Expression for Numerical Aperture of an Optical Fibre

Optical Fibre Structure: 

It is made up of transparent dielectrics (SiO₂), glass or plastics. An optical fibre consists of a central core glass (50μm) surrounded by a cladding (125 – 200 μm) which is slightly lower refractive index than cone.

The cladding is enclosed by polyurethane jacket. It act as protective layer, which is used to make the optical cable to withstand against chemical reaction with the surrounding and against crushing etc.

Total Internal Reflection (TIR):

The principle of optical fibre communication is total internal reflection takes place when it satisfies the following two conditions.

Condition 1: Light should travel from denser medium to rarer medium. ie, η₁ > η₂

η₁ – RI of core

η₂ – RI of cladding

Condition 2: The angle of incidence on core should be greater than the critical angle.

ie, Φ > Φc

Acceptance Angle:

The maximum angle at which the light can suffer total internal reflection is called acceptance angle. The cone is referred as Acceptance Cone.

Numerical Aperture:

Light gathering capability of optical fiber is called numerical aperture. It is defined as the sine of the acceptance angle of this fiber.

NA = Sinim

NA = √( η₁² – η₂²)

Fractional Refractive Index Change:

Derivation of Expression for Numerical Aperture of an Optical Fibre is explained below.

The fractional refractive index change is the fractional difference between the refractive indices of core and cladding. If η₁ is the refractive index of core and η₂ is that of cladding.

Δ = (η₁- η₂)/ η₁

It is the ratio of change of refractive indices to the refractive index of the core. It is always positive. 

For effective of propagation of light waves, Δ <<1 usually Δ = 0.01

Relation between NA, η₁, η₂ and Δ

The relations are, NA = √( η₁² – η₂²)

NA = η₁√2D

Consider a light ray AB incident at B at the edge of the core of an optic fiber from air. It is incident at an angle θ₀ with the axis of the core. This maximum angle is called acceptance angle. Since it travels from air to the core, it is refracted along BC at an angle θ_r called the critical propagation angle. This refracted ray is now incident at C at the core-cladding interface with an angle slightly greater than the critical angle θ_c. Hence the ray is undergoing total internal reflection and it is travelling along CD. At D, it is incident at an angle slightly greater than critical angle and again undergoes total internal reflection. Thus the ray is propagated through the fiber by multiple TIR.

A C, CN is drawn normal to the axis. The angle at C is taken as the limiting angle θ_c , the critical angle. Let η₀ be the refractive index of air, η₁ that of core and η₂ that of cladding.

By Snell's law, at B, Sin(θ_a/θ_r) = η₁/ η₂ ----------------- (1)

ie, η₀ Sinθ₀ = η₁Sin θ_r ------------- (2)

(η₀ for air = 1)

But, NA is Sinθ_a

ie, NA = η₁Sinθ_r ----------------- (3)

In ΔBCN, BN/BC = Cosθ_r

But BN/BC = Sinθ_c

Therefore, Cosθ_r = Sinθ_c ----------------- (4)

But, Cos²θ_r + Sin²θ_r = 1

At critical angle, considering the refraction from core to the cladding,

Sinθ_c/ Sin90 = η₂/η₁ 

Sinθ_c = η₂/ η₁ ----------------- (5)

Substitute for Sinθ_c in eqn (5)

Sin²θ_r = 1 - η₂²/η₁²

Sin²θ_r = (η₁² - η₂²)/η₁²

η₁² Sin²θ_r = η₁² - η₂² ------------------ (6)

NA² = η₁² - η₂² from equation (5)

Therefore, Numerical Aperture, NA = √ (η₁² - η₂²) --------------- (7)

NA = √ [(η₁ + η₂)(η₁ - η₂)]

η₁ ≈ η₂, So that η₁ + η₂ ≈ 2 η₁

NA = √ [2η₁ (η₁ - η₂)] ------------------ (8)

= √ [2η₁² (η₁ - η₂)/ η₁] = √(2η₁²Δ)

Δ = η₁ - η₂/ η₁

NA = η₁√(2Δ)

This represents numerical aperture in terms of η₁ and Δ.

V - Number or Normalized frequency V

V – Number is an important parameter of optic fibre. It is called the normalized frequency. V number is given by,

V = 2πa/λ. NA, where a = radius of the core

Therefore, V = πd/λ. NA, where d = core diameter

λ = wavelength of light propagation through the fibre.

NA = Numerical Aperture.
Usually, V ≤ 2.405, the fibre can support only one mode and it is called Single Mode Fibre (SMF). If V>2.405, the fibre can support many modes and it is called multi mode fibre (MMF).