Fundamentals of Optics

Jiun-You Lin

Department of Mechatronics Engineering, National Changhua University of Edu

cation

• Historically, optical theory developed roughly in the following : sequence: (1) ray optics (2) wave optics (3) electromagnetic optics (4) quantum optics

• The theory of quantum optics provides an explanation of virtually all optical phenomena. The theory of electromagnetic optics provides the most complete treatment of light within the confines of classical optics (electromagnetic optics, wave optics, ray optics)

• Wave optics is a scalar approximation of electromagnetic optics. Ray optics is the limit of wave optics when the wavelength is very short.

Classical Optics

• Wave Nature of Light

• Polarization and Modulation

of Light

Outline

Wave Nature of LightLight Waves in a Homogeneous Medium

A. Plane Electromagnetic Wave B. Maxwell’s Wave Equation and Diverging Waves

Refractive Index

Magnetic Field, Irradiance, and Poynting Vector

Snell’s Law and Total Internal Reflection

Fresnel’s EquationsA. Amplitude Reflection and Transmission

Coefficients B. Reflectance and Transmittance

Interference Principles A. Interference of Two Waves B. Interferometer

Diffraction Principles A. Diffraction B. Fraunhofer Diffraction C. Diffraction grating

Optical frequencies and wavelengths

•Ultraviolet

10nm~390nm

•Visible

390nm~760nm

•Infrared

760nm~1mm

Light Waves in a Homogeneous Medium

A. Plane Electromagnetic WaveEx

z

Direction of Propagation

By

z

x

y

k

An electromagnetic wave is a travelling wave which has timevarying electric and magnetic fields which are perpendicular to eachother and the direction of propagation, z.

oox kztEE cos

A sinusoidal wave

Ex : the electric field at position z at time t

Eo : the amplitude of the wave

: the angular frequency

k=2/ : the propagation constant

o : phase constant

(t-kz+o) : the phase of the wave

(1)

z

Ex

z

Propagation

E

B

k

E and B have constant phasein this xy plane; a wavefront

E

A plane EM wave travelling along z, has the same Ex (or By) at any point in agiven xy plane. All electric field vectors in a given xy plane are therefore in phase.The xy planes are of infinite extent in the x and y directions.

A monochromatic plane wave

Ex(z,t)=Re[Eoexp(jo)expj(t-kz)]

=Re [Ecexpj(t-kz)]

Eq. (1) can be rewritten as

where Ec= Eoexp(jo)

complex number that represents the amplitude of the wave and includes the constant phase information o

(2)

y

z

k

Direction of propagation

r

O

E(r,t )r

A travelling plane EM wave along a direction k.

E(r,t)=Eocos(t-kr+o)

where kr=kr =kxx+kyy+kzz

The relationship between time and space for a given phase, for example, that corresponding to a maximum field, according to Eq. (1):

=t-kz+o=constant The phase velocity

v=dz/dt=/k=

(3)

(4)

(5)

B. Maxell’s Wave Equation and Diverging Waves

k

Wave fronts

rE

k

Wave fronts(constant phase surfaces)

z

Wave fronts

PO

P

A perfect spherical waveA perfect plane wave A divergent beam

(a) (b) (c)

Examples of possible EM waves

There are many types of possible EM waves:

• Plane wave: the plane wave has no divergence• Spherical wave: wavefronts are spheres and k vectors diverge out

E=(A/r)cos(t-kz)

where A is a constant• Divergent beam: the wavefront are slowly bent away thereby spreading the wave

(6)

Ex. The Output from a laser

(a) Wavefronts of a Gaussian light beam. (b) Light intensity across beam crosssection. (c) Light irradiance (intensity) vs. radial distance r from beam axis ( z ).

y

x

Wave fronts

z Beam axis

r

Intensity

(a)

(b)

(c)

2wo

O

Gaussian

2w

(Gaussian beam)

