(interferometer) 1 N *3 2 ω λ k = ω/c = 2π/λ ( ) r E = A 1 e iφ1(r) e iωt + A 2 e iφ2(r) e iωt (1) φ 1 (r), φ 2 (r) r λ 2π 2 I = E 2 = A A 2 2 +

7 1 (Young) *1 *2 (interference) *1 (1802 1804) *2 2 (2005) (1993) 1

(interferometer) 1 N *3 2 ω λ k = ω/c = 2π/λ ( ) r E = A 1 e iφ1(r) e iωt + A 2 e iφ2(r) e iωt (1) φ 1 (r), φ 2 (r) r λ 2π 2 I = E 2 = A 2 1 + A 2 2 + 2A 1 A 2 cos ψ (2) ψ = φ 1 φ 2 *3 7.1 2

(2) (interference fringe) 2π ψ = φ 1 φ 2 = 2mπ I max = A 1 + A 2 2 2 (in phase) (constructive interference, ) π ψ = φ 1 φ 2 = (2m + 1)π I min = A 1 A 2 2 2 180 0 (out of phase) (destructive interference, ) M 1 BS Sc M 2 S 1 S: BS: M: Sc: 3

2.1 I max I min 90 2 2 I av I I av = I max + I min, I = I max I min 2 2 (visibility) (depth of modulation) (contrast) (3) V = I I av = I mac I min I max + I min (4) I = I av (1 + V cos ψ) (5) I max ΔI I av I min 0 π 2π 2 4

(coherent) (incoherent) 2.2 ψ(r) = φ 1 (r) φ 2 (r) 3 (Mach- Zehnder) 4 M 2 BS 2 S BS 1 M 1 3 k j = (2π/λ)t j t j φ j = k j r ψ = (k 1 k 2 ) r = 2π λ (t 1 t 2 ) r = K r = 2mπ (6) 5

K = k 1 k 2 xz z ±θ 2θ t j = (± sin θ, 0, cos θ) K = (2π/λ)(2 sin θ, 0, 0) = (4π sin θ/λ)(1, 0, 0) K Λ K = 2π Λ = 22π λ sin θ (7) Λ = λ 2 sin θ (8) 4 4 4 ABC Λ sin θ = λ/2 2θ λ A θ C λ/2 λ B Λ Λ 4 6

j r j φ j = kr j 2π r 1 r 2 = mλ ( 5) 2 1 0 1 2 2 1 0 1 2 5 3 6 d ψ 7

θ 1 θ 2 ψ = 2πd λ (sin θ 2 ± sin θ 1 ) (9) ψ = 2mπ m 1 A B C θ 1 D θ 2 d 0.8 0.6 0.4 0.2 1 N 0.5 1 1.5 2 6 7 N N I = I 0 1 + e iψ + e 2iψ + + e (N 1)iψ 2 1 cos Nψ = I 0 1 cos ψ = I sin 2 (Nψ/2) 0 sin 2 (ψ/2) (10) 7 N = 10 N 1/N N 2 I 0 N N 8

4 8 d n θ 0 θ sin θ 0 = n sin θ L = 2nd cos θ ψ ψ = 4πnd cos θ λ (11) nd 9 (fringe of equal inclination) d θ n θ 0 8 9 (etalon) t r 9

t r (Stokes) r = r, tt + r 2 = 1 (12) 1 tt tr 2 t exp( iψ) tr 4 t exp( 2iψ) I T I T = tt = 2 r 2n e niψ n=0 t 2 t 2 1 2r 2 cos ψ + r 4 = (1 r 2 ) 2 1 2r 2 cos ψ + r 4 (13) (12) 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 10 10

I T = 1 1 + F sin 2 (ψ/2), F = 4R (1 R) 2 (14) R = r 2 (Airy) 10 F = 100 (ψ = π) ψ = 0 I T = 1 1/2 ψ F ψ 2/ F ψ = π f = π ψ = π π R F = 2 1 R (15) (finesse) N 1 (16) 2 I R = 1 I T 5 11 S P SABCP P SDP L = [SABCP ] [SDP ] ψ = 4πnd cos θ λ (16) 11

S P S d D A θ C n d A θ P n B B 11 12 P S D D nd D 12 P (16) D P S θ nd cos θ (θ = 0) (nd) = λ/2 n d (fringe of equal thickness) 13 12

