Waveguiding by Total Internal Reflection
When a light ray strikes the boundary of an optically less-dense medium at a specific angle, it undergoes total internal reflection instead of passing through This phenomenon, explained in optics textbooks, is crucial in various applications, such as prisms in binoculars and camera viewfinders, and it also causes the water surface to appear mirror-like to divers.
In the context of refraction, the angle of incidence (α) and the angle of refraction (β) are defined, with β being greater than α when transitioning from a denser medium (like glass) to a less dense medium (like air) However, the angle of refraction cannot exceed 90 degrees, as demonstrated by Snell’s law, which states that the relationship between the sines of these angles is proportional to the indices of refraction of the two media.
F Mitschke,Fiber Optics, DOI 10.1007/978-3-642-03703-0 2, 15 c Springer-Verlag Berlin Heidelberg 2009
16 Chapter 2 Treatment with Ray Optics
Total internal reflection occurs when a ray of light, entering from a denser medium at an angle α, strikes the boundary with a less-dense medium If the angle α is sufficiently large, the light is not refracted but instead reflects entirely back into the denser medium.
In Case 2, the outgoing beam reaches a grazing angle, illustrating a borderline situation where the sine of angle β cannot exceed one Consequently, the critical angle α is determined by the relationship sinα crit = n A n G.
For even larger angles of incidence, the ray is reflected back into the denser medium nearly without loss This is the meaning of the term “total internal reflection.”
The same mechanism can also be used to guide light around bends In
In 1870, English scientist John Tyndall demonstrated a captivating experiment at the Royal Academy that has become a staple in physics education He used a bucket of water with a small hole at the bottom and a light source illuminating the hole from the opposite side As the water cascaded in a parabolic curve, it guided the light, causing the water column to glow dramatically in a darkened lecture hall due to light scattering from surface irregularities.
The demonstration relies on the principle that the refractive index of water (approximately 1.33) is greater than that of air (about 1.0) Additionally, most types of glass have refractive indices ranging from 1.4 to 1.8, which allows them to exhibit similar guiding effects as water when used in strands or rods.
1 Tyndall did not invent this himself The twisted but amusing story leading up to our present-day insights about light-guiding and fibers is reported in [64].
Step Index Fiber
In Boca Raton, Florida, USA, illuminated fountains glow beautifully due to total internal reflection in the water Submerged lamps light the fountain from beneath, while the water column effectively channels the light upward.
Optical fibers are long cylinders made of glass or transparent plastic, typically around 1 mm in diameter, used for specialized lighting applications to direct light into hard-to-reach areas These fibers are commonly seen in decorative lamps that bundle multiple strands together In contrast, standard optical fibers for data transmission feature a more intricate internal structure.
Step index fiber is a commonly used type of optical fiber characterized by its internal structure, which features a circular cross-section core surrounded by a ring-shaped cross-section cladding The core is made of glass with a slightly higher refractive index than the cladding, allowing light to be effectively guided within the core This two-layer design offers a significant advantage over simpler fiber structures, as the external surface of the fiber, where it meets the cladding, does not interfere with the light-guiding process Consequently, even if the fiber surface becomes soiled or comes into contact with other glass, its functionality remains intact In contrast, unstructured fibers would experience significant signal loss if exposed to substances like oil, as their refractive indices could disrupt the total internal reflection necessary for effective waveguiding.
18 Chapter 2 Treatment with Ray Optics
Figure 2.3: Sketch for calculating total internal reflection.
In step index fiber, the outer surface is irrelevant for waveguiding, allowing us to simplify the discussion by assuming an infinitely wide cladding diameter This perspective enables us to analyze the maximum angle a light ray can take relative to the fiber axis while remaining guided within the fiber.
In step index fibers, we identify three refractive indices near the fiber end face: n K for the core, n M for the cladding, and n A for the ambient air, with the relationship n K > n M ≥ n A This discussion focuses specifically on the properties of step index fibers, while noting that the equality holds true for unstructured fibers.
We apply Snell’s law: n Asinα = n Ksinβ, (2.1) n Ksinγ = n Msinδ (2.2)
When we assume the fiber axis to be perpendicular to the front face,β+γ=π/2 and hence sinβ= cosγ Then sinβ
The limiting angle for total internal refraction is defined by sinδ max= 1 ⇒ sinγ max=n M /n K (2.4)
2 These indices suggest the German words Kern (core) and Mantel (cladding), respectively.
The original German edition of this book retains the terms "core" and "cladding" in English, as they share the same initial letter and do not offer a superior alternative Additionally, the related English words "kernel" and "mantel" effectively convey the concepts of a central part and an enclosing structure, respectively.
2.2 Step Index Fiber 19 Inserting Eq (2.4) in (2.3) and that in (2.1), we obtain n Asinα max=n K
We may assumen A= 1 for air Then the limiting angleα maxfor rays to be guided is α max= arcsin n 2 K −n 2 M
The argument of arcsin bears a special name: it is called thenumerical aperture, often abbreviated asNA:
(The word “aperture” derives from Latin apertus = open and indicates some form of opening We also find it in “aperitif,” the opening of a meal, and in
“overture,” the opening of an opera or a romantic affair “Numerical” here indicates a dimensionless number.)
