Question

As a high-power laser engineer you need to focus a \(1.06-\mathrm{mm}\) diameter laser beam to a \(10.0-\mu \mathrm{m}\) diameter spot \(20.0 \mathrm{~cm}\) behind the lens. What focal length lens would you use?

Short answer

Answer: The focal length of the lens is approximately 10.5 cm.

Step-by-step solution

Identify given parameters

The given parameters are as follows: - Diameter of the incident laser beam: \(d_1 = 1.06~\text{mm}\) - Diameter of the focused laser beam: \(d_2 = 10.0~\mu\text{m}\) - Distance behind the lens: \(v = 20.0~\text{cm}\) We need to find the focal length \(f\) of the lens.

Calculate the magnification of the lens

First, we calculate the magnification (\(M\)) of the lens, which is the ratio of the diameter of the focused laser beam to the diameter of the incident laser beam: $$M = \frac{d_2}{d_1}$$ $$M = \frac{10.0\times10^{-6}\text{m}}{1.06\times10^{-3}\text{m}}=9.43\times10^{-3}$$

Calculate the object distance (u)

Next, we calculate the object distance (\(u\)) using the magnification formula: $$M= -\frac{v}{u}$$ Solving for \(u\): $$u= -\frac{v}{M}$$ $$u= -\frac{20.0\times10^{-2}\text{m}}{9.43\times10^{-3}}\approx -21.2~\text{cm}$$

Calculate the focal length (f)

Finally, we calculate the focal length (\(f\)) using the lens formula: $$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$ Solving for \(f\): $$f = \frac{1}{\frac{1}{u} + \frac{1}{v}}$$ $$f=\frac{1}{\frac{1}{-21.2\times10^{-2}\text{m}} +\frac{1}{20.0\times10^{-2}\text{m}}}$$ $$f \approx 10.5~\text{cm}$$ So, the required focal length lens is \(10.5~\text{cm}\).

Key concepts

Focal Length of Lens

The focal length of a lens is a critical parameter in the field of optics as it determines how strongly the lens converges or diverges light. For a converging lens (also known as a convex lens), the focal length is the distance at which parallel rays of light are brought to a single point after passing through the lens. On the other hand, a diverging lens (or concave lens) spreads out light rays so that they appear to originate from a single point at the focal length.

The focal length is an intrinsic property of a lens and depends on its curvature and the refractive index of the material it is made from. It is a key factor in designing optical systems like cameras, telescopes, and lasers. In our example, it determined the appropriate lens needed to focus a laser beam from its original diameter to a precise smaller spot. A shorter focal length indicates a stronger lens that is capable of bending the rays of light more sharply, thus focusing them in a shorter distance.

Magnification of Lens

Magnification in optics refers to the process of enlarging the appearance of an object via a lens or optical system. When dealing with lenses, magnification is defined as the ratio of the size of the image to the size of the object. Mathematically, it is represented as:

Magnification (M) = Image size (i) / Object size (o)

In terms of distances, for thin lenses, it can also be expressed as the negative ratio of the image distance (v, the distance from the lens to the image) to the object distance (u, the distance from the lens to the object):

Magnification (M) = -v / u

The sign convention (-) in the formula indicates that a real image formed by a converging lens is inverted. In our exercise, we determined the magnification needed to reduce the diameter of the laser beam, and this, in turn, helped us find the object distance when the image distance was known.

Lens Formula

The lens formula is a fundamental equation that relates the object distance (u), the image distance (v), and the focal length (f) of a lens. It is conventionally written as:

1/f = 1/u + 1/v

This formula is applicable to all thin lenses irrespective of whether they are converging or diverging. It assumes that the lens has a negligible thickness, allowing us to apply the formula without considering the actual thickness of the lens.

The lens formula is derived from the principles of geometrical optics and is crucial for solving problems involving image formation by lenses. It enables us to calculate any one of the three distances (u, v, f) provided the other two are known. In the context of our exercise dealing with a high-power laser beam, this formula allowed us to calculate the required focal length of the lens to achieve the precise focusing needed for the laser system.