Question

The numerical aperture is unitless

• A. false
• B. TRUE

Answer 1

Answer: (B)

TRUE

Answer 2

The numerical aperture (NA) is a dimensionless quantity that describes the range of angles over which a lens or optical fiber can accept or emit light.

The formula for numerical aperture in a medium with refractive index $n$ is given by:

$NA = n \sin(\theta)$

where $\theta$ is the half-angle of the maximum cone of light that can enter or exit the lens.

Let's analyze the units of each term:

• Refractive index ($n$): It is defined as the ratio of the speed of light in vacuum to the speed of light in the medium ($n = c/v$). Since it is a ratio of two speeds, it is a unitless quantity.
• $\sin(\theta)$: The sine function takes an angle as input and returns a ratio of lengths (opposite side / hypotenuse in a right-angled triangle). Therefore, $\sin(\theta)$ is also a unitless quantity.

Since numerical aperture is the product of two unitless quantities ($n$ and $\sin(\theta)$), the numerical aperture itself must be unitless.

Therefore, the statement "The numerical aperture is unitless" is TRUE.

Explanation:

The numerical aperture (NA) is defined as $n \sin(\theta)$, where $n$ is the refractive index (unitless) and $\sin(\theta)$ is a trigonometric ratio (unitless). Thus, NA is unitless.