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Fundamentals of Optics
Jiun-You Lin
Department of Mechatronics
Engineering, National Changhua
University of Education
• 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
Outline
• Wave Nature of Light
• Polarization and Modulation
of Light
Wave Nature of Light
Light 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 Equations
A. 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
Light Waves in a Homogeneous Medium
•Ultraviolet
10nm~390nm
•Visible
390nm~760nm
•Infrared
760nm~1mm
Optical frequencies
and wavelengths
A. Plane Electromagnetic Wave
Ex
Direction of Propagation
k
x
z
z
y
By
An electromagnetic wave is a travelling wave which has time
varying electric and magnetic fields which are perpendicular to each
other and the direction of propagation, z.
A sinusoidal wave
Ex  Eo cost  kz  o 
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)
A monochromatic plane wave
E and B have constant phase
in this xy plane; a wavefront
z
E
E
B
k
Propagation
Ex
z
A plane EM wave travelling along z, has the same E x (or By) at any point in a
given 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.
Eq. (1) can be rewritten as
Ex(z,t)=Re[Eoexp(jo)expj(t-kz)]
=Re [Ecexpj(t-kz)]
(2)
where Ec= Eoexp(jo)
complex number that represents the amplitude
of the wave and includes the constant phase
information o
y
Direction of propagation
k
E(r, t )
r
q
O
r
z
A travelling plane EM wave along a direction k
.
E(r,t)=Eocos(t-kr+o)
(3)
where kr=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
(4)
The phase velocity
v=dz/dt=/k=
(5)
B. Maxell’s Wave Equation and Diverging
Waves
Wave fronts
(constant phase surfaces)
Wave fronts
k

P

P
k

Wave fronts
E
r
O
z
A perfect plane wave
(a)
A perfect spherical wave
(b)
Examples of possible EM waves
A divergent beam
(c)
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)
(6)
where A is a constant
• Divergent beam: the wavefront are slowly bent
away thereby spreading the wave
Ex. The Output from a laser (Gaussian beam)
y
Wave fronts
(b)
x
2wo O
z Beam axis
Intensity
q
Gaussian
(c)
r
(a)
2w
(a) Wavefronts of a Gaussian light beam. (b) Light intensity across beam cross
section. (c) Light irradiance (intensity) vs. radial distance r from beam axis ( z ).
• 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
• q : divergence angle
(7)
2q=4/(2o)
In an isotropic and linear dielectric medium, these fields
must obey Maxwell’s EM wave equation,
2E 2E 2E
2E
 2  2   o r  o 2
2
x
y
z
t
o : the absolute permeability
o : the absolute permittivity
r : the relative permittivity of the medium
(8)
Refractive Index
For an EM wave traveling in a nonmagnetic dielectric
medium of relative permittivity r, the phase velocity v is
given by
v= 1/(roo)1/2
(9)
For an EM wave traveling in free space, r=1, the phase
velocity v is given by
(10)
v
= 1/(  )1/2=c=3108ms-1
vacuum
o o
The refractive index n of the medium:
n=c/v= (r
)1/2
(11)
In free space, the wavenumber:
k=2/
(12)
In an isotropic medium, the wavenumber:
kmedium=nk
(13a)
medium=/n
(13b)
Light propagates more slowly in a denser
medium that has a higher refractive index,
and the frequency  remains the same.
Magnetic Field, Irradiance and Poynting
Vector
v Dt
z
E
Area A
k
Propagation direction
B
A plane EM wave travelling alongk crosses an area A at right angles to the
direction of propagation. In timeDt, the energy in the cylindrical volumeAvDt
(shown dashed) flows throughA .
Fields in an EM wave
Ex=vBy=(c/n)By
Energy densities in an
EM wave
(1/2) ro Ex2= (1/2 o) By2 (15)
The EM power flow
per unit area
(14)
S= Energy flow per unit time
per unit area
=(AvDt) (ro Ex2)/(ADt)=v ro Ex2
= v 2ro ExBy
(16)
Poynting Vector
S= v 2ro EB
(17)
The energy flow per unit time per unit
area in a direction determined by EB
Irradiance |S|= v 2ro |EB|
(18)
Snell’s Law and Total Internal Reflection
At
A
t
kt
Refracted Light
Bt
A
y
O
qt
z
qt
B
A
n1
qi qr
qi qr
A
B
ki