The numerical aperture (NA) quantifies the refractive index difference between the core and cladding of a fiber optic The maximum acceptance angle for rays entering the fiber is represented by sinα max = NA, while within the fiber, it is given by sinα max = NA/n K In linear optics, the principle of reversibility allows the acceptance cone to also represent the exit cone of light at the opposite end of the fiber This relationship is visually illustrated in Fig 2.4, showcasing the input and output cones.
We use the opportunity to introduce another frequently used quantity which is also a measure of the index difference between core and cladding: Δ =n 2 K −n 2 M
The conversion betweenNAand Δ is given by
Usually the index difference is quite small (a few tenths of 1%) so that Δ can be simplified as Δ ≈ (n K −n M )(n K +n M ) n K(n K+n M)
This last relation justifies that Δ is callednormalized refractive index difference or normalized index step.
In single-mode fibers with a Δ of 0.3% and a refractive index (n K) of 1.46, the numerical aperture (NA) is calculated to be 0.11, corresponding to an acceptance angle of approximately ±7° Light rays entering the fiber within this acceptance cone are effectively guided through the core, while those entering at steeper angles are lost to the cladding, which typically has higher loss This results in some light being dissipated, while the remaining light eventually reaches the outer surface, where a protective plastic coating is commonly applied.
20 Chapter 2 Treatment with Ray Optics
The acceptance and exit cone of a fiber, as illustrated in Figure 2.4, is depicted schematically; however, in practice, the cone's boundaries are not as sharply defined Light rays that exit the core experience significant optical loss, leading to the conclusion that once these rays leave the core, they are irretrievably lost.
In this chapter, we explore ray optics without favoring any specific angle within the cone However, the subsequent chapter will introduce wave optics, demonstrating that only a discrete set of angles, known as modes, is physically viable within this continuum These modes are crucial for understanding the waveguiding properties of optical fibers, which we will examine in detail later Notably, as the number of modes increases, our ray-optical approximation becomes more valid, leading us to focus on multimode fibers in our ongoing analysis.
Modal Dispersion
Different modes of light, entering a fiber at various angles, travel distinct path lengths to reach the fiber's end, resulting in varied arrival times This phenomenon is referred to as modal dispersion, as illustrated in Figure 2.5.
Figure 2.5: Modal dispersion: rays propagating at an angle with the fiber axis travel a longer distance than those remaining parallel to the axis This leads to different arrival times.
In a fiber of length L, the path length of a beam propagating at an angle β with the axis is given by L = L/cosβ It is important to note that sinβ cannot exceed NA/n, where NA represents the numerical aperture and n is the refractive index Consequently, this establishes a limit on the angle β.
3 There is one subtle exception to this otherwise reliable rule: so-called whispering gallery modes will be described in Sect 7.5.
2.3 Modal Dispersion 21 approximate sinβ≈β and cosβ≈1−β 2 /2 Then we obtain
With a Δ of 0.3%, the length L is extended by 3 parts in 1000 compared to the original length Consequently, the path difference between the straight beam and the maximally inclined beam accumulates to one full wavelength after traveling a distance equivalent to 333 wavelengths.
In an interference experiment, destructive interference occurs after a path difference of half a wavelength, resulting in the cancellation of both rays over a distance of 167 wavelengths, which is just a fraction of a millimeter in fiber length However, due to the angled propagation of the rays, they do not completely cancel each other, instead creating a fringe pattern of alternating bright and dark stripes across their cross-section When averaged across that cross-section, the overall interference effect results in cancellation.
Light signals transmitted through fiber can take various paths, leading to a phenomenon known as delay distortion, which results in differing propagation times This effect is particularly pronounced with short light pulses, causing them to broaden When the pulse duration increases to the point where it overlaps with adjacent pulses, intersymbol interference occurs, potentially distorting the transmitted message to the point of being unreadable.
A rough estimate highlights a significant issue regarding light propagation in fiber optics For a step index fiber with core index nK, the propagation time τ for a length L can be calculated as τ = nK L/c Notably, the minimum travel time for a ray along the fiber's axis is τ min = nK L/c.
Meanwhile, the ray traveling at the maximal angle takes the longest time: τ max =n K L c (1 + Δ) =τ min (1 + Δ).
In comparison, the differenceδτ =τ max −τ min is δτ= n K L c Δ =τ minΔ.
This shows the simple result that the relative amount of propagation time scatter is given by Δ: δτ τ min
Let us again take Δ = 0.3 % as a typical value In a fiber of 1 km length, the arrival times will spread over ca 15 ns.
This is just a rough estimate, of course: we used approximations and we have neglected that in addition to meridional rays there can also be helical rays
4 By doing so, we momentarily ignore the distinction between phase and group velocities.
22 Chapter 2 Treatment with Ray Optics a) b)
Figure 2.6: Light guiding by total internal reflection in a fiber There are meridional and and helical rays Meridional rays (a) propagate in a plane, helical rays (b) on a twisted path.
