2. ChapterTwo wave nature light 0

(1)

Light waves in a homogeneous medium

Plane electromagnetic waves

Maxwell’s wave equation and diverging waves

Refractive index

Group velocity and group index

Snell’s law and total internal reflection (TIR)

Fresnel’s equations

Goos-Hänchen shift and optical tunneling

Diffraction

Fraunhofer diffraction

Diffraction grating

Chapter Two: Wave Nature of Light

(2)

Plane Electromagnetic Wave

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

E

_x

= E

₀

cos(

ω

t

−

kz +

φ

₀

)

propagation constant, or wave number angular frequency

phase

E

_x

(z, t) = Re[ E

₀

exp(j

φ

₀

)expj(

ω

t

−

kz)]

λ π / 2

= k

πν ω = 2

0

φ ω

(3)

Plane Electromagnetic Wave

(4)

Phase Velocity

During a time interval

δ

t, the wavefronts of contant phases move a

distance

δ

z. The phase velocity v is therefore

φ

=

ω

t

−

kz +

φ

₀

= constant

νλ

ω

δ

=

k

t

(5)

Plane Electromagnetic Wave

E(r, t) = E

₀

cos(

ω

t

−

k

⋅

r +

φ

₀

)

⋅

At a given time, the phase

difference between two points

separated by

Δ

r is simply k

⋅Δ

r. If

this phase difference is 0 or

(6)

Plane 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.

(7)

Maxwell

’

s EM Wave Equation

2 2 0 0 2 2 2 2 2 2

t

E

z

E

y

E

x

E

r

∂

=

∂

+

∂

+

∂

ε

μ

E

_x

= E

₀

cos(

ω

t

−

kz +

φ

₀

)

For a perfect plane wave:

0

2 2

=

∂

x

E

0

2 2

=

∂

y

E

( ) (

0

)

2 0

2 2

cos

ω

−

+

φ

−

=

∂

E

k

t

kz

z

E

Left side:

(

0

)

0 2 2

2

cos

ω

φ

ω

−

+

−

=

∂

E

t

kz

t

E

Right side:

2 0 0

2

ε

μ

ω

(8)

Diverging Waves

E = (A/r) cos(

ω

t

−

kr)

2 2 0 0

2 2 2

2 2

2

t

E

z

E

y

E

x

E

r

∂

=

∂

+

∂

+

∂

ε

μ

•

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.

(9)

Gaussian Beams

Gaussian beam has an exp[j(

ω

t

−

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

(10)

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 2w₀ is the spot size. The increase in beam diameter 2w with z makes an angle of 2

θ

,

which is called the beam divergence. Spot size and beam divergence are two

Gaussian Beams

(11)

Light waves in a homogeneous medium

Refractive index

Diffraction

Diffraction grating

(12)

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

ε

_r measures the ease with which the medium becomes polarized and indicates the extent of interaction between the field and the induced dipoles.

Refractive Index

0 0

1

μ

ε

=

c

In vacuum:

0 0

1

μ

ε

r

v

=

The phase velocity in a medium is:

r

c

n

=

ε

(13)

Refractive 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

k

_medium

= nk

(14)

Relative Permittivity and Refractive Index

Low frequency (LF) relative permittivity

ε

_r(LF) and refractive index n

Si 11.9 3.44 3.45 (at 2.15 μm) Electronic polarization up to optical 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 ε_r(LF) SiO₂ 3.84 2.00 1.46 (at 600 nm) Ionic polarization contributes to ε_r(LF) Water 80 8.9 1.33 (at 600 nm) Dipolar polarization contributes to Material ε_r(LF)

ε

_r(LF) n (optical) Comment

(15)

Maxwell’s wave equation and diverging waves

Refractive index

Diffraction

Diffraction grating

(16)

Group Velocity

( )

z

t

E

[

(

) (

t

k

)

z

]

