationApprox. solution of ME when n(r) is slow

rSjkerrE

0EAssume slowly varying

amplitude E and phase S

rnrS 22

Substitute into isotropic wave equation and retain lowest terms…

zkykxkrSk zyx

0 E.g. plane wave

22200 zyxkrSk

E.g. spherical wave

Countours of S(r) at multiples of 2

Sn(r)

S(r) Optical path length [m]

Ray equation – reduction of Maxwell’s equations.

“Ray” = curve to S(r)

•Design of ideal imaging systems with geometrical optics–Ray and eikonal equations

dsrnB

A


Robert McLeod 20

The eikonal equationAn equation for evolution of ray trajectory

ds

rdrnsrnrS

ˆ

s Parametric distance along ray [m]Ray trajectory [m] sr

ds

rdrn

ds

drS

ds

d

then take derivative in s,

and chain rule for ,

rSrn

rSrS

ds

rdrS

ds

d

finally apply another identity.

rnrnrn

rSrSrn

2

2

1

2

1

rnds

rdrn

ds

d


Take square-root of ray equation,

r

rdr

dss

sd

rd

ˆ

Ray

Saleh & Teich 1.3, Born & Wolf 7th edition, pg. 129


Robert McLeod 21

Ray trajectory n=constantFinally!

rnds

rdrn

ds

d


Eikonal

000

nds

rdn

ds

d

scsr

n(r) = no, a constant

where is a constant.c

Thus, we have discovered that, in homogenous materials, rays travel in straight lines!


Robert McLeod

History of geometrical optics

22

•Design of ideal imaging systems with geometrical optics–Rays and refraction

• 280BC Euclid's Catoptrics light goes in straight line. However light was emitted by eye (not)

– law of reflection was correctly formulated in Euclid's book

• Hero of Alexandria, in his Catoptrics (first century BC), also maintained that light travels with infinite speed.

– adopted the rule that light rays always travel between two points by the shortest path

• 1000 AD Arab philosopher “Alhazan” (Abu'ali al-hasan ibn al-haytham) first person to realize that light actually travels from the object seen to the eye

• 1676 Olaf Römer demonstrated that light must have a finitevelocity, using his timings of the successive eclipses of the satellites of Jupiter

• 100-170AD Claudius Ptolemy rough law of refraction was studied experimentally (only close to normal incidence)

• 1621 Dutch mathematician Willebrord Snell comes up with correct law using sines. Does not publish

• 1637 French philosopher René Descartes was the first to publish, French countries call Snell’s law Descartes law of refraction

• 1658 French mathematician Pierre de Fermat demonstrated that all three of the laws of geometric optics can be accounted for on the assumption that light always travels between two points on the path which takes the least time

Kevin Curtis OESD lecture notes


Robert McLeod 23

Pinhole camera (1/2)


• Mo Ti, China, invented 5th century BC• Aristotle observes image of eclipse cast through leaves, 300 BC

• Aristotle formulates theory of light and color (some right, some wrong)• Alhazen (Ibn Al-Haytham) invented CA 1000 AD• Della Porta invented CA 1600• Kepler named it Camera Obscura and suggested lens for efficiency, 1604

l

l’

• Magnification, M = - l’ / l• Very large depth of focus• Very small chromatic effects• Very poor power efficiency• Dr. Webster Cash at CU

funded by NASA Institute for Advanced Concepts to examine pinhole camera in space as extrasolar planet imager.

Abelardo Morell pinhole image of Manhattan


Robert McLeod

Pinhole camera (2/2)Impact of pinhole size

24


• A pinhole camera has nolens but uses a very smallhole some distance fromthe film/screen to producean image. If we assumethat light travels in straightlines, then the image of adistant point source will bea blur whose diameter isthe same as the pinhole.However, diffraction willspread the beam into theAiry disk.

• Image is sharpest when the geometrical blur (the pinholesize) is equal to the diffraction blur (Airy disk diameter).

• Design example: What pinhole diameter is optimum for avisible-light camera that is 100 mm deep?