The maximum data rate, approximately 70 MHz in our example, indicates the threshold beyond which the transit time spread can negatively impact signal integrity.
The relatively low rate of modal dispersion and the short distance of 1 km highlight the challenges faced by fiber optics in practical applications However, solutions exist to mitigate these issues, such as utilizing gradient index fibers or, for more demanding applications, single-mode fibers In this article, we will explore both options in detail.
Gradient Index Fibers
To minimize the dispersion of arrival times in optical fibers, employing a gradient index profile instead of a step index profile is effective This approach allows the refractive index to vary with radial position, expressed as n(r) = n K.
1−2Δ (r/a) α : |r| ≤a n M : |r|> a, (2.7) whereadenotes the core radius The resulting profile is sketched for selected values of the profile exponentαin Fig 2.7.
The optimal index profile minimizes transit time differences, typically achieved with α=2 In a parabolic index profile, fiber rays travel along a curved path instead of a zigzag, which, although longer geometrically than the straight path along the axis, compensates with a lower index away from the axis, resulting in an equivalent optical path.
Now one obtains for the scatter of transit times α=∞: δτ = n K L c Δ as above α= 2 : δτ = n K L c Δ 2
Mode Coupling
Figure 2.7: Some common index profiles, as described by Eq (2.7) Forα= 2 the profile is parabolic For α → 1 the profile becomes triangular, and for α→ ∞, rectangular (step index profile).
The transit time spread for a gradient index fiber with a parabolic profile has significantly improved, decreasing by approximately three orders of magnitude Consequently, modal dispersion is now limited to just a few tens of picoseconds per kilometer.
Calculating the optimum profile exponent in optical fibers is complex due to the abrupt transition from the core's index profile to the cladding's constant index Research indicates that the ideal value is not precisely α = 2 but varies based on factors such as glass type, doping material, and wavelength This variation is influenced by profile dispersion, where the refractive index difference (Δ) changes with wavelength because the dependencies of the core (nK) and cladding (nM) on wavelength differ Additionally, the optimal exponent varies for meridional and helical rays, depending on the specific combination of excited modes.
The theoretical optimum's significance diminishes due to unavoidable manufacturing tolerances in fiber production, which hinder the ability to achieve precise target values Consequently, any enhancements to the parabolic index profile by refining the index profile yield only marginal improvements.
In multimode fiber, the distribution of power among different modes can change as light travels through the fiber, particularly when the fiber is bent This bending causes mode coupling, and even slight movements or temperature variations can alter the mode partition While this alteration is typically inconsequential if the detector accurately measures the total power, inconsistencies in detector sensitivity can lead to certain modes being detected more strongly Consequently, random fluctuations in the mode partition may result in variations in the received power, a phenomenon known as mode partition noise.
As the mode partition fluctuates, the variability in transit time is reduced, making it less probable for a photon to travel the entire distance in either the fastest or slowest mode Instead, it is more likely to engage in a random walk between different modes, resulting in an averaging effect.
24 Chapter 2 Treatment with Ray Optics that the fiber length exceeds a certain minimum called the coupling length
L coupl, the temporal spread does not grow in proportion to distanceL, but only as√
L A typical value for the coupling length is on the order of 100 m for step index fibers and a few kilometers for gradient index fibers.
Mode mixing effectively reduces modal dispersion in optical fibers This enhancement can be achieved through enforced mode mixing, utilizing mechanical devices known as mode mixers that deform and bend the fiber.
Interestingly, a fiber can achieve greater bandwidth when composed of multiple segments rather than a single continuous piece While one might assume that irregularities at the joints, known as fiber splices, would hinder performance, research indicates that this is not the case.
Bending a fiber affects how light is guided within it, as the rays hit the core-cladding interface at varying angles This alteration can lead to some light loss, particularly when the angle exceeds the threshold for total internal reflection.
Shortcomings of the Ray-Optical Treatment
Current treatment methods are inaccurate, as they assume light behaves like rays reflecting at a perfect mirror on the core-cladding interface However, light is fundamentally a wave phenomenon, causing it to partially penetrate into the second medium, typically to a depth comparable to its wavelength This results in a longer ray path and introduces an additional phase shift known as the Goos–Hänchen shift.
In optical fibers with core diameters comparable to the wavelength of light, significant corrections are necessary Instead of incorporating these corrections into a ray-optical approach, we will adopt a wave-optical treatment in the following chapter Wave optics reveals that light can penetrate into the cladding and that the exit cone lacks a sharp boundary Additionally, it introduces the concept of fiber modes, which are discrete distributions of the electrical field in the fiber's cross-section, indicating that rays can only make specific angles with the axis rather than any angle within a range.
This chapter begins with Maxwell’s equations, leading to the derivation of a wave equation that is applied to fiber geometry, ultimately revealing the modal structure We will focus on step index fibers, noting that closed solutions are also available for gradient index fibers without cladding To enhance clarity, we will utilize various approximations to highlight key concepts rather than delving into intricate details.