E

_x_,₁

,

=

₀

cos

ω

−

δω

−

δ

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

−

δ

k and

k +

δ

k interfering with each other and traveling along the z direction:

( )

z

t

E

[

(

) (

t

k

)

z

]

E

_x_,₂

,

=

₀

cos

ω

+

δω

−

+

δ

(

)

⎥⎦

⎤

⎢⎣

⎡

(

+

)

⎥⎦

⎤

⎢⎣

⎡

−

=

+

B

A

B

A

B

A

2

1

cos

2

1

cos

2

cos

(17)

We obtain:

( ) ( )

[

t

k

z

]

(

t

kz

)

E

_x

=

_x_,₁

+

_x_,₂

=

2

₀

cos

δω

−

δ

cos

ω

−

Generate a wave packet containing an oscillating field at the mean

Group Velocity

(18)

The group velocity defines the speed with

which energy or information is propagated

since it defines the speed of the envelope of

( ) ( )

[

t

k

z

]

(

t

kz

)

E

_x

=

2

₀

cos

δω

−

δ

cos

ω

−

n

c

k

v

=

ω

=

Phase velocity

or

dk

d

v

_g

=

ω

The maximum amplitude

moves with a wavevector

δ

k and a frequency

δω

.

Group velocity

In vacuum:

ck

=

ω

c

dk

d

v

_g

=

ω

=

group velocity

Group Velocity

(19)

In a medium with n as a function of

λ

:

g

c

dn

c

d

v

=

ω

=

λ

dn

n

N

_g

=

−

λ

ω

π

λ

ω

λ

ω

d

dn

n

c

d

dn

d

dn

d

dn

−

=

−

=

2

n

c

n

c

ω

π

ν

λ

=

/

=

2

( )

ω

d

dn

n

d

n

d

cdk

+

=

n

ck /

=

ω

c

n

(20)

Group Index of the Medium

Pure SiO2

Both n and N_g are functions of the free space wavelength. Such a medium is called a dispersive medium. n and N_g of SiO₂ are important parameters for

For silica, N_g is minimum around 1300 nm, which means that light waves

(21)

Light waves in a homogeneous medium

Refractive index

Diffraction

Diffraction grating

(22)

Snell

’

s Law

Willebrord van Roijen Snell (1580-1626)

Incident light waves

(23)

r i

t

v

t

v

AB

θ

=

θ

=

sin

'

1 1

Reflection

Refraction

1 2

2 1

sin

'

n

v

t

v

t

v

AB

t i

=

θ

(24)

Total Internal Reflection

1

2

sin

n

t

i

=

θ

Refraction

When n

₁

> n

₂

and

θ

_t

= 90

°

, the incidence angle is

called the critical angle

θ

_c

:

1

2

sin

n

c

=

(25)

Maxwell’s wave equation and diverging waves

Refractive index

Diffraction

Diffraction grating

(26)

Fresnel

’

s Equations

(

)

(

)

(

k

r

)

r

k

r i

•

−

=

•

−

=

•

−

=

t

j

E

t

j

E

r r

i i

ω

exp

0 0

Transverse electric field (TE) waves: E_i,_⊥, E_r,_⊥, and E_t,_⊥. Transverse magnetic field (TM) waves: E_i,//, E_r,//, and E_t,//.

Augustin Fresnel (1788-1827)

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

(27)

⎪

⎫

=

⎭

⎬

⎫

=

⎭

⎬

⎫

=

⊥ ⊥

)

2

(

)

1

(

)

/

(

)

/

(

sin

tangential tangential

//

// 2 1

E

c

n

B

E

c

n

B

n

_i _t

r i

θ

Snell’s law

(28)

( )

(

)

( )

i i r

( )

r t t

(

)

(29)

(30)

(31)

[

]

[

]

[

]

⎪

⎨

⎧

−

+

=

+

=

−

+

−

=

+

−

=

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

cos

2

cos

sin

cos

sin

cos

i i i r i t i i i i i i r t i t r

E

n

E

n

E

n

E

θ

(32)

