Lloyd's mirror
This is another method for finding the wavelength of light by the division of wavefront.
Light from a slit So falls on a silvered surface at a very small grazing angle of incidence as shown in the diagram (Figure 1). A virtual image of So is formed at S1.
Interference occurs between the direct beam from So to the observer (0) and the reflected beam The zeroth fringe will be black because of the phase change due to reflection at the surface.
Application
An interesting application of this effect may be observed when a helicopter flies above the sea near a radio transmitter.The helicopter will receive two signals:
(a) one signal directly from the transmitter and
(b) a second signal after reflection from the sea
As the helicopter rises the phase difference between the two signals will alter and the helicopter will pass through regions of maxima and minima.
Lloyd's mirror Experiment
Lloyd’s Mirror is used to produce two-source interference patterns, similar to the pattern
produced by laser light passing through two slits. A diverging laser beam strikes a frontsurface mirror at a low angle, so that some of the beam reflects off the mirror to a screen,
and some shines directly on the screen. The reflected beam forms a virtual second source
that interferes with the direct beam. Varying that separation between the laser and the
mirror changes the interference pattern on the screen.
Construction of Apparatus:
This version of Lloyd’s mirror is much simpler than many previous versions, is bright enough
to be shown to a large class, and is easy to set up. It consists only of a 30 mW green laser
module with power supply, a converging lens of short focal length, and a small front-surface
mirror.
Because the laser is fairly high-powered (for improved visibility), it has been mounted in a
homemade heat sink (aluminum block) to prevent overheating in operation. The converging
lens in front of the laser causes the beam to first converge, then diverge in a wide cone.
Because the laser beam has been spread out many times its original size, the light is
rendered safe for viewing and no extensive safety precautions are needed.
To simplify adjustment of the laser-to-mirror separation, the laser has been mounted on a
small plastic jeweler’s vise. Opening and closing the vise changes the distance from the
laser to the mirror’s reflective surface in a smooth fashion, but the vise is only for
convenience and it is not difficult to achieve the effect simply by mounting the laser/lens and
the mirror on separate ring stands and sliding one past the other on a table.
A white viewing screen is the only other requirement, but a meter stick or other measuring
tool placed against the screen may be used to measure the fringe separation as a function of
distance from the mirror.
Use of Apparatus:
Lloyd’s mirror is an interesting example of two-source interference, which is similar to
but subtly different from two-slit interference.
To operate the apparatus, begin with the vise opened wide, so that the laser is as far
from the mirror as possible. Turn the laser on, then turn the screw on the vise to move
the laser in toward the mirror. At first you will see simply a bright circle of light from the
diverging beam directly striking the screen, and a second partial circle from a partial
reflection off the mirror. As the laser is brought closer to the mirror, these two spots
become more equal in size and increasingly overlap.
When the path-length difference is small enough, closely-spaced vertical fringes will
become apparent in the light on the screen. As the laser moves closer to the mirror, the
spacing of the fringes increases and the number of visible fringes decreases until they
finally disappear. At that point the laser has passed the reflecting front surface of the
mirror.
Because the two sources do not pass through narrow slits, there is no diffraction
pattern for the individual sources. Because the sources do not cause their own diffraction
patterns, Lloyd’s mirror is an example of two-source interference without the confusion of
an overlaid diffraction pattern.
It is also interesting because the light reflecting off the mirror undergoes a 180º phase
shift upon reflection, causing the fringe pattern to invert – when compared to a two-slit
pattern from slits of the same separation, there are bright fringes in one pattern where
there are dark fringes in the other, and vice-versa. Humphry Lloyd noted this
discrepancy shortly after he discovered the effect in 1834, and interpreted it as a
definitive proof that the phase of the reflected beam was being inverted.
Thus whereas the nth bright fringe for two slit interference can be shown to obey the
relationship:
xn = nλ(D/d)
The nth bright fringe for Lloyd’s Mirror obeys the relationship:
xn = (n- ½)λ(D/d)
Where:
xn = the distance from the center of the pattern to the nth bright fringe,
λ = the wavelength of the light (532 nm in this case),
D = distance to the screen, and
d = separation between the two sources (slits in one case, source and reflected
“source” in the other)
The separation between bright fringes (x) is thus found by subtracting xn from xn+1 For
both cases:
x = λ (D/d)
Thus if x, D, and d are known it is a simple matter to determine λ for the laser beam.
In a dark room, the apparatus can be moved back to 3-5 meters from the screen,
enlarging the pattern so that it is easily visible for demonstrations.