Question

Shining light through a double slit will provide such a ruler if (1) the wavelength of the light beam and slit separation is known and (2) the distance theminima/maxima appearing on the screen can be measured. But this in itself requires a physical measurement of distance on the object, which may not be practical.
In an effort to create a "laser-beam ruler" that does not require placing a physical ruler on the object, you mount a Nd:YAG laser inside a box so that the beam of thelaser passes through two slits rigidly attached to the laser. Although 1064 nm is the principal wavelength of a Nd:YAG laser, the laser is also switchable to numeroussecondary wavelengths, including 1052 nm, 1075 nm, 1113 nm, and 1319 nm.

Turning on the laser, you shine the beam on an object located nearby and observe the interference pattern with a suitable infrared camera. By switching from a 1064 nmto a 1113 nm beam, you notice that you must move the laser 1.155 cm closer to the object to align the same order maxima and minima with their original locations on theobject.
How far was the laser originally from the object in meters?* (Assume the small-angle approximation applies.)
= _______________ m
* Note that this form of ruler is not necessarily practical.

Answer 1

Using Young's Double Slit Formula of y(bright) = λLm/d can solve this problem

For this problem, it must be realized that we are asked to find where y(bright) will be the same for two different wavelengths (λ) and two differentscreen Distances (L). The order (m) and the slit distance (d) will remain the same.

The first wavelength of 1064 nm is a distance L away from the screen

The second wavelenght of 1113 nm is L - 1.155 cm away, or 1.155 cm closer

Since y(bright) for situation 1 must equal y(bright) for situation 2, you can set the equations equal to each other

λ1Lm/d = λ2(L-1.115 X 10^-2)m/d. The values of m and d will be the same on both sides of the equation and can be cancelled out leaving

(1064 X 10^-9 L) = (1113 X 10^-9) (L - 1.115 X 10^-2)

Distribute on the right side of the equation

1064 X 10^-9 L = 1113 X 10^-9 L - 1.24 X 10^-8

Solving for L provides...

1.24 X 10-8 = 4.9 X 10-8 L so L = .253 m or 25.3 cm

Answer 2

L = (2nd wavelength)(subtracted distance)/(2nd wavelength - 1st wavelength)