(33)

[

]

[

]

[

i

]

i

i i t i i i i i r

n

E

t

n

E

r

θ

cos

sin

cos

2

cos

sin

cos

sin

2 2 / 1 2 2 // , 0 // , 0 // 2 2 / 1 2 2 2 2 / 1 2 2 // , 0 // , 0 //

+

−

=

+

−

=

[

]

[

]

[

₂ ₂

]

1/2 , 0 , 0 2 / 1 2 2 2 / 1 2 2 , 0 , 0

sin

cos

2

sin

cos

sin

cos

i i i i t i i i i i r

n

E

t

n

E

r

θ

−

+

=

−

+

−

=

⊥ ⊥ ⊥ ⊥ ⊥ ⊥

(

)

(

)

(

k

r

)

r

k

r

k

t r i

•

−

=

•

−

=

•

−

=

t

j

E

t

j

E

t

j

E

t t r r i i

ω

exp

0 0 0 _r

⊥ and r// are reflection coefficients. t⊥

and t// are transmission coefficients.

Complex coefficients can only be obtained when n2 − _sin2θ

i < 0.

n

₁

> n

₂

,

θ

_i

>

θ

_c

Phase changes other than 0 or 180°

(

⊥

)

⊥

=

r

exp

−

j

r,

r

φ

(

,//

)

//

r

exp

j

r

=

−

φ

(

⊥

)

⊥

=

t

exp

−

j

t,

t

φ

(

,//

)

//

t

exp

j

t

(34)

n

=

θ

[

]

[

]

[

]

[

i

]

i

i i i i i i

n

r

n

r

θ

cos

sin

cos

sin

cos

sin

cos

2 2 / 1 2 2 2 2 / 1 2 2 // 2 / 1 2 2 2 / 1 2 2

+

−

=

−

+

−

=

⊥ 2 1 2 1 2 2 // 2 1 2 1

1

n

r

n

r

+

−

=

+

−

=

+

−

=

+

−

=

⊥

When

θ

_i

= 0 (normal incidence):

[

]

(

)

1 2 4 2 2 2 4 2 2 2 2 2 4 2 2 2 2 / 1 2 2

tan

1

cos

sin

cos

sin

cos

sin

n

p p p p p p p p p p

=

⇒

=

−

+

⇒

=

−

⇒

=

−

⇒

=

−

θ

p

is called the polarization angle or

Brewster’s angle.

(35)

Brewster

’

s Angle (Polarization Angle)

n

₁

= 1.44, n

₂

= 1.00

David Brewster (1781-1868)

°

=

°

=

98

.

43

78

.

34

c p

θ

When

θ

_i =

θ

_p, reflected light is linearly polarized and is perpendicular to the incidence plane.

When

θ

_p <

θ

_i <

θ

_c, there is a phase shift of 180° in r_//.

(36)

Example: calculate r

_⊥

and r

_//

when n

₁

= 1.44, n

₂

= 1.00, and

θ

_i

= 40

°

.

[

]

[

]

[

]

[

]

[

]

[

]

[

]

(37)

Example: calculate r

_⊥

and r

_//

when n

₁

= 1.00, n

₂

= 1.44, and

θ

_i

= 40

°

.

[

]

[

]

[

]

[

]

[

]

[

]

(38)

n

₁

= 1.00, n

₂

= 1.44

°

=

22

.

55

tan

1

2

p

n

θ

(39)

[

]

[

]

[

]

[

₂ ₂

]

1/2 2 2 / 1 2 2 2 // 2 / 1 2 2 2 / 1 2 2

sin

cos

sin

cos

sin

cos

sin

cos

n

j

n

j

n

r

n

j

n

j

r

i i i i i i

−

+

−

+

−

=

−

+

−

=

⊥

θ

( )

(

)

⎟

⎠

⎞

⎜

⎝

⎛

=

+

=

−

=

−

=

−

x

y

x

z

j

z

jy

x

z

1 2 / 1 2 2

tan

exp

φ

(

)

(

)

(

)

[

1 2

]

1 2 2 1 1 2 1

exp

φ

−

=

−

=

j

z

j

z

j

z

What are the phase changes at TIR (n

₁

> n

₂

and

θ

_i

>

θ

_c

)?

