Modern Optics IV-coherence Special topics course in IAMS Lecture speaker: Wang-Yau Cheng 2006/4.

Modern Optics IV-coherence

Special topics course in IAMS

Lecture speaker: Wang-Yau Cheng

2006/4

Outline

• Wave properties of light

• Polarization of light

• Concept of linewidth

• Coherence of light

• Special issues on quantum optics

– Spatial coherence and time coherence– Interferometer and the applications– Anti-reflection coating– Correlation function and second order

correlation

Spatial and Temporal Coherence

Beams can be coherent or only partially coherent (indeed, even incoherent) in both space and time.

Spatial andTemporalCoherence:

TemporalCoherence;Spatial Incoherence

Spatial Coherence;TemporalIncoherence

Spatial andTemporalIncoherence

The Temporal Coherence Time and the Spatial Coherence LengthThe temporal coherence time is the time over which the beam wave-fronts remain equally spaced. Or, equivalently, over which the field remains sinusoidal with a given wavelength:

The spatial coherence length is the distance over which the beam wave-fronts remain flat:

Since there are two transverse dimensions, we can define a coherence area.

The spatial coherence depends on the emitter size and its distance away.

The van Cittert-Zernike Theorem states that the spatial coherence area Ac is given by:

where d is the diameter of the light source and D is the distance away.

Basically, wave-fronts smooth out as they propagate away from the source.

Starlight is spatially very coherent because stars are very far away.

2 2

2c

D

d

What is the coherence of light?

Classical idea

Coherent length

Coherent time

lc

Phase stability is the key role! ！

Linewidth should be narrow！

A nice tool for detecting the subtle structures of our material words

Frequency-stabilized lasers is the highest coherent light source that human being ever use

The coherence time is the reciprocal of the bandwidth.

The coherence time is given by:

where is the light bandwidth (the width of the spectrum).

Sunlight is temporally very incoherent because its bandwidth isvery large (the entire visible spectrum).

Lasers can have coherence times as long as about a second,which is amazing; that's >1014 cycles!

1/c v

Orthogonal polarizations don’t interfere.

The most general plane-wave electric field is:

where the amplitude is both complex and a vector:

The irradiance is:

0, Re exp ( )E r t E i k r t

0 0 0 0, ,x y zE E E E

* * * *0 0 0 0 0 0 0 02 2 x x y y z z

c cI E E E E E E E E

Different polarizations (say x and y):

Same polarizations (say x and x, so we'll omit the x-subscripts):

Therefore: Cross term!

Orthogonal polarizations don’t interfere (cont’d)

Because the irradiance is given by:

combining two waves of different polarizations is different from combiningwaves of the same polarization.

* * * *0 0 0 0 0 0 0 02 2 x x y y z z

c cI E E E E E E E E

* *0 0 0 0 1 22 x x y y

cI E E E E I I

* * *1 1 1 2 2 22Re

2

cI E E E E E E

1 2*

1 2Rec E EI I I

– Spatial coherence and time coherence– Interferometer and the applications– Anti-reflection coating– Correlation function and second order

correlation

Mach-Zehnder Interferometer

The Mach-Zehnder interferometer is usually operated “misaligned” and with something of interest in one arm.

Mach-Zehnder Interferogram

Nothing in either path Plasma in one path

The Sagnac InterferometerThe two beams automatically take the same path around theinterferometer. The paths can differ, however, if the deviceis rotating.

The Sagnac interferometer senses rotation.

Sagnac Interferometer Math

Suppose that the beam splitter moves by a distance, d, in thetime, T, it takes light to circumnavigate the Sagnac interferometer.

As a result, one beam will travel more, and the other less distance.

If R = the interferometer radius, and = its angular velocity:

Thus, the Sagnac Interferometer's sensitivity to rotation depends on its area. And it need not be round!

2

0 0

20

exp( ) exp( )

sin ( )

outI E ikd E ikd

I kd

2

20

(2 / ) 2 ( ) / 2 /

sin (2 / )out

d R T R R c R c c

I I k c

Area

Area

Newton's Rings

Newton's RingsGet constructive interference when an integral number of half wavelengths occur between the two surfaces (that is, when an integral number of full wavelengths occur between the path of the transmitted beam and the twice reflected beam).

