Step-index fibers (STU)

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The simplest form of an optical waveguide consists of a light-conducting fiber core ... For the Cartesian field components Ex and Ey the wave equation applies to ...
Chapter 4

Step-index fibers (STU) This chapter discusses the wave propagation in step-index fibers. The field calculation in the step-index fiber as well as the chromatic dispersion are explained. The simplest form of an optical waveguide consists of a light-conducting fiber core with the refractive index n1 and a fiber cladding with the refractive index n2 < n1 , where the difference is about a few percent or even lower. The fiber core diameter is typically of the order 2a ≈ 10−100 µm and the diameter of the core and cladding is D ≈ 125 µm.

n 0= 1

Figure 4.1: Schematic of a step index fiber Typically the step-index fiber has a refractive index profile of   n1 f¨ur r ≤ a n(r) =  n f¨ur a < r ≤ D 2 2

(4.1)

θ1 is defined as the angle of the incident wave to the fiber axis in the fiber and γ1 is the angle of the incident wave coupling from the free space (n0 = 1). The wave is then guided through the core, if θ1 < θ1g applies.

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This can be obtained:

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√ sin(θ1g ) =

1 − cos2 (θ1g ) =

1 n1



n21 − n22

2

(4.2)

and with Snell’s Law the result is: √ sin(γ1g ) = n1 sin(θ1g ) = n21 − n22 = AN

(4.3)

AN is called the numerical aperture and specifies the maximum angle γ1 , for which the wave remains in the core. A typical value is AN = 0, 2 leading to a maximum angle of incidence γ1g = 11, 5◦ .

4.1 Field calculation in a step-index fiber Since the beam examination only describes the wave propagation correct for λ → 0, the question arises about the calculation of the field in the step-index fiber. Therefore the so-called eigenmodes (or normal ⃗ modes) are determined. They are given by assuming that a transverse field distribution E(x, y), propagates unaltered in z-direction with the propagation constant β. The refractive index is assumed to be independent of z. The field approach is then ⃗ ⃗ E(x, y, z) = E(x, y) exp(−j β z) (4.4) Firstly the order of β should be estimated. From the beam examination in fig. 4.1 it follows: β = k0 n1 cos(θ1 )

(4.5)

For the guided wave 0 < θ1 < θ1g can be determined, hence: k0 n2 < β < k0 n1

(4.6)

For the Cartesian field components E x and E y the wave equation applies to both the core and the cladding. △E x,y + k02 n2i E x,y = 0

(4.7)

with i = 1, 2 . The use of eq. (4.4) yields ∂ 2 E x,y = −β 2 E x,y ∂z 2

(4.8)

and consequently with eq. (4.7) the wave equation can be determined as ( ) △t E x,y + k02 n2i − β 2 E x,y = 0 with the transverse Laplace operator △t = waveguide with

∂2 ∂x2

+

∂2 . ∂y 2

(4.9)

In the following, we assume a weakly guiding

n1 − n2 ≪1 n1

(4.10)

thus leading to the following equations: |k02 n2i − β 2 | ≪ β 2

with

β = k0 · nef f

and

n2 < nef f < n1

(4.11)

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and therefore |△t | ≪ β 2

(4.12)

∂ ∂ , ≪ β ∂x ∂y

Hence

(4.13)

Considering the boundary conditions, the fields of the propagable waves from eq. (4.7) are applied. The field components E z , H z and E φ , H φ are continuous for r = a , where a is the core radius (fig. 4.2).

4.1.1

Field components of the eigenmodes of a weakly guiding step-index fiber

F a se rm a n te l

y

+ a F a se rk e rn

r

E j, H

z

-a

j

j

E z, H z

+ a x

-a

Figure 4.2: Boundary conditions of the step-index fiber Let us assume that eq. (4.9) was solved for E x and E y . The question then arises as on how to compute the ⃗ = jωµH ⃗ , we can obtain: other field components. By using the Maxwell equation −∇ × E ( ) ∂E y 1 ∂E x Hz = − − (4.14) jωµ0 ∂x ∂y ⃗ = jωε0 n2 E ⃗ it results for E x and E y : From the Maxwell equation ∇ × H i ( ) 1 ∂H z Ex = + jβH y ∂y jωε0 n2i ( ) ∂H z 1 Ey = − − jβH x ∂x jωε0 n2i

(4.15)

(4.16)

Hence, for the known E x , E y and H z from eq. (4.14), H x and H y can be determined. Assuming a weakly guiding fiber (see eq. (4.13)) ∂H z /∂y and ∂H z /∂x in eq. (4.15) and eq. (4.16) can be neglected. Therefore H x and H y can be expressed as Hx ≈ − Hy ≈

ωε0 n2i Ey β

ωε0 n2i Ex β

(4.17)

(4.18)

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E z is thus given by 1 Ez = jωε0 n2i

(

∂H y ∂H x − ∂x ∂y

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)

1 ≈ jβ

(

∂E x ∂E y − ∂x ∂y

4

) (4.19)

This shows that all six field components can be deduced from E x and E y . For weakly guiding fibers, ni can be approximated by n1 .

