Supplementary Information for Rayleigh scattering in few-mode optical fibers

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1 Supplementary Information for Rayleigh scattering in few-mode optical fibers Zhen Wang, 1, 2, Hao Wu, 1, 2, Xiaolong Hu, Ningbo Zhao, 1, 2 Qi Mo, 3 and Guifang Li 1 School of Precision Instrument and Optoelectronic Engineering, Tianjin University, Tianjin 372, China 2 Key Laboratory of Optoelectronic Information Science and Technology, Ministry of Education, Tianjin 372, China 4, 1, # 1, 2, * 3 Wuhan Research Institute of Posts and Telecommunications, Wuhan 4374, China 4 CREOL, The College of Optics & Photonics, University of Central Florida, Orlando, FL 32816, USA These authors contributed equally to this work. xiaolonghu@tju.edu.cn # li@ucf.edu 1

2 Content I. Time-dependent power of Rayleigh back-scattering II. Far fields of LP modes III. Field radiated by a dipole IV. Local capture fraction V. Overall capture fraction VI. Measurement of optical losses VII. Uncertainties of the measured intercepts, Iij VIII. Reduction of the dead zone by using the acousto-optic modulator IX. Measurement of Rayleigh backscattering on two additional fewmode optical fibers and the comparison with theory X. Measurement of inter-modal crosstalk Reference for Supplementary Information 2

3 I. Time-dependent power of Rayleigh back-scattering We derive Eq. (1) using space-time diagrams. Consider light with power P(z =, t, t ) launched into forward-propagating mode i of the few-mode fiber at its one end z =. The total length of fiber is denoted as l F. We now calculate the back-scattered power P ij BS in mode j received at z =. Figure S1 Space-time diagrams for calculating time-dependent power of Rayleigh back-scattering. We first calculate the differential energy d 2 E(, t 2 ), which is the back-scattered energy received at the time t 2 due to the light launched into the fiber at the time interval [t 1, t 1 + dt 1 ] and scattered by the scatters at the span [z s, z s + dz s ]. Fig. S1 (a) presents the spacetime diagram for this case. The differential energy d 2 E(, t 2 ) is d 2 E(, t 2 ) = P(, t 1 )dt 1 e α iz s α s (z s )dz s B ij (z s )e α jz s, (S1) where P(, t 1 )dt 1 is the differential energy launched into the fiber; e α iz s takes into account the optical loss of the fiber from z = to z = z s ; α s (z s )dz s is the ratio of total back-scattered energy to the incident energy at z = z s ; B ij (z s ) is the ratio of the energy coupled into back-propagating mode j to total scattered energy; e α jz s takes into account the optical loss of the fiber from z = z s to z = for the back-propagating mode j. Eq. (S1) can be written in the differential form: d [ de(, t 2) ] = P(, t dt 1 dz 1 )e (α i+α j )z s α s B ij (z s ), (S2) s and therefore, de(, t 2 ) dz s = P (, t 1 )e (α i+α j )z s α s B ij (z s )dt 1. (S3) For P ij BS, we have 3

4 P BS ij (t 2 ) = de(, t 2) = de(, t 2) dz s. dt 2 dz s dt 2 (S4) In Fig. S1 (a), we have cotα = v gi, cotβ = v gj, and in Fig. S1 (b) we get dt 2 = dz s cotα + dz s cotβ = dz s v gi + dz s v gj, (S5) or, equivalently, dz s = v giv gj v. dt 2 v gi + v gj (S6) In Fig. S1 (a), t 2 t 1 = z s v. (S7) From Eqs. (S4), (S6), and (S7), we obtain: P BS ij (t 2 ) = P (, t 2 z s v )e (α i+α j )z s α s (z s )B ij (z s )dz s, (S8) which is the general expression for the back-scattered power in mode j received at z = due to the excitation in mode i. Now we consider the specific case in our study. 1. The incident light is an optical pulse with a width of ΔT and a constant power P within ΔT. See Fig. S1 (c). 2. We assume α s (z) = α s and B ij (z) = B ij, i.e., they are constants along z. Then, 4