[

]

[

]

[

]

[

i

]

i

(40)

[

]

[

₂ ₂

]

1/2

(41)

[

]

[

₂ ₂

]

1/2 2 2 / 1 2 2 2 //

sin

cos

sin

cos

n

j

n

j

n

r

i i i i

−

+

−

+

−

=

θ

[

]

[

]

⎟

⎠

⎞

⎜

⎝

⎛

−

=

−

⎟

⎠

⎞

⎜

⎝

⎛

−

=

− − i i i i

n

θ

φ

π

θ

φ

cos

sin

tan

cos

sin

tan

2 2 / 1 2 2 1 2 2 2 / 1 2 2 1 1

[

]

[

]

[

i

]

(42)

Consider now:

(

k

r

)

t

⋅

−

=

E

j

t

E

_t _t₀

exp

ω

[

]

2 1/2

2 1 2 2 / 1 2 2 1 2

sin

1

2

sin

1

2

cos

sin

2

sin

2

sin

⎥

⎤

⎢

⎡

⎟⎟⎠

⎞

⎜⎜⎝

⎛

−

=

−

=

i t t t ty iz i i i t t t tz

n

k

n

k

θ

λ

π

θ

λ

π

θ

λ

π

θ

λ

π

θ

(

t

k

y

k

z

)

j

E

_t

−

_tz

(43)

Evanescent Wave

1

sin

2

2 1 2

2

1

=

⎟⎟⎠

⎞

⎜⎜⎝

⎛

>

⎟⎟⎠

⎞

⎜⎜⎝

⎛

⇒

>

c i c

i

n

θ

2 / 1

2 2

1

2

α

θ

π

n

−

=

⎥

⎤

⎢

⎡

−

⎞

⎛

−

=

At TIR:

(44)

Evanescent Wave

2 / 1

2 2

2 1 2

2

sin

1

2

⎥

⎦

⎤

⎢

⎣

⎡

−

⎟⎟⎠

⎞

⎜⎜⎝

⎛

=

i

n

θ

λ

π

α

the penetration depth

[

₂

]

1/2 2

2 2

1 2

sin

2

1

−

=

n

θ

_i

n

π

λ

α

δ

(

y

z

t

)

E

j

(

t

k

y

k

z

)

E

_t

,

=

_t₀

exp

ω

−

_ty

−

_tz

( )

k

y

j

(

t

k

z

)

j

E

_t

−

_ty

−

_iz

=

₀

exp

ω

(

y

)

j

(

t

k

z

)

E

_t

−

_iz

=

₀

exp

α

₂

exp

ω

( )

[

1

.

44

sin

50

1

.

00

]

176

nm

5

.

514

₂ ₂ ₂ 1/2

=

−

=

o −

π

δ

λ

= 514.5 nm (green light)

(45)

Total Internal Reflection Fluorescence (TIRF) Microscopy

(46)

PH

Y4320 Chapter Two

4

(47)

θi dp

48.7° 300 nm 49.7° 145 nm 51.1° 100 nm 53.9° 70 nm 57.8° 55 nm 66.5° 40 nm

[

]

/ 0

−

=

I

e

I

_z z dp

(48)

Evanescent nanometry

Achieves a resolution of ~ 1 nm.

