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PHY4320 Chapter Two 1

Light waves in a homogeneous mediumPlane electromagnetic wavesMaxwell’s wave equation and diverging waves

Refractive indexGroup velocity and group indexSnell’s law and total internal reflection (TIR)Fresnel’s equationsGoos-Hänchen shift and optical tunnelingDiffraction

Fraunhofer diffractionDiffraction grating

Chapter Two: Wave Nature of LightChapter Two: Wave Nature of Light


Plane Electromagnetic WavePlane Electromagnetic Wave

An electromagnetic wave traveling along z has time-varying electric and magnetic fields that are perpendicular to each other.

Ex = E0 cos(ωt − kz + φ0)

propagation constant, or wave numberangular frequencyphase

Ex(z, t) = Re[ E0 exp(jφ0)expj(ωt − kz)]

The optical field generally refers to the electric field.

λπ /2=kπνω 2=

0φωφ +−= kzt



A surface over which the phase of a wave is constant is referred to as a wavefront.


Phase VelocityPhase Velocity

During a time interval δt, the wavefronts of contant phases move a distance δz. The phase velocity v is therefore

φ = ωt − kz + φ0 = constant

νλωδδ

===kt

zv



E(r, t) = E0 cos(ωt − k⋅r + φ0)k⋅r = kxx + kyy + kzz

k is called the wave vector.

At a given time, the phase difference between two points separated by Δr is simply k⋅Δr. If this phase difference is 0 or multiples of 2π, then the two points are in phase.


Plane WavesPlane Waves

The plane wave has no angular separation (no divergence) in its wavevectors.E0 does not depend on the distance from a reference point, and it is the same at all points on a given plane perpendicular to k.The plane wave is an idealization that is useful in analyzing many wave phenomena.Real light beams would have finite cross sections.The plane wave obeys Maxwell’s EM wave equation.

Ex = E0 cos(ωt − kz + φ0)


MaxwellMaxwell’’s EM Wave Equations EM Wave Equation

2

2

002

2

2

2

2

2

tE

zE

yE

xE

r ∂∂

=∂∂

+∂∂

+∂∂ μεε

Ex = E0 cos(ωt − kz + φ0)For a perfect plane wave:

02

2

=∂∂

xE

02

2

=∂∂

yE

( ) ( )02

02

2

cos φω +−−−=∂∂ kztkE

zE

( )002 cos φω +−−= kztEk

Left side:

( )002

2

2

cos φωω +−−=∂∂ kztE

tE

Right side:

200

2 ωμεε rk =

00

1μεε

λνr

v ==

From EM:


Diverging WavesDiverging Waves

E = (A/r) cos(ωt − kr)

2

2

002

2

2

2

2

2

tE

zE

yE

xE

r ∂∂

=∂∂

+∂∂

+∂∂ μεε

• A spherical wave is described by a traveling wave that emerges from a point EM source and whose amplitude decays with distance r from the source.

• Optical divergence refers to the angular separation of wavevectors on a given wavefront.

• Plane and spherical waves represent two extremes of wave propagation behavior from perfectly parallel to fully diverging wavevectors. They are produced by two extreme sizes of EM wave sources: an infinitely large source for the plane wave and a point source for the spherical wave. A real EM source would have a finite size and finite power.


Gaussian BeamsGaussian Beams

Gaussian beam has an exp[j(ωt − kz)] dependence to describe propagation characteristics but the amplitude varies spatially away from the beam axis. The beam diameter 2w at any point z is defined as the diameter at which the beam intensity has fallen to 1/e2 (13.5%) of its maximum.

Many laser beams can be described by Gaussian beams.


Diffraction causes light waves to spread transversely as they propagate, and it is therefore impossible to have a perfectly collimated beam. The spreading of a laser beam is in precise accord with the predictions of pure diffraction theory. Even if a Gaussian laser beam wavefront were made perfectly flat at some plane, it would quickly acquire curvature and begin spreading out. The finite width 2w0 where the wavefronts are parallel is called the waist of the beam. w0 is the waist radius and 2w0is the spot size. The increase in beam diameter 2w with z makes an angle of 2θ, which is called the beam divergence. Spot size and beam divergence are two important parameters when choosing lasers.

Gaussian BeamsGaussian Beams


When an EM wave is traveling in a dielectric medium, the oscillating electric field polarizes the molecules of the medium at the frequency of the wave. The field and the induced molecular dipoles become coupled. The net effect is that the polarization mechanism delays the propagation of the EM wave. The stronger the interaction between the field and the dipoles, the slower the propagation of the wave. The relative permittivity εr measures the ease with which the medium becomes polarized and indicates the extent of interaction between the field and the induced dipoles.

Refractive IndexRefractive Index

00

1με

=c

In vacuum:

00

1μεε r

v =

The phase velocity in a medium is:

rvcn ε==

The refractive index of the medium is:


Refractive IndexRefractive Index

The frequency ν remains the same in different mediums.