5 z B P BS ij (t) = P e (α i+α j )z α s (z)b ij (z)dz z A z B = P α s B ij e (α i+α j )z dz = z A P α s B ij (α i + α j ) [e (α i+α j )z B e (α i+α j )z A ], (S9) where z A = (t ΔT)v and z B = tv. Thus, P ij BS (t) = = P α s B ij (α i + α j ) [e (α i+α j )tv e (α i+α j )(t ΔT)v ] P α s B ij [e (α i+α j )(t ΔT)v e (α i+α j )tv ]. α i + α j (S1) To further simplify Eq. (S1), we let α i + α j = 2α. Then P BS ij (t) = P α s B ij 2α [e 2α(t ΔT)v e 2αtv ] P α s B ij = 2α e 2αtv [e 2αΔTv 1]. (S11) If 2αv ΔT 1, i.e., the pulses are short: ΔT 1 2αv, (To estimate 1 2α v in our study, we use α =.2 db/km.4 /km and v km/s. Then, 1/(2α v ) = 125 μs. In our case, ΔT = 1 ns 1 2α v.) then e2α v ΔT 1 = 2α v ΔT, P ij BS (t) = P α s B ij v ΔTe 2α v t. (S12) For a single-mode fiber, i = j, v gi = v gj, α i = α j = α, Then P ij BS (t) = P α s B ij v 2 ΔTe 2αz (t = 2z v g ), (S13) which is the same expression for a single-mode fiber in Ref. [S1] except that we treat α s and B ij as constants along z. 5

6 II. Far fields of LP modes The far fields of LP modes can be calculated using Fraunhofer diffraction formula [S2] : ψ N ψ F (x, y, z) = ejkz exp [jk jλz 2z (x2 + y 2 )] (x, y, z = )exp [ j( x λz x + y (S14) y )] dx dy ; λz or, in a spherical coordinate system, for θ, we have z r and ψ F (r, θ, φ) = ejkr exp [jk jλr 2r (rsinθ)2 ] ψ N exp [ j( rsinθcosθ λr (R, φ ) R acosφ + rsinθsinθ R asinφ )] R ad(r a)dφ, λr (S15) where R = R a. In an optical fiber, k = k n, λ = λ n, and n n 1 n 2 for a weak-guiding fiber, then ψ F (r, θ, φ) = na2 e jk nr jλ r exp [ jk n 2 (rsin2 θ)] ψ N (R, φ )exp[ jk nar sinθ(cosφcosφ + sinφsinφ )]R dr dφ. (S16) For the far field, θ, exp [ jk n 2 rsin2 θ] 1, then, ψ F (r, θ, φ) = na2 e jk nr ψ jλ r N (r, φ )exp[ jk nar sinθcos(φ φ )]R dr dφ, (S17) where for LP 1 mode, the near field is J ψ N = ψ N1 = { (UR ) R 1 J (U)K (WR )/K (W) R 1 ; (S18) for LP 11a mode, the near field is 6

7 J ψ N = ψ N2 = { 1 (UR )cosφ R 1 J 1 (U)K 1 (WR )cosφ /K 1 (W) R 1 ; (S19) for LP 11b mode, the near field is J ψ N = ψ N3 = { 1 (UR )sinφ R 1 J 1 (U)K 1 (WR )sinφ /K 1 (W) R 1. (S2) J and J 1 are zeroth- and first-order Bessel functions of the first kind, respectively; K and K 1 are zeroth- and first-order of modified Bessel functions of the second kind, respectively. U and V are the core and cladding parameters, respectively [S3]. Define ψ N2 (R ) = ψ N2 (R, φ )/cosφ, then ψ F2 (r, θ, φ) = na2 e jk nr ψ jλ r N2 (R ) cos φ exp[ jk nar sinθcos(φ φ )]R dr dφ. (S21) Using the following relations [S4] : e jx cosφ = J (x ) + 2 j n J n (x )cos(nφ ), n=1 (S22) m n cos (mφ )cos[n(φ^ φ )]dφ = { m = n =, (S23) πcosmφ^ m = n and J 1 ( x) = J 1 (x), (S24) we get 7

