Figure 34-56 shows a beam expander made with two coaxial...

Figure $34-56$ shows a beam expander made with two coaxial converging lenses of focal lengths $f_{1}$ and $f_{2}$ and separation $d=f_{1}+f_{2}$. The device can expand a laser beam while keeping the light rays in the beam parallel to the central axis through the lenses. Suppose a uniform laser beam of width $W_{i}=2.5 \mathrm{~mm}$ and intensity $I_{i}=9.0 \mathrm{~kW} / \mathrm{m}^{2}$ enters a beam expander for which $f_{1}=12.5$ $\mathrm{cm}$ and $f_{2}=30.0 \mathrm{~cm}$. What are (a) $W_{f}$ and (b) $I_{f}$ of the beam leaving the expander? (c) What value of $d$ is needed for the beam expander if lens 1 is replaced with a diverging lens of focal length $f_{1}=-26.0 \mathrm{~cm} ?$

Key Concepts

Magnification in Afocal Systems

The magnification in a beam expander that uses two lenses is determined by the ratio of the focal lengths of the second and first lens. This ratio directly scales the beam width, meaning that if the focal length of the second lens is larger than that of the first, the beam emerges with an increased diameter. The concept is analogous to the magnification provided by a telescope and is crucial for understanding how optical components work together to alter beam properties.

Sign Conventions for Converging and Diverging Lenses

Properly applying sign conventions is vital in optical design, especially when combining converging and diverging lenses in a system. Converging lenses have positive focal lengths, while diverging lenses have negative focal lengths. These conventions affect the placement of images and the effective behavior of the system. Understanding the effect of lens polarity is important when modifying a beam expander system, such as when replacing a converging lens with a diverging one, to ensure that the desired optical performance is achieved.

Thin Lens Approximation and Lens Equation

The thin lens approximation assumes that the thickness of the lens is negligible compared to the object and image distances. This simplification allows for the use of the lens equation (1/f = 1/object distance + 1/image distance) to relate object and image positions with the focal length of the lens. In beam expander systems, this approximation facilitates the analysis and design of the optical path, ensuring that the conditions for collimation and proper beam expansion are met.

Beam Expander (Afocal Optical System)

A beam expander is an optical system that enlarges the diameter of a collimated beam without introducing angular deviation in the output rays. In such afocal systems, the input and output beams remain collimated (i.e. the rays are parallel to the optical axis), and the design typically involves two lenses placed such that their focal planes coincide. This configuration is essential in applications that require changing beam size while maintaining a uniform propagation direction.

Conservation of Energy and Intensity Scaling

When a beam’s diameter is expanded, its total power remains constant, which means that the average intensity (power per unit area) decreases. The relationship between beam width and intensity follows from the conservation of energy, as a larger cross-sectional area spreads the same amount of energy over a greater region. This principle is fundamental in optical design to achieve the desired balance between beam size and brightness.

Transcript

00:01 Hi there, so for this problem we have the figure that shows a pin expander made with two coaxial converging lenses of focal length f1 and f2 and separation d and that separation is equal to the sum of the focal lens 1 and 2 and the device can span a laser pin while keeping the light rays in the pin parallel to the central axis through the lenses.

00:33 So we need to suppose an uniform laser pin of width, of initial width, that is equal to 2 .5 millimeters, and intensity, initial intensity, that is equal to 9 kilowatts per meter square.

00:57 And that enters a pin expander for which the focal length one is also given, and that is is equal to 12 .5 centimeters.

01:10 And the focal length 2 is equal to 30 centimeters.

01:18 So for part a of this problem, we are asked about the final width, this one right here.

01:30 Now, parallel rays are bent by positive focal length lenses to their focal points f1, this one in here.

01:47 That is the focal point.

01:50 And rates that come from the focal point positions f2 in front of the positive focal lenses are made to emerge parallel.

02:00 So the key then to this type of bin expander is to have the rear focal point f1 of the first lens coincides with the front focal point f2.

02:14 So both should be equal in here.

02:19 Should be located in there.

02:22 Since the triangles that meet at the coincide focal point are similar, they're shared the same angle, they are vertex angles.

02:31 Then we are going to find the following relationship, that the final width, divided by the focal length 2 is equal to the initial width, divided by the focal length 1.

02:45 So that follows immediately.

02:48 So substituting the value, the value, given we have that solving from this equation we have that the final width is equal to the focal length 2 divided by the focal length 1 times the initial width so we will substitute those values in here we have 30 centimeters for the focal length 2 divided by 12 .5 centimeters for the focal lens 1 and this times the initial width, which is 2 .5 millimeters.

03:25 So from this we obtain a width of 6 millimeters.

03:29 So that's a solution for part a of this problem...

Please give Ace some feedback