Optics, IDC202 G R O U P

Lecture 3. Rejish Nath Contents 1. Coherence and interference 2. Youngs double slit experiment 3. Alternative setups 4. Theory of coherence 5. Coherence time and length 6. A ﬁnite wave train 7. Spatial coherence

Literature: 1. Optics, (Eugene Hecht and A. R. Ganesan) 2. Optical Physics, (A. Lipson, S. G. Lipson and H. Lipson) 3. The Optics of Life, (Sönke Johnsen) 4. Modern Optics, (Grant R. Fowles) Course Contents

1. Nature of light (waves and particles) 2. Maxwells equations and wave equation 3. Poynting vector 4. Polarization of light 5. Law of reﬂection and snell's law 6. Total Internal Reﬂection and Evanescent waves 7. Concept of coherence and interference 8. Young's double slit experiment 9. Single slit, N-slit Diffraction 10. Grating, Birefringence, Retardation plates 11. Fermat's Principle 12. Optical instruments 13. Human Eye 14. Spontaneous and stimulated emission 15. Concept of Laser

3 Coherence and interference

Optical interference is based on the superposition principle. (This is because the Maxwell’s equations are linear differential equations.)

The electric ﬁeld at a point in vacuum:

E = E1 + E2 + E3 + E4 + ...

is the vector sum of that from the different sources. The same is true for magnetic ﬁelds. Remark: This may not be generally true, deviations from linear superposition is the study of nonlinear optical phenomena or simply non-linear optics. Coherence and interference Consider two harmonic, monochromatic, linearly polarised waves

= E exp [i(k r !t + )] E1 1 1 · 1 = E exp [i(k r !t + )] E2 2 2 · 2

If the phase difference 1 2 is constant, the two sources are said to be mutually coherent.

The irradiance or the intensity at a point is provided by

I = + 2 |E1 E2| = E 2 + E 2 +2E E cos ✓ | 1| | 2| 1 · 2 ✓ = k r k r + 1 · 2 · 1 2 interference term (all the fun is due to this!) Coherence and interference

I = E 2 + E 2 +2E E cos ✓ | 1| | 2| 1 · 2 interference term ✓ = k r k r + (all the fun is due to this!) 1 · 2 · 1 2

This indicates that I can greater than or less than I1+I2.

Since � depends on position, periodic spatial variations occur (fringes).

What happens if two waves are mutually incoherent?

What happens if the polarisations are mutually orthogonal? Young’s experiment performed by Thomas young in 1802. screen

d1 P In this setup, what is so S1 crucial to have the interference pattern? d2 S

S2 Young’s experiment screen

d1 P All you have to know is the phase difference S1 of waves arriving at P from the two point sources (spherical waves). S d2

S2 = k(d d ) 2 1

Constructive interference

k(d d )= 2n⇡ 2 1 ± path difference is equal to the d d = n | 2 1| integer number of wave lengths. Young’s experiment screen d d = n P | 2 1| d1

k1 2 1/2 2 1/2 r 2 h 2 h S1 x + y + x + y = n 2 2 " ✓ ◆ # " ✓ ◆ # h d2 x y, h x This tells you where the S2 k yh 2 = n bright fringes are on the x screen.

What is the central fringe, bright or dark?

Can you displace the fringes on the screen? Alternative methods Lloyd’s single mirror experiment (1834) screen

Mirror

S0 virtual source Alternative methods Fresnel’s double mirror experiment (1816) screen

Mirror S

Mirror

S00 Home work: Fresnel’s bi-prism experiment. Basic theory of coherence

The irradiance or the intensity at a point is provided by

I = + 2 |E1 E2| = E 2 + E 2 +2E E cos ✓ | 1| | 2| 1 · 2

This equation is based on the following assumptions about the two ﬁelds: In reality the phase and 1. completely coherent the amplitude vary with 2. monochromatic time in a random manner. 3. constant in amplitude Basic theory of coherence

In reality the phase and the amplitude vary with I = E E⇤ h · i time in a random manner. 2 2 = E + E + 2Re(E E⇤) h| 1| | 2| 1 · 2 i Assuming the average quantities Time average are stationary and for the same 1 T polarization for two ﬁelds: f =lim f(t)dt h i T T !1 Z0 I = I + I + 2Re E E⇤ 1 2 h 1 2 i

path 1 S P path 2 the ﬁelds differ due to difference in their optical paths. Basic theory of coherence

path 1 S P path 2 The time difference between two paths: ⌧ The interference term can be written as

2Re E (t)E⇤(t + ⌧) = 2Re[ (⌧)] h 1 2 i 12 Mutual coherence function

11(⌧)=??? 11(0) =??? 22(0) =??? Auto-correlation or self coherence functions Basic theory of coherence

We deﬁne the normalised coherence function:

12(⌧) 12(⌧) in general a complex 12(⌧)= = 11(0)22(0) pI1I2 periodic function. complex degreep of partial coherence The intensity at a particular point in space now reads as

For an interference pattern (ﬁx the source characterised by �12), the intensity varies between Imax and Imin.