8 cos φ exp[jk nar sinθ } x cos(φ φ ) ]dφ = cos φ {J (k nar sinθ) + 2 jn J n (k nar sinθ)cos[n(φ φ )]} dφ n=1 = jcosφj 1 (k nar sinθ). } φ (S25) Therefore, ψ F2 (r, θ, φ) = k na 2 e jknr ψ r N2 (R, φ)j 1 (k nar sinθ)r dr. (S26) Similarly, ψ F3 (r, θ, φ) = k na 2 e jknr ψ r N3 (R, φ)j 1 (k nar sinθ)r dr, (S27) and ψ F1 (r, θ, φ) = k na 2 e jknr ψ r N1 (R, φ)j 1 (k nar sinθ)r dr. (S28) III. Field radiated by a dipole Consider a scatter, S, located at (r = R s, φ = φ s, z = ) in the cylindrical coordinate system, as shown in Fig. S2. Assuming that S is at the cross-section z = doesn t affect our calculation of capture fractions. 8

9 Figure S2 Coordinate systems used for calculating the far fields of LP modes and the field radiated by a dipole. If S is located at the origin (r =, φ =, z = ), then the electric field ψ s at (r, θ, φ) is [S5] ψ s (r, θ, φ) = 1 4πε k 2 r P exp( jkr)sinψ, (S29) where P = κe, κ is the permittivity, a scalar in an isotropic medium, and E is the electric field of incident light. Ψ is the angle between P and r. We assume that the intensity of the far field ψ s (r, θ, φ) doesn t change if we move the scatter from the origin to S = (R s, φ s ); however, the phase of ψ s (r, θ, φ) changes by δ(r s, φ s ): ψ s (r, θ, φ) = ψ s 1 r e j[knr+δ(r s,φ s )] sinψ, (S3) where ψ s = κe n 2 k 2, R s has been normalized by the fiber radius, a. If the dipole is 4πε polarized along x direction, then cosψ = sinθcosφ, and sinψ = (1 sin 2 θcos 2 φ) 1 2, then ψ s (r, θ, φ) = ψ s 1 r e j[knr+δ(r s,φ s )] (1 sin 2 θcos 2 φ) 1 2; (S31) if the dipole is polarized along y direction, then cosψ = sinθsinφ, and sinψ = (1 sin 2 θsin 2 φ) 1 2, then ψ s (r, θ, φ) = ψ s 1 r e j[knr+δ(r s,φ s )] (1 sin 2 θsin 2 φ) 1 2. (S32) 9

10 Now we calculate δ(r s, φ s ). Let r = r OS, then r 2 = (x x s ) 2 + (y y s ) 2 + z 2 = (x 2 + y 2 + z 2 ) 2(xx s + yy s ) + (x 2 s + y 2 s ). (S33) Using the following relations, x = rsinθcosφ, y = rsinθsinφ, x s = ar s cosφ s, and y s = ar s sinφ s, we can get r 2 = r 2 2aR s rsinθcos(φ φ s ) + (ar s ) 2. (S34) Because r ar s in the far field, we use ar s sinθcos(φ φ s ) to replace ar s, then r r ar s sinθcos(φ φ s ). (S35) So, δ(r s, φ s ) = k n(r r) k nar s sinθcos(φ φ s ). In summary, for a dipole located at (R s, φ s, z = ), if it is polarized along x direction, ψ s (r, θ, φ) = ψ s 1 r e j[knr+δ(r s,φ s )] (1 sin 2 θcos 2 φ) 1 2; (S36) if it is polarized along y direction, ψ s (r, θ, φ) = ψ s 1 r e j[knr+δ(r s,φ s )] (1 sin 2 θsin 2 φ) 1 2, (S37) where δ(r s, φ s ) = k n(r r) k nar s sinθcos(φ φ s ). (S38) We note that ψ s 2 dω = 4π 3r 2 ψ s 2. (S39) 1

