# L-8 WAVELENGTH OF LIGHT

Instructor's notes.

[THE DIFFRACTION GRATING]

As in the prism spectrometer, the student should measure the deviation of a particular line,
right and left, and divide the difference by two. This doubles the experimental precision, and
accomplishes an automatic averaging. It also tends to null out any error of placement of the
prism on the table. This is accomplished with no extra time or effort, compared to measuring
lines on only one side, with respect to the zero order.

**Results**

We have 300 line/cm and 600 line/cm gratings, mostly the latter. The grating speaces are 3.33
x 10^{-5} m^{-1} and 1.67 x10^{-5} m^{-1} respectively. That is 33,333 and 16,667 nm.

**Questions from my lab manual.**

(1) What would happen if the grating were *not* placed exactly at right angles to the collimated beam? Would there be any experimental advantage to doing this? Any disadvantages?

The tilted grating will behave as if it has a smaller grating space. This could indeed be
useful, to get higher dispersion. Orders on one side of the zeroth order would have
reduced brightness, orders on the other side would be brighter. The grating equation
becomes nλ = d(sinβ - sinα) where β is the angle of the incident light and α is the angle of the diffracted beam, both measured to the grating normal.

(2) A grating may be thought of as a regular array of transmitting "slits" with opaque space
between them. Sometimes gratings are made in which the opaque space between the slits is
exactly equal to the slit width. What effect does this have on the observed spectrum? In
particular, which "orders" will be seen and which will not be seen? You must consider the fact
that diffraction and interference are *both* happening to the light passing through a grating.

This is the principle of a "Ronchi ruling" discussed in most optics texts. Alternate (odd)
orders of the spectrum will be missing, because the minimum of the diffraction curve
occurs at the maximum of the interference curve for odd orders. Ronchi rulings are
important in computerized lens testing by Fourier methods, since their transfer function is
a square wave.

(3) The instructions stated: "If you are using a grating which is mounted on glass, be sure that the side on which the grating is mounted faces *away* from the collimator, so that the light
passes through the glass before passing through the grating." If you have previously done the
prism spectrometer experiment you observed that the spectral lines were curved. If you hadn't
followed the instructions, but instead placed the grating surface facing the collimator, you will see a slight spectral line curvature. Why does this happen?

If the collimated light passed through the grating before going through the glass, the light
emerging at an angle from the grating would be additionally refracted by the glass. As in
the case of the prism, the rays from the top and bottom of the slit will make larger angles
of incidence with the glass, and therefore the emergent rays will be refracted away from
the normal of the glass surface therefore they will be deviated more. The ends of the
spectral lines will be curved away from the zeroth order.

The explanation of the curvature essentially the same as that for the prism: the rays from
the slit ends have a greater angle of incidence on the glass surfaces than those from the slit
center, and will be deviated more due to refraction.

The experimental test of this hypothesis in the grating case is that the line curvature
should be greater for the higher orders. Another experimental test is to deliberately tilt the
grating while observing the zeroth order. This should cause even the zeroth order image
of the slit to appear curved.

So long as the grating has been properly adjusted, perpendicular to the collimated beam
both vertically and horizontally, the line curvature will not affect the accuracy of
measurements made with the instrument. Most books do not mention the correct
placement of a replica grating, but generally show it correctly placed in their diagrams.
[See the Gaertner-Peck Spectrometer Manual, for example.]

(4) It is obviously important to have the grating perpendicular to the collimated light. But what
about the placement of the grating on the spectrometer table. What if the grating were, say 1
cm displaced from the center of the table, toward the collimator? How much error would this
introduce into your measurements? Be quantitative. Remember that the telescope rotates
around the center of the table, and this is also the center of the scale with which you measure
angles.

Quantitatively, this will have zero effect, *provided* that both telescope and collimator are
properly collimated! So long as the deviated light passes into the telescope, the
measurements of the angle of the collimated beam will be correct. Now if the grating were
*too* far removed, it would not be in the line of sight of the telescope; much of the light
from the grating would miss the telescope and the spectrum's brightness would be greatly
reduced, but the angle measurements would still be correct. This emphasizes again the
importance of collimating the system properly.

(Much of the detail given in this question was deliberate misdirection.)

(5) Why are the higher orders less bright? Before you answer, consider the fact that some
gratings are made in which the second order is brighter than the first order spectrum. How
could this be accomplished?

The diffraction intensity curve modulates the interference pattern. If the first order
spectrum occurs where the diffraction curve has its first minimum, the brightness of the
first order will be greatly reduced, and the other orders will be brighter. The total energy
(all orders) is conserved. In fact, if the opaque and transparent areas of the grating are of
equal width, the even orders will be eliminated.

(6) Suppose the effective area of the grating were reduced, say by covering all but the center with opaque paper. How would this affect your measurements?

The precision of the measurements would be decreased, just as if you reduced the aperture
of the telescope or collimator. The standard formula [given in the GP manual] is λ/Δlambda; = mn where m is the number of grating lines and n is the spectral order.

**Questions from Wilson, Exp. 50, Diffraction.**

(1) If a grating with more lines per unit length were used, how would the observed angles or
spread of the spectra be affected?

The grating equation, nλ = d sin(θ) shows that both the observed deviation angles and the spread (dispersion) of the spectra would be increased as d is made smaller. The sine of the deviation angle changes in inverse proportion to the groove spacing, d. The dispersion, Δλ, also increases as d is made smaller.

(2) Was there any difference in the accuracy of the determination of the wavelengths of
the mercury lines for the different order spectra? If so, give an explanation.

In the spectrometer you used, the absolute error in reading the angle is
constant. The relation between wavelength λ and the angle θ is

θ = sin^{-1}(nλ/d),

so the spectrum is spread out more in angle in higher orders, and therefore the error in determining the wavelengths is smaller in higher orders.

However, this assumes that you can *see* the lines equally clearly. The brightness of lines is less in the higher orders, and so some lines may be too faint to see there.

(3) Is it possible for the first-order spectrum to overlap the second-order spectrum? Explain, and assume a continuous spectrum.

No, it is not possible. Lines from two orders will have the same angle of deviation only
when:

n_{1}λ_{1} = d sin(θ) = n_{2}λ_{2}

So, n_{1}/n_{2}= λ_{2}/λ_{1}

The extremes of the visible spectrum are at approximately 4000 Å and 7000 Å, in ratio 7/4.
The longest wavelength is deviated the most. Overlap would occur for the smallest
wavelength in the spectrum with larger n and the largest wavelength in the spectrum with
smaller n. So when the n values are, say 2:1, and the wavelengths are in ratio less than that,
overlap will not occur. 7/4 is less than 2/1 so overlap will not occur between first and second
order. But it *can* happen between second and third order, which has ratio 3/2, and all higher orders.

(4) Is there a theoretical limit to the order of the spectrum one would be able to observe? Justify your answer mathematically.

Yes, there's a limit set by nλ = 90°, since a transmission grating can't transmit backward!

nλ = d sin(90°) = d

So, nλ < d, or n < d/λ

This represents the highest order of transmission spectra observable.

However, a transmission grating can also produce a reflection spectrum (generally very dim, and of little use). There the orders are numbered from the reflection normal, and also limited to nλ < 90°