Maxwell’s Equations
In MKS units of measurement, Maxwell’s equations are [75]
J current density (A/m 2 ) ρ charge density (As/m 3 )
Many textbooks focus on processes in a vacuum, but this approach is not practical for our needs To accurately describe processes occurring within a material, it is essential to utilize quantities derived from the material's properties.
F Mitschke,Fiber Optics, DOI 10.1007/978-3-642-03703-0 3, 25 c Springer-Verlag Berlin Heidelberg 2009
26 Chapter 3 Treatment with Wave Optics
Polarization and magnetization refer to the alteration of atomic orbitals caused by electromagnetic fields Conductivity, which involves the movement of electric charges—since magnetic charges do not exist—typically manifests as a tensor in general cases.
0 vacuum permittivity (dielectric constant of free space), μ 0 vacuum permeability (permeability constant of free space).
The numerical values are given by
Two combinations have special relevance: the product μ 0 0= 1/c 2 , wherec= 2.99792458×10 8 m/s is the speed of light in vacuum, and the ratio μ 0 / 0 4πc
Z 0 ≈ 377 Ω is the vacuum impedance and denotes the amplitude ratio of the electric and the magnetic part of the electromagnetic wave:
In air and glass we may simplify as follows: ρ= 0 There are no free charges (Approximation 1)
J= 0 There are no currents (Approximation 2)
M= 0 There is no magnetization (Approximation 3)
Wave Equation
Hence, of all properties of the material, we retain only the ones which influence the polarization Using these approximations, Maxwell’s equations are reduced to
We rearrange the RHS using Eqs (3.10) and (3.5) and obtain
We thus find the wave equation
A fully analogous equation can be derived for the magnetic field.
The relationship between polarization (P) and electric field strength (E) is influenced by the material properties We assume that polarization responds instantaneously to changes in field strength, surpassing any other relevant time scales Consequently, we can express polarization as a function of the electric field strength.
In this section, we present an additional approximation by assuming that the polarization of the material is consistently parallel to the electric field strength (Approximation 5) This assumption is justified, as in a homogeneous medium, the susceptibility tensor χ (i) simplifies to a scalar While some optical crystals exhibit nonhomogeneous properties, glass is considered homogeneous due to its structural characteristics (refer to Sect 6.1.2).
In a fiber, the homogeneity is only slightly perturbed due to the refractive index
28 Chapter 3 Treatment with Wave Optics profile On the other hand, wave guiding essentially occurs parallel to the axis.
In optical arrangements, the paraxial approximation allows for the assumption that light propagation occurs at small angles relative to the axis This results in the minimal impact of the index change between the core and cladding, as both the electric (E) and magnetic (H) fields remain perpendicular to the interface and are proportional to each other The proportionality constant, known as the impedance, is defined as Z₀ in free space.
This book primarily employs scalar approximation due to its simplicity compared to the more complex vectorial treatment, which has minimal impact on results We will also highlight the differences between the modes derived from scalar treatment and the hybrid modes obtained from vectorial calculations.
We return to the wave equation, in which we can now introduce a simplifi- cation Given that nowEP and thusDE, it follows that∇ ãD=∇ ãE= 0.
On the LHS of the wave equation, the term with∇ ãE then disappears and it remains
Linear and Nonlinear Refractive Index
Linear Case
In many situations, it is well justified to truncate the series expansion of
Eq (3.18) after the linear term
This is the linear approximation (Approximation 6); it is valid for low light intensities Due to Eq (3.5) we then get
The expression inside the bracket is the relative dielectric constant
In this study, we focus on the propagation of light in highly pure, low-loss glass, where the refractive index is denoted as 'n' and Beer’s coefficient of absorption is represented by 'α' We aim to justify the use of the low-loss approximation, which assumes that α is approximately zero.
7), this is it Then, is real and is given by
On the RHS of Eq (3.19), we insert the relation (3.20) betweenE andP and then obtain
3.3 Linear and Nonlinear Refractive Index 29
This is the linear wave equation, as it is obtained directly from Maxwell’s equa- tions using Approximations 1–7 An analogous equation
∂t 2 H (3.24) can be found by similar procedure for the magnetic component of the wave.From now on we will drop vector symbols (arrows) for convenience.
Nonlinear Case
To fully understand complex physical processes relevant for advanced applications, it is essential to extend the serial expansion beyond the linear term When the electric field (E) is sufficiently large, the limitations of the linear approximation become apparent, leading us into the field of nonlinear optics, where significant experimental observations can be made.
The primary focus of this article is the interaction of light with glass, a material characterized by its isotropic statistical structure and inversion symmetry, leading to χ(2) = 0 Consequently, the first significant term in the series expansion is χ(3), while higher-order terms can generally be disregarded due to their small coefficients, which only become relevant at extremely high intensities Therefore, our discussion will center on the χ(3) term, which can have a substantial impact on the behavior of light in glass.
As before, we keep the low-loss approximation, so that the only conceivable effect is a modification of the refractive index In the linear case we had
To clarify, we will refer to the linear case as "linear" and denote the refractive index in this context as \( n_0 \), which will be termed the small signal refractive index In contrast, the nonlinear case will be addressed separately.
