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Answered: - Diff raction from a Compact Disk There is a part in this l


Diffraction from a Compact Disk

There is a part in this lab report where I am to derive the grating equation from what is given. Please derive this for me and show the work, thank you



Di?raction from a Compact Disk

 


 

The track spacing of a compact disk is measured by using it as a di?raction grating.

 

The wavelength of a second laser is then determined.

 


 

Introduction

 

Your ?rst year physics text is a useful general reference to the di?raction grating. This

 

experiment allows you to devise your own procedure. On the bench there are two lasers

 

and a compact disk (CD). Please be careful with these components, they are rather fragile.

 

Lasers are sources of coherent, monochromatic light. The basic idea is to use the CD as

 

a di?raction grating and to ?rst measure the track spacing (?1.6 ?m) using one of the

 

lasers, the wavelength of which is known. You will then determine the wavelength of the

 

second laser. The wavelength of both lasers will be given and you will have to pretend one

 

is unknown. There should be 2 of 3 possible lasers. The RED laser has a wavelength of

 

632.8 nm, the GREEN is 543.5 nm and the YELLOW/ORANGE is 594.0 nm.

 

CAUTION! Never look directly into a laser beam.

 

The aim is to obtain as precise and as accurate an estimate as possible of the track

 

spacing on the compact disk, and to then use this to measure the wavelength of the second

 

laser. You can probably get an approximate value of track spacing within 30-40 minutes, so

 

you have the rest of the afternoon to think about how to con?gure the experiment and make

 

your measurements to satisfy the goal. A paper entitled The compact disk as a di?raction

 

grating (Kettler, Am. J. Phys. 59 (1991)) has been included and provides some useful

 

background theory. Note the wavelength of the He-Ne laser is 6328 ?(10 ? = 1 nm).

 

A

 

A

 

There are two possible geometries which are brie?y outlined below. Examine both to

 

determine which gives the highest accuracy with your equipment.

 

You will have to think about questions like (these may not all be relevant!): How is the

 

zero point, from which distances are to be measured, located? The optical bench is ?at, but

 

is it necessary to assume the wooden table is ?at in the setup? Is it necessary for the laser

 

beam to be parallel to the optical bench, and how accurately can this be attained? How

 

accurately is the optical bench perpendicular to the walls and what inaccuracy is likely to

 

be important? Do di?erent locations on the CD give the same results? Is the CD ?at?

 


 

Vertical Geometry

 

With the CD aligned vertically, the grating equation is

 

n? = d(sin ?n ? sin ?0 ).

 


 

(1)

 


 

This geometry is shown in Fig. 1. If we let the distance from the grating to the wall be L

 

1

 


 

Figure 1: Geometry for Eqn. (1). The zeroth order di?raction

 

(n = 0) at ?0 is basically a simple re?ected beam. Note that the

 

sketch assumes the CD is parallel to the wall.

 


 

and the distances of the zeroth and nth order di?raction spots from the horizontal place be

 

Y0 and Yn respectively, then

 

sin ?0 =

 


 

Y0

 

Y02 + L2

 


 

and

 


 

sin ?n =

 


 

Yn

 

.

 

2

 

Yn + L2

 


 

(2)

 


 

You could think about aligning the laser beam normal to the di?raction grating. Then

 

?0 = 0 and the grating equation simpli?es to

 

n? = d sin ?n =

 


 

dYn

 

.

 

2

 

Yn + L2

 


 

(3)

 


 

However, the full expression Eqn. (1) must still be used for error calculations. N.B. Do not

 

assume that the beams are necessarily parallel to the ?boxes? containing the lasers.

 


 

Horizontal Geometry

 

The CD ?at can be placed horizontally on the holder provided. This geometry is pictured in

 

Fig. 1 of The compact disk as a di?raction grating. Be careful to label your angles correctly.

 

In the grating equation, ? is the angle between the normal to the disk and the laser beam,

 

but you will likely make your measurements relative to the plane of the disk, say ?, so that

 

cos ? = sin ?.

 


 

Formal Report

 

Compare the two geometries. Identify the major sources of error and how you tried to

 

minimize them. Be sure to derive the grating equation.

 


 

2

 


 

 


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