• 2w: the beam diameter at any point z

• w2: the cross sectional area at z point contains

85% of the beam power

• 2wo: the beam diameter at point O = the waist

of the beam =the spot size

• wo: the waist radius

• : divergence angle

2=4/(2o)(7)

In an isotropic and linear dielectric medium, these fields must obey Maxwell’s EM wave equation,

2

2

2

2

2

2

2

2

t

E

z

E

y

E

x

Eoro

o : the absolute permeability

o : the absolute permittivity

r : the relative permittivity of the medium

(8)

Refractive IndexFor an EM wave traveling in a nonmagnetic dielectric medium of relative permittivity r, the phase velocity v is given by

v= 1/(roo)1/2

For an EM wave traveling in free space, r=1, the phase velocity v is given by

vvacuum= 1/(oo)1/2=c=3108ms-1

The refractive index n of the medium:

n=c/v= (r)1/2

(9)

(10)

(11)

In free space, the wavenumber:

k=2/

In an isotropic medium, the wavenumber:

kmedium=nk

medium=/n

Light propagates more slowly in a denser medium that has a higher refractive index, and the frequency remains the same.

(12)

(13a)

(13b)

Magnetic Field, Irradiance and Poynting Vector

z

Propagation direction

E

B

k

Area A

vt

A plane EM wave travelling along k crosses an area A at right angles to thedirection of propagation. In time t, the energy in the cylindrical volume Avt(shown dashed) flows through A .

Ex=vBy=(c/n)ByFields in an EM wave

Energy densities in an EM wave

(1/2) ro Ex2= (1/2 o) By

2

The EM power flow per unit area

S= Energy flow per unit time

per unit area

=(Avt) (ro Ex2)/(At)=v ro Ex

2

= v 2ro ExBy

(14)

(15)

(16)

S= v 2ro EBPoynting Vector

The energy flow per unit time per unit area in a direction determined by EB

Irradiance |S|= v 2ro |EB|

(17)

(18)

Snell’s Law and Total Internal Reflection A

n2z

y

O

i

n1

Ai

ri

Incident Light BiAr

Br

t t

t

Refracted Light

Reflected Light

kt

At

Bt

BA

B

A

Ar

ki

kr

A light wave travelling in a medium with a greater refractive index (n1 > n2) suffersreflection and refraction at the boundary.

• Ai and Bi are in phase

Ar and Br must still be be in phase

BB=AA=v1t

AB=v1t/sin i =v1t/sinr

ir

• Ai and Bi are in phase

A and B must still be be in phase

BB=v1t=ct/n1 ; AA=v2t =ct/n2

AB= BB /sin i = AA /sin t

AB= BB /sin i = AA /sin t

= v1t/sin i = v2t/sin t

Snell’s law : sin i /sint=v1/v2=n2/n1 (19)

n2

i

n1 > n2

i

Incidentlight

t

Transmitted(refracted) light

Reflectedlight

kt

i>c

c

TIR

c

Evanescent wave

ki

kr

(a) (b) (c)

Light wave travelling in a more dense medium strikes a less dense medium. Depending onthe incidence angle with respect to c, which is determined by the ratio of the refractiveindices, the wave may be transmitted (refracted) or reflected. (a) i < c (b) i = c (c) i

> c and total internal reflection (TIR).

When n1>n2 and the refraction angle t reaches 90

Critical angle: sin c =n2/ n1 (20)

Fresnel’s Equations A. Amplitude Reflection and Transmission Coefficients

k i

n2

n1 > n2

t =90Evanescent wave

Reflectedwave

Incidentwave

i r

Er,//

Er,Ei,

Ei,//

Et,

(b) i > c then the incident wavesuffers total internal reflection.However, there is an evanescentwave at the surface of the medium.

z

y

x into paper i r

Incidentwave

t

Transmitted wave

Ei,//

Ei,Er,//

Et,

Et,

Er,

Reflectedwave

k t

k r

Light wave travelling in a more dense medium strikes a less dense medium. The plane ofincidence is the plane of the paper and is perpendicular to the flat interface between thetwo media. The electric field is normal to the direction of propagation . It can be resolvedinto perpendicular () and parallel (//) components

(a) i < c then some of the waveis transmitted into the less densemedium. Some of the wave isreflected.