In non-crystalline materials (glasses and liquids), the material structure is the same in all directions and n does not depend on the direction. The refractive index is isotropic.Crystals, in general, have anisotropic properties. The refractive index seen by a propagating EM wave will depend on the direction of the electric field relative to crystal structures, which is further determined by both the propagation direction and the electric field oscillation direction (polarization).

kmedium = nk

λmedium = λ/n


Relative Permittivity and Refractive IndexRelative Permittivity and Refractive Index

Low frequency (LF) relative permittivity εr(LF) and refractive index n

Si 11.9 3.44 3.45 (at 2.15 μm) Electronic polarization up tooptical frequencies

Diamond 5.7 2.39 2.41 (at 590 nm) Electronic polarization up to UV light

GaAs 13.1 3.62 3.30 (at 5 μm) Ionic polarization contributes to εr(LF)

SiO2 3.84 2.00 1.46 (at 600 nm) Ionic polarization contributes to εr(LF)

Water 80 8.9 1.33 (at 600 nm) Dipolar polarization contributes toεr(LF), which is large

Material εr(LF) )(LFrε n (optical) Comment

The relative permittivity for many materials can be vastly different at high and low frequencies because different polarization mechanisms operate at these frequencies. Al low frequencies, all polarization mechanisms (dipolar, ionic, electronic) contribute to εr, whereas at high (optical) frequencies only the electronic polarization can respond to the oscillating field.


Group VelocityGroup Velocity

( ) ( ) ( )[ ]zkktEtzEx δδωω −−−= cos, 01,

Since there are no perfect monochromatic waves in practice, we have to consider the way in which a group of waves differing slightly in wavelength travel along the same direction. For two harmonic waves of frequencies ω − δω and ω + δω and wavevectors k − δk and k + δk interfering with each other and traveling along the z direction:

( ) ( ) ( )[ ]zkktEtzEx δδωω +−+= cos, 02,

( ) ( )⎥⎦⎤

⎢⎣⎡ +⎥⎦

⎤⎢⎣⎡ −=+ BABABA

21cos

21cos2coscos

Using


We obtain:

( ) ( )[ ] ( )kztzktEEEE xxx −−=+= ωδδω coscos2 02,1,

Generate a wave packet containing an oscillating field at the mean frequency ω with the amplitude modulated by a slowly varying field of frequency δω.



The group velocity defines the speed with which energy or information is propagated since it defines the speed of the envelope of the amplitude variation.

( ) ( )[ ] ( )kztzktEEx −−= ωδδω coscos2 0

nc

kv ==

ωPhase velocityor

dkdvgω

=

The maximum amplitude moves with a wavevector δk and a frequency δω.

Group velocityIn vacuum:

ck=ω

cdkdvg ==ω

group velocity

phase velocity



In a medium with n as a function of λ:

Ng is the group index of the medium.g

g Nc

ddnn

cdkdv =

−==

λλ

ωλ

λddnnNg −=

λλ

ωπ

λω

ωλ

λω

ωω

ddn

nc

ddn

dd

ddn

ddn

−=−== 2

2nc

nc

ωπνλ 2/

==

( )ω

ωωω

ω ddnn

dnd

dcdk

+==

nck /=ω

cnk /ω=


Group Index of the MediumGroup Index of the Medium

Pure SiO2

Both n and Ng are functions of the free space wavelength. Such a medium is called a dispersive medium. n and Ng of SiO2 are important parameters for optical fiber design in optical communications.

For silica, Ng is minimum around 1300 nm, which means that light waves with wavelengths close to 1300 nm travel with the same group velocity and do not experience dispersion.


SnellSnell’’s Laws Law

Willebrord van Roijen Snell(1580-1626)

Incident light waves Ai and Bi are in phase. Reflected waves Ar and Br and refracted waves Atand Bt should also be in phase, respectively.


ri

ri

tvtvAB

θθθθ

=

==sinsin

' 11

Reflection

Refraction

1

2

2

1

21

sinsin

sinsin'

nn

vv

tvtvAB

t

i

ti

==

==

θθ

θθ


Total Internal ReflectionTotal Internal Reflection

1

2

sinsin

nn

t

i =θθ

RefractionWhen n1 > n2 and θt = 90°, the incidence angle is called the critical angle θc:

1

2sinnn

c =θ


FresnelFresnel’’s Equationss Equations

( )( )( )rk

rkrk

t

r

i

•−=•−=•−=

tjEEtjEEtjEE

tt

rr

ii

ωωω

expexpexp

0

0

0

Transverse electric field (TE) waves: Ei,⊥, Er,⊥, and Et,⊥.Transverse magnetic field (TM) waves: Ei,//, Er,//, and Et,//.

Augustin Fresnel(1788-1827)

The incidence plane is the plane containing the incident and reflected light rays.

Our goal is to find Er0 and Et0 relative to Ei0, including any phase changes.


