Physical Methods for Cultural Heritages Radiometry and Fotometry G. Valentini - Tel. 6071 -...

Physical Methods for Cultural Heritages

Radiometry and FotometryG. Valentini - Tel. 6071 - [email protected]

Robert W. BoydRadiometry and the Detection of Optical radiationJohn Wiley & Sons

Gianluca Valentini

• Electromagnetic wave

Electromagnetic waves

c = wavelenght

= frequency

c = vacuum light speed

T = period

= wavelenght

= frequency

c = vacuum light speed

T = period

1

T

)kztcos(H)t,z(H

)kztcos(E)t,z(E

y0y

x0x

2 /2k

In a transparent medium

n= refraction index

In a transparent medium

n= refraction index

n

ncc

kc

n=1,5

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The electromagnetic spectrum

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• Reflection:

• Refraction:

Foundamental laws of optics (1)

• Dispersion

(n2 depends on

11

21

2

2

1 nnn

sinsin

22

22 (nn

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5Foundamental laws of optics (2)

• Diffraction • Interference

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6Foundamental laws of optics (3)

• Polarization • Diffusion

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7Real and virtual images

• An optical system forms a real image of an object when the light exiting any point M of the object (spherical wave) is focused to a point M’ of a plane called “conjugated plane”

The eye forms a real image of the observed objects to the retina A photographic lens

forms a real image of the scene to the film (CCD)

• An optical system forms a virtual image of an object when the spherical wave exiting any point of the object O is converted to a spherical wave exiting a new point O’

Optical devices designed for direct eye observation(microscope, telescope, binoculars, etc.) form virtual images

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8Light diffraction

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Elements of Radiometry

• Radiometry refers to the measure of the energetic content of a radiation field and study the energy propagation in free space or in an optical system

• Radiometry mainly deals with incoherent radiation sources and assumes that the optical field propagates according to the laws of the geometrical optics

Radiometric quantitiesQuantity Symbol Definition Unity of meas.

Radiant energy Q J

Radiant energy density u dQ/dV J·m-3

Radiant flux dQ/dt W

Radiant exitance M d/dA W·m-2

Irradiance E d/dA W·m-2

Radiance L d2/(dAprd W·m-2· sr-1

Radiant intensity I d/d W· sr-1

dt


Radiometric quantities

• Radiant flux Power carried by a radiation field [W] Foundamental radiometric quantity

• Radiant exitance Radiant flux emitted by an extended source per unit of area [W·m-2]

• Irradiance Radiant flux incident onto a surface per unit area [W·m-2]

• Radiance Radiant flux emitted by an extended source

per unit of solid angle and unit of projected area [W·m-2·sr-1]

• Radiant intensity Radiant flux emitted toward the direction (θ,φ) per unit of solid angle

(useful for pointlike radiant sources) [W ·sr-1]

dAdM

dAdM

ddA

dL

proj .

2

ddI

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r

d0 d1 dA1dA0

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Law of conservation of the radiance(free space propagation)

• Let’s suppose that the radiation field propagates in a non-absorbing homogneous medium from a source to a receiver

• We define:

• The flux transferred from dA0 to dA1 is:

• The radiance measured on surface dA1 in direction r is given by:

211

0r

cosdAd

Solid angle subtended by dA1 for an observer in dA0

200

1r

cosdAd

200000

2 dELdcosdALd L0 = radiance measured on

surface A0

Solid angle subtended by dA0 for an observer in dA1

0002 dcosdAdE E = Etendue of the optical system

000

2

211

11

000

111

2

1 coscos

coscos

cosL

dAr

r

dAdAdA

LddA

dL

EddL 22


Lambertian sources

• A lambertian source is a source whose radiance does not depend on the observation angle L() = constant = L0

• The radiant intensity (W/sr) emitted by a lambertian source of small area A0 and radiance L0 in a generic direction is given by:

The radiant intensity changes with the cosine of the observation angle The change of I as a function of depends on the variation of the

apparent area of the source

• Let’s consider a small size lambertian source (dA0) The radiant flux that impinges on dA1 is:

L() is independent on angle • The radiant exitance of a lambertian source is:

cosIcosALdAcosLdd

I 000.Sorg 0

002 dcosId

2

0 00

2

002 00 L21

2LdsincosdLdcosLM

dA0

dA1d1

d0

001

2

21

0011

2

LcosdAdA

r

r

dAcosdAL

ddAd

L


Disk-like lambertian source

• Irradiance produced by a disk-like lambertian source of radius R on a surface dA1 located at distance z

• A ring element on the disk has area given by:

• The solid angle d0 subtended by dA1 from any point on dA0 is given by:

• The radiant flux transferred from dA0

to dA1 is given by:

• The irradiance on the surface dA1 is then:

dcossin

z2dtancos

z2drr2dA

32

2

2

0

)(cos

dzdr

tanzr

2

21

21

0cosz

cosdAhcosdA

d

22

2

00 212

001 zR

RLsinLdcossinL2

dAd

E21

dsincosdAL2

zcosdA

cosdcossin

z2LdcosdALd 102

31

32

00002

z

220 zRLE

00 MLE z0

z∞


Aplanatic image forming system

• Within paraxial approximation the angle between any ray and the optical axis is small ( << 1) The images are exact replica of the objects because the spherical

aberration is negligible

• An optical system is called stigmatic for the two axial points P and P’ if P’ is a perfect image of P (no aberrations)

• For a system stigmatic for P e P’ to be aberration free for points slightly off axis it is required that the Abbe condition is satisfied:

• A system stigmatic for P and P’ that also obeys the Abbe condition is said to be aplanatic for the two points and is not affected by coma

sinhnsinnh For any ray leaving P under any angle

h, h’ small , ’ arbitrary

n n’


Radiance Theorem

• The radiance L of a radiation field is conserved as the beam propagates through a uniform lossless medium or through an aplanatic optical system

• Let’s demonstrate that the radiance (L/n2) is conserved when the beam crosses the interface between two media with different refraction indexes The radiant flux carried by a beam falling onto the surface element dA

from the solid angle d1 is:

• According to Snell law:

• Diefferentiating the previous equation:

• The ratio between solid angle d1 e d2

in polar coordinates is:

1112 cos ddALd

121

222

222

111

2

1

cosncosn

ddsinddsin

dd

21 dd 2211 sinnsinn

222111 dcosndcosn

22

221

1

nL

nL

21

22

11

21

222

2

11

22

111

22

2

2 nn

Lcosncosn

coscos

LdcosdAdcosdA

LdcosdA

dL

n1 n2


Radiance of an image

• Let’s calculate the radiance of the image of a light source produced by an aplanatic optical system

• The radiant flux produced by the source(dx0, dy0) within solid angle d0 at the entrance pupil of the optical system is:

• In a lossless optical system the flux d impinges onto the image element (dx1,dy1) and produces the radiance in direction (1,1) given by:

• Using the Abbe condition:

• The image of a lambertian source is still lambertian and has the same radiance of the source Given a light source, the radiance (L/n2) of its image can never be

greater than that of the source

ddsincosdydx,Ld 00000002

111

2

111 cos),(

ddA

dL

111000 sindxnsindxn 11110000 dcosdyndcosdyn x→y,diff

20

21

011111

000000011 n

nL

ddsincosdydxddsincosdydx

),(L),(L

n0 n1


Irradiance of an image

• Let’s calculate the irradiance given by an optical system in the image plane• The flux transferred from a lambertian surface dA to a ring element dis:

• The flux collected by the optical system taking into account its aperture is:

• The irradiance of the image is then:

• Using the Abbe condition one gets:

• Introducing the definition of focal ratio (f#):

• The irradiance of the image depends on the radiant exitance of the source and on the aperture of the optical system

dsin2cosdALdcosdALd 000002

02

000 00 sindALdcossindAL2d0

02

1

00

11 sin

dAdA

LdAd

E

12

20

21

01 sinnn

LE

2#eff

22

21

02#eff

22

21

01f

1nn

41

Mf

1nn

4LE

Df

f # 11

#eff sin2

1tan2

1f

Irradiance given on dA1 by the exit pupil having uniform irradiance:

20

2101 nnLL

#eff1

2 f41sin

n0 n1

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The spectral sensitivity of the human eye

• The visual stimulus produced by a radiation depends on its spectral power density according to the spectral sensitivity of the human eye

• The vision process is triggered by the isomerisation reaction of Rhodopsin

• Photopic vision It is characterized by activation

of cones Gives a clear perception of

colours Can be experienced during

daylight vision Mainly corresponds to the

maximum visual acuity (macula)

• Scotopic vision It is characterized by the activation of rods Can be experienced during night vision Chromatic sensitivity is very low It is more effective in the peripheral region of the retina

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Spectrophotometric sensitivity of the eye

• Through experiment made with bipartite colour fields it has been possible to measure the Spectral Luminous Efficiency for the Standard Observer

• Photopic vision (CIE, 1924) → V ()

• Scotopic vision (CIE, 1951) → V’()

Luminous flux Fv [lm]

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ba

bmbt

dVKmv


The photometric quantities

• Photometry deals with the measurement of the visual response caused by radiation fields with wavelength within the visible range (380-700 nm)

• Photometric quantities stem from the analogous radiometric quantities “weighted” by the spectral response of the eye of a normal observer (i.e. not affected by ocular diseases)

Photometric quantities

Quantity Symbol Definition Meas. Unit Symbol

Luminous energy Qv talbot lm·s

Luminous density uv dQv/dV lumen·s·m-3 lm·s·m-3

Luminous flux v dQv/dt lumen lm

Luminous exitance Mv dv/dA lux lm·m-2 = lx

Illuminance Ev dv/dA lux lm·m-2 = lx

Luminance Lv

dv2/

(dAprd nit lm·m-2·sr-1 = nt

Luminous intensity* Iv dv/d candle lm·sr-1 = cd

dtv

1 candle = luminous intensity produced by a source emitting monocromatic radiation @ = 540 1012 Hz ( = 555 nm) with a Radiant intensity of 1/683 W/sr

