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Optics, IDC202 G R O U P 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 + φ )] E1 1 1 · − 1 = E exp [i(k r !t + φ )] E2 2 2 · − 2 If the phase difference φ 1 φ 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 + φ φ 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 + φ φ (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 ∆φ = k(d d ) 2 − 1 Constructive interference k(d d )= 2n⇡ 2 − 1 ± path difference is equal to the d d = nλ | 2 − 1| integer number of wave lengths. Young’s experiment screen d d = nλ P | 2 − 1| d1 k1 2 1/2 2 1/2 r 2 h 2 h S1 x + y + x + y = nλ 2 − − 2 " ✓ ◆ # " ✓ ◆ # h d2 x y, h x This tells you where the S2 k yh 2 = nλ 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 S Mirror S0 virtual source Alternative methods Fresnel’s double mirror experiment (1816) screen Mirror S S0 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[Γ (⌧)] h 1 2 i 12 Mutual coherence function Γ11(⌧)=??? Γ11(0) =??? Γ22(0) =??? Auto-correlation or self coherence functions Basic theory of coherence We define the normalised coherence function: Γ12(⌧) Γ12(⌧) in general a complex γ12(⌧)= = Γ11(0)Γ22(0) pI1I2 periodic function. complex degreep of partial coherence The intensity at a particular point in space now reads as I = I + I +2 I I Re[γ (⌧)] 0 γ12(⌧) 1 1 2 1 2 12 | | p γ (⌧) =0 (completely incoherent) | 12 | 0 < γ (⌧) < 1 (partially coherent) | 12 | γ (⌧) =1 (completely coherent) | 12 | Basic theory of coherence For an interference pattern (fix the source characterised by �12), the intensity varies between Imax and Imin. I = I + I +2 I I γ max 1 2 1 2| 12| p I = I + I 2 I I γ min 1 2 − 1 2| 12| p We define the fringe visibility or contrast: I I 2pI I γ = max − min = 1 2| 12| V Imax + Imin I1 + I2 If I1 = I2, is zero if the fields are incoherent, = γ 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? φ(t) a quasi monochromatic field i!t+iφ(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 + ⌧) γ(⌧)=h i self coherence function E 2 h| | i Coherence time and coherence length E(t)E⇤(t + ⌧) γ(⌧)=h i self coherence function E 2 i!t+iφ(t) h| | i E(t)=E0e− i!⌧ i[φ(t) φ(t+⌧)] γ(⌧)= e e − D T E i!⌧ 1 i[φ(t) φ(t+⌧)] = e lim e − dt T T !1 Z0 ⌧ < ⌧ Take 0 φ(t) φ(t + ⌧) | − | ⌧0 ⌧0 1 i[φ(t) φ(t+⌧)] ⌧ e − dt =??? 0 ⌧ 0 Z0 t ⌧ ⌧ Coherence time and coherence length ⌧0 1 i[φ(t) φ(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. ⌧ γ(⌧)= 1 ei!⌧ (⌧ < ⌧ ) − ⌧ 0 ✓ 0 ◆ (⌧ ⌧ ) =0 ≥ 0 Plot γ(⌧) vs ⌧. | | Coherence time and coherence length γ(⌧) | | 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. φ(t) a quasi monochromatic field i!t+iφ(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 <t< − 2 2 =0 otherwise 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, λ2 l = c ∆λ Ordinary light sources Δλ ∼ 5000 Å lc ∼ 2mm What if we take a LASER fields? Spatial coherence P2 S 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⇡ ∆φ = k 2d = 2d ⇥ λ ⇥.
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