Coherence and Source Requirements

•• CoherenceCoherence TimeTime •• CoherenceCoherence LengthLength •• PartialPartial CoherenceCoherence •• TemporalTemporal CoherenceCoherence •• SpatialSpatial CoherenceCoherence

James C. Wyant

Coherence

• Temporal Coherence – A source is never strictly monochromatic • Spatial Coherence – A source is never truly a point source

James C. Wyant

Interference and Interferometry - James C. Wyant Page 1 Coherence Time

Maximum transit time difference for good contrast fringes

E = A cos(kx − 2πνt) ∆ν = Frequency Bandwidth

For good contrast fringes 2πν ∆t − 2π()ν +∆ν ∆t<2π 1 ∆t<∆ν Coherence time ∆ = τ = 1 t c ∆ν

James C. Wyant

Coherence Length

Distance light travels during coherence time

Coherence length = τ = c = λ 2 lc c c ∆ν ∆λ

James C. Wyant

Interference and Interferometry - James C. Wyant Page 2 Partial Coherence

* I = ()E1 + E2 ()E1+E2 =2+2+*+* E1E2E1E2E1E2 =++ * I1I22Re E1E2 = I1 + I2 +2 I1I2 Re[]γ12(τ)

φ τ γ12(τ) = γ12(τ) ei 12 ( ) Let φ12(τ) = α12(τ) − ω τ Then

I = I1 + I2 + 2 I1I2 γ12(τ) cos()α12(τ) − ω τ

James C. Wyant

Degree of Coherence

γ12 (τ) is called degree of coherence

Effect of γ12(τ) is to reduce fringe contras

γ12(τ) = 1, complete coherence 0 < γ12(τ) < 1, partial coherence γ12(τ) = 0, complete incoherence

Effect of φ12(τ) is to shift the fringes

James C. Wyant

Interference and Interferometry - James C. Wyant Page 3 Visibility.nb Optics 505 - James C. Wyant 1

Fringe Visibility

!!!!!!!!!!!!! i = i1 + i2 + 2 i1 i2 Abs@g12@tDD Cos@a12@tD -wtD; imax - imin visibility = þþþþþþþþþþþþþþþþþþþþþþþþþþþþþ; imax + imin !!!!!!!!!!!!! i1 i2 visibility = 2 þþþþþþþþþþþþþþþþþþþ Abs@g12@tDD; i1 + i2 Let i1 = i2 then

visibility = Abs@g12@tDD;

Irradiance 1

0.8 g=1

0.6 g=0.5

0.4

g=0.0 0.2

2 4 6 8 10 12 Temporal Coherence

Source Distribution I(ν)

∞ ∫ I()νcos(2πντ)dν 0 Re[]γ ()τ = ∞ ∫I()ν dν 0

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Fringe Visibility as Function of Path Difference for Laser Having N Longitudinal Modes

x axis is path difference in units of the laser cavity length

James C. Wyant

Interference and Interferometry - James C. Wyant Page 4 VanCittertZernike.nb Optics 505 - James C. Wyant (1998) 1

5.4.2 Van Cittert ZernikeTheorem Broad Source - Narrow Spectrum

η y r4 ξ r v ξ ,η 2 y2 P 1 1 x r3 r1 y 1 u x 2 x 1 zo d !!!!!!!!!!!! i@pD = i1 + i2 + 2 i1 i2 Cos@k Hr1 - r2L + k Hr3 - r4LD "############################################################### H -h L2 H -x L2 2 2 2 y1 1 x1 1 r1 = Hy1 -h1L + Hx1 -x1L + zo ; r1 zo + þþþþþþþþþþþþþþþþþþþþþþþþ + þþþþþþþþþþþþþþþþþþþþþþþþ 2 zo 2 zo "############################################################### H -h L2 H -x L2 2 2 2 y2 1 x2 1 r2 = Hy2 -h1L + Hx2 -x1L + zo ; r2 zo + þþþþþþþþþþþþþþþþþþþþþþþþ + þþþþþþþþþþþþþþþþþþþþþþþþ 2 zo 2 zo

2 2 if h1 << zo l and x1 << zo l then for general source point x, h

2 2 2 2 y1 - 2 y1 h x1 - 2 x1 x y2 - 2 y2 h x2 - 2 x2 x r1 zo + þþþþþþþþþþþþþþþþþþþþþþþþþþþ + þþþþþþþþþþþþþþþþþþþþþþþþþþþ; r2 zo + þþþþþþþþþþþþþþþþþþþþþþþþþþþ + þþþþþþþþþþþþþþþþþþþþþþþþþþþ 2 zo 2 zo 2 zo 2 zo i@pD = i1 + i2 + 2 2 2 2 !!!!!!!!!!!! x1 - x2 Hx1 - x2L x y1 - y2 Hy1 - y2L h 2 i1 i2 CosAk J þþþþþþþþþþþþþþþþþþþþþ - þþþþþþþþþþþþþþþþþþþþþþþþþþ + þþþþþþþþþþþþþþþþþþþþþ - þþþþþþþþþþþþþþþþþþþþþþþþþþN + k Hr3 - r4LE 2 zo zo 2 zo zo Let

x 2 - x 2 Hx - x L x y 2 - y 2 Hy - y L h a = þþþþþþþþþþþþþþþþ1 2þþþþþ - þþþþþþþþþþþþþþþþ1 þþþþþþþþþþ2 + þþþþþþþþþþþþþþþþ1 2þþþþþ - þþþþþþþþþþþþþþþþ1 þþþþþþþþþþ2 2 zo zo 2 zo zo