Ei,

• Ei, Er ,and Et: transverse electric field (TE)

Ei//, Er// ,and Et// : transverse magnetic field (TM)

• Incident wave, reflected wave, and transmitted wave

Ei=Eioexpj(t-kir)

Er=Eroexpj(t-krr)

Et=Etoexpj(t-ktr)

(21a)

(21b)

(21c)

According to Maxwell’s EM wave equations and Boundary conditions

r=Ero, /Eio,=

t=Eto, /Eio,=1+ r

r//=Ero,// /Eio,//=

t//=Eto,// /Eio,//=(1/n)(1+ r//)

where n=n2/n1

Reflection and transmission coefficients for E and E//

(22a)

(22b)

(22c)

(22d)

Internal reflection: (a) Magnitude of the reflection coefficients r// and rvs. angle of incidence i for n1 = 1.44 and n2 = 1.00. The critical angle is

44? (b) The corresponding phase changes // and vs. incidence angle.

//

(b)

60

120

180

Incidence angle, i

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40 50 60 70 80 90

| r// |

| r |

c

p

Incidence angle, i

(a)

Magnitude of reflection coefficients Phase changes in degrees

0 10 20 30 40 50 60 70 80 90

c

p

TIR

0

60

20

80

• Polarization angle =Brewser’s angle:

r//=0 tan p =n2/ n1

• Normal incidence:

r//=r = (n1 n2)/( n1 +n2) (23)

(24)

Reflectance and Trasmittance

Reflectance:

R=Ero, 2/ Eio, 2= r2

R//=Ero,// 2/ Eio,// 2= r//2

Transmittance:

T=(n2Eto, 2)/(n1 Eio, 2)= (n2/n1)t2

T//=(n2Eto,// 2)/(n1Eio,// 2=(n2/n1) t//2)

(25)

(26)

(27)

(28)

Interference Principles

superposed

andwhere

(29)

(30)

If

1.2.3.

(31)

B. Interferometer

Delay by a distance d

1. d= m, m=0, 1, 2,….,

2. d= m/2, m=1, 3,….,

3. = 2I0

(32)

Diffraction Principles

A. Diffraction

bright rings (called Airy rings). If the screen is far away from the aperture, this would be a

Light intensity pattern

Incident light wave

Diffracted beam

Circular aperture

A light beam incident on a small circular aperture becomes diffracted and its light

intensity pattern after passing through the aperture is a diffraction pattern with circular

Fraunhofer diffraction pattern.

Fresnel and Fraunhofer diffraction region

Fresnel region(near field)

Zi

aperture

D

Fraunhofer region(far field)

Fraunhofer region

Fresnel region

2DZ i

2DZ i

B. Fraunhofer Diffraction

(a) Huygens-Fresnel principles states that each point in the aperture becomes a source ofsecondary waves (spherical waves). The spherical wavefronts are separated by . The newwavefront is the envelope of the all these spherical wavefronts. (b) Another possible

wavefront occurs at an angle to the z-direction which is a diffracted wave.

Incident plane wave

Newwavefront

A secondarywave source

(a) (b)

Another newwavefront (diffracted)

z

Huygens-Fresnel principle: every unobstructed point of a wavefront, at a given instant time, serves as a source of spherical secondary waves. The amplitude of the optical field at any point beyond is the superposition of all these wavelets.

