Flat disk of glass with an index, n(r), that varies as a function of distance r from the center is an example of a

GRIN lens .

Based on the observation that rays (wave-fronts) slow down in an optically dense region and speed-up in less dense regions.

The center of cylindrical lens has n = nmax along its optical axis. So, along the optical axis the optical path length (OPL) is given as (OPL)O = nmaxd

At a height r, (OPL)r n(r)d.

In order for the rays to converge at a focus, the planar wavefront must bend into a spherical wavefront, which defines surfaces of constant phase.

Graded Index Optical Systems)a(

)b(

In order to match the phase at all points on the wavefront we must require

fd

rn

d

ffrfnrnThen

frAssumed

ffrnrn

dnffrdrn

fAFABTherefore

frAFalso

dnABdrnand

OPLABOPL Or

2

/1)(

)(

)(

)(

)()(

2

max

2/122

max

22

max

max22

22

max

n(r)

r

nmax

fd

rnrn

2)(

2

max

Similar to a multimode-graded index core optical fiber

The most common device is a GRIN cylinder a few millimeters in diameter. They are usually fabricated using an ionic diffusion process in which a homogeneous glass is immersed in a molten salt bath for many hours.

The focal length is determined by the index change n < ~0.10. The profile is usually expressed as

n(r) = nmax(1 – ar2/2)

Rays striking the surface in a plane of incidence that contains the optical axis travel in a sinusoidal path have spatial period T = 2/a1/2 where a = a().

It is possible to create real erect images by changing the object distance or length L of the lens.

Common Applications: Laser printers, photocopiers, fax machines (i.e. devices requiring image transfer between surfaces.)

Used in a copy machine

Radial GRIN lenses are often specified in terms of their pitch. A 1.0 pitch rod represents L = T = 2/a1/2 (full sine wave). A pitch of 0.25 has a length of quarter sine wave T/4.

Note that the block of glass is formed so that n = n(z). The block can then be grounded and polished into the shape of a lens. The result is a reduced n for marginal rays.

It is possible to use this approach to significantly reduce the effects of spherical aberration in comparison to a regular lens in (c) to the right.

Superposition of Electromagnetic (E-M) Waves

Any E-M wave satisfies the wave equation:2

2

22 1

tv

The equation is linear since any linear combination will also satisfy this equation: ),(),(

1

trctr i

N

ii

This is called the principle of superposition.

Consider the addition of E-M waves possessing the same frequency but having different phases:

202101

202101

120201202

201

20

021

20221011

coscos

sinsintan

cos2

sin

sinsin

EE

EEand

EEEEEwhere

tEEEEThen

tEEandtEE

Note that we can define a phase difference, , such that = 2 - 1.

The composite wave is harmonic with the same frequency, but the amplitude and phase are different.

Let = -(kx + ) = 2 - 1 = (2/)(x1 - x2), where we let 1 = 2 for now.

x1 and x2 are the distances from the sources of the two waves to the point of observation and is index dependent, and so = o/n . Then, we can write

)(#

)(;

22

21

21

21

mediuminwavesxx

thatNote

OPDDifferencePathOpticalxxnwhere

kxxnxk

o

oo

o

If 1 = 2 = const. the two E-M waves are said to be coherent.

Suppose that we have a superposition of two waves that travel a small difference in distance (x):

xkandxxktxkEEEE

xkxxkkxandABBABAUse

xkEEletand

kxtEEandxxktEE

)2/(sin2/cos2

2/)2/()cos()sin()cos()sin()sin(

;

sin)(sin

0121

120201

022011

Consider two special cases: (1) x = (n + 1/2) (out-of-phase) and (2) x = n (in-phase); n = 0, 1, 2, 3… ,

nxknnxk

Case

nxknnxk

Case

)2(2

12

22

)2/1(2)2/1()2/1(2

12

21

Destructive Interference E = 0, which is a minimum in Intensity.

Constructive Interference E0 = 2E01, which is a maximum in Intensity .

In both cases n = 0, 1, 2, 3…

For the more general case in which E01 E02

Case 1: Partial destructive interference = 2(n + 1/2)

Case 2: Constructive interference = 2n

Fig. 7.3 Waves out-of-phase by kx radians.

Fig. 7.4 The French fighter Rafale uses active cancellation to confound (frustrate) radar detection. It sends out a nearly equal signal that is out-of-phase by /2 with the radar wave that it reflects. Therefore, the reflected and emitted waves cancel in the direction of the enemy receiver.

In general, the sum of N such E-M waves is

N

iii

N

iii

j

N

ij

N

iiji

N

ii

i

N

ii

EEand

EEEEwhere

tEtEE

10

10

100

1

20

20

01

0

cossintan

)cos(2

coscos

Suppose that we have N random sources (e.g. a light bulb).