4.1.2 Continuity of the tangential field components The tangential field components of the fiber E φ , H φ , E z and H z have to be continuous at r = a . E φ and H φ are thus of the form E φ = E y cos(φ) − E x sin(φ) (4.20) H φ = H y cos(φ) − H x sin(φ)

(4.21)

As a result of eq.(4.17), eq. (4.18), eq. (4.20) and eq. (4.21) with ni ≈ n1 , it can be deduced that E x and E y have to be continuous (at r = a), if E φ and H φ are also to be continuous. Since this continuity should ∂E

∂E

apply for all φ, ∂φx and ∂φy at r = a must also be continuous. As already shown in eq. (4.14) and eq. (4.19) E z and H z can be determined from E x and E y , in which ∂ ∂ E x and E y are derived with respect to x and y, respectively. The derivatives of ∂x and ∂y are in cylindrical coordinates ∂ 1 ∂ ∂ = cos(φ) − sin(φ) ∂x ∂r r ∂φ ∂ ∂ 1 ∂ = sin(φ) + cos(φ) ∂y ∂r r ∂φ Inserted into eq. (4.14) and eq. (4.19) yields ( ) ∂E y ∂E y 1 ∂E x 1 ∂E x 1 cos(φ) − sin(φ) + sin(φ) + cos(φ) Ez = jβ ∂r r ∂φ ∂r r ∂φ ( ) ∂E y ∂E y 1 1 ∂E x 1 ∂E x Hz = − cos(φ) − sin(φ) − sin(φ) − cos(φ) jωµ0 ∂r r ∂φ ∂r r ∂φ Since

∂E x ∂φ

and

∂E y ∂φ

(4.22) (4.23)

(4.24)

(4.25)

at r = a, due to the continuity of E φ and H φ , have to be continuous, according to eq. ∂E

∂E

(4.24) and eq. (4.25), ∂rx and ∂ry at r = a also have to be continuous. The condition of continuity of E φ , H φ , E z and H z can be replaced by the requirement of continuity of E x , Ey,

∂E x ∂r

and

∂E y ∂r

.

4.1.3 Linearly polarized LP-waves Since E x and E y for weakly guided fibers are neither coupled by the wave equation (4.9) nor for the boundary condition at r = a , eigenmodes can be found, where e.g. E y = 0 . These eigenmodes are called linearly polarized LP-wave, which are defined as e.g. E x = ψ(r, φ) exp(−jβz)

(4.26)

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and

{ ψ(r, φ) =

ψ1 (r, φ) f¨ur r ≤ a

5

(4.27)

ψ2 (r, φ) f¨ur r > a . From the wave equation (4.9), the scalar wave equation can be deduced ( ) △t ψi + k02 n2i − β 2 ψi = 0

(4.28)

With i = 1, 2 and the boundary conditions ψ1 |r=a = ψ2 |r=a ∂ψ2 ∂ψ1 |r=a = |r=a . ∂r ∂r Initially, the following normalizations are introduced: √ Fiber parameter: V = k0 a n21 − n22 = k0 · a · AN ( ) ( ) β β β2 2 β − n · + n − n 2 2 2 k0 k0 k2 k0 − n2 Normalized propagation constant: B = 02 = ≈ (n1 − n2 ) · (n1 + n2 ) n1 − n2 n1 − n22 √ √ Core parameter: u = V 1 − B = a k02 n21 − β 2 √ √ Cladding parameter: v = V B = a β 2 − k02 n22

(4.29) (4.30)

(4.31) (4.32) (4.33) (4.34)

Inserting these normalizations in eq. (4.28), the result is a2 △t ψ1 + u2 ψ1 = 0

for r ≤ a

(4.35)

a △t ψ2 − v ψ2 = 0

for r ≥ a

(4.36)

2

2

Solutions of these differential equations are

( u ) {cos(l · φ)} ψ1 (r, φ) = A1 Jl r (4.37) a sin(l · φ) ( v ) {cos(l · φ)} (4.38) ψ2 (r, φ) = A2 Kl r a sin(l · φ) Here Jl is a Bessel function and Kl is a modified Hankel function of integer order. For low orders, they are shown in figure 4.3 and 4.4. Their derivatives with respect to the argument are given by l l Jl (x) = Jl−1 (x) − Jl (x) x x

dJl (x) dx

= Jl′ (x) = −Jl+1 (x) +

dKl (x) dx

= Kl′ (x) = −Kl+1 (x) +

l l Kl (x) = −Kl−1 (x) − Kl (x) x x

(4.39)