11 IV. Local capture fraction The local capture fraction, b j, of mode j is defined as the ratio of the back-scattered power coupled into mode j over the total scattered power by the scatter located at (R s, φ s ). b j can be calculated by the following overlap integral [S1] : b j (R s, φ s ) = 1 2 ψ Fj ψ s dω 2 ψ Fj 2. (S4) dω ψ s 2 dω where j = 1, 2, 3. For LP 11a, we first calculate the numerator of Eq. (S4): ψ Fj ψ s dω 2 = k 2 n 2 a 4 2 ψ π s r 4 2 ψ N2 (R )J 1 (k nar sinθ) R dr cos φ(1 sin 2 θcos 2 θ) 1 2e jk nar s sinθcos(φ s φ) dφsinθdθ. (S41) For n 1 n 2 n in the weakly-guiding few-mode fiber, the far-field distribution only has a significant intensity for small θ. Therefore, sinθ θ, 1 sin 2 θcos 2 θ 1. This approximation means that we have neglected polarizations. The upper limit of the integral over θ in Eq. (S41) can be extended to infinity. Let η = k naθ, then, ψ Fj ψ s dω 2 = a2 2 ψ s k 2 n 2 r 2 ψ N2 (R )J 1 (ηr )R dr cosφj 1 (R s η)ηdη 2. (S42) Using Hankel transforms [S6], f 2 (r 2 ) = f 1 (r 1 )J m (r 1 r 2 )r 1 dr 1, (S43) and f 1 (r 1 ) = f 2 (r 2 )J m (r 2 r 1 )r 2 dr 2, (S44) we obtain 11

12 ψ Fj ψ s dω 2 = 4π2 2 ψ s k 2 n 2 r [ψ 4 N2(R s, φ s )] 2. (S45) Now we calculate the denominator: ψ F2 2 dω = k 2 n 2 a 4 π 2 r 2 cos 2 φ dφ ψ N2 (R )J 1 (k nar sinθ)r dr 2 sinθdθ. (S46) We use the same approximation as we just used for calculating of the numerator: ψ F2 2 dω = πa2 r 2 ψ N2 (R )J 1 (R η)r dr 2 πa 2 = r 2 ψ N2 (R ) 2 R dr. ηdη (S47) Using Eqs. (S45), (S47) and (S39), we obtain 3 ψ N2 2 b 2 (R s, φ s ) = 2k 2 n 2 a 2 ψ N2 (R ) 2 R dr. (S48) Similarly, 3 ψ N3 2 b 3 (R s, φ s ) = 2k 2 n 2 a 2 ψ N3 (R ) 2 R dr, (S49) and 3 ψ N1 2 b 1 (R s, φ s ) = b 1 (R s ) = 4k 2 n 2 a 2 ψ N1 (R ) 2 R dr. (S5) 12

$Figure S3 Justification of the approximations used for calculating local caption fractions.$

13 Figure S3 Justification of the approximations used for calculating local caption fractions. (a), (b), and (c) are local caption fractions calculated without using approximations for LP 1, LP 11a, and LP 11b modes, respectively. (d), (e), and (f) are the absolute value of the difference between the calculations with and without using the approximations for LP 1, LP 11a, and LP 11b modes, respectively. To confirm the validity of the approximation used above, we calculate b j using Eq. (S4) without approximations. b j and b j b j is presented in Fig. S3. The maximum difference of b j b j with and without approximations are , and for LP 1, LP 11a and LP 11b modes, respectively. The maximum relative difference, b j b j b j, in the core is 6%. V. Overall capture fraction The overall capture fraction, B ij, is defined as the back-scattered power coupled into backpropagating mode j over the total scattered power at z = z s. We can explicitly write down B ij (i = 1, 2, 3; j = 1, 2, 3) from Eq. (3): 13

