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Chapter 35: Q.24 (page 1017)

A scientist needs to focus a helium-neon laser beam (λ=633nm)to a 10-μm-diameterspot 8.0cmbehind a lens

(a) what focal-length lens should she use?

(b) what minimum diameter must the lens have?

Short Answer

Expert verified

(a) Focal-length lens she uses is S'=f=8cm.

(b) The minimum diameter is D=0.012m→1.2cm.

Step by step solution

01

Part (a) Step 1: Given information 

We need to find the focal length.

02

Part (a) Step 2: Simplification 

Parallel rays always focus at focal length,

Therefore, S'=f=8cm.

Here fis focal length.

03

part (b) Step 1: Given Information

We need to find the minimum diameter of the lens.

04

part (b) step 2: Calculation

By using,

ω=2.44λfD (By multiplying both sides with D.)

D×ω=D×2.44λfD

D×ω=2.44λf (Divide both the sides by ω.)

D×ωω=2.44λfω

D=2.44λfω (substitute values in equation.)

D=2.44×6.33×109×0.0810×10-6

D=0.012m→1.2cm.

Here, Dis the diameter, fis focal length and λis the wavelength.

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Most popular questions from this chapter

A beam of white light enters a transparent material. Wavelengths for which the index of refraction is n are refracted at angle θ2. Wavelengths for which the index of refraction is n+δn, where δn<<n , are refracted at angle θ2+δθ.

a). Show that the angular separation of the two wavelengths, in radians, is δθ=-(δnn)tanθ2

b). A beam of white light is incident on a piece of glass at 30.0°. Deep violet light is refracted 0.28° more than deep red light. The index of refraction for deep red light is known to be 1.552. What is the index of refraction for deep violet light?

The lens shown in FIGURE CP35.49 is called an achromatic doublet, meaning that it has no chromatic aberration. The left side is flat, and all other surfaces have radii of curvature R.

a. For parallel light rays coming from the left, show that the effective focal length of this two-lens system is f=R/12n2-n1-12, where localid="1648757054673" n1and n2are, respectively, the indices of refraction of the diverging and the converging lenses. Don’t forget to make the thin-lens approximation.

b. Because of dispersion, either lens alone would focus red rays and blue rays at different points. Define ∆n1 and ∆n2 as nblue-nred for the two lenses. What value of the ratio ∆n1/∆n2makes fblue=fredfor the two-lens system? That is, the two-lens system does not exhibit chromatic aberration.

c. Indices of refraction for two types of glass are given in the table. To make an achromatic doublet, which glass should you use for the converging lens and which for the diverging lens? Explain

nblue nred

Crown glass 1.525 1.517

Flint glass 1.632 1.616

d. What value of R gives a focal length of 10.0cm?

The center of the galaxy is filled with low-density hydrogen gas that scatters light rays. An astronomer wants to take a picture of the center of the galaxy. Will the view be better using ultraviolet light, visible light, or infrared light? Explain.

The resolution of a digital camera is limited by two factors:

diffraction by the lens, a limit of any optical system, and the fact

that the sensor is divided into discrete pixels. Consider a typical

point-and-shoot camera that has a 20-mm-focal-length lens and

a sensor with 2.5@mm@wide pixels.

a. First,ass ume an ideal, diffractionless lens. At a distance of

100 m, what is the smallest distance, in cm, between two

point sources of light that the camera can barely resolve? In

answering this question, consider what has to happen on the

sensor to show two image points rather than one. You can use

s′ = f because s W f.

b. You can achieve the pixel-limited resolution of part a only if

the diffraction width of each image point is no greater than

1 pixel in diameter. For what lens diameter is the minimum

spot size equal to the width of a pixel? Use 600 nm for the

wavelength of light.

c. What is the f-number of the lens for the diameter you found in

part b? Your answer is a quite realistic value of the f-number

at which a camera transitions from being pixel limited to

being diffraction limited. For f-numbers smaller than this

(larger-diameter apertures), the resolution is limited by the

pixel size and does not change as you change the aperture. For

f-numbers larger than this (smaller-diameter apertures), the

resolution is limited by diffraction, and it gets worse as you

“stop down” to smaller apertures

Two converging lenses with focal lengths of 40 cm and 20 cm are 10 cm apart. A 2.0-cm-tall object is 15 cm in front of the 40-cm-focal-length lens. a). Use ray tracing to find the position and height of the image. Do this accurately using a ruler or paper with a grid, then make measurements on your diagram.

b). Calculate the image position and height. Compare with your ray-tracing answers in part a.
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