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

Multiple-beam interference: The Fabry-Perot Interferometer or Etalon

A Fabry-Perot interferometer is a pair of parallel surfaces that reflect beams back and forth. An etalon is a type of Fabry-Perot etalon, and is a piece of glass with parallel sides.The transmitted wave is an infinite series of multiply reflected beams.

Transmitted wave:

Incident wave: E0

Reflected wave: E0r

= round-trip phase delay inside medium

2 2 2 2 2 2 2 2 30 0 0 0 0( ) ( ) ...i i i

tE t E t r e E t r e E t r e E

Transmitted wave: E0t

r, t = reflection, transmission coefficients from glass to air2

0t E2 2

0it r e E

2 2 20( )it r e E

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

The Etalon (cont'd)

The transmitted wave field is:

2 2 2 2 2 2 2 2 30 0 0 0 0

2 2 2 20

( ) ( ) ...

1 ( ) ( ) ...

i i it

i i

E t E t r e E t r e E t r e E

t E r e r e

2 20 0 / 1 i

tE t E r e

2

1

1 sin / 2T

F

2

2

2

1

rF

r

The transmittance is:2 22 4

02 2 2

0 1 (1 )(1 )t

i i i

E t tT

E r e r e r e

4 2 2 2 2

4 4 2 2 2 4 2 2

(1 ) (1 )

{1 2cos( )} {1 2 [1 2sin ( / 2)]} {1 2 4 sin ( / 2)]}

t r r

r r r r r r

where:

Dividing numerator and denominator by 2 2(1 )r

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, the widths of the high-transmission regions become very narrow.

Transmission maxima occur when:

2L/ = 2m

or:

/L m

The Etalon Free Spectral Range

FSR

FSR =Free Spectral

Range

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

2 2 2 22 2

[1 / ]FSR FSR

L L L L

22 [1 / ] 1 /2 1 12FSR FSR

FSR

LL

L L

Etalon Linewidth and Finesse

The Linewidth LW is a transmittance peak's full-width-half-max (FWHM).

Setting equal to LW/2 should yield T = 1/2:

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

The Finesse, F, is the ratio of the Free Spectral Range and the Linewidth:

Substituting we have:

The Finesse is the number of wavelengths the interferometer can resolve.

2

1

1 sin / 2T

F

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

2/ 4 1/ 4 /LW LWF F

2

24 /

F

F

F

2

2

2

1

rF

r

2/ [1 ]r F

= 2 corresponds to one FSR

taking 1r

Applications of Fabry-Perot interferometers and etalons

To frequency filter a beam (this is often done inside a laser).

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 thickness of the interferometer and watch for the transmission vs. thickness).

Money is now coated with interferometric inks to help foil counterfeiters.Notice the shade of the“20,” which is shown from two different angles.

– Spatial coherence and time coherence– Interferometer and the applications– Anti-reflection coating– Correlation function and second order

correlation

Anti-reflection Coating

Notice that the center of the round glass plate looks like it’s missing.It’s not! There’s an “anti-reflection coating” there (on both the front and back of the glass).

Anti-reflection Coating Math

Consider a beam incident on a piece of glass (n = ns) with a layer of material (n = nl) if thickness, h, on its surface.

It can be shown that the Reflectance is:2 2 2 2 2 2

0 02 2 2 2 2 2

0 0

2 20

2 20

( ) cos ( ) ( ) sin ( )

( ) cos ( ) ( ) sin ( )

At normal incidence, and if / 2 (i.e., / 4)

( )

( )

l s s l

l s s l

s l

s l

n n n kh n n n khR

n n n kh n n n kh

kh h

n n nR

n n n

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

Multilayer coatingsTypical laser mirrors and camera

lenses use many layers.

The reflectance and transmittance

can be tailored to taste!

– Spatial coherence and time coherence– Interferometer and the applications– Anti-reflection coating– Correlation function and second order

correlation