14 B 11 = B 12 = B 13 = B 21 = B 22 = B 23 = B 31 = B 32 = B 33 = 3 [ψ N1 (R )] 4 R dr 4k 2 n 2 a 2 { [ψn1 (R )] 2 R dr } 2, 3 [ψ N1 (R )] 2 [ψ N2 (R )] 2 R dr 4k 2 n 2 a 2 [ψ N1 (R )] 2 R dr [ψ N2 (R )] 2 R dr, 3 [ψ N1 (R )] 2 [ψ N2 (R )] 2 R dr 4k 2 n 2 a 2 [ψ N1 (R )] 2 R dr [ψ N2 (R )] 2 R dr, 2 3 [ψ N2 (R )] [ψ N1 (R )] 2 R dr 4k 2 n 2 a 2 [ψ N2 (R )] 2 3 R dr [ψ N1 (R )] [ψ N2 (R )] R dr 8k 2 n 2 a 2 { [ψ N2 (R )] 2 R dr } 2, 3 8k 2 n 2 a 2 4k 2 n 2 a 2 [ψ N2 (R )] 2 4 [ψ N2 (R )] R dr { [ψ N2 (R )] 2 R dr } 2, 2 [ψ N2 (R )] [ψ N1 (R )] 2 R dr R dr [ψ N1 (R )] [ψ N2 (R )] R dr 8k 2 n 2 a 2 { [ψ N2 (R )] 2 R dr } 2, 9 8k 2 n 2 a 2 4 [ψ N2 (R )] R dr { [ψ N2 (R )] 2 R dr } 2. R dr, R dr, (S51) Eq. (S51) shows the following relations: B 12 = B 21 = B 13 = B 31, (S52) and B 22 = B 33 = 3B 23 = 3B 32. (S53) 14

15 VI. Measurement of optical losses In order to compare the theoretical and experimental results, we carefully measured optical losses. Table S1 lists measured additional losses, L ij, other than the optical attenuation losses in the few-mode fiber. The additional losses includes the insertion losses of two circulators (for laser light), the insertion losses of the spatial mode-multiplexer/demultiplexer (for both laser light and Rayleigh back-scattering), coupling loss from free space to the few-mode fiber (for laser light), coupling loss from free space to the singlemode fiber (for Rayleigh back-scattering), the insertion loss of the AOM (for Rayleigh back-scattering), and the insertion loss of one (if i j) or two (if i = j) circulators (for Rayleigh back-scattering). We further corrected the experimental curves by taking into account L ij : 1 2 (P ij BS /P )[db] = αz [1log 1(α s vδt) + 1log 1 B ij L ij ][db] (S54) The optical loss, α i, of each mode in the few-mode fiber is obtained by fitting the OTDR data. Each OTDR curve shows a slope of α instead of 2α; that is where the factor 1 in Eq. 2 (S62) comes from. We plot 1 [P 2 ij BS P ][dbm] in Fig. 4 to compare with experimental data. Table S1 Additional optical losses measured for each combination of excitation and Rayleigh back-scattering Loss (db) j = LP 1 j = LP 11a j = LP 11b i = LP ± ± ±.8 i = LP 11a 22.5± ± ±.9 i = LP 11b 21.83± ± ±.8 VII. Uncertainties of the measured intercepts, Iij We analyze the uncertainties of the measured intercepts of the OTDR curves, Iij. The analytical expression of Iij can be obtained by simply setting z = in Eq. (S4). The dominant source of uncertainties is the uncertainties on the measured insertion losses, L ij. Other two terms in Eq. (S4) were measured by OTDR averaging over 3 seconds; their uncertainties are negligibly small compared with the uncertainties of L ij. The least-square fittings of OTDR curves generate errors of Iij on the order of approximately 1-4 db, which can also be neglected. Thus, the uncertainties of the measured intercepts of the OTDR curves, Iij, is.5δl ij, where δl ij is the uncertainty of L ij, as we include in Table S1. In Fig. 4 (g) in the main text, the error bar on experimental I11-I12, is therefore calculated by ±.5 (δl 11 ) 2 + (δl 12 ) 2. The error bars for other cases are similarly calculated. 15

