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General Physics part 1 / PhysPraktikum / Woan G. The Cambridge Handbook of Physics Formulas (CUP, 2000)(ISBN 0521573491)(230s)_PRef_.pdf

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Chapter 8 Optics

8.1Introduction

Any attempt to unify the notations and terminology of optics is doomed to failure. This is partly due to the long and illustrious history of the subject (a pedigree shared only with mechanics), which has allowed a variety of approaches to develop, and partly due to the disparate ﬁelds of physics to which its basic principles have been applied. Optical ideas ﬁnd their way into most wave-based branches of physics, from quantum mechanics to radio propagation.

Nowhere is the lack of convention more apparent than in the study of polarisation, and so a cautionary note follows. The conventions used here can be taken largely from context, but the reader should be aware that alternative sign and handedness conventions do exist and are widely used. In particular we will take a circularly polarised wave as being right-handed if, for an observer looking towards the source, the electric ﬁeld vector in a plane perpendicular to the line of sight rotates clockwise. This convention is often used in optics textbooks and has the conceptual advantage that the electric ﬁeld orientation describes a right-hand corkscrew in space, with the direction of energy ﬂow deﬁning the screw direction. It is however opposite to the system widely used in radio engineering, where the handedness of a helical antenna generating or receiving the wave deﬁnes the handedness and is also in the opposite sense to the wave’s own angular momentum vector.

162	Optics

8.2 Interference Newton’s ringsa

	rn2					rn		radius of nth ring
nth dark ring	rn2	= nRλ0			(8.1)	n		integer (≥ 0)
		= n+		Rλ0		R		lens radius of curvature
nth bright ring	rn2		1		(8.2)	λ	0	wavelength in external
								medium
			2					medium

aViewed in reﬂection.

Dielectric layersa

η1

ηa

η1

N × { ηb

single layer η2

multilayer

η3

1 − R

η3

1 − RN

ﬁlm thickness

Quarter-wave

a =

m λ0

(8.3)

thickness integer

condition

η2

(m ≥ 0)

η2

ﬁlm refractive index

λ0

free-space wavelength

η1

η3

−

η22

power reﬂectance

(m odd)

coe cient

Single-layer

η1η3 +

η2

reﬂectanceb

R =

(8.4)

η1

entry-side refractive

η1

η3

−

(m even)

index

η1 + η3

η3

exit-side refractive

index

Dependence of

max if

(−1)m(η1 − η2)(η2 − η3) > 0

(8.5)

R on layer

min if

(−1)m(η1 − η2)(η2 − η3) < 0

(8.6)

thickness, m

R = 0

η2

= (η1η3)

1/2

and m odd

(8.7)

multilayer reﬂectance

number of layer pairs

Multilayer

ηa

refractive index of top

η1 − η3(ηa/ηb)

(8.8)

reﬂectance

η1 + η3(ηa/ηb)2N

layer

ηb

refractive index of

bottom layer

aFor normal incidence, assuming the quarter-wave condition. The media are also assumed lossless, with µr = 1. bSee page 154 for the deﬁnition of R.

cFor a stack of N layer pairs, giving an overall refractive index sequence η1ηa,ηbηa ...ηaηbη3 (see right-hand diagram). Each layer in the stack meets the quarter-wave condition with m = 1.

8.2 Interference	163

Fabry-Perot etalona

eiφ

e2iφ

e3iφ

incident rays

incremental phase di erence

φ = 2k0hη cosθ

(8.9)

free-space wavenumber (= 2π/λ0)

Incremental

= 2k0hη 1

1/2

cavity width

ηsinθ

fringe inclination (usually

phase

−

(8.10)

internal angle of refraction

di erence

cavity refractive index

= 2πn

for a maximum

(8.11)

external refractive index

fringe order (integer)

Coe cient of

F =

(8.12)

coe cient of ﬁnesse

ﬁnesse

(1 − R)

interface power reﬂectance

F =

F1/2

(8.13)

ﬁnesse

Finesse

λ0

free-space wavelength

(8.14)

cavity quality factor

η h

−

R)2

I(θ) =

(8.15)

Transmitted

1 + R

− 2R cosφ

transmitted intensity

intensity

(8.16)

incident intensity

1 + F sin2(φ/2)