kr
Ai
Incident Light
n2
Br
Bi
Ar
Reflected Light
A light wave travelling in a medium with a greater refractive index ( n1 > n2) suffers
reflection 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 qi =v1t/sinqr
• 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 qi = AA /sin qt
qiqr
AB= BB /sin qi = AA /sin qt
= v1t/sin qi = v2t/sin qt
Snell’s law : sin qi /sinqt=v1/v2=n2/n1
(19)
qt
ki
qi q
i
Incident
light
Transmitted
(refracted) light
kt
n2
n 1 > n2
kr
Evanescent wave
qc qc
TIR
Reflected
light
(a)
qi >qc
(b)
(c)
Light wave travelling in a more dense medium strikes a less dense medium. Depending on
the incidence angle with respect toq c, which is determined by the ratio of the refractive
indices, the wave may be transmitted (refracted) or reflected. (a)qi < qc (b) q i = qc (c) q i
> q c and total internal reflection (TIR).
When n1>n2 and the refraction angle qt reaches 90
Critical angle: sin qc =n2/ n1
(20)
Fresnel’s Equations
A. Amplitude Reflection and Transmission Coefficients
Et,//
y
qt
z
x into paper
Ei,//
ki
qi qr
Ei,^
Incident
wave
Er,//
Transmitted wave
kt
Et,^ Evanescent wave
Et,^
Er,^
(a) qi < qc then some of the wave
is transmitted into the less dense
medium. Some of the wave is
reflected.
qi qr
Ei,//
Ei,^
kr
Reflected
wave
n2
q t =90
Incident
wave
Er,^
n1 > n 2
Er,//
Reflected
wave
(b) qi > q c then the incident wave
suffers total internal reflection.
However, there is an evanescent
wave at the surface of the medium.
Light wave travelling in a more dense medium strikes a less dense medium. The plane of
incidence is the plane of the paper and is perpendicular to the flat interface between the
two media. The electric field is normal to the direction of propagation . It can be resolved
into perpendicular (^ ) and parallel (//) components
• 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-kir)
(21a)
Er=Eroexpj(t-krr)
(21b)
Et=Etoexpj(t-ktr)
(21c)
According to Maxwell’s EM wave equations
and Boundary conditions
Reflection and transmission coefficients
for E ^ and E//
r^=Ero,^ /Eio,^=
(22a)
t^=Eto,^ /Eio,^=1+ r^
(22b)
r//=Ero,// /Eio,//=
(22c)
t//=Eto,// /Eio,//=(1/n)(1+ r//)
(22d)
where n=n2/n1
Magnitude of reflection coefficients
Phase changes in degrees
qc
1
0.9
0.8
(a)
(b)
120
0.7
0.6
0.5
0.4
0.3
0.2
0.1
TIR
180
60
qp
0
| r^ |
 60
qp
| r // |
0
Incidence angle, q i
//
qc
1 20
1 80
0 10 20 30 40 50 60 70 80 90
^
0
10 20 30 40 50 60 70 80 90
Incidence angle, q i
Internal reflection: (a) Magnitude of the reflection coefficients r// and r^
vs. angle of incidence q i for n1 = 1.44 and n2 = 1.00. The critical angle is
44? (b) The corresponding phase changes // and ^ vs. incidence angle.
• Normal incidence:
r//=r^ = (n1 n2)/( n1 +n2)
(23)
• Polarization angle =Brewser’s angle:
r//=0
tan qp =n2/ n1
(24)
Reflectance and Trasmittance
Reflectance:
R^=Ero,^ 2/ Eio,^ 2= r^2
(25)
R//=Ero,// 2/ Eio,// 2= r//2
(26)
Transmittance:
T^= (n2Eto,^ 2)/(n1 Eio,^ 2)= (n2/n1)t^2 (27)
T//= (n2Eto,// 2)/(n1Eio,// 2=(n2/n1) t//2) (28)
Interference Principles
superposed
(29)
where
and
(30)
If
1.
2.
3.
(31)
B. Interferometer
Delay by a distance d
(32)
1. d= m, m=0, 1, 2,….,
2. d= m/2, m=1, 3,….,
3.
= 2I0
Diffraction Principles
A. Diffraction
Light intensity pattern
Diffracted beam
Incident light wave
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
bright rings (called Airy rings). If the screen is far away from the aperture, this would be a
Fraunhofer diffraction pattern.
Fresnel region Fraunhofer region
(near field)
(far field)
aperture
D
Zi
Fresnel and Fraunhofer diffraction region
Zi 
D2
Zi 
D2