16 VIII. Reduction of the dead zone by using the acousto-optic modulator Figure S4 presents the OTDR data measurement with and without using the acousto-optic modulator (AOM). Without using the AOM, the giant peak due to reflection generates a dead zone approximately from to 1.5 km. Clearly, the AOM reduced the reflection and the dead zone. Relative backscattering power / db without AOM with AOM Distance, z / km Figure S4 Optical time-domain reflectometry using and without using an acousto-optical modulator (AOM). The AOM reduced the dead zone that was generated by the giant peak due to reflection at the fiber facet. IX. Measurement of Rayleigh backscattering on two additional few-mode optical fibers and the comparison with theory Using the same method described in the main text, we measured Rayleigh backscattering on two additional few-mode optical fibers and compared the experimental and theoretical results. Additional few-mode optical fiber 1: This sample is a step-index optical fiber with three guiding modes (LP1, LP11a, and LP11b). Its length is 3 km. The radius of its core is 6.8 m, its numerical aperture is.135, and our measurement was done at the normalized frequency of Figure S5 presents the experimental and theoretical results. 16

17 Expriment Theory j=lp 1 j=lp 11a j=lp 11b -3 (a) i=lp 1 (b) i=lp 1 Relative back-scattered power [2 db / div] (c) i=lp 11a (e) i=lp 11b (d) i=lp 11a (f) i=lp 11b Relative back-scattered power / db Distance, z / km 2 db Theory Expriment (g) 1 db db I 11 -I 12 I 11 -I 13 I 21 -I 22 I 21 -I 23 I 31 -I 32 I 31 -I 33 Figure S5 Optical-time-domain-reflectometer measurement of Rayleigh back-scattering in additional few-mode optical fiber 1 and the corresponding theoretical results. (a) experimental results with LP 1 excitation; (b) theoretical results with LP 1 excitation; (c) experimental results with LP 11a excitation; (d) theoretical results with LP 11a excitation; (e) experimental results with LP 11b excitation; (f) theoretical results with LP 11b excitation; (g) comparison of the experimental and theoretical intercepts relative to I i1. Additional few-mode optical fiber 2: This sample is a graded-index optical fiber with three guiding modes (LP1, LP11a, and LP11b). Its length is 1 km. Figure S6 presents the distribution of its refractive index at the wavelength of 155 nm. For calculating the capture fractions, the modal near fields were simulated by finite-element method using COMOSL Multiphysics. Figure S7 presents the experimental and theoretical results. 17

18 Optical refractive index, n Radius / m Figure S6 The distribution of its refractive index, at the wavelength of 155 nm, of additional fewmode optical fiber 2. Discussion: Table S2 lists the difference between theoretical and experimental results for the fiber reported in the main text (fiber ), additional fiber 1, and additional fiber 2. For all three samples, in absence of strong coupling (I11-I12, I11-I13, I21-I22, and I31-I33), the experimental results match the theoretical results, evidencing the validity of our theory; if inter-modal strong coupling exists, as we measured I23 and I32, the measured I23 and I32 are larger than what our theory predicts because the theory does not take into account the inter-modal coupling of the excitation. These analysis has also been presented in detail in the main text. 18

19 Expriment Theory j=lp 1 j=lp 11a j=lp 11b -3 (a) i=lp 1 (b) i=lp 1-32 Relative back-scattered power [2 db / div] (c) i=lp 11a (d) i=lp 11a (e) i=lp 11b (f) i=lp 11b Distance, z / km Relative back-scattered power / db 2 db Theory Expriment (g) 1 db db I 11 -I 12 I 11 -I 13 I 21 -I 22 I 21 -I 23 I 31 -I 32 I 31 -I 33 Figure S7 Optical-time-domain-reflectometer measurement of Rayleigh back-scattering in additional few-mode optical fiber 2 and the corresponding theoretical results. (a) experimental results with LP 1 excitation; (b) theoretical results with LP 1 excitation; (c) experimental results with LP 11a excitation; (d) theoretical results with LP 11a excitation; (e) experimental results with LP 11b excitation; (f) theoretical results with LP 11b excitation; (g) comparison of the experimental and theoretical intercepts relative to I i1. Table S2 The differences between theoretical and experimental results. The shaded two columns are the cases that inter-modal strong coupling exists in our measurement. Difference(dB) I 11 I 12 I 11 I 13 I 21 I 22 I 21 I 23 I 31 I 32 I 32 I 33 Fiber Fiber Fiber

