1

1

2

be greater and hence the line DE, would be more inclined, making the point P, farther from the center. While if the waves were shorter the bands would be closer together, as shown at V1V2 in the third row of figure 583. Consequently when white light shines upon the grating, having all wave lengths present, each wave length produces a bright band at the appropriate distance from the center, and therefore there results the spectra of the different orders represented in the lower part of the figure, the violet end of each order being toward the center, showing that wave lengths increase from the violet toward the red end of the spectrum.

937. Effect of Removing Grating.—If the grating is removed the side spectra all vanish, leaving only the central image at O. For the wavelets which were cut out by the bars of the grating now interfere with the wavelets which formed the side spectra. This is easily seen by a consideration of figure 584. Let ac be drawn from the edge of one grating space so that bc, its distance from the corresponding edge of the next space, is one wave length, then be will be the direction in which the first order spectrum is formed. Imagine the grating bar between a and b

e

λ

C

removed so that light may now proceed from all points between a and b. Let e be a point half-way between a and b, then ef is one-half a wave length, and corresponding to any point h between a and e there is a point g between e and b which is just a half wave length farther from ac. Waves therefore which start in the same phase from h and g must reach ac in opposite phases, and consequently light going out in the direction bc from points between e and b will be exactly interfered with and neutralized by light from points between e and a, and there will therefore be no first order spectrum. In a similar way it may be shown that if the grating bars are removed there will be no side spectra of any order.

FIG. 584.

938. Resolving Power of Grating.-A grating should have a large number of bars and spaces for two reasons. First, the brightness of the diffraction spectra will be greater the larger the number of grating spaces. And, second, the power of a grating to give a sharply defined spectrum is proportional to the total number of grating spaces, other

things being equal. For let AB, figure 585, represent a grating of 1000 spaces, and let AD be so drawn that its distance from the first grating space next to A is one wave length, from the second space its distance is 2 wave lengths, etc., from the 500th space at C its distance is 500 wave lengths represented by CE, and from the 1000th space at B its distance BD is 1000 wave lengths. Wavelets from all the openings of the grating therefore reach AD in the same phase and therefore conspire to form the bright first order spectrum in the direction EF. But now suppose the direction of AD to be slightly changed so that BD = 1001 wave lengths, then CE will equal 500 wave lengths, and light from B and C will therefore reach AD in opposite phases; so also light from the next opening above B will reach AD in opposite phase to that from the next above C, and so on, light from the openings between B and C opposing that from the corresponding openings between C and A. There will therefore be no light of the given wave length in that direction.

C

B

E

D

FIG. 585.

It thus appears that when BD is 1000 wave lengths there is a bright band of the first order in the direction EF, but this bright band must be exceedingly narrow for so slight a change in direction of EF as will change BD to 1001 or to 999 wave lengths will take us beyond its limits. Hence the more lines there are in the grating the narrower will be the bright image due to any one wave length and the closer together two spectrum lines may be and yet be separately distinguishable.

939. Measurement of Wave Length of Light.-Diffraction gratings afford one of the most convenient means of measuring the wave length of light. The grating may be mounted on a spectrometer, as shown in figure 586, so that light from the slit S passes through the lens of the collimator and falls upon the grating at G in plane waves. The observer adjusts the telescope T so that the image of some line in the first order spectrum falls on the cross-hairs in the telescope. The telescope may then be moved into the position T" shown by the dotted lines, so that the central bright image (Fig. 579) comes on the cross-hairs, then the angle between these two positions of the telescope, which is read from the graduated circle, is the angle dce or x in the small diagram and this is equal to the angle bac. But the triangle acb is rightangled at b, and bc is equal to the wave length which is to be

determined, while ac is known from the measurement of the grating and is called the grating space. Representing ac by s we have cb=ac. sin. x, or

From this formula the wave length may be determined when x has been measured as above described; since s is known from the

where n is the number of lines per millimeter in

941. Concave Gratings. It was discovered by Rowland that when a grating is ruled on a polished concave mirror surface instead of on a flat one very perfect diffraction spectra may be formed without the intervention of any lenses whatever. This was a capital discovery, for it was thus made possible to focus

2nd order

the spectra directly on a sensitive plate, and so obtain a photographic map of the lines in the spectrum free from the errors and absorption that lenses introduce. The most perfect maps of the solar spectrum are those made in this way by Rowland.

The mode of mounting a concave grating is shown in figure 587. Two rails AS and SG are fixed at right angles to each other and a diagonal bar AG, having a length just equal to the radius of curvature of the grating, is mounted on carriages at

1st order

1st order

FIG. 587.-Concave grating spectroscope.

A and G so that G may be moved toward or away from S along the rail while A moves along the other rail. The grating is mounted on the diagonal bar at G facing toward the eye-piece or photographic plate holder which is attached to the other end of the bar at A. At S is the slit through which light falls on the grating G. The central bright image of the slit will be found in focus at 0, on the circle of which AG is the diameter, and on each side of O the various orders of spectra are formed in focus on the same circle. In the diagram it will be seen that the instrument is in position for examining the second order spectrum. By sliding A toward S the first order spectrum may be brought in front of the eye-piece.

An important advantage of the spectrum photographs made with this apparatus is that the distances between spectrum lines are proportional to the differences in their wave lengths, so that a scale of equal parts may be made, which when applied to the photograph will give the wave length of every line on the plate.

Problems.

1. Two flat pieces of glass touching at one edge and separated at the other by a thin piece of tinfoil show 30 bright interference bands when examined in sodium light reflected perpendicularly from the thin air film. What is the thickness of the tinfoil? 2. A narrow slit illuminated by light of wave length 600 μ gives rise to diffraction bands on a screen 2 meters behind the slit. The two dark bands, one on each side of the central bright band and nearest to it, are just one centimeter apart. Find the width of the slit.

3. A glass transmission diffraction grating has 50 lines to the millimeter. How far will the first order spectra of sodium light be from the central line when the screen is 6 meters distant?

4. What orders of diffraction spectra will be absent in the spectra produced by a transmission grating in which the bars are exactly equal in width to the spaces between them? See §937.

POLARIZED LIGHT.

942. Polarization by Tourmaline. If two plates of tourmaline, cut parallel to the axis of the crystal and of suitable thickness, are placed one upon the other with their axes parallel,

light will be transmitted through both plates; but if one is gradually turned on the other the transmitted beam will become fainter until when the two are crossed at right angles there is complete extinction. It is thus seen that light after coming through the first plate of tourmaline is different from ordinary light; for the second plate must have its axis in a particular direction in order to transmit the beam, while in case of ordinary light the transmitted beam is equally intense whatever may be the direction of the crystal axis.

A beam of light having this characteristic is said to be