=n 2 = 1 +χ (1) +χ (3) E 2 = linear+χ (3) E 2 (3.26) This is the same as
Since the nonlinear contribution to the refractive index is a small correction, we obtain n=n 0
30 Chapter 3 Treatment with Wave Optics with ¯ n 2=χ (3)
The numerical value of ¯n 2 for fused silica is slightly influenced by frequency and dopants, but these effects are minimal, allowing us to use a standard value of 10 − 22 m 2 /V 2 The intensity (I) of a light field, which represents power per area, is proportional to the square of the field amplitude Consequently, it is common to express the refractive index as n = n 0 + n 2 I, where I is defined as (n 0 /Z 0)E 2 and n 2 is approximately 3×10 − 20 m 2 /W.
We see that inclusion of theχ (3) term results in a modification of the refrac- tive index: The index always depended on wavelength, but now it also depends on intensity.
Under conditions that one would consider “reasonable”, this modification is tiny indeed: Even an irradiated power of 1 kW, focused down to a spot of
100μm 2 , will result in an increase of the index of only n 2 I= 3×10 − 20 m 2 /W 10 3 W
The change in the index difference is minor compared to the core-cladding index difference of a fiber, making it negligible when analyzing the field distribution Equation (3.23) holds true; in the linear scenario, n can be equated with n0, while in the nonlinear case, n(I) is represented as n0 + n2 I This nonlinearity's significance will be further explored in Chapter 9, particularly in relation to the phase evolution of a light wave.
Separation of Coordinates
In this section, we simplify our analysis by leveraging the unique geometry of fibers, characterized by a circular cross-section and an elongated form along the longitudinal axis This approach naturally leads us to employ cylindrical coordinates, specifically r, φ, and z, with the positive z direction indicating the propagation path Notably, the Laplacian operator in cylindrical coordinates is well-established and plays a crucial role in our calculations.
We introduce the following ansatz for the optical field of the light wave:
N =N(r, φ) is the field amplitude distribution in the plane normal to thez-axis,
3.4 Separation of Coordinates 31 denotes a running wave with wave numberβ, and
The monochromatic wave is represented as T = T(t) = e^(iωt), where ω is the angular frequency This separation is valid due to the linearity of the system, allowing us to factor out E₀, and the paraxial approximation, which ensures that the electric and magnetic field components are predominantly perpendicular to the direction of propagation, thereby decoupling longitudinal and transverse processes We denote the propagation constant as β, distinguishing it from the wave vector's longitudinal component to facilitate propagation analogous to rays that form an angle with the fiber axis By employing cylindrical coordinates and this approach, the wave equation is expressed in a specific form.
(3.36) Obviously, all terms contain the constant factorE 0, which is thus cancelled out. The physical reason, again, is the linearity assumed here.
Partial derivatives act differently uponN,Z, andT The first term can be rewritten as
, which can be simplified to yield
On the RHS we obtain
We will denote the vacuum wave number byk 0 Inside a medium with refractive indexn, we will writek=nk 0=nω/c Then, the RHS becomes
The factor ZT is common to all terms and is therefore eliminated, resulting from the problem's homogeneity in both space and time Consequently, we focus on the field amplitude distribution in the plane perpendicular to the propagation direction Given that typical fibers have a circular cross-section, further factorization is advantageous.
32 Chapter 3 Treatment with Wave Optics
When we now reinsert all terms and multiply withr 2 /RΦ, we obtain r R
∂φ 2 Φ−r 2 β 2 =−k 2 r 2 , (3.38) which after sorting of terms becomes
The left-hand side (LHS) contains Φ and not R, while the right-hand side (RHS) features the opposite Consequently, both sides must equal a constant, which we will refer to as m² This leads us to two independent equations governing the azimuthal and radial components of the field amplitude distribution.
R=Rm 2 , which, by using the abbreviationκ 2 =k 2 −β 2 , is written in simpler form as r 2 ∂ 2
To understand the significance of the quantities κ, k, and β, we can refer to the ray-optical description, which involves rays propagating at a small angle relative to the axis In this context, the propagation constant k is associated with the wave, while β represents its component in the direction of propagation Consequently, κ can be viewed as the transverse component of the wave.
Modes
The azimuthal structure equation, Eq (3.40), has a general solution represented as Φ = c₀cos(mφ + φ₀), where c₀ and φ₀ are constants It is essential for Φ and its derivative ∂Φ/∂φ to remain continuous at φ₀ and φ₀ + 2π, which necessitates that m be an integer, thereby limiting the number of potential solutions for the radial structure equation, Eq (3.41) When substituting y = R and x = r, Eq (3.41) takes the form x²y + xy + (κ²x² - m²)y = 0, which is recognized as Bessel’s differential equation For integer values of m, this equation is solved by y = c₁Jₘ(κx) + c₂Nₘ(κx).
