34. Even more Interference EffectsThe Fabry-Perot interferometer

Thin-film interference

Anti-reflection coatings

Single- and multi-layer

Advanced topic: Photonic crystals

Natural and artificial periodic structures

Interference and Interferometers

2

*2

1

1Rec E E

I I I

Different wavelengths

Different polarizations

Same wavelength

Same polarizations

1 2I I I

1 2I I I 1 2I I I

Interference only occurs when waves have the same wavelength and polarization.

beam-splitter

inputbeam

delay

mirror

mirror

output

Interferometers (like the Michelson) produce fringes but only if L is less than the coherence length: the interferogram

Incident wave: E0

Reflected wave: E0r

= round-trip phase between the two surfaces

Transmitted wave: E0t

20t E

2 20

jt r e E

2 2 20( ) jt r e E

2 2 30( ) jt r e En n = 1n = 1

A Fabry-Perot interferometer is a pair of parallelsurfaces that reflect beams back and forth.

An etalon is one particular type of Fabry-Perot: a thin piece of glass with parallel sides.

The transmitted and reflected waves are both an infinite series of multiply reflected beams.

Multiple-reflection interference: The Fabry-Perot Interferometer

Maurice Paul Auguste Charles

Fabry (1867-1945)

Jean-BaptisteAlfred Perot (1863-1925)

e.g., for normal incidence, = 2nkd

d

2 2 2 20 0 1 ( ) ( ) ...j j

tE t E r e r e

The field of the transmitted wave is an infinite sum:2 2 2 2

0 0 0 0 02 2 2 2 3( ) ( ) ...j

tj jr e r eE t E t E t t ErE e

Here we can use the identity:

2 3 11 ...1

x x xx

The Fabry-Perot Interferometer

2

0 021

t j

tE Er e

The Fabry-Perot interferometer

then the transmittance is:2 22 4

02 2 2

0 1 (1 )(1 )

tj j j

E t tTE r e r e r e

2 2

2 4 2 2

(1 )1 2 4 sin ( / 2)]

rr r r

Dividing numerator and denominator by 2 2(1 )r

2

0 021

t j

tE Er e

So, if the transmitted E-field is:

2

11 sin / 2

TF

2

2

21

rFr

where:

F is known as the ‘coefficient of finesse’.

Etalon Transmittance vs. Thickness, Wavelength, or Angle

The transmittance varies significantly with thickness or wavelength.We can also vary the incidence angle, which also affects .

As the reflectance of each surface (r2) approaches 1, F approaches infinity, and the widths of the transmission peaks become very narrow.

Transmission maxima occur when sin(/2) = 0, or when /2 = m

But = kL = 2L/(L = round trip length between the two reflecting surfaces, which depends on propagation angle)

So max = L / m

trans

mitt

ance

= 2L/

F = 0.2

F = 1

F = 20 F = 200

2

2

21

rFr

The Etalon Free Spectral Range

FSR =“Free Spectral

Range”

The Free Spectral Range is the wavelength range between transmission maxima.

2 2 2 FSR

L L

2

But usually L >> , so FSR L

2 FSR

2

FSR L

FSR

trans

mitt

ance

= 2L/

F = 200

Etalon Linewidth and Finesse 2

11 sin / 2

TF

2 21 1 sin / 2 / 2 2 or sin / 4 1/LW LWF FT

2/ 4 1/ 4 /LW LWF F

224 /

F

F

= 2 corresponds to one FSR

Which means: setting equal to LW/2 should yield T = 1/2.

For << 1, we can make the small argument approximation:

2

2

21

rFr

2/ [1 ] r rSubstituting we have:

The Finesse, , is the ratio of the phase corresponding to one Free Spectral Range to the phase corresponding to one linewidth:

The Linewidth LW is the value of at which a transmittance peak has dropped to half of its largest value: the full-width at half-max (FWHM).

trans

mitt

ance

= 2L/

Finesse = 100Building an etalon with a finesse of 100 requires mirror reflectivities greater than 98%.

A finesse of 100 is quite high, but can be achieved.

Etalon Linewidth and FinesseThe Finesse tells us the resolving power of the interferometer.

A high-finesse etalon placed inside of a multi-mode laser cavity can force the laser to operate on a single mode only.

Other uses of Fabry-Perot interferometers and etalons To frequency filter a beam (e.g., for multi-frequency telecommunications systems such as wavelength division multiplexing)

To measure the wavelength or spectrum of a beam (but you must know it in advance to within a Free Spectral Range, and you must scan the length of the interferometer and watch for the transmission vs. length).

Money is now coated with interferometric inks to help foil counterfeiters.Notice the shade of the“100,” which is shown from two different angles.

Thin film interference

trans

mitt

ance

= 2L/

Finesse = 100FSR

For very thin films (where Lapproaches ), the free spectral range becomes large.

If it is large enough, then only one visible color can be transmitted through the film at a given angle.

This effect causes the colors in bubbles and oil films on puddles.

2

FSR L

A single layer on a surfaceConsider a beam incident on a piece of glass (n = ns) with a layer of material (n = nc) of thickness, h, on its surface.