Then cos(i-j)t = 0 E02t =NE2

01 if each atom emits waves of equal E01.

This result is for an incoherent source of emitters.

For a coherent source, we have i = j and the sources are in-phase which gives

201

2

2

10

20

20 ENEEE

N

iit

Each atom emits waves of equal E01.

Complex Method for Phasor Additon

111011

1011011

~Reexp

~coscos

EEwithtiEE

kxtEtkxEE

The addition of N E-M waves becomes

N

j

ij

i

tiitiN

j

ij

N

jj

j

j

eEeE

eeEeeEEE

100

01

01

~~

The complex amplitude can be expressed as a vector in the complex plane, and is known as a phasor. The resultant complex amplitude is the sum of all constituent phasors.

*0020

ii eEeEE Which can be used to calculate the resulting irradiance from the complex amplitudes of the constituent waves.

E1

Real Axis

Imaginary Axis

Phasor Addition

Consider the sum of two E-M waves: E = E1 + E2

202101

202101

120201202

201

20

021

20221011

coscos

sinsintan

cos2

sin

sinsin

EE

EEand

EEEEEwhere

tEEEEThen

tEEandtEE

From the law of cosines we can easily calculate E0

2 and further analysis of the geometry gives tan .

Fig. 7.7 The phasor sum of E1, E2, E3, E4 and E5.

E1=5sint

E2=10sin(t+45º)

E3=sin (t-15º)

E4= 10sin(t+120º)

E5=8sin(t+180º)

Consider the addition of these five E-M waves using the phasor addition below.

The summation of two sinusoidal functions of the same frequency using phasor additon. Here E1 is taken as the reference phasor, and since E2 leads E1 (i.e. its peak occurs at an earlier location) the angle is positive. Thus is positive and the resultant E also leads to E1.

50, 1045, 1-15, 10120, and 8180

In electrical engineering, these phasors can also be written with the following notation:

)sin(

)sin(

022

011

tkxEE

tkxEE

04030201

044

033

022

011

)4sin(

)3sin(

)2sin(

)sin(

EEEE

tkxEE

tkxEE

tkxEE

tkxEE

Standing Waves: Consider reflection of E-M waves of a mirror

kxtEtkxE

tkxtkxEEEE

tkxEEandtkxEE

RI

RI

sincos2cossin2

sinsin

sinsin

00

0

00

Description of standing wave:

Nodes: x = 0, /2, , 3/2, 2...

Anti-Nodes: x = /4, 3/4, 5/4.…Standing wave: Time-varying amplitude with sinusoidal spatial variation (see previous slide).

Now, consider the addition of two E-M waves having different frequencies:

2

2

1

12121

2201211011

,,

coscos

kkkvkk

txkEEandtxkEE

ph

2/;2/

2/2/

coscos2

2cos

2cos2coscos

coscos

2121

2121

01

221101

kkkkkkand

andwhere

txktxkEEThen

withand

txktxkEEThen

m

m

mm

The resultant wave is a traveling wave of frequency and wave number: k,

txkEtxEwithtxktxEEThen mm cos2,cos, 0100

)modulated or time varying amplitude()traveling wave(

Note that the Irradiance is given by the following

)(2

22cos12cos4

21

201

2201

20

frequencybeat

txkEtxkEEI

m

mmmm

2

1

Detector with fast response )(

2 21

frequencybeat

mB

Beats can also be observed through the superposition of E-M waves possessing different amplitudes, as well as different frequencies. The phasor method can be used to help illustrate the formation of beats.

Heterodyne PrincipleHeterodyning is a method for transferring a broadcast signal from its carrier to a fixed local intermediate frequency in the receiver so that most of the receiver does not have to be retuned when you change channels. The interference of any two waves will produce a beat frequency, and this technique provides for the tuning of a radio by forcing it to produce a specific beat frequency called the "intermediate frequency" or IF.

The carrier wave exhibits a high frequency

The phase velocity is given by

The group velocity is given by

2/21 c

kvph /

kkkkv

m

mg

21

21

This is the rate at which the modulation envelope or energy of the wave advances or propagates. For a general dispersion = (k)

dk

dvg and this speed is usually less than the speed of light c.

Using = vk and v = c/n

dk

dn

n

kv

dk

dn

n

ck

n

c

dk

dn

n

ckv

ncdk

dkv

dk

dvkvvk

dk

d

dk

dvg

1

)/()(

22

For normal dispersion, dn/dk > 0 and therefore vg < v.