Geometrical blur

Diffraction blur

D/2

D/2

L=100 mm

mm 37.0

1001055.44.2

44.2

/2/22.1

sin22.1

3

LD

LDD

Kevin Curtis OESD lecture notes


Robert McLeod

Evolution of the eye

25


Wikipedia

Lens evolved to improve radiometric efficiency of an otherwisepretty good imaging system.


Robert McLeod 26

Fermat’s principleAnother (important) form of the eikonal

• Propagation time = S / c• Hero of Alexandria (in Catoptrica, ca 50

AD): “Light travels in straight lines”• Pierre de Fermat (ca 1650): “Light travels the

path which takes the minimum time.”• Correct: “The time of travel is stationary:”

– Example 1: Concave mirror with radius of curvature < ellipse

– Example 2: Lifeguard problem– See Born and Wolf for a derivation from

Maxwell’s equations

• Why do we care? At an image point, all OPL must be equal.

dsrnSB

A

lengthpath Optical

0 dsrnB

A

P. Mouroulis & J. Macdonald, Geometrical Optics and Optical Design, pg 11-12, Oxford, 1997R. Feynman, Lectures on Physics

•Design of ideal imaging systems with geometrical optics–Fermat’s principle


Robert McLeod

The lifeguard problem

27

•Design of ideal imaging systems with geometrical optics–Fermat’s principle

A

BSand

n

Watern´

Problem: The lifeguard atpoint A needs to get to thedrowning swimmer at B inminimum time. What is theoptimum point at which toenter the water given that thelifeguard runs at speed V/n onsand and swims at speed V/n´in water?

Solution 1: Write equation fortravel time with entry point asparameter, minimize time,solve for entry point.

Example: If lifeguard onshore, this is a problem oftotal internal reflection, so ´is the critical angle:

´

A

BSand

n

Watern´

´

n

n

1-sin90°

Solution 2: Use Fermat’sprinciple which says thatangles obey Snell’s Law, usethis constraint to solve forpath.


Robert McLeod 28

RadiometryReview of terminoligy & inverse square law

•Design of ideal imaging systems with geometrical optics–Radiometry

O’Shea chapter 3, Mouroulis and Macdonald 5.3

Q Energy [J] Power (flux) [J/s=W]I Intensity [W/sr]E Irradiance [W/m2]L Radiance (photometric “brightness”) [W/(sr m2)]

2R

A Solid angle is area of sphere

subtended A over radius of sphere R2

I Intensity I of point source is power emitted into solid angle

2R

IA

E

Irradiance E on surface is

power per unit area A

Irradiance of surface by a point sourceis given by intensity I of point source

over distance to surface R2

AL

Radiance of surface normal area A illuminated by a point source of power

radiating into a solid angle


Robert McLeod

Examples

29


100 W

2 m 222 m

W25

m 2

W100

R

IE

What is irradiance 2 m from 100 W light bulb?

2 m

What is brightness of a typical 100 W light bulb?

Sr4mareafilament

W2

L W/m2/Sr] for a bulb

with a 1 mm2 filament

What is brightness of a typical 1 W laser?

Lots of power, moderate area, but huge radiation angle = low brightness.

2

22220

2220

1

radm

W

radangle divergencem

W

0

MMw

MwL

w

W/m2/Sr] for aperfect 1 W laser at =1 um

Less power and similar area to the lamp filament, but MUCH smaller radiation angle = high brightness. Besides the potentially narrow spectrum, brightness is what sets lasers apart from lamps.


Robert McLeod 30

Radiometry via rays


Consider a finite source radiating into a cone

A1 A2

R2

R1

24 RAE

Now launch a set of rays inside this cone and examine the ray density as a function of radius

ER

N

A

N

24

A general proof that ray density is proportional to irradiance can be found in Born and Wolf.

Irradiance E on surface Agiven power into cone

Mouroulis and Macdonald 1.4


Robert McLeod 31

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 it

• Rays satisfy Snell’s laws of refraction and reflection

• The irradiance at any point is proportional to the ray density at that point

•Design of ideal imaging systems with geometrical optics–Foundations

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•Design of ideal imaging syst ems with geometrical …ecee.colorado.edu/.../lecturenotes/FoundationsOfGeometricalOptics.pdf · that all three of the laws of geometric optics can