Notice that R = 0 if: 20c sn n n

2 20

2 20

( ) ( )

s c

s c

n n nRn n n

If h is chosen so that kh = /2 (i.e., h = / 4), then this becomes:

glass, index ns

coating, index nc & thickness h

air, index n0

2 2 2 2 2 20 0

2 2 2 2 2 20 0

( ) cos ( ) ( ) sin ( )( ) cos ( ) ( ) sin ( )

c s s c

c s s c

n n n kh n n n khRn n n kh n n n kh

It can be shown that the reflectance from the coated surface, for a beam impinging at normal incidence, is:

If the index of the coating is the geometric mean of n0 and ns, then the reflectance is zero! -an “anti-reflection coating”

A coating like this, which has a broadband anti-reflection, requires multiple layers.

Coatings with hundreds of layers are not unusual.

coatingno coating

Anti-reflection CoatingAn “anti-reflection coating”reduces the reflection from this piece of glass to nearly zero.

Deposition of multiple layersThere are numerous techniques for forming multi-layer structures like these. Common ones include chemical vapor deposition (CVD), physical vapor deposition (PVD), molecular beam epitaxy (MBE), and pulsed laser deposition (PLD).

These techniques can be used to deposit thin films of almost any material. For optical components, commonly used dielectrics include titanium dioxide, magnesium fluoride, zinc sulfide, and silicon dioxide.

A plasma CVD chamber

The opposite of Anti-reflectionMulti-layer coatings can also be engineered to provide highreflection, rather than anti-reflection. In this case, we have a mirror which contains no metal!

This is known as a Bragg mirror, after William H. Bragg and William L. Bragg, who discovered x-ray crystallography.

more layers = better reflectivity

Multilayer coatingsTypical laser mirrors and cameralenses use many coating layers.

The reflectance and transmittancecan be tailored to taste!

In the X-ray region of the spectrum, multilayer coatings are the only optical components that work.

transmission electron micrograph of a cross-section of a silicon-molybdenum multilayer mirror for x-ray optics

Multiple layers can be used to make dielectric mirrors that are much better than the best polished metal surface. But they usually work over a smaller range of wavelengths.

dielectric mirror reflectance:metal mirror reflectance:

1 inch mirror: $50 1 inch mirror: $175

Multilayer coatings - the best mirrors

Interference filtersAn interference filter is any multi-layer structure which has a narrow band pass, and which therefore can be used to filter the spectrum of a light wave.

An etalon is one example of an interference filter. But often multiple dielectric layers are used to narrow or tailor the filter function.

More layers are added to narrow the band pass.

This filter has a 10 nm band pass.

Many many layers: a 1D crystal!

dielectricconstant 1

dielectricconstant 2

Approximate the dielectric of this structure using a periodic dielectric function, of the form:

0

0

cos

2jGz jGz

Gz

e e

where G = 2/(period)

The field inside the medium may generally contain an incident and a reflected wave:

1 2

j k G zjkzE C e C e

Plug these into the one-dimensional wave equation:2 2

2 2 2d E d Edz c dt

distance z

1

2

exact dielectric profile

0

our approximation

Many many layers: a 1D crystal!

221 2

2

0 1 22 2 2

j k G zjkz

j k G zjkz jkz jkz

k C e k G C e

e e C e C ec

Wave equation:

Match up terms with the same exponents:

2 22

0 1 22 2

2 22

1 0 22 2

02

02

k C Cc c

C k G Cc c

Two equations in two unknowns: this only has a solution if the determinant vanishes, which gives a condition on k:

2 2 2 42 2

0 02 2 42 4G Gk G

c c c

Note: it is no longer true that k = n/c - a new dispersion relation for k()!

Dispersion of a 1D photonic crystal2 2 2 4

2 20 02 2 4

2 2

0 2

2 4

2 4

G Gk Gc c c

G Gc

2 42 2

0 2 4Gc c where

Note that is always positive. What if it is large?

2 2

0 2 0,4

Gc When

then the k vector has an imaginary component.

“Photonic band gap”

For these values of , k is complex.

0.8

0.9

1.0

1.1

1.2

0.40 0.50 0.60

Freq

uenc

y

Re k G

0

in units of 2cG

0kc

At frequencies in the gap, the propagating wave contains a term of the form exp[Im{k} z], which attenuates the wave.

One-dimensional photonic band gap

In this one-dimensional example, the device is equivalent to a Bragg mirror, with high reflection in a narrow band.

Photonic band gaps in 2D, 3D

What is required for a photonic band gap?

• A periodic variation of the refractive index, with a period close to the wavelength.• A large enough difference between the maximum and minimum values of the index (i.e., the value of )

This can be achieved in two or even three dimensional structures!

Eli Yablonovich1946 -

A cartoon of a 3D photonic crystal

the end face of a photonic crystal optical fiber

A 2D photonic crystal with an integrated waveguide

Natural photonic crystals

Butterfly wings are iridescent and brightly colored, not because of a pigment, but because of diffraction and interference!

Butterfly wings are natural photonic crystals.

increasingly close-up views of a butterfly wing

the blue morpho butterfly

Other photonic crystals

Scanning electron microscope (SEM) image of a natural opal

Opals are photonic crystals.

At certain angles, the reflected light at a particular wavelength experiences constructive interference, giving rise to colored reflections.

Artificial opal thin film

2 m

SEM image of an artificial opal

Natural opal