Today’s summary - MITweb.mit.edu/2.710/Fall06/2.710-wk7-b-sl.pdf · • Reflection and refraction at a dielectric interface: – wave approach to derive Snell’s law – reflection

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MIT 2.71/2.710 Optics10/19/05 wk7-b-1

Today’s summary

• Energy / Poynting’s vector• Reflection and refraction at a dielectric interface:

– wave approach to derive Snell’s law– reflection and transmission coefficients– total internal reflection (TIR) revisited

• Two-beam interference

MIT 2.71/2.710 Optics10/19/05 wk7-b-2

Energy

MIT 2.71/2.710 Optics10/19/05 wk7-b-3

The Poynting vector

E

B

S

BEBES ×=×= 02

0

1 εµ

c

so in free space

kS ||

S has units of W/m2

so it representsenergy flux (energy perunit time & unit area)

MIT 2.71/2.710 Optics10/19/05 wk7-b-4

Poynting vector and phasors (I)

20

02

1 ESEEBEkB

BESε

ωω

εc

ck

c=⇒

==⇒×=

×=

For example, sinusoidal field propagating along z

( ) ( )tkzEctkzE ωεω −=⇒−= 22000 coscosˆ SxE

Recall: for visible light, ω~1014-1015Hz

MIT 2.71/2.710 Optics10/19/05 wk7-b-5

Poynting vector and phasors (II)

Recall: for visible light, ω~1014-1015Hz

So any instrument will record the averageaverage incident energy flux

∫+

=Tt

t

tT

d1 SS where T is the period (T=λ/c)

S is called the irradianceirradiance, aka intensityintensityof the optical field (units: W/m2)

MIT 2.71/2.710 Optics10/19/05 wk7-b-6

Poynting vector and phasors (III)

21d)(cos)(cos 22 =−=− ∫

+

ttkztkzTt

t

ωω

2002

1 Ecε=S

For example: sinusoidal electric field,

Then, at constant z:

( ) ( )tkzEctkzE ωεω −=⇒−= 22000 coscosˆ SxE

MIT 2.71/2.710 Optics10/19/05 wk7-b-7

Relationship between E and B

EB

k

( )

EkB

k

xEBE rk

×=⇒

−≡∂∂

×≡∇×⇒

=∂∂

−=×∇ −⋅

ω

ω

ω

1

and

eˆ where 0

it

i

Et

ti

Vectors k, E, B form aright-handed triad.

Note: free space or isotropic media only

MIT 2.71/2.710 Optics10/19/05 wk7-b-8

Poynting vector and phasors (IV)

Recall phasor representation:

( ) ( )( ) ( ) ( )

e : phasor""or amplitudecomplex sincos,ˆ

cos,

φ

φωφω

φω

iAtkziAtkzAtzf

tkzAtzf

−

−−+−−=

−−=

Can we use phasors to compute intensity?

MIT 2.71/2.710 Optics10/19/05 wk7-b-9

Poynting vector and phasors (V)

Consider the superposition of two two fields of the samesame frequency:

( ) ( )( ) ( )φω

ω−−=

−=tkzEtzEtkzEtzE

cos,cos,

202

101

( ) ( )φεε cos22

...d 2010220

210

0221

0 EEEEctEET

c Tt

t

++==+= ∫+

S

Now consider the two corresponding phasorsphasors:

φiE

E−e20

10

( )φεε φ cos22

...e2 2010

220

210

02

20100 EEEEcEEc i ++==+ −

and the quantity

MIT 2.71/2.710 Optics10/19/05 wk7-b-10

Poynting vector and phasors (V)

Consider the superposition of two two fields of the samesame frequency:

( ) ( )( ) ( )φω

ω−−=

−=tkzEtzEtkzEtzE

cos,cos,

202

101

( ) ( )φεε cos22

...d 2010220

210

0221

0 EEEEctEET

c Tt

t

++==+= ∫+

S

Now consider the two corresponding phasorsphasors:

φiE

E−e20

10

( )φεε φ cos22

...e2 2010

220

210

02

20100 EEEEcEEc i ++==+ −

and the quantity

= !!