(a) The aperture is divided into N number of point sources each occupying y withamplitude y. (b) The intensity distribution in the received light at the screen far awayfrom the aperture: the diffraction pattern

A

ysin

y

Y

y

zy

ScreenIncidentlight wave

R = Large

c

b

Light intensity

a

y

y

z

(a) (b)

Incidentlight wave

The wave emitted from point source at y: Eyexp(-jkysin)

All of these waves from point source from y=0 to y=a interfere at the screen and the field at the screen is their sum

The resultant field E( ):

ay

yjkyyCE

0)sinexp()(

sin21

sin21

sinsin

2

1

ka

kaaCekaj

(33)

• The pattern has bright and dark regions, corresponding to

constructive and destructive interference of waves from

the aperture.• The zero intensity points:

sin=m/a ; m=1, 2,……

For a circular aperture

The intensity I at the screen E( )2

sin

2

1);(sin)0(

sin2

1

sin2

1sin

)( 2

2

kacIka

kaaC

I

sin=1.22/D

(34)

(35a)

(35b)angular radius of Airy disk

The rectangular aperture of dimensions a b on the leftgives the diffraction pattern on the right.

a

b

C. Diffraction Grating

dz

y

Incidentlight wave

Diffraction grating

One possiblediffracted beam

a

Intensity

y

m = 0m = 1

m = -1

m = 2

m = -2

Zero-order

First-order

First-order

Second-order

Second-order

Single slitdiffractionenvelope

dsin

(a) (b)

(a) A diffraction grating with N slits in an opaque scree. (b) The diffracted lightpattern. There are distinct beams in certain directions (schematic)

All waves from pairs of slit will interfere constructively when this a multiple of the whole wavelength,

Grating Equation: dsinm=m ; m=0, 1, 2,……

When the incident beam is not normal to the diffraction grating, the diffraction angle m for the m-th mode:

Grating Equation: d(sinm-sini)=m ; m=0, 1, 2,……

(36)

(37)

Incidentlight wave

m = 0

m = -1

m = 1Zero-order

First-order

First-order

(a) Transmission grating (b) Reflection grating

Incidentlight wave

Zero-orderFirst-order

First-order

(a) Ruled periodic parallel scratches on a glass serve as a transmission grating. (b) Areflection grating. An incident light beam results in various "diffracted" beams. Thezero-order diffracted beam is the normal reflected beam with an angle of reflection equalto the angle of incidence.

Polarization and Modulation of Light

Polarization

A. State of Polarization

Birefringent Optical Devices

A. Birefringence B. Retarding Plates C. Compensator D. Birefringent Prisms

Electro-Optic Effects

A. Definitions B. Pockels Effect C. Kerr Effect

Acousto-Optic Effects

A. Definitions

Polarization

(a) A linearly polarized wave has its electric field oscillations defined along a lineperpendicular to the direction of propagation, z. The field vector E and z define a plane ofpolarization . (b) The E -field oscillations are contained in the plane of polarization. (c) Alinearly polarized light at any instant can be represented by the superposition of two fields Ex

and Ey with the right magnitude and phase.

x

y

z

Ey

Ex

yEy

^

xEx

^

(a) (b) (c )

E

Plane of polarization

x

y

EE

A. State of Polarization

kztEE xox cos

kztEE yoy cos

where is the phase difference between Ey and Ex

By choosing Eyo = Exo and = , the field in the wave:

kztEykztExEyEx yoxoyx cosˆcosˆˆˆE

kzt cosEo

where yoxoo EyEx ˆˆE

the vector Eo at –45 to the x-axis: linear polarization

(38a)

(38b)

(39)

z

Ey

Ex

EE

= kz

z

z

A right circularly polarized light. The field vector E is always at rightangles to z , rotates clockwise around z with time, and traces out a fullcircle over one wavelength of distance propagated.