⎪⎭

⎪⎬⎫

=

=⎭⎬⎫

==

⎭⎬⎫

==

⊥

⊥

)2()1(

)2()1(

)/()/(

sinsin

tangentialtangential

tangentialtangential

//

//

21

BB

EE

EcnBEcnB

nn ti

ri

θθθθ Snell’s law

Maxwell’s equations

Boundary conditions


( ) ( ) ( )

( ) ( ) ( ) ttrrii

ttrrii

ttrrii

tritri

tri

EcnEcnEcnBBB

EEEEcnEcnEcnBBB

EEE

θθθθθθ

θθθ

cos/cos/cos/coscoscos

coscoscos///

,2,1,1

//,//,//,

//,//,//,

//,2//,1//,1,,,

,,,

⊥⊥⊥

⊥⊥⊥

⊥⊥⊥

=−

⇒=−

−=−−

−=+−⇒−=+−

−=−−

Incidence plane


⎪⎪⎩

⎪⎪⎨

⎧

++

=

++−

=

⇒

⎩⎨⎧

=+

−=−⇒

⎩⎨⎧

=−

−=−−

⊥⊥

⊥⊥

⊥⊥⊥

⊥⊥⊥

⊥⊥⊥

⊥⊥⊥

,12

11,

,12

12,

,1,2,1

,,,

,2,1,1

,,,

coscoscoscoscoscoscoscos

coscoscos

coscoscos

irt

rit

irt

itr

iittrr

itr

ttrrii

tri

EnnnnE

EnnnnE

EnEnEnEEE

EnEnEnEEE

θθθθθθθθ

θθθ

θθθ

⎪⎪⎩

⎪⎪⎨

⎧

−−

=

−−

=⇒

⎩⎨⎧

=+=+

bcadceafy

bcadbfdex

fdycxebyax


⎪⎪⎩

⎪⎪⎨

⎧

−−−−

=

−−+−

=

⇒

⎩⎨⎧

−=−

=+⇒

⎩⎨⎧

−=−−

−=+−

//,21

11//,

//,21

21//,

//,//,//,

//,1//,2//,1

//,//,//,

//,2//,1//,1

coscoscoscoscoscoscoscos

coscoscos

coscoscos

irt

rit

irt

itr

iittrr

itr

ttrrii

tri

EnnnnE

EnnnnE

EEEEnEnEn

EEEEnEnEn

θθθθθθθθ

θθθ

θθθ

⎪⎪⎩

⎪⎪⎨

⎧

−−

=

−−

=⇒

⎩⎨⎧

=+=+

bcadceafy

bcadbfdex

fdycxebyax


[ ][ ]

[ ]⎪⎪

⎩

⎪⎪

⎨

⎧

−+=

++

=

−+

−−=

++−

=

⊥⊥⊥

⊥⊥⊥

,2/122,12

11,

,2/122

2/122

,12

12,

sincoscos2

coscoscoscos

sincossincos

coscoscoscos

i

ii

ii

rt

rit

i

ii

iii

rt

itr

En

EnnnnE

EnnE

nnnnE

θθθ

θθθθ

θθθθ

θθθθ

[ ] [ ] 2/1222/1222

21

12

sinsincos

sinsinsinsin

/

itt

ti

ti

ri

nnnn

nnn

nnn

θθθ

θθθθ

θθ

−=−=⇒

=⇒⎪⎩

⎪⎨

⎧

===


[ ] [ ] 2/1222/1222

21

12

sinsincos

sinsinsinsin

/

itt

ti

ti

ri

nnnn

nnn

nnn

θθθ

θθθθ

θθ

−=−=⇒

=⇒⎪⎩

⎪⎨

⎧

===

[ ][ ]

[ ]⎪⎪

⎩

⎪⎪

⎨

⎧

+−=

−−−−

=

+−

−−=

−−+−

=

//,22/122//,21

11//,

//,22/122

22/122

//,21

21//,

cossincos2

coscoscoscos

cossincossin

coscoscoscos

i

ii

ii

rt

rit

i

ii

iii

rt

itr

Enn

nEnnnnE

EnnnnE

nnnnE

θθθ

θθθθ

θθθθ

θθθθ


[ ][ ]

[ ] ii

i

i

t

ii

ii

i

r

nnn

EE

t

nnnn

EE

r

θθθ

θθθθ

cossincos2

cossincossin

22/122//,0

//,0//

22/122

22/122

//,0

//,0//

+−==

+−

−−==

[ ][ ]

[ ] 2/122,0

,0

2/122

2/122

,0

,0

sincoscos2

sincossincos

ii

i

i

t

ii

ii

i

r

nEE

t

nn

EE

r

θθθ

θθθθ

−+==

−+

−−==

⊥

⊥⊥

⊥

⊥⊥

( )( )( )rk

rkrk

t

r

i

•−=•−=•−=

tjEEtjEEtjEE

tt

rr

ii

ωωω

expexpexp

0

0

0 r⊥ and r// are reflection coefficients. t⊥and t// are transmission coefficients.