MIT 2.71/2.710 Optics10/19/05 wk7-b-11

Poynting vector and irradiance

SummarySummary

20

0

;1 ESBES εµ

c=×=

∫+

=Tt

t

tT

d1 SS

Poynting vector

Irradiance (or intensity)

220 phasor or phasor2

∝= SS εc

(free space or isotropic media)

MIT 2.71/2.710 Optics10/19/05 wk7-b-12

Reflection / RefractionFresnel coefficients

MIT 2.71/2.710 Optics10/19/05 wk7-b-13

Reflection & transmission @ dielectric interface

MIT 2.71/2.710 Optics10/19/05 wk7-b-14

Ei ki kiEi


MIT 2.71/2.710 Optics10/19/05 wk7-b-15


I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidence

( )[ ]( )[ ]txykiE

tiE

iiii

iii

ωθθω

−+−==−⋅=

)sincos(expˆ expˆ

0

0

zrkzE

Incident electric field:

( )[ ]( )[ ]txykiE

tiE

rrir

rrr

ωθθω

−++==−⋅=

)sincos(expˆ expˆ

0

0

zrkzE

Reflected electric field:

( )[ ]( )[ ]txykiE

tiE

tttt

ttt

ωθθω

−+−==−⋅=

)sincos(expˆ expˆ

0

0

zrkzE

Transmitted electric field:

where:vacuumvacuum

2 ,2λπ

λπ t

ti

inknk ==

MIT 2.71/2.710 Optics10/19/05 wk7-b-16



( )[ ]( )[ ]( )[ ]txykiE

txykiEtxykiE

EEE

tttt

rrir

iiii

t

ri

ωθθωθθωθθ

−+−=−+++−+−⇒

==+

)sincos(exp )sincos(exp

)sincos(expl)(tangentia

l)(tangential)(tangentia

0

0

0

Continuity of tangential electric fieldat the interface:

But at the interface y=0 so( )[ ]( )[ ]

( )[ ]txkiEtxkiEtxkiE

ttt

rir

iii

ωθωθωθ

−=−+−

sinexp sinexp

sinexp

0

0

0

MIT 2.71/2.710 Optics10/19/05 wk7-b-17


I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidenceContinuity of tangential electric fieldat the interface:

( )[ ]( )[ ]


ttt

rir

iii

ωθωθωθ

−=−+−

sinexp sinexp

sinexp

0

0

0

Since the exponents must be equalfor all x, we obtain

tt

ii

ttii

ri

nnkk θλπθ

λπθθ

θθ

sin2sin2sinsin

and

vacuumvacuum

=⇔=

=

MIT 2.71/2.710 Optics10/19/05 wk7-b-18


I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidenceContinuity of tangential electric fieldat the interface:

ri θθ =

ttii nn θθ sinsin =

law of reflection

Snell’s law of refraction

so wave description is equivalentto Fermat’s principle!! ☺

MIT 2.71/2.710 Optics10/19/05 wk7-b-19



( )[ ]txykiE iiiii ωθθ −+−= )sincos(expˆ 0zE

Incident electric field:

( )[ ]txykiE rrirr ωθθ −++= )sincos(expˆ 0zEReflected electric field:

( )[ ]txykiE ttttt ωθθ −+−= )sincos(exp0zETransmitted electric field:

Need to calculate the reflected and transmitted amplitudes E0r, E0t

i.e. need twotwo equations

MIT 2.71/2.710 Optics10/19/05 wk7-b-20


I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidenceContinuity of tangential electric fieldat the interface gives us one equation:

( )[ ]( )[ ]


ttt

rir

iii

ωθωθωθ

−=−+−

sinexp sinexp

sinexp

0

0

0

which after satisfying Snell’s law becomes

tri EEE 000 =+

MIT 2.71/2.710 Optics10/19/05 wk7-b-21



l)(tangentia l)(tangential)(tangentia

t

ri

BBB=

=+

The second equation comes from continuity of tangential magnetic fieldat the interface:

Recall

0000cossinˆˆˆ

1

1

Ekk θθ

ω

ωzyx

EkB

=

×=

MIT 2.71/2.710 Optics10/19/05 wk7-b-22


I. Polarization normal to plane of incidenceI. Polarization normal to plane of incidenceSo continuity of tangential magnetic field Bx at the interface y=0 becomes:

tttrriiii

tttrriiii

EnEnEnEkEkEk

θθθθθθ

coscoscoscoscoscos

000

000

=−⇔=−

MIT 2.71/2.710 Optics10/19/05 wk7-b-23



ttii

ii

i

t

ttii

ttii

i

r

nnn

EEt

nnnn

EEr

θθθ

θθθθ

coscoscos2

coscoscoscos

0

0

0

0

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

⊥

⊥

⊥

⊥

Solving the 2×2 system of equations:

tttrriiii EnEnEn θθθ coscoscos 000 =−tri EEE 000 =+

we finally obtain

MIT 2.71/2.710 Optics10/19/05 wk7-b-24

tiit

ti

i

t

tiit

tiit

i

r

nnn

EEt

nnnn

EEr

θθθ

θθθθ

coscoscos2

coscoscoscos

||0

0||

||0

0||

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

Following a similar procedure ...