By choosing Eyo = Exo =A and =/2 , the field in the wave:

kztAykztAxEyEx yx sinˆcosˆˆˆE

222x AEE y : circularly polarized

• =/2: right circularly polarized (clockwise)

• =-/2: left circularly polarized (counterclockwise)

(40)

E

y

x

Exo = 0Eyo = 1 = 0

y

x

Exo = 1Eyo = 1 = 0

y

x

Exo = 1Eyo = 1 = /2

E

y

x

Exo = 1Eyo = 1 = /2

(a) (b) (c) (d)

Examples of linearly, (a) and (b), and circularly polarized light (c) and (d); (c) isright circularly and (d) is left circularly polarized light (as seen when the wavedirectly approaches a viewer)

E

y

x

Exo = 1Eyo = 2 = 0

Exo = 1Eyo = 2 = /4

Exo = 1Eyo = 2 = /2

y

x

(a) (b)E

y

x

(c)

(a) Linearly polarized light with Eyo = 2Exo and = 0. (b) When = /4 (45 ), the light isright elliptically polarized with a tilted major axis. (c) When = /2 (90), the light isright elliptically polarized. If Exo and Eyo were equal, this would be right circularlypolarized light.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Birefringent Optical Devices

A. Birefringence

• Opically anisotropic crystal : the refractive index n of

a crystal depends on the direction of the electric field

in the propagating light beam.

• Birefringence : optically anisotropic crystals are

called birefringence because an incident light beam

may be doubly refracted

A line viewed through a cubic sodium chloride (halite) crystal(optically isotropic) and a calcite crystal (optically anisotropic).

B. Retarding Plates

x = Fast axis

z = Slow axis

E//

E

E//

E

E

L

y

no

ne = n3

Optic axis

L

y

no

ne = n3

A retarder plate. The optic axis is parallel to the plate face. The o- and e-waves travelin the same direction but at different speeds.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Uniaxial crystal ne>no

For E : a phase change kL=(2/)no

For E// : a phase change k//L= (2/)ne

Lnn oe )(2

Relative phase through retarder plate

(41)

• Half-wave plate retarder: =, corresponding

to a half of wavelength (/2).

• Quarter-wave plate retarder: =/2,

corresponding to a half of wavelength (/4)

x

= arbitrary

(b)

Input

z

xE

z

x

(a)

Output

Optic axis

Half wavelength plate: = Quarter wavelength plate: = /2

x

< 45

E

z

x

E

E

x

z z

= 45

45 陣

Input and output polarizations of light through (a) a half-wavelengthplate and (b) through a quarter-wavelength plate.

C. Compensator: is adjustable

Optic axis

Optic axisd

D

Wedges can slide

Plate

E1

E2

Soleil-Babinet Compensator© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

)(2

1 Dndn oe

)(2

2 Dndn eo

The phase difference :

))((2

12 dDnn oe

(42a)

(42b)

(43)

A Soleil-Babinet compensator

D. Birefingent Prisms:

Optic axis

e-ray

o-rayA

B

Optic axis

e-ray

o-ray

Optic axis A

B Optic axis

E1

E2

E1

E1

E2

E2

The Wollaston prism is a beam polarization splitter. E1 is orthogonal to the plane ofthe paper and also to the optic axis of the first prism. E2 is in the plane of the paperand orthogonal to E1.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Commercial Wollaston Prisms

Electro-Optic Effects

A. Definition

Electro-Optic Effects : changes in the refractive index of a material induced by the application of an external electric field, which therefore modulates the optical properties.

Field induced refractive index:

n=n+a1E+a2E2+…

(a) Pockel effect: n=a1E

(b) Kerr effect : n=a2E2

(44)

B. Pockels Effect

Outputlight

z

x

Ex

d

EyV

z

Ex

Eyy

Inputlight Ea

Tranverse Pockels cell phase modulator. A linearly polarized input lightinto an electro-optic crystal emerges as a circularly polarized light.

