A laser beam of diameter d₁ = 1.4 mm is directed along the optical axis of a thin lens of focal length +5.8 cm (see figure below). (a) How far from the lens will the beam be focused? (b) A second positive lens is placed to the right of the first. Light emerges from the second lens in a parallel beam of diameter d₂ = 3.9 mm. Thus the combination of lenses acts as a beam expander. Find the focal length of the second lens. Find the distance between the lenses.
Ray Optics
Optics is the study of light in the field of physics. It refers to the study and properties of light. Optical phenomena can be classified into three categories: ray optics, wave optics, and quantum optics. Geometrical optics, also known as ray optics, is an optics model that explains light propagation using rays. In an optical device, a ray is a direction along which light energy is transmitted from one point to another. Geometric optics assumes that waves (rays) move in straight lines before they reach a surface. When a ray collides with a surface, it can bounce back (reflect) or bend (refract), but it continues in a straight line. The laws of reflection and refraction are the fundamental laws of geometrical optics. Light is an electromagnetic wave with a wavelength that falls within the visible spectrum.
Converging Lens
Converging lens, also known as a convex lens, is thinner at the upper and lower edges and thicker at the center. The edges are curved outwards. This lens can converge a beam of parallel rays of light that is coming from outside and focus it on a point on the other side of the lens.
Plano-Convex Lens
To understand the topic well we will first break down the name of the topic, ‘Plano Convex lens’ into three separate words and look at them individually.
Lateral Magnification
In very simple terms, the same object can be viewed in enlarged versions of itself, which we call magnification. To rephrase, magnification is the ability to enlarge the image of an object without physically altering its dimensions and structure. This process is mainly done to get an even more detailed view of the object by scaling up the image. A lot of daily life examples for this can be the use of magnifying glasses, projectors, and microscopes in laboratories. This plays a vital role in the fields of research and development and to some extent even our daily lives; our daily activity of magnifying images and texts on our mobile screen for a better look is nothing other than magnification.
![### Laser Beam Focusing and Beam Expansion Using Thin Lenses
**Problem Statement:**
A laser beam of diameter \( d_1 = 1.4 \) mm is directed along the optical axis of a thin lens of focal length \( +5.8 \) cm (see figure below).
**Figure Description:**
The diagram illustrates a laser beam, initially of diameter \( d_1 \), converging through a thin lens and eventually expanding to a diameter \( d_2 \) after passing through a second lens. The beam initially converges to a focal point before diverging again.
#### Questions
**(a) How far from the lens will the beam be focused?**
\[ \boxed{\_\_\_\_\_\_\_\_ \text{ cm}} \]
**(b) A second positive lens is placed to the right of the first. Light emerges from the second lens in a parallel beam of diameter \( d_2 = 3.9 \) mm. Thus, the combination of lenses acts as a beam expander. Find the focal length of the second lens.**
\[ \boxed{\_\_\_\_\_\_\_\_ \text{ cm}} \]
#### Find the distance between the lenses.
\[ \boxed{\_\_\_\_\_\_\_\_} \]
**Explanation of Concepts:**
1. **Focal Distance and Beam Convergence:**
- The focal length of the first lens determines where the beam converges to a point. Given the focal length \( f_1 = 5.8 \) cm, the distance from the lens to the focal point can be computed directly.
2. **Beam Expander:**
- For the second lens, we need to form a collimated (parallel) beam of diameter \( d_2 \). This setup requires a thorough understanding of the lens-to-lens setup distance involving both focal lengths and the beam diameter before and after each lens interaction.
- The magnification (M) of the beam expander is given by the ratio of the output beam diameter to the input beam diameter (\( M = \frac{d_2}{d_1} \)).
- Using the magnification and the focal length of the first lens, the focal length of the second lens can be determined using the formula \( f_2 = M \cdot f_1 \).
3. **Distance Between Lenses:**
- The separation distance between](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2379bb0-49c5-44f7-ac7a-000baa150133%2F819f8052-b833-42a2-b7b7-54691fdde42c%2F3fcsrgv_processed.jpeg&w=3840&q=75)
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