Let i1 = i2 and have an extended source i1 = i@x, hD

ji ¾ i@x, hD Cos@ka+ k Hr3 - r4LD Çx Çh zy i@pD = J2 Å i@x, hD Çx ÇhN j1 + þþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþz k ¾ i@x, hD Çx Çh { i@pD = J2 Å i@x, hD Çx ÇhN H1 + Re@g12@tDDL

H + H - LL H < L @ D *@ +tD > ¾ i@x, hD EI ka k r3 r4 Çx Çh g @tD = þþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþe 1 t e2 þþþþþþþþþþþþþþþþt þþþþþþþ = þþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþ 12 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!* * < e1 e1 > ¾ i@x, hD Çx Çh

Let x h x = x2 - x1; y = y2 - y1; ax = þþþþþþþ; ay = þþþþþþþ; zo zo

H 2- 2+ 2- 2L Ha +a L Ik x1 x2 y1 y2 2 zo ¾ @a a D Ik x x y y Ça Ça Ik Hr -r L E i x, y E x y g12@tD = E 3 4 þþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþ ¾ i@ax, ayD Çax Çay VanCittertZernike.nb Optics 505 - James C. Wyant (1998) 2

I f @tD g12@tD = Abs@g12@tDD E 12 f12@tD =a12@tD -w t

Ik Hx 2-x 2+y 2-y 2L2 z Ik Ha x+a yL E 1 2 1 2 o ¾ i@ax, ayD E x y Çax Çay m12 =g12@0D = þþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþþ ¾ i@ax, ayD Çax Çay Spatial Coherence

I(ξ, η) is source distribution θx and θy are angular distance between points being interfered as measured from the plane of the source

Points θ being interfered Source

Fringe Visibility ik ξθ +ηθ ∫∫ I(ξ,η)e ( x y )dξdη V = γ12 = ∫∫ I(ξ,η)dξdη

James C. Wyant

Interference Fringes obtained Using Partially Coherent Light

V=0.593 V=0.361 V=0.146

Source size fixed, distance between points interfered varied.

From B.J. Thompson and E. Wolf, J. Opt. Soc. Am. 47, 895 (1957)

James C. Wyant

Interference and Interferometry - James C. Wyant Page 5 Michelson Stellar Interferometer

Fringe Plane

h d

Fringe spacing depends on d. Fringe visibility depends on h.

Fringe localization James C. Wyant University of Arizona, Optical Sciences Center, Tucson, Arizona 85721. Received 31 March 1978. 0003-6935/78/0615-1853$0.50/0. © 1978 Optical Society of America. The technique for locating the region of fringe localization for an interferometer used with a spatially incoherent source is well known. For each light ray going into an interferometer, two or more rays will emerge. The fringes are lo- calized in the region where these emerging rays derived from a single input ray intersect. The proof most often given for Fig. 1. Illustrating the van Cittert-Zernike theorem. the above result involves a large amount of geometry and al- gebra. 1,2 The purpose of this short Letter is to point out that the proof follows directly from the van Cittert-Zernike theorem. The van Cittert-Zernike theorem3 states that for a quasi- monochromatic spatially incoherent source, the magnitude viewing surface for an interferometer should be the surface of the degree of spatial coherence |µ | between two points that is the locus of points of intersection of the rays which s originate from one incident ray coming from the source. That P(x1,y1) and P(x2,y2) is given by the magnitude of the nor- malized Fourier transform of the intensity distribution of the is, for maximum fringe visibility, light propagating in the di- source. That is, rection of a single ray should interfere with itself. If this surface for ray intersection depends upon the particular ray coming from the source, there may be no region for which good visibility fringes exist. However, if the surface does not de- pend upon the ray selected, good visibility fringes will exist if the fringes are observed on a surface conjugate to this ray intersection surface. where I(x',y') is the intensity distribution of the source, q = x References (x2 - x1/R, and q y = (y2 - y 1)/R. x1, y1,, x2, y2, and R are illustrated in Fig. 1. In arriving at Eq. (1) it is assumed that 1. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, the separation R is much greater than both the extent of the 1964), p. 291. 2 2 1/2 source and [(x2 - x1) + (y2 - y1) ] . 2. M. Francon, Optical Interferometry (Academic, New York, 1966), It can be easily shown that the magnitude of the coherence p. 66. 4 function is a maximum when q X = q y = 0. If in an interfer- 3. R. J. Collier, L. Burckhardt, and L. Lin, Optical Holography ometer the light at P1 is interfered with the light at P2, the (Academic, New York, 1971), p. 140. fringe visibility is equal to the value of |µs| given by Eq. (1). 4. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics

Since for maximum fringe visibility q X = q y = 0, the fringe (Wiley, New York, 1978), p. 174.