Fraunhofer region
Fresnel region
B. Fraunhofer Diffraction
Incident plane wave
A secondary
wave source
Another new
wavefront (diffracted)
New
wavefront
(a)
q
z
(b)
(a) Huygens-Fresnel principles states that each point in the aperture becomes a source of
secondary waves (spherical waves). The spherical wavefronts are separated by . The new
wavefront is the envelope of the all these spherical wavefronts. (b) Another possible
wavefront occurs at an angle q to the z-direction which is a diffracted wave.
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.
Incident
light wave
Y
dy
y
Incident
light wave
Screen
R = Large
q
y
b
a
q
A
c
y
z
dy
q
ysinq
z q0
q
Light intensity
(a)
(b)
(a) The aperture is divided intoN number of point sources each occupying d y with
amplitude  dy. (b) The intensity distribution in the received light at the screen far away
from the aperture: the diffraction pattern
The wave emitted from point source at y:
dEdyexp(-jkysinq)
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(q ):
y a
E (q )  C  dy exp(  jky sin q )
y 0
Ce

1
 j ka sinq
2
1

a sin  ka sin q 
2

1
ka sin q
2
(33)
The intensity I at the screen  E(q )
2
2


1


 C a sin  2 ka sin q  
1


2
  I (0) sin c (  );   ka sin q
I (q )  
1
2


ka sin q
(34)