Fluorescence intensity

(49)

Intensity, Reflectance, and Transmittance

2 0 2 0

2

1

nE

E

v

I

=

ε

_r

ε

_o

∝

Light intensity 2 2 , 0 2 , 0 ⊥ ⊥ ⊥

⊥

=

r

E

R

i r 2 // 2 // , 0 2 // , 0 //

r

E

R

i r

=

Reflectance 2 // 2 2 2 // , 0 2 //

t

n

E

n

T

t

⎟⎟⎠

⎞

⎜⎜⎝

⎛

=

2 2 2 2 , 0 2 ⊥ ⊥

⊥

=

⎜⎜⎝

⎛

⎟⎟⎠

⎞

t

(50)

[

]

[

₂ ₂

]

1/2

2 / 1 2 2 , 0 , 0

sin

cos

sin

cos

i i i i i r

n

E

r

θ

−

+

−

=

⊥ ⊥ ⊥

[

]

[

i

]

i

i i i r

n

E

r

θ

cos

sin

cos

sin

2 2 / 1 2 2 2 2 / 1 2 2 // , 0 // , 0 //

+

−

=

[

₂ ₂

]

1/2 , 0 , 0

sin

cos

2

i i i i t

n

E

t

θ

−

+

=

⊥ ⊥ ⊥

[

i

]

i

i i t

n

E

t

θ

cos

sin

cos

2

2 2 / 1 2 2 // , 0 // , 0 //

+

−

=

2 2 1 //

⎟⎟⎠

⎞

⎜⎜⎝

⎛

+

−

=

_⊥

n

R

When

θ

_i

= 0 (normal incidence):

(

)

2

2 1 //

4

n

T

+

=

_⊥ 2 1 2 1

1

n

+

−

=

+

−

=

2 1 2 1 2 2

n

+

−

=

+

−

=

2 1 1

2

1

2

n

=

+

(51)

Example: A ray of light traveling in a glass medium of refractive index n

₁

=

1.450 becomes incident on a less dense glass medium of refractive index n

₂

=

1.430. Suppose the free space wavelength (

λ

) of the light ray is 1

μ

m. (a)

What is the critical angle for TIR? (b) What is the phase change in the

reflected wave when

θ

_i

= 85

°

and 90

°

? (c) What is the penetration depth of the

evanescent wave into medium 2 when

θ

_i

= 85

°

and 90

°

?

°

=

⇒

=

/

1

.

430

/

1

.

450

80

.

47

sin

θ

_c

n

₂

n

₁

θ

_c

Solution:

[

]

₌ _⇒ ₌ _° ° ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ° = − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⊥

⊥ 1.614 116.45

85 cos 450 . 1 430 . 1 85 sin cos sin 2 1 tan 2 / 1 2 2 2 / 1 2 2 φ θ θ φ i i n

[

]

₌ _⇒ ₌ ₋ _° ° ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ° = − = ⎟ ⎠ ⎞ ⎜ ⎝

⎛ ₊ ₁_.₆₆₀ ₆₂_.₁₃

85 cos 450 . 1 430 . 1 450 . 1 430 . 1 85 sin cos sin 2 1 2 1

tan ₂ _//

2 / 1 2 2 2 2 / 1 2 2 // φ θ θ π φ i i n n

θ

i

= 90

°

φ

_⊥

= 180

°

(52)

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

.

0

5

.

1

0

.

1

5

.

1

0

.

1

4

04

.

0

5

.

1

0

.

1

5

.

1

0

.

1

5

.

1

0

.

1

2 2 2 1 2 1 2 2 2 1 2 1 2 1

n

T

n

R

n

Solution:

(

) (

)

⎪

⎩

⎪

⎨

⎧

=

+

×

=

+

=

⎟

⎠

⎞

⎜

⎝

⎛

+

−

=

⎟⎟⎠

⎞

⎜⎜⎝

⎛

+

−

=

⇒

⎩

⎨

⎧

=

96

.

0

.

1

5

.

1

0

.

1

5

.

1

4

04

.

0

.

1

5

.

1

0

.

1

5

.

1

0

.

1

5

.

1

2 2 2 1 2 1 2 2 2 1 2 1 2 1

n

T

n

R

n

(53)

Antireflection Coatings

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

(

)

(

)

(

1

.