For a LiNbO3 crystal

Ea=0 : the refractive indices are equal no

Ea0 : Pockel effect

aoo

aoo

Ernnn

Ernnn

223

2

223

1

2

12

1

(45a)

(45b)

)2

1(

22:

)2

1(

22:

2232

2

2231

1

d

Vrnn

LLnE

d

Vrnn

LLnE

ooy

oox

Vd

Lrno 22

321

2

Transverse Pockels Effect

(46a)

(46b)

(47)

Transmission intensity

V

Io

Q

0 V

V

Inputlight

P ADetector

Crystal

zx

y

QWP

Left: A tranverse Pockels cell intensity modulator. The polarizer P and analyzer A havetheir transmission axis at right angles and P polarizes at an angle 45 to y-axis. Right:Transmission intensity vs. applied voltage characteristics. If a quarter-wave plate ( QWP)is inserted after P, the characteristic is shifted to the dashed curve.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Intensity modulator

tE

ytE

x oo cos2

ˆcos2

ˆE

The total field at the analyzer (transmission axis=45°):

The intensity I of the detected beam:

2/

22

2sin

2

1sin

V

VIII oo

where V/2: half-wave voltage

(48)

(49)

Integrated optical modulatorV(t)

Ea

Cross-section

LiNbO3

d

Thin buffer layerCoplanar strip electrodes

EO Substratez

y

x

Polarizedinputlight

WaveguideLiNbO 3

L

Integrated tranverse Pockels cell phase modulator in which a waveguide is diffusedinto an electro-optic (EO) substrate. Coplanar strip electrodes apply a transversefield Ea through the waveguide. The substrate is an x-cut LiNbO3 and typically thereis a thin dielectric buffer layer (e.g. ~200 nm thick SiO2) between the surfaceelectrodes and the substrate to separate the electrodes away from the waveguide.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Vd

Lrno 22

32

0.5-0.7 (50)

V(t)

LiNbO3 EO Substrate

A

BIn

OutC

DA

B

Waveguide

Electrode

An integrated Mach-Zender optical intensity modulator. The input light issplit into two coherent waves A and B, which are phase shifted by theapplied voltage, and then the two are combined again at the output.

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Ti diffused lithium noibate electro-optic (Pockels effect) modulator

C. Kerr Effect:

z

x

yEa

Outputlight

Ez

Inputlight

Ex

E

An applied electric field, via the Kerr effect, induces birefringences in anotherwise optically istropic material. .

Phase modulator

n=a2E2=KEa2 K: Kerr coefficient (51)

Acousto-Optic Effects

A. Definition

Acousto-Optic Effects : changes in the refractive index of a material induced by a strain (S), which therefore modulates the optical properties Photoelastic effect

Photoelastic effect:

pSn

2

1(52)

Interdigitally electrodedtransducerModulating RF voltage

Piezoelectriccrystal

Acousticwavefronts

Induced diffractiongrating

Incidentlight

Diffracted light

Through light

Acoustic absorber

Traveling acoustic waves create a harmonic variation in the refractive indexand thereby create a diffraction grating that diffracts the incident beam throughan angle 2.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

A

B

Incident optical beam Diffracted optical beam

O

O'

P Q

B'

A'

sin sin Acoustic

wave fronts

nmax

nmax

nmin

nmin

nmin n

ma x

x

nmin

nma x

x

nn

Simplified Actual

Acousticwave

vacoustic

Consider two coherent optical waves A and B being "reflected" (strictly,scattered) from two adjacent acoustic wavefronts to become A' and B'. Thesereflected waves can only constitute the diffracted beam if they are in phase. Theangle is exaggerated (typically this is a few degrees).

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Bragg condition : 2sin=/n

The condition that gives the angle for a diffracted beam to exist is,

If is the frequency of the acoustic wave, the diffracted beam has a Dopper shifted frequency:

Doppler shift :

where is the angular frequency of the incident optical wave

(53)

(54)

參考書目

1. S. O. Kasap, Optoelectronics and Photonics: Principles and Practices, Prentice-Hall, Inc., 2001.

2. B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, John Wiley & Sons, Inc. 1991.