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Light sources - the sun

• Spectral irradiance of the sun (W·m-2·m-1) outside the atmosphere and at earth’s surface

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Daylight

• Daylight corresponds to the direct illumination by the sun + light from the sky on a horizontal surface: Colour temperature Tc = 5.000 – 7.000 K (temperature of the solar

corona Ts 5.780 K)

• Overcast sky Colour temperature is Tc = 5.000 – 7.000 K

• Bright sky without direct sun light (shadow) Colour temp Tc > 7.000 K up to 40.000 K for bright sky in north

direction

• Solar disk with “atmospheric filter” Colour temperature Tc ≈ 5000 K

• The conventional colour temperature of the daylight is Tc = 6.500 K

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Light sources - Daylight

• Relative spectral power distribution for different phases of the daylight

Fraunhofer absorption lines in the solar spectrum

(H, Na,etc.)

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Light sources - Colour temperature of the daylight

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Light sources - The Planck blackbody

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“Planckian locus” and colour temperature

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Light sources - Daylight

• Colorimetric coordinates of daylight (cromaticy diagram)

Cromaticy of the blackbody at differen

temperatures

(Plankian locus)

White light

x=y=1/3

Daylight locus: 275.0x870.2x000.3y 2DD

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Light emission mechanisms

• The major light emission mechanisms are:

Thermal emission:• High temperature bodies

• The emission spectrum is continouos

Emission by excited electronic states• Discharge lamps

• The emission spectrum is characterisedby sharp lines and bands

Emission by semiconductor• The emission occurs via inter-band

transitions (electron- hole recomb.)

• The emission spectrum shows a band 20-30 nm wide

knr

S0

S1

S1’

hAhF

kf

e-

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Incandscence lamps

• The emission is produced by a tungsten filament at temperature from 2200 to 3400 °K

• The emission spectrum is similar to that of a blackbody with temperature about 40 K lower than that of the filament

• The emission obeys the following physical laws:

• The quality of the light and theefficiency of the lamp increases with the temperature of the filament(distribution temperature)

• Halogen lamps have a temperaturehigher than that of normal lamps The presence of iodine makes the

evaporated tungsten to come backto the filament

Stefan-Boltzmann law)m/W(T1067.5W 248

Wien law3

max 108978.2T

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Blackbody spectral power density

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Discharge lamps

• Emission is mainly produced by electronic treansitions

• The atom excitation is achieved by an electric discharge ia a gas or in an ionized vapour

• Emission spectrum is made by lines that undergo collision widening up to a quasi-continuous spectrum

• Direct emission lamps The emitted light comes directly from electronic transitions The quartz or glass bulb absorbs only the harmful UV radiation

• Lamp with wavelength conversion

The radiation emitted by electronic transitions (typically UV) is converted to visible light by a phosphor layer covering the internal wall of the glass bulb

The phosphor emission takes place by the luminescence/fluorescence effect the name “fluorescent lamps”

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Discharge lamp mechanism

• The active medium is a gas or vapour in a glass or quartz bulb or tume

• The current flows into the lamp through two electrodes(anode + e cathode – )

• The electron – and ion + flux causes furtherionization and excitation of the atoms that emitvia radiative transitions

• When the current reaches high density, an arc established between the electrodes The gas reaches high temperature and is

almost completely ionized The thermal emission contributes to the light generation in addition to

atomic transition• Efficiency goes down but the light quality greatly improves

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Discharge lamp types

• Sodium lamps Low pressure ( > 120 lm/W)

• High efficiency, almost monochromatic light street lighting

High pressure ( 50 lm/W)• Good efficiency and higher chromatic quality

• Xenon lamps ( = 30...50 lm/W) High chromatic quality, high efficiency Emission spectrum close to the solar one

• Metal halide lamps ( > 90 lm/W) High efficiency and chromatic CRI=90 Low cost

• Mercury lamps Direct emission

• High UV emission special uses Wavelength conversion by phosphors

• Good efficiency, but low light quality

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High pressure xenon lamp

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Flashtube xenon lamp

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Fluorescent lamps

• They are made by low pressure mercury lamps

A small quantity of noble gas (neon) assists the discharge setup

Mercury emission is mainly in the UV range

Light is produced by a mixture of phosphors with high quantum efficiency( = 60...100 lm/W)

A higher number of phosphor improves the light quality, but lower the efficiency

Different light “tones” can be achieved by changing the phosphor recipe(cool white, warm white, daylight, etc.)

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Physical Methods for Cultural Heritages Radiometry and Fotometry G. Valentini - Tel. 6071 -...

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