0

3

.

5

)

0

.

31

5

.

3

0

.

1

2 2 2 2 1 2 2 1

=

+

−

=

+

−

=

n

R

Air Coating (Si3N4) Si (solar cell)

1.0 1.9 3.5

( )

⎟⎟⎠

⎞

⎜⎜⎝

⎛

=

⇒

=

Δ

2 2

4

2

n

m

d

m

d

n

d

k

_c c

λ

π

λ

π

φ

r

_12,_⊥

= r

_12,//

=

−

0.31

r

₁₂

= r

₂₃

in order to obtain efficient

destructive interference.

3 1 2 3 2 3 2 2 1 2

1 _n _n _n

(54)

Dielectric Mirrors

0

1 2

21

2 1

12

2 1

>

+

−

=

<

+

−

=

<

n

r

n

r

n

0

2

0

2

1 2

2 21

2 1

1 12

>

+

=

>

+

=

n

t

n

t

We can find that A, B, and C

waves are in phase with each

other. They interfere with each

other constructively.

(55)

Photonic Crystals

1-D

periodic in one direction

2-D

periodic in two directions

3-D

periodic in three directions

(56)

Defects in Photonic Crystals

(57)

Low

-

Pumping

-

Current Lasers

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju,

(58)

Low

-

Pumping

-

Current Lasers

radii of air holes:

0.28a (I)

0.35a (II)

0.385a (III)

electric field intensity profile

(59)

Low

-

Pumping

-

Current Lasers

These low threshold current, highly efficient lasers might function as

(60)

Wavelength Channel Drop Devices

(61)

Refractive index

Diffraction

Diffraction grating

(62)

Goos

-

H

ä

nchen

Shift

λ

= 1

μ

m

n

₁

= 1.45

n

₂

= 1.43

θ

i

= 85

°

δ

= 0.78

μ

m

Δ

z

≈

18

μ

m

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.

(63)

Optical 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

(64)

Beam Splitters Based on Frustrated TIR

(65)

Light waves in a homogeneous medium

Refractive index

Diffraction

Diffraction grating

(66)

Diffraction

Diffraction 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

(67)

Diffraction

Diffraction can be understood using the Huygens-Fresnel principle. Every

(68)

Diffraction Pattern through a 1D Slit

The one-dimensional (1D) slit has a width of a. Divide the width a into N (a large number) coherent sources

δ

y (= a/N). Because the slit is uniformly illuminated, the

amplitude of each source will be proportional to

δ

y. 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 kysin

θ

.

( ) (

δ

θ

)

δ

E

∝

y

exp

−

jky

sin

( )

=

C

∫

a

y

(

−

jky

)

(69)

Let x = y

−

a/2, we obtain

( )

=

C

∫

a

y

(

−

jky

)

E

0

δ

exp

sin

θ

(

)

[

]

∫

− − − −

=

+

−

=

2 / 2 / sin sin 2 1 2 / 2

/

exp

/

2

sin

a a jkx ka j a a

dx

e

Ce

dx

a

x

jk

C

θ θ

θ

⎟⎟⎠

⎞

⎜⎜⎝

⎛

−

=

− − −

∫

θ θ θ

θ

sin 2 sin 2 2 / 2 / sin

sin

1

jka jka

a a jkx

e

jk

dx

e

θ

1

sin

2

1

sin

2

1

sin

2

a

ka

(70)

( ) ( )

2 2

sin

2

1

sin

2

1

sin

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛

=

θ

ka

Ca

E

I

( )

θ

sin

2

1

sin

2

1

sin

sin 2 1

ka

a

Ce

E

ka j

⎟

⎠

⎞

⎜

⎝

⎛

=

−

Bright and dark regions

The central bright region is larger than the slit width (the transmitted beam is

diverging)

The zero intensity occurs when:

π

θ

λ

π

θ

a

m

ka

sin

=

sin

=

2

1

a

m

λ

θ

=

sin

L

,

2

,

1

±

=

m

(71)

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

D

λ

θ

1

.

