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Coherence ( )

Temporal Coherence Michelson Interferometer Spatial Coherence Young s Interference Spatiotemporal Coherence

參料 [1] Eugene Hecht, Optics, Addison Wesley Co., New York 2001 [2] W. Lauterborn, T. Kurz, M. Wiesenfeldt, Coherent Optics, Fundamentals and Applications. Springer Verlag, Berlin 2003 [3] http://www.ltyz.gx.cn/wuli/lzwl/kejian/kejian.htm

Wave Optics

說復行行復數率行復離說度率狀

x' = x vt One-Dimensional Waves ψ ( xt, ) ψ ( xt, ) = f( xt, ) t=0 ψ ( x, t) = f( x,0) = = f( x) t 0 行 t=t ψ = f( x') ψ ( x, t) = f( x vt) 行 t = t+ t ψ ( xt, ) = f( x vt) = f [( x+ v t) v( t+ t)] = f( x vt) 不

Harmonic Waves k : 數 (Wave Number) λ : (Wave Length) ω : 率 (Angular Frequency) ν : 率 (Frequency) τ : (Period) ε : (Initial Phase) ψ = Asin( kx ωt+ ε) 2π k = λ 2π = = ω τ 2πν

F = qe = ma E E F = qvb= ma B B E = cb F E E c = = = a E F vb v a B B c v a a E B

Coherence 1. 兩 2.

兩兩若兩兩不兩 Coherent

數 ψ ω ε (,) zt = Asin( kz t+ ) ψ (,) z t E(,) z t ( a a ) E B

Harmonic Waves: Euler Formula: 數 Ezt (,) = Asin( kz ωt) with Linear Waves:

Intensity of a Light Wave

Intensity of a Light Wave Energy Density: (Energy in Volume) Plane Wave:

Intensity Intensity: u = ε E 0 2 with E cos R = E0 kz ER Real

Intensity T : Measuring Time m

Intensity 量 Harmonic wave has constant intensity Intensity is defined as:

Temporal Coherence

Temporal Coherence Temporal Coherence ( 數 ) 兩兩論兩數異

Temporal Coherence Fig. The Michelson interferometer, Coherent length 2d is used for characterizing the interference properties of light.

Coherent length=2d Temporal Coherence E2() t = E1( t+ τ ) E1() t = E2( t τ ) Superposition: Intensity: Complex Self Coherence Function: Γ( τ )

數 E = x + iy 1 1 1 E = x + iy 2 2 2 E = x iy * 1 1 1 E = x iy * 2 2 2 EE = ( x + iy )( x iy ) = xx ixy + ix y + y y * 1 2 1 1 2 2 1 2 1 2 2 1 1 2 EE = ( x + iy )( x iy ) = xx + ixy ix y + y y * 2 1 2 2 1 1 1 2 1 2 2 1 1 2 EE + EE = 2xx + 2yy = 2Re EE * * * 1 2 2 1 1 2 1 2 1 2

Complex Self Coherence Function * Complex Self Coherence Function: τ Γ ( ) = E1E2 E () t = E ( t+ τ ) 2 1 Intensity: { } I = EE = 2I + 2Re E E * * 1 1 2 來度

Example: Temporal Coherence Harmonic wave: The self coherence function harmonically depends on the time delay τ.

Complex Self Coherence Function Γ ( τ ) = I exp( iωτ ) 1

Normalized complex self coherence function Normalized complex self coherence function Γ ( τ ) = I exp( iωτ ) 1 γ ( τ ) γτ ( ) Γ( τ ) = Γ (0) I( τ ) = 2I + 2Re Γ( τ ) 1 { }

Contrast of Fringes I( τ ) = 2I + 2Re Γ( τ ) Intensity: { } 1 Maximum Minimum = I I max min K I max + I min

Contrast of Fringes The Contrast of Fringes: K depends on the time shift between the light waves: [ τ, τ ] 1 2 τ > τ 2 1