II. Polarization parallel to plane of incidenceII. Polarization parallel to plane of incidence

MIT 2.71/2.710 Optics10/19/05 wk7-b-25


n=1.5

“Brewsterangle”

MIT 2.71/2.710 Optics10/19/05 wk7-b-26

Reflection & transmission of energyenergy@ dielectric interface

Recall Poynting vector definition:

20 ES εc=

different on the two sides of the interface

incvacuum

tncvacuum

22

0

0

22

0

0

coscos

coscos t

nn

EE

nnT

rEER

ii

tt

i

t

ii

tt

i

r

θθ

θθ

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

MIT 2.71/2.710 Optics10/19/05 wk7-b-27

Energy conservation

1coscos i.e. ,1 2t2 =+=+ t

nnrTR

ii

t

θθ

MIT 2.71/2.710 Optics10/19/05 wk7-b-28

Reflection & transmission of energyenergy@ dielectric interface

MIT 2.71/2.710 Optics10/19/05 wk7-b-29

Normal incidence

tiit

ti

i

t

tiit

tiit

i

r

nnn

EEt

nnnn

EEr

θθθ

θθθθ

coscoscos2

coscoscoscos

||0

0||

||0

0||

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

ttii

ii

i

t

ttii

ttii

i

r

nnn

EEt

nnnn

EEr

θθθ

θθθθ

coscoscos2

coscoscoscos

0

0

0

0

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

⊥

⊥

⊥

⊥

0 and 0 == ti θθ

it

i

it

it

nnntt

nnnnrr

+==

+−

==

⊥

⊥

2||

||

( )2||

2

||

4

it

it

it

it

nnnnTT

nnnnRR

+==

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

==

⊥

⊥

Note: Note: independent of polarization

MIT 2.71/2.710 Optics10/19/05 wk7-b-30

Brewster angle

.0 ),tan( 2

When . allfor 0 , If

)tan()tan(

coscoscoscos

)sin()sin(

coscoscoscos

||ti

||

===−≠≠

+−

+=+−

=

+−

−=+−

=

⊥

⊥

rnnrnn

nnnnr

nnnnr

i

tiiti

ti

ti

tiit

tiit

it

ti

ttii

ttii

θπθθθ

θθθθ

θθθθ

θθθθ

θθθθ

Recall Snell’s Law ttii nn θθ sinsin =

This angle is known as Brewster’s angle. Under such circumstances, for an incoming unpolarized wave, only the component polarized normal to the incident plane will be reflected.

MIT 2.71/2.710 Optics10/19/05 wk7-b-31

Why does Brewster happen?

elementaldipoleradiatorexcited bythe incidentfield

MIT 2.71/2.710 Optics10/19/05 wk7-b-32


MIT 2.71/2.710 Optics10/19/05 wk7-b-33


MIT 2.71/2.710 Optics10/19/05 wk7-b-34


MIT 2.71/2.710 Optics10/19/05 wk7-b-35

Turning the tables

rt

1

r’ t’

1

Is there a relationship between r, t and r’, t’ ?

MIT 2.71/2.710 Optics10/19/05 wk7-b-36

Relation between r, r’ and t, t’

a

arat

air glass

a’

a’r’

a’t’

12 =′+

−=′

ttrrr Proof: algebraic from the Fresnel coefficients

or using the property of preservation of thepreservation of thefield properties upon time reversalfield properties upon time reversal

glass air

Stokes relationships

MIT 2.71/2.710 Optics10/19/05 wk7-b-37

Proof using time reversal

a

arat

air glassatt’

arat

air glass

atr’

artar2

( )( ) 1

0 22 =′+⇒=′+

′−=⇒=′+

ttrattrarrtrra

⇔

MIT 2.71/2.710 Optics10/19/05 wk7-b-38

Total Internal Reflection

no energy transmitted

Happens when1sin >iin θ

Substitute into Snell’s law

1sinsin >= it

it n

n θθ

ok if θt complex

y

MIT 2.71/2.710 Optics10/19/05 wk7-b-39



Propagating component[ ]ttt yikE θcosexp∝

wherey

1sincos 2

22

−±=t

iit n

ni θθ

so

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−∝ 1sinexp 2

22

t

iitt n

nykE θ

MIT 2.71/2.710 Optics10/19/05 wk7-b-40



y⎥⎥⎦

⎤

⎢⎢⎣

⎡−−∝ 1sinexp 2

22

t

iitt n

nykE θ

Pure exponential decay≡≡ evanescentevanescent wave

It can be shown that:

0≈tS

MIT 2.71/2.710 Optics10/19/05 wk7-b-41

Phase delay upon reflection

Phase delay is 0

Phase delay is π

ni=1 (air)nt=1.5 (glass)

MIT 2.71/2.710 Optics10/19/05 wk7-b-42

Phase delay upon TIR

⊥== ⊥δδ ii rr e and e ||

||

y

E0i

E0r= rE0i

where

it

tiii

nnnn

θθδ

cossin

2tan 2

222|| −

−=

ii

tii

nnn

θθδ

cossin

2tan

222 −−=⊥

MIT 2.71/2.710 Optics10/19/05 wk7-b-43

Phase delay upon TIR

y

E0i

E0r= rE0i

δ||

δ⊥

MIT 2.71/2.710 Optics10/19/05 wk7-b-44

Interference

MIT 2.71/2.710 Optics10/19/05 wk7-b-45

Optical path delay

z

t = 0

z

t = T/4z = 2.875λ

MIT 2.71/2.710 Optics10/19/05 wk7-b-46

Optical path delay

z

t = 0

z

t = T/4

0E 022 E

022 E−

0

z = 2.875λ

MIT 2.71/2.710 Optics10/19/05 wk7-b-47

Optical path delay

z

t = T/2

z

t = 3T/4

0E− 022 E−

022 E

0

z = 2.875λ

MIT 2.71/2.710 Optics10/19/05 wk7-b-48

Optical path delay

z

t( )tE ω−cos0 ⎟

⎠⎞

⎜⎝⎛ +−

4cos0

πωtE

z = 2.875λ

t

MIT 2.71/2.710 Optics10/19/05 wk7-b-49

Optical path delay

z

t( )tE ω−cos0 ⎟

⎠⎞

⎜⎝⎛ +−

47cos0πωtE

z = 2.875λ

t

( ) ⎟⎠⎞

⎜⎝⎛⇒−

λπω ziEtkzE 2expcos 00In general,

phasor dueto propagation(path delay)

MIT 2.71/2.710 Optics10/19/05 wk7-b-50

Plane wave propagation

z=0

plane of observation

x path delay increaseslinearly with xx

θz

λ

MIT 2.71/2.710 Optics10/19/05 wk7-b-51


zz=0


x path delay increaseslinearly with xx

θ⎟⎠⎞

+⎜⎝⎛

θλ

π

θλ

π

cos2

sin2exp0

zi

xiEλ

MIT 2.71/2.710 Optics10/19/05 wk7-b-52


x

path delayφ

at fixed z

π2π3π

– π–2π– 3π

λ 2λ 3λ– 3λ – 2λ –λ

( ) θλ

πθλ

πϕϕ cos2sin2 whereexp0zxiE +=

λθπ sin2slope

MIT 2.71/2.710 Optics10/19/05 wk7-b-53

Spherical wave propagation

z=0


x path delay increaseswith x x as

z

λ ( )

zx

zxz

2

222

λπλπ

≈

−+

for x << x << zz

MIT 2.71/2.710 Optics10/19/05 wk7-b-54


z=0


x path delay increaseswith x x as

z

λ ( )

zx

zxz

2

222

λπλπ

≈

−+

quadraticquadraticnear the z axis

MIT 2.71/2.710 Optics10/19/05 wk7-b-55


z=0


x

z

λ

⎟⎠⎞

+⎜⎜⎝

⎛

λπ

λπ

zi

zxiE

2

exp2

0

MIT 2.71/2.710 Optics10/19/05 wk7-b-56


x

path delayφ

at fixed z

π2π3π

– π–2π– 3π

λ 2λ 3λ– 3λ – 2λ –λ

( )zyxziE

λπ

λπϕϕ

22

0 2 whereexp ++=

MIT 2.71/2.710 Optics10/19/05 wk7-b-57

Optical path delays matter

L

L

nn

path delay inmaterial ofindex n :

λπ nL2

compare withfree space

propagation :

λπ L2

MIT 2.71/2.710 Optics10/19/05 wk7-b-58

Optical path delays matter

L

L

nn path differencedifference:

( )λ

π Ln 12 −

MIT 2.71/2.710 Optics10/19/05 wk7-b-59

Phase delays matter

• Direct measurement does not work: light waves oscillate too fast for any instrument to follow

• We need an indirect method• Solution: interferometersinterferometers “map” phase onto light

intensity which can be measured directly

Can we measure them and how?