I = I + I +2 I I max 1 2 1 2| 12| p I = I + I 2 I I min 1 2 1 2| 12| p We deﬁne the fringe visibility or contrast:

I I 2pI I = max min = 1 2| 12| V Imax + Imin I1 + I2

If I1 = I2, is zero if the ﬁelds are incoherent, = V | 12| and maximum if they are coherent. Coherence time and coherence length How the degree of partial coherence is related to the source characteristics? (t) a quasi monochromatic ﬁeld i!t+i(t) ⌧0 ⌧0 E(t)=E0e

⌧0 ⌧0

t The phase of the field is making some random jumps. Suppose I divide the field into two equal fields and now the interference pattern will be governed by

E(t)E⇤(t + ⌧) (⌧)=h i self coherence function E 2 h| | i Coherence time and coherence length

E(t)E⇤(t + ⌧) (⌧)=h i self coherence function E 2 i!t+i(t) h| | i E(t)=E0e

i!⌧ i[(t) (t+⌧)] (⌧)= e e

D T E i!⌧ 1 i[(t) (t+⌧)] = e lim e dt T T !1 Z0 ⌧ < ⌧ Take 0 (t) (t + ⌧) | | ⌧0 ⌧0 1 i[(t) (t+⌧)] ⌧ e dt =??? 0 ⌧ 0 Z0 t ⌧ ⌧ Coherence time and coherence length

⌧0 1 i[(t) (t+⌧)] ⌧0 ⌧ ⌧ i e dt = + e ⌧ ⌧ ⌧ 0 Z0 0 0

First term is same for all intervals. The second term averages to zero over long time.

⌧ (⌧)= 1 ei!⌧ (⌧ < ⌧ ) ⌧ 0 ✓ 0 ◆ (⌧ ⌧ ) =0 0

Plot (⌧) vs ⌧. | | Coherence time and coherence length (⌧) | | 1 ⌧0 is the coherence time.

⌧0 ⌧ provides the fringe visibility (or the coherence of a quasi-monochromatic source) the path difference between two beams must not exceed

lc = c⌧0

to observe the interference fringes. This length is called the coherence length or the length of an uninterrupted wave. Coherence time and coherence length

the path difference between two beams must not exceed

lc = c⌧0

to observe the interference fringes. This length is called the coherence length or the length of an uninterrupted wave.

(t) a quasi monochromatic ﬁeld i!t+i(t) ⌧0 ⌧0 E(t)=E0e

⌧0 ⌧0

t In reality there is a statistical distribution for coherence lengths (or time). A ﬁnite wave train

i! t ⌧0 ⌧0 f(t)=e 0 for

Obtain its Fourier transform:

1 1 i!t f(t)= g(!)e d! p2⇡ Z1

1 1 g(!)= f(t)ei!tdt p2⇡ Z1

2 sin[(! !0)⌧0/2] G(!)= g(!) 2 g(!)= | | ⇡ ! !0 Power spectrum r A ﬁnite wave train f(t)

t ⌧ 0 coherence time

2 sin[(! ! )⌧ /2] g(!)= 0 0 ⇡ ! ! r 0

!0 ! 2⇡ ! = 2⇡ 2⇡ ⌧ ! !0 + 0 0 ⌧ ⌧0 0 2! A ﬁnite wave train 2⇡ 1 ! = ⌫ = ⌧0 ⌧0 In other words “the frequency width or line width” gives you the corresponding coherence time and also the coherence length.

Exercise: Show that the coherence length can be expressed as,

2 l = c

Ordinary light sources Δλ ∼ 5000 Å

lc ∼ 2mm What if we take a LASER ﬁelds? Spatial coherence

P 1 P3

The coherence between the ﬁelds at P1 and P3 measures the longitudinal spatial coherence of the ﬁeld.

The coherence between the fields at P1 and P2 measures the transversal spatial coherence of the field. Interference Methods: Classifications Interference by 1. division of wave front. A single point like source emitting waves in different directions. Theses waves are then brought together by means of mirrors, prisms and lenses.

2. division of amplitude.

A single beam (or a wave) of light is divided into two or more beams by partial reﬂection. e.g. Michelson interferometer Michelson Interferometer (1880)

mirror D

compensating plate

S A B C partially reﬂecting mirror mirror path difference: 2d= 2(CA-DA) E Fringes phase difference: 2⇡ = k 2d = 2d ⇥ ⇥