14 13 14 (Newton ring) *4 13 (Fizeau) d 6 (optical flat) 15 *4 13

(λ/2) λ/4 I(x) = I 0 (x) + I 1 (x) cos [ Kx + ψ(x) ] (17) I 0 I 1 K Λ K = 2π/Λ ψ(x) K *5 ψ 16 17 0 (K = 0) 18 19 OF S 15 *5 14

1.0 0.5 0.0 2 0.5 2 0 0 2 2 16 17 18 19 7 cos SN 15

(17) (x) I 0, I 1 ψ (1) (2) (3) 7.1 (heterodyne) v λ/2 f = 2v/λ = 2fv/c (17) K = 0 I(x, t) = I 0 (x) + I 1 (x) cos [ ft + ψ(x) ] (18) x = 0 ψ(x) ψ(0) 16

7.2 (fringe scan method) N (17) I 0, I 1, ψ K = 0 ψ N = 4 λ/8 λ/8 π/2 I(x, 0) = I 0 (x) + I 1 (x) cos ψ(x) I(x, π/2) = I 0 (x) + I 1 (x) cos [ ψ(x) + π/2 ] = I 0 (x) I 1 (x) sin ψ(x) I(x, π) = I 0 (x) + I 1 (x) cos [ ψ(x) + π ] = I 0 (x) I 1 (x) cos ψ(x) I(x, 3π/2) = I 0 (x) + I 1 (x) cos [ ψ(x) + 3π/2 ] = I 0 (x) + I 1 (x) sin ψ(x) tan ψ(x) = I(x, 3π/2) I(x, π/2) I(x, 0) I(x, π) (19) arctan 7.3 (Fourier transform method) K = 0 K 17

(fast Fourier transform, FFT) cos cos θ = [exp(iθ) + exp( iθ)]/2 (17) I(x) = I 0 (x) + 1 2 I 1(x)e i(kx+ψ) + 1 2 I 1(x)e i(kx+ψ) (20) K 20 3 I F T (x) = 1 2 I 1(x)e i(kx+ψ) (21) exp( ikx) log ( I F T e ikx) = log(i 1 /2) + iψ (22) λ/100 A 18

F[I 1 e -i(kx+ψ) ] F[I 0 ] F[I 1 e i(kx+ψ) ] K 0 K k 20 A.1 21 (Michelson) (Twyman - Green) M 1 M 2 M 1 PZT M 2 22 ( ) ( ) ( ) A.2 13 15 (Fizeau) A.3 3 19

M 2 S BS M 1 PZT D 21 L SM 22 A.4 (Sagnac) 23 20

M 2 S BS M 1 D 23 A.5 4 ( ) 2 (Fabry - Pérot) B 13 r R m r m π 0 21

r 0 = 0 r d (r, R d) R r 2 + (R d) 2 = R 2 R d d = r 2 /2R m mλ/2 rm 2 = mλr (23) C (coherence) 24 D ω D x S S z 1 d A B x P P z 2 24 22

d z 1 z 2 S S x x S x x P A B L = SBP SAP d z 1 x S + d z 2 x P (24) z 1 z 2 I(x P, x S ) I(x P, x S ) = I 0 { 1 + cos ( β2 x P + β 1 x S )} (25) β j = kd/z j = 2πd/λz j P I(x P ) I(x P ) = D/2 D/2 I(x P, x S )dx S = I 0 D[1 + γ cos(β 2 x P )] (26) γ ( ) πdd γ = sinc λz 1 23 (27)

sinc(x) = sin(x)/x γ (degree of coherence) D z 1 d (27) (27) d d = 0 1 d = λz 1 /D 0 d λz 1 /D D D/z 1 d γ π 3 (26) D 25 (Michelson) S(k) k = ω/c ω S BS M 1 M 2 BS D L = 2(L 2 L 1 ) k I(k) = S(k)[1 + cos(kl)] 24

M 2 S L 2 BS L 1 M 1 D 25 I(L) I(L) = 0 [1 + cos(kl)]s(k)dk = I 0 [1 + γ(l)] (28) I 0 = 0 S(k)dk (29) γ(l) γ(l) = 1 cos(kl)s(k)dk (30) I 0 0 L k 0 = ω 0 /c 2 = 2Γ/c γ(τ) = cos(ω 0 τ) sinc(γτ) (31) τ = L/c τ 1/Γ 25

c L c = cτ 26 S 26 E L (28) (interferogram) 27 0 26

0 0 L 27 27