Airy function

= I0A(θ)

(8.17)

Fringe

∆φ = 2arcsin(F−1/2)

(8.18)

∆φ

phase di erence at half intensity

intensity

2F−1/2

(8.19)

point

proﬁle

Chromatic

λ0

R1/2πn

δλ 1

−

= nF

(8.20)

δλ

minimum resolvable wavelength

resolving

di erence

power

2Fhη

(θ

(8.21)

λ0

Free spectral

δλf = Fδλ

(8.22)

δλf

wavelength free spectral range

rangec

δνf =

(8.23)

δνf

frequency free spectral range

2η h

aNeglecting any e ects due to surface coatings on the etalon. See also Lasers on page 174. bBetween adjacent rays. Highest order fringes are near the centre of the pattern.

cAt near-normal incidence (θ 0), the orders of two spectral components separated by < δλf will not overlap.

164	Optics

8.3 Fraunhofer di raction

Gratingsa

coherent plane waves

Young’s

kDs

double

I(s) = I0 cos2

(8.24)

slitsb

N equally

sin(Nkds/2)

spaced

I(s) = I0

N sin(kds/2)

(8.25)

narrow slits

Inﬁnite

nλ

∞

grating

I(s) = I0 n=

−∞

−

(8.26)

Normal

nλ

sinθn =

(8.27)

incidence

Oblique

sinθn + sinθi =

nλ

(8.28)

incidence

Reﬂection

sinθn − sinθi =

nλ

(8.29)

grating

Chromatic

resolving

= Nn

(8.30)

δλ

power

Grating

∂θ

(8.31)

dispersion

∂λ

dcosθ

Bragg’s

2asinθn = nλ

(8.32)

lawc

I(s) di racted intensity

I0 peak intensity

θdi raction angle

s= sinθ

Dslit separation

λwavelength

Nnumber of slits

kwavenumber (= 2π/λ)

dslit spacing

ndi raction order

δDirac delta function

θn	angle of di racted
	maximum
θi	angle of incident
	illumination

δλ di raction peak width

aatomic plane spacing

aUnless stated otherwise, the illumination is normal to the grating. bTwo narrow slits separated by D.

cThe condition is for Bragg reﬂection, with θn = θi.

θi

θn

θi θn

θn a

8.3 Fraunhofer di raction	165

Aperture di raction

)

(

x,y

coherent plane-wave

illumination,

normal

to the xy plane

∞

di racted wavefunction

General a1-D

ψ(s) −∞ f(x)e−iksx dx

(8.33)

di racted intensity

aperture

I(s)

ψψ (s)

(8.34)

di raction angle

= sinθ

aperture amplitude

General 2-D

transmission function

aperture in

ik(sxx+sy y)

x,y

distance across aperture

(x,y) plane

ψ(sx,sy) f(x,y)e−

dxdy

(8.35)

wavenumber (= 2π/λ)

(small angles)

∞

deﬂection xz plane

I(s) = I0

sin2(kas/2)

(8.36)

peak intensity

Broad 1-D

(kas/2)2

slit width (in x)

slitb

(8.37)

≡ I0 sinc (as/λ)

wavelength

Sidelobe

intensity

(n > 0)

(8.38)

nth sidelobe intensity

(2n+ 1)2

Rectangular

2 asx

2 bsy

aperture width in x

aperture

sinc

(8.39)

I(sx,sy) = I0 sinc λ

aperture width in y

(small angles)

Circular

2J1(kDs/2)

ﬁrst-order Bessel function

aperturec

I(s) = I0

(8.40)

aperture diameter

kDs/2

First

s = 1.22

(8.41)

wavelength

minimumd

First subsid.

s = 1.64

(8.42)

maximum

Weak 1-D

f(x) = exp[iφ(x)] 1 + iφ(x)

(8.43)

φ(x) phase distribution

phase object

i2 =

−

Fraunhofer

(∆x)2

distance of aperture from

(8.44)

observation point

limite

∆x

aperture size

aThe Fraunhofer integral.

bNote that sincx = (sinπx)/(πx).

cThe central maximum is known as the “Airy disk.”

dThe “Rayleigh resolution criterion” states that two point sources of equal intensity can just be resolved with di raction-limited optics if separated in angle by 1.22λ/D.

ePlane-wave illumination.