2


• The pattern has bright and dark regions, corresponding to
constructive and destructive interference of waves from
the aperture.
• The zero intensity points:
sinq=m/a ; m=1, 2,……
(35a)
For a circular aperture
(35b)
sinq=1.22/D
angular radius of Airy disk
b
a
The rectangular aperture of dimensionsa  b on the left
gives the diffraction pattern on the right.
C. Diffraction Grating
y
One possible
diffracted beam
y
Incident
light wave
Single slit
diffraction
envelope
a
m=2
Second-order
m=1
First-order
m=0
Zero-order
m = -1 First-order
d
q
m = -2 Second-order
z
dsin q
Diffraction grating
(a)
Intensity
(b )
(a) A diffraction grating with N slits in an opaque scree. (b) The diffracted light
pattern. 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: dsinqm=m ; m=0, 1,
2,……
(36)
When the incident beam is not normal to the diffraction
grating, the diffraction angle qm for the m-th mode:
Grating Equation: d(sinqm-sinqi)=m ; m=0, 1,
2,……
(37)
First-order
Zero-order
Incident
light wave
(a) Transmission grating
Incident
light wave
m = 1 First-order
m = 0 Zero-order
First-order
m = -1 First-order
(b) Reflection grating
(a) Ruled periodic parallel scratches on a glass serve as a transmission grating. (b) A
reflection grating. An incident light beam results in various "diffracted" beams. The
zero-order diffracted beam is the normal reflected beam with an angle of reflection equal
to 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. State of Polarization
Plane of polarization
y
E
x
E
^
y
y
^
x
E
E
z
(a )
(b)
^
xE
x
x
^
yE
y
E
(c )
(a) A linearly polarized wave has its electric field oscillations defined along a line
perpendicular to the direction of propagation, z . The field vector E and z define a plane of
polarization . (b) The E -field oscillations are contained in the plane of polarization. (c) A
linearly polarized light at any instant can be represented by the superposition of two fields E x
and E y with the right magnitude and phase.
Ex  Exo cost  kz
E y  E yo cost  kz   
(38a)
(38b)
where  is the phase difference between Ey and Ex
By choosing Eyo = Exo and  = , the field in the wave:
E  xˆE x  yˆE y  xˆE xo cost  kz  yˆE yo cost  kz
 E o cost  kz
where E o  xˆE xo  yˆ E yo
(39)
the vector Eo at –45 to the x-axis: linear polarization
Dz
q = k Dz
E
z
Ey
E
z
Ex
A right circularly polarized light. The field vector E is always at right
angles to z , rotates clockwise around z with time, and traces out a full
circle over one wavelength of distance propagated.
By choosing Eyo = Exo =A and  =/2 , the field in the
wave:
E  xˆE x  yˆE y  xˆA cost  kz  yˆA sin t  kz (40)
E  E  A : circularly polarized
2
x
2
y
2
•  =/2: right circularly polarized (clockwise)
•  =-/2: left circularly polarized (counterclockwise)
y
y
(a)
y
(b)
(c)
x
x
Exo = 0
Eyo = 1
=0
y
Exo = 1
Eyo = 1
=0
(d)
E
Exo = 1
Eyo = 1
 = /2
x
E
Exo = 1
Eyo = 1
 = /2
Examples of linearly, (a) and (b), and circularly polarized light (c) and (d); (c) is
right circularly and (d) is left circularly polarized light (as seen when the wave
directly approaches a viewer)
x
y
y
(a)
(b)
y
E
x
x
Exo = 1
Eyo = 2
=0
(c)
E
=1
=2
 = /4
Exo
Eyo
x
=1
=2
 = /2
Exo
Eyo
(a) Linearly polarized light with Eyo = 2Exo and  = 0. (b) When  = /4 (45), the light is
right elliptically polarized with a tilted major axis. (c) When  = /2 (90), the light is
right elliptically polarized. If Exo and Eyo were equal, this would be right circularly
polarized 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
B. Retarding Plates
z = Slow axis
Optic axis
E //

E //
E
ne = n3

y
E^
E^
no
LL
x = Fast axis
A retarder plate. The optic axis is parallel to the plate face. The o- and e-waves travel
in the same direction but at different speeds.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Uniaxial crystal ne>no
For E^ : a phase change k^L=(2/)no
For E// : a phase change k//L= (2/)ne
Relative phase through retarder plate 

2

(ne  no ) L
(41)
• Half-wave plate retarder: =, corresponding
to a half of wavelength (/2).
• Quarter-wave plate retarder: =/2,
corresponding to a half of wavelength (/4)
Half wavelength plate: = 
Optic axis

Quarter wavelength plate: = /2
z
z
E

x
Input
 = arbitrary
x
x
0 < 45 
 = 45 
z
z
E
45陣
z

Output
x
(a)
E
E
x
(b)
Input and output polarizations of light through (a) a half-wavelength
plate and (b) through a quarter-wavelength plate.
x
C. Compensator:  is adjustable
E1
E2
Wedges can slide
Optic axis
d
D
Plate
Optic axis
Soleil-Babinet Compensator
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
1 
2
(ne d  no D )

2
2 
(no d  ne D)

(42a)
(42b)
The phase difference :
  2  1 
2

(ne  no )( D  d )
(43)
A Soleil-Babinet compensator
D. Birefingent Prisms:
e-ray
Optic axis
A
B
o-ray Optic axis
A
E1
e-ray
E1
E1
E2
E2
E2 q B
Optic axis
Optic axis
o-ray
The Wollaston prism is a beam polarization splitter. E1 is orthogonal to the plane of
the paper and also to the optic axis of the first prism. E2 is in the plane of the paper
and 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: Dn=a1E
(b) Kerr effect : Dn=a2E2
(44)
B. Pockels Effect
Ey
V
45
Input
light
y Ey
d
Ea x
D
Ex
z
z Output
light
Ex
Tranverse Pockels cell phase modulator. A linearly polarized input light
into an electro-optic crystal emerges as a cir cularly polarized light.
For a LiNbO3 crystal
Ea=0 : the refractive indices are equal no
Ea0 : Pockel effect
1 3
n1  no  no r22 Ea
2
1 3
n2  no  no r22 Ea
2
(45a)
(45b)
2n1L
2L
1 3 V
E x : 1 

