Question

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

Short answer

Answer: The required focal length for the lens is approximately 19.95 mm or 19952.12 μm.

Step-by-step solution

Write down the given information and convert into same units.

We have the following given information: - Diameter of the laser beam (object size): 1.06 mm or 1060 \(\mu\)m. - Diameter of the focused spot (image size): 10.0 \(\mu\)m. - Distance behind the lens (image distance): 20.0 cm or 20000 \(\mu\)m.

Calculate the magnification.

Magnification is the ratio of the size of the image to the size of the object. The formula for magnification is: $$ M = \frac{h_i}{h_o} $$ Where \(M\) is the magnification, \(h_i\) is the image size or focused spot diameter, and h_o is the object size or laser beam diameter. Plug in the values we know: $$ M =\frac{10.0 \mu m}{1060 \mu m} = \frac{1}{106} $$

Calculate object distance.

The magnification formula also relates the distances from the lens to the image and object: $$ M = -\frac{d_i}{d_o} $$ Where \(M\) is the magnification, \(d_i\) is the image distance or the 20000 \(\mu\)m, and \(d_o\) is the object distance we're trying to find. Since we know the magnification from step 2, we have : $$ \frac{1}{106} = -\frac{20000 \mu m}{d_o} $$ Now, solve for the object distance, \(d_o\): $$ d_o = -20000 \mu m \cdot 106 = -2120000 \mu m $$ The negative sign indicates that the object is on the same side of the lens as the image, which is typical for laser focusing systems.

Use the lens formula to find the focal length.

The lens formula is given by: $$ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $$ Where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance. We can plug in the known values and solve for the focal length, \(f\): $$ \frac{1}{f} = \frac{1}{-2120000 \mu m} + \frac{1}{20000 \mu m} $$ Now, solve for \(f\): $$ f = \frac{1}{\frac{1}{-2120000 \mu m} + \frac{1}{20000 \mu m}} \approx 19952.12 \mu m $$ So, the required focal length lens is approximately 19.95 mm or 19952.12 μm.

Key concepts

Laser Focusing

Laser focusing involves concentrating a laser beam into a smaller, spot-like area to increase the intensity and precision for applications such as cutting or engraving. In this process, a lens is essential in guiding and converging the light to the desired point. Focusing transforms a relatively broad beam into a tiny spot with high energy density. This is crucial for achieving fine details or penetrating hard materials.

When setting up a laser focusing system, usually, the distance from the lens to the focused spot and the laser beam parameters are pivotal in determining the lens characteristics, like its focal length. Understanding the relationship between beam diameter and focused spot size aids in selecting proper lens attributes to ensure effective focusing.

Lens Formula

The lens formula is a fundamental equation in optics, facilitating calculations involving distances and focal lengths. It's expressed as:

• \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)

where:

• \( f \) is the focal length of the lens,
• \( d_o \) is the distance from the object to the lens, and
• \( d_i \) is the distance from the lens to the image.

This formula assumes that all measurements are taken from the same side of the lens and are generally used for thin lenses. By rearranging and solving this equation, you can determine unknown variables such as the focal length, crucial for designing optical systems like laser setups.

In scenarios involving a lens, understanding this formula allows you to calculate how a lens will modify light paths and affect image formation, which is vital for configuring and troubleshooting optical instruments effectively.

Magnification

Magnification is a measure of how much larger or smaller an image is compared to the object producing it. It is represented by the equation:

• \( M = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \)

where:

• \( M \) represents magnification,
• \( h_i \) is the height or size of the image,
• \( h_o \) is the height or size of the object,
• \( d_i \) is the distance from the image to the lens, and
• \( d_o \) is the distance from the object to the lens.

This concept helps describe how the lens manipulates the size relations between the original object and its image. A positive magnification indicates an upright image, while a negative one suggests the image is inverted. In laser applications, predicting the magnification helps to ensure that focused laser spots are correctly sized for the task, emphasizing precision and accuracy in processes like engraving.

Object and Image Distance

Object and image distances are key elements in evaluating the performance of a lens in focusing light. Object distance \( d_o \) is the space from the original object to the lens, while image distance \( d_i \) refers to the span from the lens to the formed image.

These distances are interconnected through the lens formula, affecting how sharp and clear the resultant focus is. In optical systems, adjusting these distances allows technicians and engineers to manipulate image properties for alignment, clarity, and definition.

Understanding these distances, and how they interplay, informs decisions on lens placement and system extent—a fundamental aspect when configuring any precision optics tools, including those employed in laser focusing practices. Correctly calculated distances enable you to achieve the desired focus, a critical factor in sharp imaging.