3.5 Modes 33 wheneverκxis real (i.e.,κ 2 x 2 ≥0), or by y=c 3I m (κx) +c 4K m (κx), (3.44) wheneverκx is imaginary (i.e., κ 2 x 2 < 0) Here, functions J m (κx), N m (κx),
I m (κx), and K m (κx) are Bessel functions Properties of Bessel functions are given in Appendix C.
To determine the coefficients c1 to c4, it is essential to define the specific geometry of the fiber Previously, we assumed a cylindrical geometry with a minimal refractive index difference; however, we now specify that we are focusing on a step index fiber This choice is not only straightforward but also realistic In a step index fiber, the refractive index is defined as nK for r ≤ a (within the core) and nM for r > a (in the cladding).
In waveguiding, the refractive index in the core (K) is greater than that in the cladding (M), which is essential for effective light confinement To differentiate between the refractive indices and wave numbers in these two regions, we use the indices "K" for the core and "M" for the cladding, as previously defined in Chapter 2.2 The index "0" represents the corresponding values in a vacuum, such as the relationship k K = n K k 0.
In scenarios where the wavelength is significantly smaller than the geometric dimensions of the fiber's cross-section, light behaves according to ray optics principles, primarily guiding within the core Consequently, we focus on solutions that concentrate the majority of the light wave in the core, while the field amplitudes in the cladding diminish with distance from the center, ensuring they do not play a significant role in the guided wave.
For effective wave propagation, the wave number (κr) must be real within the core and imaginary in the cladding Specifically, in the core, the condition k ≥ β must be satisfied, indicating that the wave number is greater than or equal to its longitudinal component Conversely, in the cladding, the scenario may be reversed, leading to solutions of Bessel’s equation characterized by transverse standing waves in the core and radially decaying waves in the cladding.
The field amplitude distribution should not exhibit singularities, which necessitates that the coefficient in the N m term for the core equals zero Similarly, in the cladding, the coefficient of the I m term must also be zero This aligns with physical expectations, as we anticipate the field amplitude to decay at least as quickly as 1/r away from the core; otherwise, the integral of power across the entire transverse plane would become infinite.
To achieve κr real in the core, the condition atr ≤ a must be met, alongside (κr)² ≥ 0 or k²K - β²r² ≥ 0 Conversely, in the cladding where atr > a, the requirement is (κr)² ≤ 0, leading to k²M - β²r² ≤ 0 Collectively, these conditions indicate that kK must be greater than or equal to β, which in turn must be greater than or equal to kM.
The range of possible wave numbers for propagation down the fiber is thus constrained by the requirement of waveguiding.
Once again we introduce abbreviations: The transversal components of the wave number in core and cladding are given by κ 2 K = k K 2 −β 2 , κ 2 M = − k M 2 −β 2
In wave optics, it is common to express certain quantities in relation to the core radius \( a \), represented by the equations \( u = \kappa K a \) and \( w = \kappa M a \) Here, \( u \) and \( w \) are dimensionless, positive real numbers frequently referenced in academic literature.
The variable \( u \) represents the radial phase constant, while \( w \) denotes the radial decay constant, which together illustrate the progression of phase and the transverse decay of amplitude.
In order to establish a relation between these somewhat abstract quantities and measurable quantities, we use the following relation betweenuandw: u 2 +w 2 k 2 K −β 2 a 2 − k M 2 −β 2 a 2 (3.48) k 2 K −k 2 M a 2 (3.49)
It is clear thatu 2 +w 2 equals a constant This constant is of central importance and is callednormalized frequencyor simplyV number It is given by
A step index fiber is defined by its core radius (a) and the refractive indices (nK and nM), or alternatively, by its numerical aperture The description of the experimental situation is completed by either the wavelength or the vacuum wave number The critical factor for determining the number of modes that a fiber can support is the value of V.
Using uand w, the general solution of the wave equation for a step index
fiber can be written as:
A ray-optical approach suggests a rectangular field distribution, with a constant amplitude of 100% in the core and 0% in the cladding However, it is clear that the actual behavior deviates from this idealized expectation.
For a seamless connection between both solutions, it is essential that the transition remains continuous and differentiable, avoiding any discontinuities This requirement necessitates identical angular dependence for both solutions, which is why we have specified the same φ₀ and m in Eq (3.52).
The conditions for smooth transition are
Solutions for m = 0
In the second of these equations we can introduce
∂(ur/a) (3.57) and thus use the argumentur/athroughout; then we can write the derivative atr=aas
(the prime ( ) denotes the derivative with respect to the argument).
For the existence of a solution, it is required that the determinant of coeffi- cients be zero:
The recursion relations for Bessel functions, expressed as \( uJ_m(u) = mJ_m(u) - uJ_{m+1}(u) \) and \( wK_m(w) = mK_m(w) - wK_{m+1}(w) \), allow for the immediate elimination of \( C K C M \) Consequently, this approach facilitates the removal of derivatives effectively.
= K m (w) mJ m (u)−uJ m+1(u) wJ m (u)K m+1(w) = uK m (w)J m+1(u) or
The relationship between the variables \( u \) and \( w \) allows us to identify the permissible solutions for the fundamental mode of the fiber It is evident that the functions \( K_m(w) \) on the right-hand side are consistently positive, while the functions \( J_m(u) \) on the left-hand side often fluctuate in sign This observation indicates specific combinations of argument values, and consequently \( V \) numbers, for which viable solutions exist.