(no  no r22 ) (46a)


2
d
2n2 L 2L
1 3 V
E y : 2 

(no  no r22 ) (46b)


2
d
Transverse Pockels Effect
2
L
D  1  2 
nr
V

d
3
o 22
(47)
Intensity modulator
QWP
Transmission intensity
V
y
P
Io
A
Detec tor
Input
light
45
Crystal
Q
x
z
0
V /
V
Left: A tranverse Pockels cell intensity modulator. The polarizer P and analyzer A have
their 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)
The total field at the analyzer (transmission axis=45°):
Eo
Eo
E   xˆ
cos t  yˆ
cost  D 
2
2
(48)
The intensity I of the detected beam:
V 
1 
2

I  I o sin  D   I osin  
2 
 2 V / 2 
2
where V/2: half-wave voltage
(49)
Integrated optical modulator
V(t)
Coplanar strip electrodes
Thin buffer layer
d
Polarized
input
light
L
LiNbO 3
Ea
EO Subs trate
Waveguide
x
z
y
LiNbO 3
Cross-section
Integrated tranverse Pockels cell phase modulator in which a waveguide is diffused
into an electro-optic (EO) substrate. Coplanar strip electrodes apply a transverse
field E a through the waveguide. The substrate is an x-cut LiNbO 3 and typically there
is a thin dielectric buffer layer ( e.g. ~200 nm thick SiO 2) between the surface
electrodes and the substrate to separate the electrodes away from the waveguide.
2
© 1999 S.O. Kasap,Optoelectronics (Prentice Hall)
L
D  
nr
V

d
3
o 22
0.5-0.7
(50)
V(t)
Electrode
C
In
B
B
A
A
Out
D
Waveguide
LiNbO 3
EO Substrate
An integrated Mach-Zender optical intensity modulator. The input light is
split into two coherent waves A and B, which are phase shifted by the
applied 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:
Phase modulatorz
Ez
E
D
Input
light
Ea
y
Output
light
Ex
x
An applied electric field, via the Kerr effect, induces birefringences in an
otherwise optically istropic material. .
Dn=a2E2=KEa K: Kerr coefficient (51)
2
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:
 1
D 2   pS
n 
(52)
Acoustic absorber
Induced diffraction
grating
Incident
light
Acoustic
wave
fronts
q
Diffracted light
q
Through light
Piezoelectric
crystal
Modulating RF voltage
Interdigitally electroded
transducer
Traveling acoustic waves create a harmonic variation in the refractive index
and thereby create a diffraction grating that diffracts the incident beam through
an angle 2 q.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Incident optical beam
A
Diffracted optical beam
A'
B
nmin
nmax
nmin
nmax
Acoustic
wave
x
x
B'
q

O
q
Q
P
si nq
si nq
Acoustic
wave fronts
O'
vacoustic
nmin nma x
Simplified
n
nmin
nma x
n
Actual
Consider two coherent optical waves A and B being "reflected" (strictly,
scattered) from two adjacent acoustic wavefronts to become A' and B'. These
reflected waves can only constitute the diffracted beam if they are in phase. The
angle q is exaggerated (typically this is a few degrees).
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
The condition that gives the angle q for a diffracted
beam to exist is,
Bragg condition : 2sinq=/n
(53)
If  is the frequency of the acoustic wave, the
diffracted beam has a Dopper shifted frequency:
Doppler shift : 
(54)
where  is the angular frequency of the incident
optical wave
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.