The velocity of light in the core of a step index fibre is 2 × 108 ms−1 ,and the critical angle at the core–cladding interface is 70°. Determine thenumerical aperture and the acceptance angle for the fibre in air, assumingit has a core diameter suitable for consideration by ray analysis. ( 3 marks)The velocity of light in a vacuum is 3 × 108 ms−
Question
The velocity of light in the core of a step index fibre is 2 × 108 ms−1 ,and the critical angle at the core–cladding interface is 70°. Determine thenumerical aperture and the acceptance angle for the fibre in air, assumingit has a core diameter suitable for consideration by ray analysis. ( 3 marks)The velocity of light in a vacuum is 3 × 108 ms−
Solution
To solve this problem, we need to use the following formulas:
- The numerical aperture (NA) is given by the formula: NA = sin(acceptance angle)
- The acceptance angle is the inverse sine of the numerical aperture.
- The refractive index (n) is given by the formula: n = c/v, where c is the speed of light in a vacuum and v is the speed of light in the medium.
Step 1: Calculate the refractive index of the core. n_core = c/v = (3 × 10^8 m/s) / (2 × 10^8 m/s) = 1.5
Step 2: Calculate the refractive index of the cladding using Snell's law. n_cladding = n_core * sin(critical angle) = 1.5 * sin(70°) = 1.5 * 0.9397 = 1.4095
Step 3: Calculate the numerical aperture. NA = sqrt(n_core^2 - n_cladding^2) = sqrt(1.5^2 - 1.4095^2) = sqrt(0.225) = 0.4749
Step 4: Calculate the acceptance angle. Acceptance angle = arcsin(NA) = arcsin(0.4749) = 28.07°
So, the numerical aperture of the fibre is 0.4749 and the acceptance angle is 28.07°.
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