Explicit solutions are best obtained numerically However, for our present purpose, we can inspect some special cases which will give us insight without invoking a computer.
Let us first consider the case ofm= 0, which stands for rotationally symmetric field distributions over the fiber’s cross-section Then the equation is reduced to J0(u) uJ1(u) = K0(w) wK1(w) (3.63)
For a survey of possible solutions, we use the following table in which LHS andRHS of Eq (3.63) are juxtaposed:
36 Chapter 3 Treatment with Wave Optics u J0 J1 J0 /uJ1 K0 /wK1 w
The analysis reveals an alternation between ranges with solutions and those without, as indicated by the algebraic sign in the table However, we can enhance our understanding by calculating the values associated with the solution branches and visualizing them in the (u, w) plane, as demonstrated in Fig 3.1 The resulting curves, which represent the ratios of two Bessel functions, exhibit a shape akin to that of a tangent function This observation aligns with our findings in Appendix C, where we note that the Bessel function J0 resembles a cosine function, while J1 resembles a sine function.
Figure 3.1: Solutions of the eigenvalue equation form = 0 in the u–w plane.
Asuincreases, there is an alternation between regimes with existing (e.g., 0≤ u ≤ 2.405) or nonexisting (e.g., 2.405 ≤ u ≤ 3.832) solutions Labels at the branches denote indicesmpof the LP mp modes, as explained in Sect 3.9.
We can graph the locus of all points associated with a specific V number, which results in circular segments in the (u, w) plane Each fiber produces circles of varying radii that represent experiments conducted at different wavelengths.
Solutions for m = 1
other way to look at it is that for a given wavelength, different radii correspond to different core diameters.
The intersection points between the tan-like branches and the circle segments indicate potential solutions, representing combinations of u and w for a fixed value of V The first solution branch is present for any V value ranging from zero to infinity, while the second branch is only valid above a minimum V of 3.832, determined by the first zero of the Bessel function J1 Similarly, all other branches are characterized by a minimum V, which is also defined by the zeros of J1.
We might proceed by directly inserting m = 1 into the eigenvalue equation
To better understand the situation without using a computer, we can utilize an alternative recursion relation of Bessel functions that incorporates m−1 instead of m+1 This approach allows us to derive an equation for m=1 that solely involves J0 and J1, simplifying our analysis and enhancing clarity.
So let us use uJ m (u) = −mJ m (u)−uJ m − 1(u) and wK m (w) = −mK m (w)−wK m−1 (w) in Eq (3.59) to obtain
In precise analogy to the procedure shown above we can again write a table to locate the possible branches of solutions: u J1 J0 −J1 /uJ 0 K1 /wK0 w
The analysis reveals an alternating pattern of permitted and forbidden regimes, with transitions occurring at the zeroes of Bessel functions Unlike the case for m = 0, the roles of these regimes are reversed here, leading to a new branch of solutions that starts at V = 2.405, representing a second mode in addition to the fundamental mode Figure 3.2 presents a comprehensive overview of all solutions identified to date, including those for m = 0 and m = 1.
38 Chapter 3 Treatment with Wave Optics
Figure 3.2: Solutions of the eigenvalue equation form = 0 andm = 1 in the u–w plane for m = 1 Now there are branches of solutions where there were gaps in Fig 3.1.
Solutions for m > 1
At larger m values one again finds that allowed and forbidden ranges alter- nate, with the transitions occurring whereV equals zeroes of Bessel functions. Figure 3.3 shows all modes up toV = 8.
We wrap up what we have learned:
ForV λ 0 ,D is positive His- torically, the visible range was investigated first and therefore the trend observed there was considered “normal.” Then, the case D 0 is called
“anomalous dispersion.” If the fiber is used in the second window near 1.3μm, there is a minimum of the dispersion (D≈0), while in the third window around 1.5μm there is anomalous dispersion.
This article focuses exclusively on group velocity dispersion, a specific type of dispersion Similar terminology applies to the dispersion of the refractive index, which also distinguishes between "normal" and other categories.
“anomalous” dispersion “Normal” refers to the case that the index decreases
Impact of Dispersion
In the transparent range of most materials, dispersion typically occurs toward longer wavelengths, while anomalous dispersion arises near atomic resonance frequencies However, some authors may not clearly specify the type of dispersion they are discussing, leading to potential confusion.
The waveguide contribution to the total dispersion is negative through- out the visible and near infrared A typical value for standard fibers is
At long wavelengths, the dispersion in standard fibers contrasts with material dispersion, resulting in a shift of the zero-dispersion wavelength toward longer wavelengths by approximately 20–30 nm compared to bulk fused silica The CCITT standard, established in 1984, specifies that the dispersion of fibers used for telecommunication must adhere to certain limits.
|D| ≤3.5 ps/nm km for 1,285 nm≤λ≤1,330 nm,
|D| ≤20 ps/nm km atλ= 1550 nm.