Complex coefficients can only be obtained when n2 − sin2θi < 0.

n1 > n2 ,θi > θcPhase changes other than 0 or 180°occur only when there is TIR.1//// =+ ntr

⊥⊥ =+ tr 1

( )⊥⊥⊥ −= ,exp rjrr φ

( )//,//// exp rjrr φ−=

( )⊥⊥⊥ −= ,exp tjtt φ

( )//,//// exp tjtt φ−=


2sinnn

c =θ

[ ][ ]

[ ][ ] ii

ii

ii

ii

nnnnr

nnr

θθθθ

θθθθ

cossincossin

sincossincos

22/122

22/122

//

2/122

2/122

+−

−−=

−+

−−=⊥

21

212

2

//

21

21

11

nnnn

nnnnr

nnnn

nnr

+−

=+−

=

+−

=+−

=⊥

When θi = 0 (normal incidence):

[ ]( )

1

2

422242

2

2

2

242222/122

tan

tantan1cossin

cos

cossincossin

nn

nnnnnnnn

p

ppp

p

p

pppp

=⇒

=−+⇒=−⇒

=−⇒=−

θ

θθθθ

θ

θθθθ

θp is called the polarization angle or Brewster’s angle.

When r// = 0 (reflected light is polarized):


BrewsterBrewster’’s Angle (Polarization Angle)s Angle (Polarization Angle)n1 = 1.44, n2 = 1.00

David Brewster(1781-1868)

°=

°=

98.43

78.34

c

p

θ

θ

When θi = θp, reflected light is linearly polarized and is perpendicular to the incidence plane.

When θp < θi < θc, there is a phase shift of 180° in r//.

|r⊥| and |r//| increases with θi when θi > θp.

There are phase shifts (other than 0 or 180°) in r⊥ and r// when θi > θc.


Example: calculate r⊥ and r// when n1 = 1.44, n2 = 1.00, and θi = 40°.

[ ][ ][ ][ ]

[ ][ ]

[ ][ ] 170.0

40cos694.040sin694.040cos694.040sin694.0

cossincossin

490.040sin694.040cos40sin694.040cos

sincossincos

694.044.100.1

22/122

22/122

22/122

22/122

//

2/122

2/122

2/122

2/122

1

2

−=°+°−

°−°−=

+−

−−=

=°−+°

°−−°=

−+

−−=

===

⊥

ii

ii

ii

ii

nnnnr

nnr

nnn

θθθθ

θθθθ

Solution:


Example: calculate r⊥ and r// when n1 = 1.00, n2 = 1.44, and θi = 40°.

[ ][ ][ ][ ]

[ ][ ]

[ ][ ] 104.0

40cos44.140sin44.140cos44.140sin44.1

cossincossin

255.040sin44.140cos40sin44.140cos

sincossincos

44.100.144.1

22/122

22/122

22/122

22/122

//

2/122

2/122

2/122

2/122

1

2

−=°+°−

°−°−=

+−

−−=

−=°−+°

°−−°=

−+

−−=

===

⊥

ii

ii

ii

ii

nnnnr

nnr

nnn

θθθθ

θθθθ

Solution:


n1 = 1.00, n2 = 1.44

°=

=

22.55

tan1

2

p

p nn

θ

θ


[ ][ ]

[ ][ ] 2/1222

2/1222

//

2/122

2/122

sincossincos

sincossincos

njnnjnr

njnjr

ii

ii

ii

ii

−+

−+−=

−+

−−=⊥

θθθθ

θθθθ

( )( )

⎟⎠⎞

⎜⎝⎛=

+=

−=

−=

−

xy

yxz

jzzjyxz

1

2/122

tan

exp

φ

φ

( )( )

( )[ ]212

1

22

11

2

1

exp

expexp

φφ

φφ

−−=

−−

==

jzz

jzjz

zzz

What are the phase changes at TIR (n1 > n2 and θi > θc)?

[ ][ ]

[ ][ ] ii

ii

ii

ii

nnnnr

nnr

θθθθ

θθθθ

cossincossin

sincossincos

22/122

22/122

//

2/122

2/122

+−

−−=

−+

−−=⊥

1

1

// =

=⊥

r

r


[ ][ ] 2/122

2/122

sincossincos

njnjr

ii

ii

−+

−−=⊥

θθθθ [ ]

[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

−

−

i

i

i

i

n

n

θθφ

θθφ

cossintan

cossintan

2/1221

2

2/1221

1

[ ]

[ ]i

i

i

i

n

n

θθφ

θθφφφ

cossin

21tan

cossintan2

2/122

2/1221

21

−=⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=−=

⊥

−⊥


[ ][ ] 2/1222

2/1222

//sincossincos

njnnjnr

ii

ii

−+

−+−=

θθθθ

[ ]

[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

−

−

i

i

i

i

nn

nn

θθφ

πθ

θφ

cossintan

cossintan

2

2/1221

2

2

2/1221

1

[ ]