20 X. Measurement of inter-modal crosstalk We measured the inter-modal crosstalk of the three-mode optical fiber and made comparison with the crosstalk due only to Rayleigh forward scattering obtained from our theory. Using the method in Ref. S7, the mode-coupling ratio from mode i to mode j where the excitation is in mode i can be obtained by η ij = P ij BS P ii BS = 2h ij l F + κ (S 55) where h ij is the mode-coupling coefficient, l F, as defined before, is the length of the fiber, and κ is a constant. Figure S8 presents the raw data of η ij and the linear fittings to obtain inter-modal crosstalk 1log(h ij l F ). We found that the inter-modal crosstalk of LP 1 LP 11a, LP 11a LP 1, LP 1 LP 11b, and LP 11b LP 1 are db, db, db, and db, respectively. Because the dispersion relations of LP 11a and LP 11b modes are degenerate, these two modes experience strong coupling. Thus, the analysis of the mode coupling using (S 55) that assumes weak coupling is no longer valid. Indeed, the net mode coupling LP 11a LP 11b obtained using this method was only db; the net mode coupling LP 11b LP 11a was not measurable. Compared with the inter-modal crosstalk due to Rayleigh forward scattering that is -37 db at its maximum, as predicted by our theory in the main text, the total crosstalk due to random mode coupling and fiber imperfections is dominantly large. 2

21 hl F = hl 12 F =.1574 (a) 1log(hl 21 F ) = db 1log(hl F )= db (b) Mode coupling ratio hl F = log(hl F ) = db 2hl F = log(hl F ) = db (c) (e) 31 2hl F = log(hl F ) = db 32 2hl F = log(hl F ) : NA (d) (f) Mode coupling ratio Distance, z / km Figure S8 Measure of inter-modal crosstalk of the three-mode optical fiber, using optical timedomain reflectometry. (a), (b), (c), and (d) show the mode coupling of LP 1 LP 11a, LP 11a LP 1, LP 1 LP 11b,and LP 11b LP 1, respectively. (e) and (f) show the raw data and fittings for LP 11a LP 11b and LP 11b LP 11a, respectively. 21

22 Reference for Supplementary Information S1. Hartog, A. H. & Gold, M. P. On the theory of back-scattering in single-mode optical fibers. J. Lightwave Technol. 2, (1984). S2. Born, M. & Wolf, E. Principles of Optics. (Cambridge University Press, 1999). S3. Snyder, A. W. & Love, J. Optical Waveguide Theory. (Springer, 1983). S4. Riley, K. F., Hobson, M. P. & Bence, S. J. Mathematical Methods for Physics and Engineering. (Cambridge University Press, 26). S5. Griffiths, D. J. Introduction to Electrodynamics. (Addison Wesley, 1999). S6. Debnath, L. & Bhatta, D. Integral Transforms and Their Applications. (CRC press, 214). S7. Nakazawa, M., Yoshida, M. & Hirooka, T. Measurement of mode coupling distribution along a few-mode fiber using a synchronous multi-channel OTDR. Opt. Express 22, (214). 22

OPTI510R: Photonics. Khanh Kieu College of Optical Sciences, University of Arizona Meinel building R.626

OPTI510R: Photonics. Khanh Kieu College of Optical Sciences, University of Arizona Meinel building R.626 OPTI510R: Photonics Khanh Kieu College of Optical Sciences, University of Arizona kkieu@optics.arizona.edu Meinel building R.626 Announcements Homework #4 is assigned, due March 25 th Start discussion