Reference

Introduction to Electrodynamics

By D. J. Griffith

Reflection and Transmission at Normal incidence

Reflection and Transmission at Normal incidence

0 1

0 11

ˆ( , ) exp

1 ˆ( , ) exp

I I

I I

E x t E i k x t j

B x t E i k x t kv

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Suppose yz plane forms the boundary between two linear media. Aplane wave of frequency ω traveling in The x direction (from left) and polarizedalong y direction, approaches the interface from left (see figure)

In medium 1 following reflected wave travels back

0 1

0 11

ˆ( , ) exp

1 ˆ( , ) exp

R R

R R

E x t E i k x t j

B x t E i k x t kv

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�������������� -sign in BR is because Poynting vector mustaim in the direction of propagation

Incident wave

Transmitted wave

0 2

0 22

ˆ( , ) exp

1 ˆ( , ) exp

T T

T T

E x t E i k x t j

B x t E i k x t kv

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In medium 2 we get a transmitted wave:

At x=o the combined fields to the left

EI+ER and BI+BR, must join the fields to the right ET and BT in accordance to the boundary condition.

Since there are no components perpendicular to the surface so boundary conditions (i) and (ii) are trivial. However last two [(iii) & (iv)] yields:

0 0 0

0 0 01 1 1 2 2

1 1 1 1 1

R I T

I R T

E E E

E E Ev v v

(1)

00

00

xBxB

xDxD

00

00////

////

xHxH

xExE

(i)

(ii)

(iii)

(iv)

The reflected wave is in phase if v2>v1 or n1>n2

and out of phase if v2<v1.or n1<n2

Using (1) and (2)

0 0 0

1 1 1 2 1 2

2 2 2 1 2 1

0 0 0 0

1r

2

2 1 20 0 0 0

2 1 2 1

,

where

1 2and

1 1

vIf 1(nonmagnetic media) then =

v

2thus we have, and

I R T

R I T I

R I T I

or E E E

v n

v n

E E E E

v v vE E E E

v v v v

(2)

2 1 20 0 0 0

2 1 2 1

1 2 10 0 0 0

2 1 2 1

The real amplitudes are related by

2and

cin terms of refractive index n=

v

2and

R I T I

R I T I

v v vE E E E

v v v v

n n nE E E E

n n n n

Reflected wave is 180o out of phase when reflected froma denser medium. This fact was encountered by you during

Last semester optics course. Now you have a proof!!!

Reflection coefficient (R) and

Transmission coefficient (T)• Intensity (average power per unit area is given by):

• If μ1= μ2 = μ0, i.e μr=1 , then the ratio of the reflected

intensity to the incident intensity is

20

1

2I vE

2 2

0 1 2

0 1 2

RR

I I

EI n nR

I E n n

Where as the ratio of transmitted intensity to incident intensity is 2 2

02 2 2 1 1 22

1 1 0 1 1 2 1 2

2 4

( )TT

I I

EI v n n n nT

I v E n n n n n

NOTE: R+T=1 => conservation of energy

Use ε α (n)2

Reference: Chapter-21, “OPTICS” by Ajoy Ghatak

Oblique Incidence

Oblique Incidence-1

Prove that: (1) Angle of Incidence = Angle of Reflection(2) Snell’s Law

Oblique Incidence-2

Derive the expressions for reflection coefficient and transmission coefficient

Z

X

Y

Interface of two medium

k1 k3

k2

Plane of incidence

Note: k1,k2,k3 and E1,E2,E3 lies in X-Z Plane. Understand 3-D picture but work With 2-D Fig for calculations

E1

As x=0

k1z=k2z=k3z

Zero

Interface

Incident wave

Refracted wave

Reflected wave

Using fig.

Identity: If

A e(iax)+Be(ibx)=Ce(icx)

Then a=b=c (Ref: Griffith)

Using identity

Snell’s Law

-------------(X)

-------------(Y)

Simplify equation (X) and (y), substitute the value of E20 from (X) to (Y)

BC—(iii)

BC—(i)

x

zy

As Θ1= Θ3

Divide equation (X) by E10 and substitute the value of E30/E10,

Case II: If E is perpendicular to the plane of incidence (do it yourself)

All these four equations are known as Fresnel’s equation

Snell’s Law

ε = n2

Case II: E is perpendicular to plane of incidence.

……(YY)

Since the Y-axis is tangential or parallel to the interface, the y-component of E must be continuous across the interface.

E10 + E30 = E20 ……….(XX)

Using (XX) and (YY), we can get

[Boundary Condition (iv)]

900

Polarized light

(b)

(A)

(A)

More to do…..

• (C) Phase Change in reflection( Role of Brewster’s angle). Ref: Page-21.4 & 21.6 , “Ghatak”

• (D) Total Internal Reflection– Ref: Page-21.5 & 21.6 , “Ghatak”

If θ2=900 and θ1= θC , determine r//, t// and r┴ & t┴ .

Numerical to do…

(1) For an Air glass Interface (n1=1.0 and n2=1.5)

(a) determine r// and t// for normal incidence.

(b) determine Brewster’s angle.

(c) if incidence angle is 300 determine r// and t// for oblique incidence.

Example: 21.5, 21.6, 21.7 Page-21.11, “Ghatak” 3rd edition.

Problem: 21.4 Page21.17 “Ghatak” 3rd edition