[ ]

[ ]i

i

i

i

i

i

nn

nn

nn

θθπφ

θθπφ

πθ

θφφφ

cossin

21

21tan

cossintan2

cossintan2

2

2/122

//

2

2/1221

//

2

2/1221

21//

−=⎟

⎠⎞

⎜⎝⎛ +

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=+

−⎟⎟⎠

⎞⎜⎜⎝

⎛ −=−=

−

−


Consider now: ( )rkt ⋅−= tjEE tt ωexp0

[ ]2/1

22

2

12

2/122

12

sin12sin12cos

sinsin2sin2sin

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=−==

=====

itttty

iziiittttz

nnnnkk

kknnkk

θλπθ

λπθ

θθλπθ

λπθ

( )zkyktjE tzt −−= ty0 exp ω


Evanescent WaveEvanescent Wave

1sinsin 22

2

122

2

1 =⎟⎟⎠

⎞⎜⎜⎝

⎛>⎟⎟

⎠

⎞⎜⎜⎝

⎛⇒> cici n

nnn θθθθ

2

2/1

22

2

12 1sin2 αθ

λπ j

nnnjk ity −=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

At TIR:

Why do we put a minus here?


Evanescent Wave2/1

22

2

122 1sin2

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= in

nn θλπα

the penetration depth[ ] 2/122

221

2

sin2

1 −−== nn iθπ

λα

δ

( ) ( )zkyktjEtzyE tztt −−= ty0 exp,, ω

( ) ( )zktjykjE iztyt −−= ωexpexp0

( ) ( )zktjyE izt −−= ωα expexp 20

( )[ ] nm17600.150sin44.12

5.514 2/1222 =−=−o

πδ

λ = 514.5 nm (green light)n1 = 1.44 (glass)n2 = 1.00 (air)θi = 50° (>θc = 43.98°)


Total Internal Reflection Fluorescence (TIRF) MicroscopyTotal Internal Reflection Fluorescence (TIRF) Microscopy

Carl Zeiss

TIR illumination can greatly improve signal-to-noise ratios in applications requiring imaging of minute structures or single molecules in specimens having large numbers of fluorophoreslocated outside of the optical plane of interest, such as molecules in solution in Brownian motion, vesicles undergoing endocytosis or exocytosis, or single protein trafficking in cells.


θi dp48.7° 300 nm49.7° 145 nm51.1° 100 nm53.9° 70 nm57.8° 55 nm66.5° 40 nm

[ ] 2/122

221

/0

sin4

−

−

−=

=

nnd

eII

ip

dzz

p

θπλ


Evanescent nanometry

Achieves a resolution of ~ 1 nm.Requires highly photostable fluorescent materials.

Fluorescence intensity

Distance


Intensity, Reflectance, and TransmittanceIntensity, Reflectance, and Transmittance

20

202

1 nEEvI or ∝= εεLight intensity

22

,0

2,0

⊥

⊥

⊥⊥ == r

E

ER

i

r 2//2

//,0

2//,0

// rE

ER

i

r ==

Reflectance

2//

1

22

//,01

2//,02

// tnn

En

EnT

i

t⎟⎟⎠

⎞⎜⎜⎝

⎛==

2

1

22

,01

2,02

⊥

⊥

⊥⊥ ⎟⎟

⎠

⎞⎜⎜⎝

⎛== t

nn

En

EnT

i

t

Transmittance


[ ][ ] 2/122

2/122

,0

,0

sincossincos

ii

ii

i

r

nn

EE

rθθθθ

−+

−−==

⊥

⊥⊥

[ ][ ] ii

ii

i

r

nnnn

EE

rθθθθ

cossincossin

22/122

22/122

//,0

//,0//

+−

−−==

[ ] 2/122,0

,0

sincoscos2

ii

i

i

t

nEE

tθθ

θ−+

==⊥

⊥⊥

[ ] ii

i

i

t

nnn

EE

tθθ

θcossin

cos222/122

//,0

//,0//

+−==

2

21

21// ⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

=== ⊥ nnnnRRR

When θi = 0 (normal incidence):

( )221

21//

4nnnnTTT

+=== ⊥

21

21

11

nnnn

nn

+−

=+−

=

21

212

2

nnnn

nnnn

+−

=+−

=

21

121

2nn

nn +=

+=

21

12

22nn

nnnn

+=

+=


Example: A ray of light traveling in a glass medium of refractive index n1 = 1.450 becomes incident on a less dense glass medium of refractive index n2 = 1.430. Suppose the free space wavelength (λ) of the light ray is 1 μm. (a) What is the critical angle for TIR? (b) What is the phase change in the reflected wave when θi = 85° and 90°? (c) What is the penetration depth of the evanescent wave into medium 2 when θi = 85° and 90°?