Harmonic Waves: Intensity of light Complex self coherence function Intensity of interference with harmonic waves Coherent length Temporal Coherence Normalized complex self coherence function Ezt (, ) = Eexp[ ikz ( ωt)] 0 I * = EE Γ ( τ ) = I exp( iωτ ) 1 I( τ ) = 2 I (1+ cos ωτ ) l c = 1 cτ c Γ( τ ) γτ ( ) = Γ (0)

Quasimonochromatic Light ω 1 ω

Harmonic Waves: Harmonic Waves

Completely Incoherent Light γ ( τ) = 0, τ 0 γ (0) = 1

Mercury-Vapor Lamp

Two-Mode Laser

Two-Mode Laser Consider two harmonic waves with the same amplitude: Self coherence function: Γ ( τ ) = E E * 1 2

Two-Mode Laser [ ] Γ ( τ ) = E exp( iωτ) + exp( iωτ) 2 0 1 2 Normalized complex self coherence function γτ ( ) Γ( τ ) = 2 Γ (0) = I Γ (0) 1 = 2 E0

Two-Mode Laser The Contrast of Fringes:

Complex Light Field Sum of many harmonic waves with different frequency [ ] Γ ( τ ) = E exp( iωτ) + exp( iωτ) 2 0 1 2 ω = 2πν dω = 2πdν

Complex Light Field ω = 2πν dω = 2πdν Self coherence function: Power spectrum of the complex light field: W ( ν ) = E ( ν ) 0 2

Spatial Coherence

Spatial Coherence Spatial Coherence ( 數 ) 兩兩論兩數異

Young s Interference Spherical Waves: E ( s, t) = A( s )exp[ i( ks ωt+ ϕ )] 1 1 1 1 1 E ( s, t) = A( s )exp[ i( ks ωt+ ϕ )] 2 2 2 2 2

Young s Interference

Spatial Coherence Spherical Waves: E ( s, t) = A( s )exp[ i( ks ωt+ ϕ )] 1 1 1 1 1 E ( s, t) = A( s )exp[ i( ks ωt+ ϕ )] 2 2 2 2 2 Superposition: Intensity:

數 E E E E = A( s ) e 1 1 = A( s ) e 2 2 = A( s ) e * 1 1 = A( s ) e * 2 2 i( ks ωt+ ϕ ) 1 1 i( ks ωt+ ϕ ) 2 2 i( ks ωt+ ϕ ) 1 1 i( ks ωt+ ϕ ) 2 2 EE = As ( ) e As ( ) e = As ( ) As ( ) e * i( ks1 ωt+ ϕ1) i( ks2 ωt+ ϕ2) i( ks1 ωt+ ϕ1) i( ks2 ωt+ ϕ2) 1 2 1 2 1 2 ik [ ( s1 s2) + ϕ1 ϕ2] ik s2 s1 + ϕ2 ϕ1 = As ( ) As ( ) e = A( 1 2 2 1 2 1 1 2 [ ( ) ] * i( ks2 ωt+ ϕ2) i( ks1 ωt+ ϕ1) i( ks2 ωt+ ϕ2) i( ks1 ωt+ ϕ1) 2 1 2 1 1 2 1 2 ik [ ( s s) + ϕ ϕ ] s ) A( s ) e EE = As ( ) e As ( ) e = As ( ) As ( ) e = As ( ) As ( ) e EE EE As ( ) As ( ) e As ( ) As ( ) e ik [ ( s2 s1) + ϕ2 ϕ1] ik [ ( s2 s1) + ϕ2 ϕ1 + = + ] * * 1 2 2 1 1 2 1 2 = 2 As ( ) As ( )cos[ ks ( s) + ϕ ϕ ] 1 2 2 1 2 1

Young s Interference Assumptions: Observation Area: d (,0, zl) 2 ( xy,,0) (0,0, z L ) d (,0, zl) 2