(4.40)

Hence, following from eq. (4.29) and eq. (4.30), they can be written as A1 Jl (u) = A2 Kl (v)

(4.41)

u v A1 Jl′ (u) = A2 Kl′ (v) a a

(4.42)

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Figure 4.3: Bessel function of integer order

Figure 4.4: Modified Bessel- and Hankel function of 0th and 1st order

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Dividing now the two equations by themselves, the characteristic equation to determine the propagation constants β is obtained (u and v contain only β as unknown variable). u · Jl′ (u) v · Kl′ (v) = Jl (u) Kl (v)

(4.43)

By using eq. (4.39) and eq. (4.40), it yields −

u · Jl+1 (u) v · Kl+1 (v) + =0 Jl (u) Kl (v)

(4.44)

with V 2 = u2 + v 2 . For a given V and a given circumferential order l, the propagation constant from eq. (4.44) can be determined numerically. The equation generally has multiple solutions, which are numbered with p = 1, 2, 3.... p denotes the number of the field maxima in the radial direction. Therefore, the term LPlp -wave is chosen, with l for the circumferential and p for the radial order. (LP field distributions of some LPl p-waves are shown in figure 4.5.) With the given dimension (a, λ, n1 , n2 ) V is implied. Thus u and v can be deduced from eq. (4.44). With eq. (4.33) and eq. (4.34) β and therefore from eq. (4.37) and eq. (4.38) the field distribution ψ(r, φ) can be derived. The solution of eq. (4.44) is shown in fig. 4.6. It shows the normalized phase constant B (see eq. (4.32)) as a function of V .

4.1.4 Single mode range and condition In fig. 4.6 it is obvious, that only the LP01 -wave for arbitrarily small V ( V ∼ frequency) is capable of propagation. This wave is also called the fundamental mode. The characteristic equation of the LP01 -wave is, in accordance with eq. (4.44): u · J1 (u) v · K1 (v) − =0 J0 (u) K0 (v)

(4.45)

This equation leads to solutions even for arbitrarily small V s. In this case, however, the wave propagates essentially in the fiber cladding since B ≈ 0 and therefore β ≈ k0 n2 . Hence the wave is badly guided. For a better guidance V should be something bigger, e.g. V > 1.5 . For the fiber parameter 1.5 < V < 2.5 eq. (4.45) is approximately solved by: v = 1, 1428 · V − 0, 996

(4.46)

In the single-mode range the LP11 -wave (compare with fig. 4.6) is not guided. It follows that the characteristic equation of the LP11 -wave from eq. (4.43) with l = 1 is: u · J0 (u) v · K0 (v) + =0 J1 (u) K1 (v)

(4.47)

The limit of the propagation ability of the LP11 -wave is reached when the cladding parameter is v = 0. Taking this condition the core parameter uc can be determined from eq. (4.47). uc J0 (uc ) =0 J1 (uc )



J0 (uc ) = 0

(4.48)

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Figure 4.5: Field distributions of some LP modes. From the top: LP01 , LP11 , LP25 and LP73 (from: Voges/Petermann, Handbuch Optische Kommunikationstechnik)

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Figure 4.6: Normalized phase constant B of LPlp -wave in weakly guided step-index fibers (from: Voges/Petermann, Handbuch Optische Kommunikationstechnik) The core parameter uc corresponds to the first zero of the Bessel function J0 (x). Hence the core parameter is √ calculated to uc (LP11 ) = 2.405 . With V = u2 + v 2 it follows that Vc = 2.405 . This means, that for a fiber parameter V < 2.405 , a single-mode fiber exits (although the remaining LP01 -wave is still propagable in two polarizations (E x and E y )). Typically, a fiber parameter V > 1.5 is chosen because otherwise the wave is too weakly centered on the fiber core. For example these fiber parameters are given: Fiber diameter: Relative refractive index difference: Numerical aperture: Fiber parameter:

2a = 8 µm n1 − n2 ∆= = 3 · 10−3 n1 AN = 0, 116 µm V = 2, 9 λ

With this fiber a single-mode operation is possible for the wavelength range of 1, 2 µm < λ < 1, 9 µm . ψ(r) of the fundamental mode is often approximated by a Gaussian distribution: ( 2) r ψ(r) = A0 exp − 2 . (4.49) w Here w corresponds to the spot radius. Assuming a step-index fiber with V > 1.2 , then wa ≈ 0.65 + 1.619 + 2.879 . Thus with increasing V the V6 V 3/2 spot radius decreases. This corresponds to an increasing concentration of the field in the fiber core.