166	Optics

8.4 Fresnel di raction

Kirchho ’s di raction formulaa

									y
										S	x
							ρ		d	S
	dA						ρ		d
	dA						source			sˆ
		r		θ			source			sˆ
				θ			P
	ψ0
	ψ0
										r	z
											z
(source at inﬁnity)											P
								ψP	complex amplitude at P
		i	K(θ)		eikr			λ	wavelength
Source at		i			eikr			k	wavenumber (= 2π/λ)
Source at	ψP = − λψ0					dA	(8.45)	k	wavenumber (= 2π/λ)
inﬁnity	ψP = − λψ0				r	dA	(8.45)	ψ0	incident amplitude
		plane						θ	obliquity angle
								θ	obliquity angle
where:								r	distance of dA from P ( λ)
Obliquity								dA	area element on incident
Obliquity		1							wavefront
factor		1							wavefront
factor	K(θ) =	2 (1 + cosθ)					(8.46)	K	obliquity factor
(cardioid)								dS	element of closed surface
								dS	element of closed surface
		iE0		eik(ρ+r)				ˆ	unit vector
Source at	ψP = −	iE0		eik(ρ+r)			· rˆ) − cos(sˆ · ρˆ )] dS	s	vector normal to dS
ﬁnite	ψP = −	λ		2ρr	[cos(sˆ		· rˆ) − cos(sˆ · ρˆ )] dS	r	vector from P to dS
distanceb	closed surface						(8.47)	ρ	vector from source to dS
							(8.47)	E0	amplitude (see footnote)
								E0	amplitude (see footnote)

aAlso known as the “Fresnel–Kirchho formula.” Di raction by an obstacle coincident with the integration surface can be approximated by omitting that part of the surface from the integral.

bThe source amplitude at ρ is ψ(ρ) = E0eikρ/ρ. The integral is taken over a surface enclosing the point P .

Fresnel zones

		y
source	z1	z2	observer

E ective aperture	1		1			1
distancea			=			+
		z		z1			z2
Half-period zone	yn = (nλz)1/2
radius
Axial zeros (circular	zm =				R2
aperture)					2mλ

	z	e ective distance
(8.48)	z1	source–aperture distance
	z2	aperture–observer distance
	n	half-period zone number
(8.49)	λ	wavelength
	yn	nth half-period zone radius
	zm	distance of mth zero from
(8.50)		aperture
	R	aperture radius

aI.e., the aperture–observer distance to be employed when the source is not at inﬁnity.

8.4 Fresnel di raction	167

Cornu spiral

0.8

√2

0.6

Cornu Spiral

√3

√5

Edge di raction

∞

2.5

)

0.4

0.2

−2

intensity

1.5

+)w

−

1 2

−0.2

C(w)

CS(

−0.4

√5

√3

−1

−∞

−0.6

−

0.5

0.8

−√2

0.2 0

0.2

0.4

0.6

0.8

−

0.6

0.4

−

πt2

Fresnel

C(w) = 0

cos 2

(8.51)

Fresnel cosine integral

integrals

πt2

Fresnel sine integral

S(w) =

sin 2

(8.52)

CS(w) = C(w) + iS(w)

(8.53)

Cornu spiral

v,w

length along spiral

CS(±∞) = ± 2 (1 + i)

(8.54)

ψ0

ψP

complex amplitude at P

ψ0

unobstructed amplitude

ψP = 21/2 [CS(w) + 2 (1 + i)]

(8.55)

wavelength

Edge di raction

1/2

distance of P from

where

w = y

(8.56)

aperture plane [see (8.48)]

λz

position of edge

ψ0

coherent

Di raction

ψP = 21/2 [CS(w2) − CS(w1)]

(8.57)

plane waves

from a long

1/2

slitb

where

wi = yi λz

(8.58)

ψ0

ψP =

2 [CS(v2) − CS(v1)] ×

(8.59)

Di raction

[CS(w2) − CS(w1)]

(8.60)

positions of slit sides

from a

1/2

positions of slit

rectangular

where

vi = xi

(8.61)

aperture

λz

top/bottom

wi = yi

1/2

and

(8.62)

λz

aSee also Equation (2.393) on page 45. bSlit long in x.

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