This title slide presents the topic "Derivation of Acceptance Angle and Numerical Aperture of Optical Fiber." It is subtitled under "Engineering Physics – Photonics and Fiber Optics."

Derivation of Acceptance Angle and Numerical Aperture of Optical Fiber

Engineering Physics – Photonics and Fiber Optics

This introductory slide explains light propagation in optical fibers via total internal reflection. It describes the core with higher refractive index n₁, surrounded by cladding with n₂ < n₁, and external air with n₀ ≈ 1.

Introduction

• Light propagates in optical fibers via total internal reflection.
• Core: central region with refractive index n₁ (higher).
• Cladding: surrounding layer with refractive index n₂ < n₁.
• External medium (air): refractive index n₀ ≈ 1.

The slide depicts the acceptance angle concept in optical fibers, showing incident ray AB entering the core at angle θ₀ from air. The maximum θ₀ allows total internal reflection at the core-cladding interface for light guidance.

Acceptance Angle Concept

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• Incident ray AB enters core at angle θ₀.
• θ₀: air-core incidence angle for acceptance.
• Internal reflections at core-cladding interface.
• Maximum θ₀ enables total internal reflection.

Source: Wikipedia Numerical aperture

This slide applies Snell's Law at the air-core interface for an incident ray, expressed as n₀ sin θ₀ = n₁ sin(90° - c). It simplifies to n₀ sin θ₀ = n₁ cos c, the key equation governing ray entry into the fiber core.

Snell’s Law at Air-Core Interface

• Apply Snell’s Law at air-core interface for incident ray.
• n₀ sin θ₀ = n₁ sin(90° - c)
• sin(90° - c) = cos c, so n₀ sin θ₀ = n₁ cos c
• Key equation governing ray entry into fiber core

The slide derives cos c using the Pythagorean identity sin²c + cos²c = 1, yielding cos c = √(1 - sin²c). It then substitutes this into Snell's law as n₀ sin θ₀ = n₁ √(1 - sin²c).

cos c from Trigonometric Identity

• • Apply identity: sin²c + cos²c = 1
• • Isolate: cos²c = 1 - sin²c
• • Derive: cos c = √(1 - sin²c)
• • Substitute: n₀ sin θ₀ = n₁ √(1 - sin²c)

Source: Derivation of Acceptance Angle and Numerical Aperture of Optical Fiber

The slide applies Snell's Law at the core-cladding interface: n₁ sin c = n₂ sin 90°. Simplifying with sin 90° = 1, it derives sin c = n₂ / n₁ and sin² c = (n₂ / n₁)².

Snell’s Law at Core-Cladding

• Snell's Law: n₁ sin c = n₂ sin 90°
• sin 90° = 1, so n₁ sin c = n₂
• sin c = n₂ / n₁
• sin² c = (n₂ / n₁)²

Source: n₁ sin c = n₂ sin 90° = n₂ → sin c = n₂ / n₁ → sin²c = (n₂ / n₁)².

The slide's left column shows the equation after substituting sin²c = n₂²/n₁², yielding n₀ sin θ₀ = n₁ √(1 - n₂²/n₁²). The right column simplifies it to sin θ₀ = (1/n₀) √(n₁² - n₂²), noting that √(n₁²(1 - n₂²/n₁²)) = √(n₁² - n₂²).

Substituting and Simplifying

| From previous: n₀ sin θ₀ = n₁ √(1 − sin²c) Substitute sin²c = n₂² / n₁²: n₀ sin θ₀ = n₁ √(1 - n₂²/n₁²) | Divide both sides by n₀: sin θ₀ = (1/n₀) √(n₁² - n₂²)

Note: √(n₁²(1 - n₂²/n₁²)) = √(n₁² - n₂²) |

Source: Engineering Physics – Photonics and Fiber Optics

The slide presents the acceptance angle formula for air (n₀ = 1): θ₀ = sin⁻¹(√(n₁² - n₂²)), representing the maximum angle for total internal reflection. It depicts the acceptance cone, illustrating the cone of light acceptance and the limit for incident ray entry angles.

Acceptance Angle Formula

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• For air (n₀ = 1): θ₀ = sin⁻¹(√(n₁² - n₂²))
• Maximum angle for total internal reflection
• Acceptance cone shows cone of light acceptance
• Defines limit of incident ray entry angle

Source: Image from Wikipedia article "Numerical aperture"

Numerical Aperture (NA) is defined as sin θ₀, the sine of the maximum acceptance angle, and calculated as √(n₁² - n₂²) using refractive indices. It measures a fiber's light-gathering capability and efficiency, with higher NA enabling a larger cone of accepted light.

Numerical Aperture (NA)

• NA defined as sin θ₀, maximum acceptance angle.
• NA = √(n₁² - n₂²), derived from refractive indices.
• Measures fiber's light-gathering capability and efficiency.
• Higher NA enables larger cone of accepted light.

Source: Slide 9: Definition and significance

When \( n1 \approx n2 \), the slide defines \( \Delta = (n1 - n2)/n1 \) and expresses \( n2 = n1 (1 - \Delta) \). It then approximates \( n1^2 - n2^2 \approx 2 n1^2 \Delta \), yielding \( \text{NA} = \sqrt{n1^2 - n2^2} \approx n_1 \sqrt{2\Delta} \).

Simplified NA Expression

• When n₁ ≈ n₂, define Δ = (n₁ - n₂)/n₁
• Express n₂ = n₁ (1 - Δ)
• Approximate n₂² ≈ n₁² (1 - 2Δ)
• n₁² - n₂² ≈ 2 n₁² Δ
• NA = √(n₁² - n₂²) ≈ n₁ √(2Δ)

The slide displays a diagram (Fig 5.16) graphing Numerical Aperture (NA) versus acceptance angle variation. It highlights the direct relationship NA = sin θ₀, where a smaller NA reduces θ₀ and complicates light launch.

Graphical Representation

!Image

• Diagram (Fig 5.16): NA vs. acceptance angle variation
• NA = sin θ₀ shows direct relationship
• Smaller NA reduces θ₀, harder light launch

Source: Numerical aperture optical fiber

The slide summarizes key equations for the acceptance angle θ₀ = sin⁻¹(√(n₁² - n₂²)) and numerical aperture NA = √(n₁² - n₂²) = n₁ √(2Δ), essential for optical communications efficiency. It stresses that mastering NA optimizes fiber systems and urges applying these derivations in designs.

Summary

**Key Equations:

• θ₀ = sin⁻¹(√(n₁² - n₂²))
• NA = √(n₁² - n₂²) = n₁ √(2Δ)

Vital for optical communications efficiency.**

Mastering NA optimizes fiber systems.

Apply these derivations in your designs!

Source: Engineering Physics – Photonics and Fiber Optics