22

sin

=

(72)

Diffraction

-

Limited Resolution

Consider two neighboring coherent sources with an angular separation of Δθ

examined 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.

(

θ

)

1

.

22

λ

sin

Δ

=

(73)

Diffraction

-

Limited Resolution

(74)

Diffraction

-

Limited Resolution

( )

α

sin

NA

=

n

⋅

(75)

Diffraction

-

Limited Resolution

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

NA

61

.

0

r

=

λ

(76)

Overcome Diffraction

-

Limited Resolution

(77)

(78)

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,

Overcome Diffraction

(79)

(80)

Diffraction Grating

A 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.

λ

Bragg diffraction condition

(maximum intensity):

m

λ

θ

=

±

(81)

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

sin

2

1

sin

2

1

sin

sin 2 1 slit

ka

a

Ce

E

ka j

⎟

⎠

⎞

⎜

⎝

⎛

=

−

α

sin

1 j

e

C

−

=

⎟

⎠

⎞

⎜

⎝

⎛

=

θ

α

sin

2

1

ka

Ca

C

For the diffraction from the grating:

( )

θ

( )

θ

( )

θ

( )

θ

( )

θ

( 1) sinθ

slit sin 2 slit sin slit slit grating kd N j kd j jkd

e

E

e

E

e

E

=

+

−

+

−

+

L

+

− −

( )

θ

[

+

− θ

+

− θ

+

− ( − ) θ

]

(82)

( )

θ θ

θ

sin sin slit grating

1

jkd jNkd

e

E

₋ −

−

=

( )

sin 2

sin 2 2 1 2 grating grating

1

sin

θ θ

α

θ

jkd jNkd

e

C

E

I

₋ −

−

⎟

⎠

⎞

⎜

⎝

⎛

=

(

)

(

)

(

θ

)

(

θ

)

θ

(

θ

)

θ θ

sin

cos

2

sin

cos

2

sin

cos

1

sin

cos

1

2 2 2 sin sin

kd

Nkd

kd

j

kd

Nkd

j

Nkd

e

jkd jNkd

−

=

+

−

+

−

=

−

− −

(

)

(

)

2 2 2

sin

2

1

sin

2

sin

2

1

sin

2

sin

cos

1

sin

cos

1

⎟⎟⎠

⎞

⎜⎜⎝

⎛

=

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

=

−

=

β

θ

N

kd

Nkd

kd

Nkd

⎟

⎠

⎞

⎜

⎝

⎛ =

β

sin

θ

2

1

kd

( )

2 2 2

1 grating

sin

⎟⎟⎠

⎞

⎜⎜⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

=

β

α

θ

C

N

(83)

⎟

⎠

⎞

⎜

⎝

⎛ =

β

sin

θ

2

1

kd

⎩⎨

⎧

=

0

sin

0

sin

β

N

β

When is maximum. 2 2

sin

N

=

⎟⎟⎠

⎞

⎜⎜⎝

⎛

β

d

m

λ

θ

π

β

=

sin

m

=

±

1

,

±

2

,

L

⎩⎨

⎧

≠

=

0

sin

0

sin

β

N

β

When

0

sin

2

=

⎟⎟⎠

⎞

⎜⎜⎝

⎛

β

N

d

N

k

m

λ

θ

⎟

⎠

⎞

⎜

⎝

⎛ +

=

sin

⎩⎨

⎧

−

=

1

±

,

2

,

±

,

1

(84)

( )

2 2 2 1

grating

sin

⎟⎟⎠

⎞

⎜⎜⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

=

β

α

θ

C

N

I

⎪

⎩

⎪

⎨

⎧

=

θ

β

θ

α

sin

2

1

sin

2

1

kd

ka

The amplitude of the diffracted

beam is modulated by the

diffraction amplitude of a single

slit since the latter is spread

(85)

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

(86)

Reading Materials