Near 1.55μm, a value of D = 18 ps/nm km is typical (According to
Eq (4.27) this corresponds to β 2 = −23 ps 2 /km.) This value will generate a propagation time difference between two wavelength components that are
At a distance of 10 km, a separation of 1 nm results in a time delay of 180 ps The zero-dispersion wavelength, located near 1300 nm, minimizes propagation time spread in the second window Unlike the second-order term, third-order dispersion shows less variation with wavelength, typically measured at S(λ 0) = 0.085 ps/nm² km, which corresponds to β 3(λ 0) = -0.08 ps³/km.
The intentional adjustment of the zero-dispersion wavelength through waveguide dispersion enables the creation of custom-designed fibers These specialized fibers can operate in the infrared spectrum, extending beyond the zero-dispersion point of pure fused silica.
Consider the propagation of a light pulse which we think of as being generated by taking a monochromatic oscillation
Eˆcos(ωt−βz) and multiply (modulate) it with anenvelopefunction For the latter a reasonable choice is a Gaussian: e − t 2 2T 2
The temporal profile of the intensity (irradiance) or power of a pulse so generated is
The International Telecommunication Union's Telecommunication Standardization Sector (ITU-T), formerly known as the International Telegraph and Telephone Consultative Committee, is a specialized agency of the United Nations focused on information and communication technology matters.
In optical pulse characterization, the peak intensity (I₀) and pulse duration (T₀) are critical parameters While T₀ is defined as the time interval from the peak intensity to the point where it decreases to 1/e of its maximum value, many practitioners prefer to use the full width at half maximum (FWHM), denoted as τ, which measures the duration between points where the intensity reaches half of its peak value For Gaussian pulses, the relationship between τ and T₀ is given by τ = 2√ln(2)T₀ It is important to note that both pulse duration and peak power are altered after the pulse propagates over a fiber length (L).
|β 2 | (4.33) is a characteristic length called the dispersion length After distance L D, the pulse duration has increased by √
2 After considerably longer distance, the pulse duration grows in proportion to distance as
The relationship between diffraction and dispersion highlights how a narrow fan of light rays spreads transversely, while a short light pulse disperses longitudinally In far-field diffraction, known as Fraunhofer diffraction, the spread of light increases with distance at a constant divergence angle, with the shape of the light rays determined by the Fourier transform of the initial form In contrast, near-field diffraction, or Fresnel diffraction, presents a more complex scenario; however, Gaussian shapes maintain their form, aside from scale factors, demonstrating their unique behavior during this transformation.
This close analogy becomes especially clear when we replace the Gaussian envelope of Eq (4.30) with a rectangular envelope
While the proposition may not be entirely realistic, it closely resembles diffraction at a slit Initially, undulations form near steep slopes and, as they propagate, they begin to spread out After some oscillation and interference, the pulse shape ultimately approaches the functional form of (sin(x)/x)² This highlights the clear connection to diffraction at a slit and illustrates the transition from near field to far field.
In Eq (4.33) we used β 2 and T 0 However, experimentalists and techni- cians often prefer to use the dispersion parameter D and the full width at
4.4 Impact of Dispersion 57 time position power
The dispersive broadening of a Gaussian pulse is illustrated in Figure 4.5, where a pulse with an initial full width at half maximum (FWHM) of 0.5 picoseconds propagates over 21 meters, resulting in an increase in width As the pulse travels, its peak height diminishes due to the conservation of energy, with fiber dispersion parameters set at β2 = -18 ps²/km and β3 = 0.
Dispersive broadening of a rectangular pulse demonstrates how steep slopes of the initial shape are significantly altered by dispersion This academic example highlights the relationship between the pulse width and the half maximum duration, represented by τ The conversion formula τ₀ = τ(L = 0) = 2√ln(2)T₀ is essential for understanding this phenomenon.
|β 2 |=|D|λ 2 /(2πc) as given above, we can write the relevant term in Eq (4.32) as follows:
The first fraction on the right-hand side indicates the fiber parameters (L, D) along with the light signal characteristics (λ, τ₀) The second fraction consists solely of constants, making it situation-independent, with a value of 1.4709×10⁻⁹ s/m By substituting L into the equation, relevant calculations can be performed.
58 Chapter 4 Chromatic Dispersion km,Din ps/nm km,λinμm, andτ 0in ps, units combine to give an additional numerical factor of 10 9 , and we can write τ L =τ 0
The equation features directly measurable quantities in standard technical units, making it practically valuable The "magic number" of 1.47 is specifically applicable to Gaussian pulses, while other pulse shapes, like the hyperbolic secant squared (sech²), commonly found in solitons, necessitate a different value of 1.87.
Dispersive broadening restricts the information-carrying capacity of optical fibers, as pulses must be sufficiently spaced apart The optimal capacity occurs at the lowest dispersion, specifically at the zero-dispersion wavelength, which is why many installed fibers operate around the 1.3μm range However, this conclusion is based on a linear approximation, applicable only at low powers or intensities Nonlinear effects can alter this outcome, potentially maximizing capacity under different conditions.