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4.2 Chromatic dispersion Even with a single-mode fiber it has to be taken into account that the group delay of the LP01 -wave is wavelength dependent (chromatic dispersion), which affects the transmission characteristics. The group delay of the fundamental mode per length is: dβ dω With eq. (4.32) the propagation constant β can be described as: τ=

(4.50)

β = k0 (B(n1 − n2 ) + n2 )

(4.51)

d(k0 (B(n1 − n2 ) + n2 )) (4.52) dω For simplicity, the assumption is made, that the refraction indices n1 and n2 are equally dependent upon ω: ⇒

τ=

dn1 dn2 = dω dω d(n1 − n2 ) =0 ⇒ dω ( ) √ 2 − n2 d n 1 2 dAN ⇒ = ≈0 dω dω Thus the equation of the group delay eq. (4.52) can be simplified to: d(k0 B) d(k0 n2 ) n1 − n2 d(V · B) 1 + = · + N2 dω dω c dV c The chromatic dispersion is the derivative of the group delay with respect to the wavelength: τ = (n1 − n2 )

dτ n1 − n2 V d2 (V · B) =− · dλ | c · λ {z dV 2 } ∧

DW =wave guide dispersion

λ d2 n2 − · 2 | c {zdλ }

(4.53) (4.54) (4.55)

(4.56)

(4.57)



DM =material dispersion

Thus the chromatic dispersion consists essentially of two parts (fig. 4.7): 1. the waveguide dispersion DW and 2. the material dispersion DM . While the material dispersion was discussed in the chapter GRU in detail, the waveguide dispersion arises mainly from the curvature of the B(V ) characteristics. For illustration, the normalized phase constant B, the term d(V · B)/dV , and the term V · d2 (V · B)/dV 2 are shown in fig. 4.8 as a function of V for the LP01 -wave of a single-mode fiber. For single mode fibers with the fiber parameter V < 2, 4 the derivative V ·d2 (V ·B)/dV 2 becomes positive and therefore the wave guide dispersion is negative. At wavelengths of λ > 1, 3 µm the material dispersion DM is positive. This can be used to adjust the dimensions of the fiber, so that the zero of the total dispersion is adjusted to the wavelengths λ > 1, 3 µm . In particular, the zero point of the entire chromatic dispersion for λ ≈ 1, 55 µm can be moved to the point, where the minimum attenuation is achieved. Examples:

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Figure 4.7: waveguide dispersion DW and material dispersion DM of a standard single mode fiber (from: Voges/Petermann, Handbuch Optische Kommunikationstechnik)

Figure 4.8: Dispersion values of the LP01 -fundamental wave with weakly guided step-index fiber (from: Voges/Petermann, Handbuch Optische Kommunikationstechnik)

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1. Standard single-mode fiber for example a standard single-mode fiber has the following dimensions: a = 4 µm , 2 ∆ = n1n−n = 3 · 10−3 , respectively n1 − n2 = 4.5 · 10−3 . For a wavelength of λ = 1.55 µm , a 1 value of V = 1.88 follows. According to fig. 4.8 V · d2 (V · B)/dV 2 = 0.58 . Using these values and ps eq. (4.57), a waveguide dispersion of DW = −5.6 km·nm follows. This dispersion can compensated partially by the material dispersion. For illustration the single components of the chromatic dispersion of such a fiber are shown in fig. 4.7. 2. Dispersion-shifted (single-mode) fibers By varying the fiber parameters, higher fiber dispersion values can be obtained, for example to comps pensate and even to overcompensate for the material dispersion DM = 20 km·nm at the given wavelength λ = 1, 55 µm (e.g. for a so called dispersion-compensating fibers). According to eq. (4.57), higher values of the waveguide dispersion for an increased refractive index difference and lower V 2 values are achievable. For example with the fiber dimensions a = 2.4 µm , ∆ = n1n−n = 5 · 10−3 , 1 respectively n1 −n2 = 7.5·10−3 with λ = 1.55 µm V = 1.46 and V ·d2 (V ·B)/dV 2 = 1.13 is obps tained. Thus a significantly larger magnitude of the fiber dispersion results in DW = −18, 2 km·nm , with which the material dispersion at λ = 1.55 µm can be compensated substantially in order to achieve an improved wave guidance for the required small V -values. The highly light-guiding fiber core is, again, usually surrounded by a refractive index-ring, as shown schematically in fig. 4.9. n

a

r

Figure 4.9: Schematic of the refraction profile for a dispersion displaced fiber