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Optical Engineering

Part 24: Photometry, color theory

Herbert Gross

Summer term 2020

Introduction

Solid angle

Flux transport

Lambert characteristic

Optical systems

Color theory

Color triangle

2

Contents

Radiometric vs Photometric Units

Quantity Formula Radiometric Photometric

Term Unit Term Unit

Energy Energy Ws Luminous Energy Lm s

Power

Radiation flux

W

Luminous Flux Lumen Lm

Power per area and solid angle

Ld

d dA

2

cos

Radiance W / sr /

m2

Luminance cd / m

2

Stilb

Power per solid angle

dAL

d

dI

Radiant Intensity W / sr

Luminous Intensity Lm / sr,

cd

Emitted power per area

dLdA

dE cos

Radiant Excitance W / m2

Luminous Excitance Lm / m2

Incident power per area

dLdA

dE cos

Irradiance W / m2

Illuminance Lux = Lm / m

2

Time integral of the power per area

H E dt

Radiant Exposure Ws / m2

Light Exposure Lux s

3

Photometric Quantities

Radiometric quantities:

Physical MKSA units, independent of receiver

Photometric quantities:

Referenced on the human eye as receiver

Conversion by a factor Km

Sensitivity of the human eye V(l)

for photopic vision (daylight)

ll l )(VKmV

W

LmKm 673

V(l )

l400 450 500 550 600 650 700 750

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Illuminance description

1 Lux just visible

50 - 100 Lux coarse work

100 Lux projection onto

screen

100 - 300 Lux fine work

1000 Lux finest work

100000 Lux sunlight on paper

4

Solid Angle

ddA

r

dA

r

cos

2 2

2D extension of the definition of an angle:

area perpendicular to the direction over square of distance

Area element dA in the distance r with inclination

Units: steradiant sr

Full space: = 4p

half space: = 2p

Definition can be considered as

cartesian product of conventional angles

source point

d

rdA

n

yxr

dy

r

dx

r

dAd

2

5

Solid Angle: Special Cases

Cone with half angle j

Thin circular ring on spherical surface

j jp cos12

jjpjjp

dr

drrd

sin2

sin22

j

dj

r

ring

surfacep1cosj

r

z

x

y

j

d

dj

6

Irradiance

Irradiance: power density on a surface

Conventional notation: intensity

Unit: watt/m2

Integration over all incident directions

Only the projection of a collimated beam

perpendicular to the surface is effective

dLdA

dE cos

cos)( 0 EE

A

A

E()

Eo

7

d

s

dAS

S

n

Differential Flux

Differential flux of power from a

small area element dAs with

normal direction n in a small

solid angle dΩ along the direction

s of detection

L radiance of the source

Integration of the radiance over

the area and the solid angle

gives a power

S

SS

S

AdsdL

dAdL

dAdLd

cos

2

PdA

A

8

Fundamental Law of Radiometry

Differential flux of power from a

small area element dAS on a

small receiver area dAR in the

distance r,

the inclination angles of the

two area elements are S and

R respectively

Fundamental law of radiometric

energy transfer

The integration over the geometry gives the

total flux

ESES

ES

dAdAr

L

dAdAr

Ld

coscos2

2

2

z

s

s

xs

ys

source

receiver

xR

yR

zR

AS

r

ns

AR

nR

S

R

9

Radiance independent of space coordinate

and angle

The irradiance varies with the cosine

of the incidence angle

Integration over half space

Integration of cone

Real sources with Lambertian

behavior:

black body, sun, LED

constLsrL

,

Lambertian Source

jpj 2sin)( ALLam

coscos oEALE

LAdEHR

Lam p )(

E()

x

z

L

x

z

10

Radiation Transfer

Basic task of radiation transfer problems:

integration of the differential flux transfer law

Two classes of problems:

1. Constant radiance, the integration is a purely geometrical task

2. Arbitrary radiance, a density function has to be integrated over the geometrical light tube

Special cases:

Simple geometries, mostly high symmetric , analytical formulas

General cases: numerical solutions

- Integration of the geometry by raytracing

- Considering physical-optical effects in the raytracing:

1. absorption

2. reflection

3. scattering

ESESES dAdAr

LdAdA

r

Ld coscos

22

2

11

Transfer of Energy in Optical Systems

Conservation of energy

Differential flux

No absorption

Sine condition fulfilled

d d2 2 '

jddudAuuLd cossin2

T 1

y

dA dA's's

EnP ExP

n n'

F'F

y'

u u'

'sin''sin uynuyn

12

Natural Vignetting: Setup with Rear Stop

Stop behind system:

exact integration possible

Special case on axis

Approximation small aperture:

Classical cos-to-the-fourth-law

2/1

222

222

'tan'cos1

'tan'cos411

'

2)'(

uw

uw

n

nLwE

p

'sin'

'sin')0(' 2

2

2 uLn

nuLE

pp

'cos)0()'( 4 wEwE

AP

u'w'

rw

ro

w'

13

Change of color perception:

bleaching of chemical receptors

Effect of Bezold:

the color perception depends

in addition on the environmental

color

Subjective Color Perception with the Eye

Mixing of colors:

1. additive: RGB = red gree blue

2. subtractive: CMY = cyan magenta yellow

Mixing of Colors: Additive - Subtractive

Additive mixing of color: RGB Subtractive mixing of color: CMY

Color perception values of the eye:

spectral integration over the three receptors with sensitivity and stimulus j(l)

Spectral signal over all receptors

(color valence)

Color Perception with the Human Eye

LLMMSSF

nm

nm

dlL

780

380

)()( lllj

nm

nm

dmM

780

380

)()( lllj

nm

nm

dsS

780

380

)()( lllj

relativesensitivity

l400 500 550 600 650 700 750

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

450

450

545 558

)(ll

)(lm

)(ls

R

G

B

1

1

1

r+g+b = 1

direction of

the hue

B

G

R

F

r

b

g

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.0

0.2

0.1

1.00.90.80.70.60.50.40.30.0 0.20.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.0

0.2

0.1

According to Maxwell, the color can be described by an equal sided triangle

The 3 corners represent the basic color types

A point inside the triangle defines an arbitrary color

by the barycentric values

(foot point projections)

In general the triangle is a cross section area of a

plane in the cartesian coordinate system of the

three colors

The distance from the origin describes the hue

Maxwells Color Triangle

There are different possibilities for spectral sensitivity curves

The systems are convertable by matrix algorithms

The most important

systems are:

1. LMS eye cons

2. RGB

3. XYZ standard

The observed color

perception is given

by

The power density is given by

(law of Abney)

Spectral Sensitivity Curves

)()()()( llll zZyYxXF

ZYXF LZLYLXL )(l

l400 500 550 600 650 700 750

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

m

l

s

450

450

545 558

lin

nm400 500 600 700-0.1

0

0.1

0.2

0.3

0.4

b( l )

g( l )

r( l )

lin

nm400 450 500 550 600 650 700 750

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y( l )

z( l )

x( l )

XYZ

RGB

LMS

linear

conversion

Correspondences of colors areas in the

classical color triangle to conventional

names

Wavelength ranges of spectral colors

Conventional Colors

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

0.9

y

green

yellow

orange

purple

blue

whitered

Color l in nmred 750 ... 640orange 640 ... 600yellow 600 ...555green 555 ... 485blue 485 ... 430violet 430 ... 375