°=⇒== 47.80450.1/430.1/sin 12 cc nn θθSolution:

[ ]°=⇒=

°⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−°

=−

=⎟⎠⎞

⎜⎝⎛

⊥⊥ 45.116614.185cos

450.1430.185sin

cossin

21tan

2/122

2/122

φθ

θφi

i n

[ ]°−=⇒=

°⎟⎠⎞

⎜⎝⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−°

=−

=⎟⎠⎞

⎜⎝⎛ + 13.62660.1

85cos450.1430.1

450.1430.185sin

cossin

21

21tan //2

2/122

2

2/122

// φθ

θπφi

i

nn

[ ] [ ] mmnn i μπμθ

πλδ 780.0430.185sin450.1

21sin

22/12222/12

222

1 =−°=−=−−

θi = 90°

φ⊥ = 180°

φ// = 0°

δ = 0.663μm


Example: Consider a light ray at normal incidence on a boundary between a glass medium of refractive index 1.5 and air of refractive index 1.0. (a) If light is traveling from air to glass, what are the reflectance and transmittance? (b) If light is traveling from glass to air, what are the reflectance and transmittance? (c) What is the Brewster’s angle when light is traveling from air to glass?

( ) ( )⎪⎪

⎩

⎪⎪

⎨

⎧

=+××

=+

=

=⎟⎠⎞

⎜⎝⎛

+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=⇒

⎩⎨⎧

==

96.05.10.1

5.10.144

04.05.10.15.10.1

5.10.1

2221

21

22

21

21

2

1

nnnnT

nnnnR

nn

Solution:

°=⇒== 31.560.15.1tan

1

2pp n

n θθ

( ) ( )⎪⎪

⎩

⎪⎪

⎨

⎧

=+××

=+

=

=⎟⎠⎞

⎜⎝⎛

+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=⇒

⎩⎨⎧

==

96.00.15.1

0.15.144

04.00.15.10.15.1

0.15.1

2221

21

22

21

21

2

1

nnnnT

nnnnR

nn

Light will lose 4% of its intensity whenever it travels through an interface between air and glass.


Antireflection CoatingsAntireflection CoatingsCurrent commercial solar cells are made of silicon (n ≈ 3.5 at wavelengths of 700–800 nm).

( )( )

( )( )

31.05.30.15.30.1

2

2

221

221 =

+−

=+−

=nnnnR Air Coating (Si3N4) Si (solar cell)

1.0 1.9 3.5

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒

===Δ

2

2

4

222

nmd

mdndkcc

λ

πλπφ

r12,⊥ = r12,// = −0.31r23,⊥ = r23,// = −0.30

r12 = r23 in order to obtain efficient destructive interference.

31232

32

21

21 nnnnnnn

nnnn

=⇒+−

=+−

nmnmn

d 929.14

7004 2

=×

==λ

87.15.30.1312 =×== nnn


Dielectric MirrorsDielectric Mirrors

0

0

12

1221

21

2112

21

>+−

=

<+−

=

<

nnnnr

nnnnr

nn

02

02

12

221

21

112

>+

=

>+

=

nnnt

nnnt

We can find that A, B, and C waves are in phase with each other. They interfere with each other constructively.The total reflectance will be close to unity.


Photonic CrystalsPhotonic Crystals1-D

periodic in one direction

2-D

periodic in two directions

3-D

periodic in three directions

Photonic crystals are ordered structures in which two media with different dielectric constants or refractive indices are arranged in a periodic form, with lattice constants comparable to the wavelength of light. The light propagation is completely forbidden within photonic crystals if they are properly designed (refractive index contrast, crystal structure, and lattice constant).


Defects in Photonic CrystalsDefects in Photonic Crystals

cavity waveguide


LowLow--PumpingPumping--Current LasersCurrent Lasers

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, J.-K. Yang, J.-H. Baek, S.-B. Kim, Y.-H. Lee, Science 2004, 305, 1444.

SEM image



radii of air holes:0.28a (I)0.35a (II)

0.385a (III)0.4a (IV)0.41a (V)

electric field intensity profileDesigned according to FDTD (finite difference time domain) calculations.



These low threshold current, highly efficient lasers might function as single photon sources, which are useful for cavity quantum electrodynamics and quantum information processing.


Wavelength Channel Drop DevicesWavelength Channel Drop Devices

B.-S. Song, S. Noda, T. Asano, Science 2003, 300, 1537.


GoosGoos--HHäänchennchen ShiftShift

λ = 1 μmn1 = 1.45n2 = 1.43θi = 85°δ = 0.78 μmΔz ≈ 18 μm

Careful experiments show that the reflected wave at TIR appears to be laterally shifted from the incidence point at the interface. This lateral shift is known as the Goos-Hänchen shift.The reflected wave appears to be reflected from a virtual plane inside the optically less dense medium. The spacing between the interface and the virtual plane is approximately the penetration depth of the evanescent wave δ when n1 is very close to n2.

iz θδ tan2=Δ


Optical TunnelingOptical Tunneling

At TIR, we shrink the thickness d of the medium B. When B is sufficiently thin, an attenuated beam emerges on the other side of B in C. This phenomenon, which is forbidden by simple geometrical optics, is called optical tunneling or frustrated total internal reflection. Optical tunneling is due to the evanescent wave, the field of which penetrates into medium B and reaches the interface BC.