Polarization of Light

Linear Polarization: Begin by defining individual components:

componentsphaseofouttkzEjEiE

nnIfEEE

componentsphaseintkzEjEiE

nnIfEEE

tkzEjtzEandtkzEitzE

yx

yx

yx

yx

yyxx

"180"cosˆˆ

...,3,2,1,0,12

""cosˆˆ

...,3,2,1,0,2

cosˆ),(cosˆ),(

00

00

00

It is therefore possible to define any polarization orientation with a constant vector in the x-y plane for the case of linear polarization.

For linear polarization, the state of polarization is often referred to as a P - state. This is the symbol for a script P.

.

sinˆcosˆ

...,3,2,1,0,22/

cosˆ),(cosˆ),(

0

200

00

constEE

EEEwithtkzjtkziEE

nnIfEEE

tkzEjtzEandtkzEitzE

yx

yyxx

Circular polarization:

E0

The electric field vector clearly rotates clockwise while looking back at the source from the direction of propagation. The frequency of rotation is with a period of T = 2/. This is the case of “right-circularly polarized light”. It is often expressed as an R – state. This is a script R.

.sinˆcosˆ

...,3,2,1,0,22/

00 constEEtkzjtkziEE

nnIf

E0

The electric field vector clearly rotates counter-clockwise while looking back at the source from the direction of propagation. The frequency of rotation is with a period of T = 2/. This is the case of “left-circularly polarized light”. It is often expressed as an L– state. This is a script L.

Left circular polarization, i.e. an L– state.

Right circular polarization (R – state)

It is easy to understand that for general parameters E0x, E0y, and , we have elliptical polarization.

yx

yyxx

EEE

tkzEjtzEandtkzEitzE

cosˆ),(cosˆ),( 00

EEE LR

Actual polarizers. Irradiance is independent of the rotation angle for the conversion of natural light

(unpolarized) to linear polarization .

In the figure below, only the component E01cos is transmitted I() = I(0)cos2, which is known as Malus’s Law.

If Iu = I (natural or unpolarized light), then I(0) = Iu <cos2>t = Iu/2.

Dichroism –selective absorption of one of two orthogonal E components.

1 .Absorption of E-field in the y-direction causes e’s to flow.

2 .Re-radiation of waves that cancel incident waves polarized in

the y-direction .

This results in transmission of waves with E-fields perpendicular to the wires (i.e., along the x-direction).

Polaroid sheet (H-Sheet), most commonly used linear polarizer.

Contains a molecular analogue of the wire grid.

1 .Sheet of clear polyvinyl alcohol is heated and stretched.

2 .Then it is dipped in an ink solution rich in Iodine.

3 .Iodine is incorporated into straight long-chain polymeric molecules allowing electron conduction along the chain, simulating a metal wire.

HN-50 is the designation of a hypothetical, ideal H-sheet that transmits 50% of the incident natural light while absorbing the other 50%. In practice, about 4% of the light will be reflected at each surface leaving a maximum transmittance of 92% for linearly polarized light incident on the sheet.

Thus, HN-46 would transmit 46% of incident natural light, and might be the optimal polarizer. In general, for HN-x, the irradiance of polarized light transmitted would be I=Io(x/50), where Io is the irradiance for the ideal case. In practice, it is possible to purchase HN-38, HN-32, and HN-22 in large quantities for reasonable prices, each differing in the amount of iodine present.

Crystal System: Hexagonal (trigonal) Habit: As well-formed, elongate, trigonal prisms, with smaller, second order prism faces on the corners. Prism faces are often striated parallel to direction of elongation (c-axis). The rounded triangular cross-sectional shape of tourmaline crystals is diagnostic; no other gem mineral has such a shape .Hardness: 7-7.5 Cleavage: none

High birefringence (Two differenct indices of refraction)Strong Dichroism Any transparent gem having a mean R.I. of 1.63 and a birefringence of 0.015-0.020 is tourmaline .

Tourmaline is widespread in metamorphic, igneous and sedimentary rocks. Gem Elbaite is, however, nearly restricted to pegmatites. Literally thousands of tourmaline-bearing pegmatites are known; only a few hundred apparently contain gem quality material in mineable quantities .

Found in Brazil, Sri Lanka, U.S., Southern California

Tourmaline

Tourmaline) Boron Silicates)

Chemical Formula

XY3Al6B3Si6OH

(X = Na or Ca)

Y = Mg, Li, Al or Fe

Hardness7 - 7.5

Specific Gravity

 3.0 - 3.3Refractive

Index1.62-1.65

1 (There is a specific direction within the crystal known as the principal or

optic axis .

2 (The E-field component of an incident wave that is perpendicular to the optic axis is strongly absorbed.

3 (The thicker the crystal the more complete will be the absorption.

4 (A plate cut from a tourmaline crystal parallel to its principle axis and several mm thick will serve as a linear polarizer.

5 (Absorption depends on .

6 (Advantages over H-sheet polarizers with regard to maximum irradiance permitted and can be used with high power lasers.