Three Assumptions 1. Assumptions: 2. Observation Area: L1 L1 3. The spherical waves leaving and be of the same intensity. d (,0, zl) 2 ( xy,,0) (0,0, z L ) d (,0, zl) 2

Spatial Coherence Binomial Theorem 略

Binomial Theorem 2 ( 1) (1 ) 1 2! m n mm x mx x R m R + = + + + + 1 2 2 2 1,2 2 2 2 2 2 1 1 1 2 1 2 2 L L L L L L d x y s z z z d x y z z z = + + = + + 略

Difference of Optical Length 2 2 1 2 1 1 2 1 2 2 L L L d x y s z z z = + + 2 2 2 2 1 1 2 1 2 2 L L L d x y s z z z + = + + 2 1 2 2 1 2 2 2 L L s s d d x x z xd z = + =

Assumption of Wave Amplitude: Spatial Coherence The amplitude of the two spherical waves are constant and equal across the screen. I = A ( s ) + A ( s ) + 2 A( s ) A( s )cos( k( s s ) + ϕ ϕ ) 2 2 1 2 1 2 2 1 2 1 Fringe Pattern: d s2 s1 = x z L k = 2π λ

Spatial Coherence Spherical Waves: E ( s, t) = A( s )exp[ i( ks ωt+ ϕ )] 1 1 1 1 1 E ( s, t) = A( s )exp[ i( ks ωt+ ϕ )] 2 2 2 2 2 Intensity of light Spatial coherence function Intensity of Young s interference with spherical waves I * = EE Γ ( r, r,0) = E( r, t) E ( r, t) * 1 2 1 2 I( x, y,0) 2 2π d = 2 A ( zl) 1+ cos( x+ ϕ2 ϕ1) λzl

Fringe Pattern: Spatial Coherence

Spatial Coherence Fringe Separation: First Maximum:

Fringe Separation Harmonic Wave : ψ ( xt, ) = Acos( kx ωt+ ε) = λ k = 2π λ Fringe Separation = a k = 2π 2π d kx = x = x λzl a = a λzl d 2π a

First Maximum Harmonic Wave : ψ ( xt, ) = Acos( kx ωt+ ε) = λ 2π ε = λ x m Fringe Separation = a x m z = a = 2π d 2 ϕ ϕ λ ϕ ϕ 1 2 L 1 2 π 2π ϕ ϕ = 1 2 a x m

Harmonic Waves sinθ = sin( θ ± 2 π) sin x= sin( x± λ)

Harmonic Waves - continued k, λ > 0

Spatial Coherence with incoherent Light Source

Spatial Coherence with incoherent Light Source Q ( L, R,0) 1 ( d L,0,0) 1 2 Q 0 (0, R,0) ( d L,0,0) 2 2

Spatial Coherence with incoherent Light Source Simultaneously: r1 = QL 1 1 r2 = Q2L2 ϕ ϕ1 = kr ω 1 t ϕ ϕ = kr ωt 2 2 k = 2π λ 1 2 xm a ϕ = ϕ 2π a xm = ( r1 r2) λ

Spatial Coherence with incoherent Light Source The condition for interference fringes to be visible: a xm = ( r1 r2) λ

Spatial Coherence with incoherent Light Source

Spatial Coherence with incoherent Light Source Binomial Theorem The condition of spatial coherence with incoherent light source.

Spatial Coherence Spatial coherence function:

Spatiotemporal Coherence

Spatiotemporal Coherence Intensity of light I * = EE Mutual coherence function Normalized mutual coherence function Γ ( r, r, t, t ) = E( r, t+τ ) E ( r, t) * 1 2 1 2 1 2 γ 12 ( τ) Γ ( τ ) 12 = Γ 11 Γ 22 (0) (0)

Spatiotemporal Coherence Mutual (complex) Coherence Function: Normalized Mutual Coherence Function:

Spatiotemporal Coherence Intensity:

Spatiotemporal Coherence The degree of mutual coherence obeys two wave equations. r = ( x, y, z ) j j j j j =1, 2

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