Beam Splitters Based on Frustrated TIRBeam Splitters Based on Frustrated TIR

The extent of light intensity division between the two beams depends on the thickness of the thin layer B and its refractive index.


DiffractionDiffractionDiffraction occurs when light waves encounter small objects or apertures. For example, the beam passed through a circular aperture is divergent and exhibits an intensity pattern (called diffraction pattern) that has bright and dark rings (called Airy rings).

Two types of diffraction: (1) In Fraunhofer diffraction, the incident light is a plane wave. The observation or detection screen is far away from the aperture so that the received waves also look like plane waves. (2) In Fresnel diffraction, the incident and received light waves have significant wavefront curvatures. Fraunhofer diffraction is by far the most important.


DiffractionDiffractionDiffraction can be understood using the Huygens-Fresnel principle. Every unobstructed point of a wavefront, at a given instant in time, serves as a source of spherical secondary waves (with the same frequency as that of the primary wave). The amplitude of the optical field at any point beyond is the superposition of all these wavelets (considering their amplitudes and relative phases).


Diffraction Pattern through a 1D SlitDiffraction Pattern through a 1D SlitThe one-dimensional (1D) slit has a width of a. Divide the width a into N (a large number) coherent sources δy (= a/N). Because the slit is uniformly illuminated, the amplitude of each source will be proportional to δy. In the forward direction (θ = 0°), they will be in phase and constitute a forward wave along the z direction. They can also be in phase at some other angles and therefore give rise to a diffracted wave along such directions.

The intensity of the received wave at a point on the screen will be the sum of all waves arriving from all the sources in the slit. The wave Y is out of phase with respect to A by kysinθ.

( ) ( )θδδ sinexp jkyyE −∝

( ) ( )∫ −=a

jkyyCE0

sinexp θδθ


Let x = y − a/2, we obtain

( ) ( )∫ −=a

jkyyCE0

sinexp θδθ

( )[ ]

∫∫

−

−−−

=

+−=2/

2/

sinsin21

2/

2/sin2/exp

a

a

jkxkaj

a

a

dxeCe

dxaxjkC

θθ

θ

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

−

−

−∫θθθ

θsin

2sin

22/

2/

sin

sin1 ajkajka

a

jkx eejk

dxe

θ

θ

θ

θ

sin21

sin21sin

sin

sin21sin2

ka

kaa

jk

kaj ⎟⎠⎞

⎜⎝⎛

=−

⎟⎠⎞

⎜⎝⎛−

=


( ) ( )

2

2

sin21

sin21sin

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡⎟⎠⎞

⎜⎝⎛

==θ

θθθ

ka

kaCaEI

( )θ

θθ

θ

sin21

sin21sin

sin21

ka

kaaCeE

kaj⎟⎠⎞

⎜⎝⎛

=

−

Bright and dark regionsThe central bright region is larger than the slit width (the transmitted beam is diverging)

The zero intensity occurs when:

πθλπθ maka == sinsin

21

amλθ =sin

L,2,1 ±±=m

λ = 1.3 μma = 100 μmdivergence 2θ = 1.5°


The diffraction patterns from two-dimensional (2D) apertures such as rectangular and circular apertures are more complicated to calculate but they use the same principle based on the interference of waves emitted from all point sources in the aperture.The diffraction pattern from a circular aperture of diameter D has a central bright region. The divergence of this bright region is determined by

Dλθ 22.1sin =

Why is the central bright region rotated relative to the rectangular aperture?

Diffraction from rectangular apertures


DiffractionDiffraction--Limited ResolutionLimited ResolutionConsider two neighboring coherent sources with an angular separation of Δθexamined through an imaging system with an aperture of diameter D. The aperture produces a diffraction pattern of the sources. As the sources get closer, their diffraction patterns overlap more.

( )Dλθ 22.1sin min =Δ

According to Rayleigh criterion, the two spots are just resolvable when the principle maximum of one diffraction pattern coincides with the minimum of the other, which is given by the condition:


DiffractionDiffraction--Limited ResolutionLimited Resolution

Objectives are key components in an optical microscope.



( )αsinNA ⋅= nThe numerical aperture (NA) of a microscope objective is a measure of its ability to gather light and resolve fine specimen detail at a fixed object distance.



The resolution of an optical microscope is defined as the shortest distance between two points on a specimen that can still be distinguished by the observer or camera system as separate entities. It is the most important feature of the optical system and influences the ability to distinguish between fine details of a particular specimen.