MIT 2.71/2.710 Optics10/19/05 wk7-b-60

Wave interference

λ1λ2

observation screen

MIT 2.71/2.710 Optics10/19/05 wk7-b-61

Interference: extreme cases

+ =Waves in-phase

Constructiveinterference

+ =Waves out-of-phase

Destructiveinterference

MIT 2.71/2.710 Optics10/19/05 wk7-b-62

Interference vs phase delay & contrast

φ

Intensity

averageintensity

fringevisibility

0 π 2π 3π

22

21

212aa

aam+

=

( ) ( )φφ cos10 mII +=

22

210 aaI +=

( )( )φωω

+==

tata cos 2 Field

cos 1 Field

2

1

0I

m akacontrast

relative phase delay

I

MIT 2.71/2.710 Optics10/19/05 wk7-b-63

Interference vs phase delay & contrast

Intensity

Intensity

φ

φ

a1=a2perfect contrast

a1<<a2no interference

Intensity

φ

a1≠a2imperfect contrast

m=1 0<m<1

m≈0Highest contrast / fringe visibility is obtained

by interfering beams ofequal amplitudes

MIT 2.71/2.710 Optics10/19/05 wk7-b-64

Polarization and interference

+φ

-polarized wavesinterfere

-polarized wavesdo not interfere

+φ

I

I

MIT 2.71/2.710 Optics10/19/05 wk7-b-65

Michelson interferometer

incominglaserbeam

l1

l2

( )2122 ll −=λπφ

path difference:

MIT 2.71/2.710 Optics10/19/05 wk7-b-66

Michelson with variable phase-delay

incominglaserbeam

l1

l1

Gas cell

( ) cell 122 tn −∆=λπφ

∆n

MIT 2.71/2.710 Optics10/19/05 wk7-b-67

Mach-Zehnder interferometer

incominglaserbeam

incominglaserbeam

interferencepattern

spatialperiod ~ θ

λsin

θ

brightfringe

(matchedpaths)

MIT 2.71/2.710 Optics10/19/05 wk7-b-68

Young interferometer

incominglaserbeam opaque

screen

x

d1

d2

l

a

MIT 2.71/2.710 Optics10/19/05 wk7-b-69

Young interferometer

incomingplane wave

opaquescreen

x´

d1

d2

l

a

intensity

alλ

=Λx=–a/2

x=a/2

MIT 2.71/2.710 Optics10/19/05 wk7-b-70

Two point sources interfering: math…

x´

intensity

alλ

=Λ

( ) ( )

( ) ( ).2cos12cos4

cos42exp2),(),(:Intensity

.cos42exp2

22exp

22exp42exp1

2/exp2/exp2exp1),(

:Amplitude

22

2

2

22

2

2

22

2

22

2

2222

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧ ′⎟

⎠⎞

⎜⎝⎛+=⎟

⎠⎞

⎜⎝⎛ ′

=

=⎟⎠⎞

⎜⎝⎛ ′

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧ ′++′+=′′=′′

⎟⎠⎞

⎜⎝⎛ ′

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧′++′

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧ ′

+⎭⎬⎫

⎩⎨⎧ ′−

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧′++′

+=

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧ ′++′

+⎭⎬⎫

⎩⎨⎧ ′+−′

⎭⎬⎫

⎩⎨⎧−=′′

xl

all

xal

lxa

l

yaxili

liyxeyxI

lxa

l

yaxili

li

lxai

lxai

l

yaxili

li

lyaxi

lyaxili

liyxe

λπ

λλπ

λ

λπ

λπ

λπ

λ

λπ

λπ

λπ

λ

λπ

λπ

λπ

λπ

λ

λπ

λπ

λπ

λ

Paraxial analysis:

Documents

Today’s summary - MITweb.mit.edu/2.710/Fall06/2.710-wk7-b-sl.pdf · • Reflection and refraction at a dielectric interface: – wave approach to derive Snell’s law – reflection