NA61.0r λ

=resolution


Overcome DiffractionOvercome Diffraction--Limited ResolutionLimited ResolutionIf a sufficient number of photons are collected from a single fluorescent molecule, the center of this molecule can be determined up to 1.5-nm resolution by curve-fitting the fluorescence image to a Gaussian function.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, P. R. Selvin, Science 2003, 300, 2061.


Discernable 30-nm and 7-nm steps are readily observed upon moving the cover slip, either at a constant rate or a Poisson-distributed rate.

Overcome DiffractionOvercome Diffraction--Limited ResolutionLimited Resolution


Myosin (肌球蛋白，肌凝蛋白) V is a cargo-carrying motor that takes 37-nm center of mass steps along actin (肌动蛋白) filaments. It has two heads held together by a coiled-coil stalk. Hand-over-hand and inchworm models have been proposed for its movement along actin filaments. For a single fluorescent molecule attached to the light chain domain of myosin V, the inchworm model predicts a uniform step size of 37 nm, whereas the hand-over-hand model predicts alternating steps of 37 – 2x, 37 + 2x, where x is the in-plane distance of the dye from the midpoint of the myosin.



The dye is 7 nm from the center of mass along the direction of motion.This result indicates that myosin V moves in a hand-over-hand fashion along actin filaments.



Diffraction GratingDiffraction GratingA diffraction grating in its simplest form has a periodic series of slits. An incident beam of light is diffracted in certain well-defined directions that depend on the wavelength and the grating periodicities.

dm λθ =sin

Bragg diffraction condition(maximum intensity):

amλθ =sin L,2,1 ±±=m

Zero diffraction intensity condition for a single slit:

L,2,1 ±±=m


Let’s try to derive the actual diffraction intensity from the grating by assuming a plane incident wave and that there are N slits.

For the diffraction from a single slit:

( )θ

θθ

θ

sin21

sin21sin

sin21

slit

ka

kaaCeE

kaj⎟⎠⎞

⎜⎝⎛

=

−

ααα sin1

jeC −

= ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

=

θα sin21

1

ka

CaC

For the diffraction from the grating:

( ) ( ) ( ) ( ) ( ) ( ) θθθ θθθθθ sin1slit

sin2slit

sinslitslitgrating

kdNjkdjjkd eEeEeEEE −−−− ++++= L

( ) ( )[ ]θθθθ sin1sin2sinslit 1 kdNjkdjjkd eeeE −−−− ++++= L

A geometrical progression


( ) ( ) θ

θ

θθ sin

sin

slitgrating 11

jkd

jNkd

eeEE −

−

−−

=

( ) ( )2

sin

sin22

1

2

gratinggrating 11sin

θ

θ

ααθθ jkd

jNkd

eeCEI −

−

−−

⎟⎠⎞

⎜⎝⎛==

( ) ( )( ) ( )

( )( )θ

θθθ

θθθ

θ

sincos22sincos22

sinsinsincos1

sinsinsincos111

2

22

sin

sin

kdNkd

kdjkd

NkdjNkdee

jkd

jNkd

−−

=+−

+−=

−−

−

−

( )( )

2

2

2

sinsin

sin21sin2

sin21sin2

sincos1sincos1

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=−−

=ββ

θ

θ

θθ N

kd

Nkd

kdNkd

⎟⎠⎞

⎜⎝⎛ = θβ sin

21 kd

( )22

21grating sin

sinsin⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

ββ

ααθ NCI


⎟⎠⎞

⎜⎝⎛ = θβ sin

21 kd

⎩⎨⎧

==0sin

0sinββNWhen

is maximum.22

sinsin NN

=⎟⎟⎠

⎞⎜⎜⎝

⎛ββ

dm

mλθ

πβ

=

=

sinL,2,1 ±±=m

⎩⎨⎧

≠=0sin

0sinββNWhen

0sin

sin2

=⎟⎟⎠

⎞⎜⎜⎝

⎛ββN

dNkm λθ ⎟⎠⎞

⎜⎝⎛ +=sin

⎩⎨⎧

−=±±=

1,,2,1,2,1Nk

mL

L

N−1 zero points, N−2 maxima


( )22

21grating sin

sinsin⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

ββ

ααθ NCI

⎪⎩

⎪⎨

⎧

=

=

θβ

θα

sin21

sin21

kd

ka

The amplitude of the diffracted beam is modulated by the diffraction amplitude of a single slit since the latter is spread substantially than the former.


The diffraction grating provides a means of deflecting an incoming light by an amount that depends on its wavelength. This is of great use in spectroscopy.

In reality, any periodic variations in the refractive index would serve as a diffraction grating.

dmimλθθ =− sinsin

dmimλθθ =+ sinsinL,2,1 ±±=m


Reading MaterialsReading Materials

S. O. Kasap, “Optoelectronics and Photonics: Principles and Practices”, Prentice Hall, Upper Saddle River, NJ 07458, 2001, Chapter 1, “Wave Nature of Light”.

chapter two wave nature lightstaff.uny.ac.id/sites/default/files/2. chaptertwo wave... ·...

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