Dr Searle, Experiments with a plane diffraction grating 99 



Part II. Non-parallel Light. 



§ 1. Introduction. In the following experiments the incident 

 light does not form a parallel beam. The diffraction now not 

 merely changes the direction of the axial ray of the beam, but 

 also, in general, introduces astigmatism into, or changes the astig- 

 matism of, the incident beam. The exception is when the incident 

 and diffracted axial rays are perpendicular to the rulings and the 

 deviation is a minimum. The diffracted rays will, in general, pass 

 through two focal lines when the aperture is small. If the aperture 

 is increased, aberration will appear and all the rays will not pass 

 accurately through the two lines. Aberration can be minimised 

 by keeping the aperture small, but astigmatic effects are inseparable 

 from the diffraction in the general case. 



The formulae for the general case are easily obtained, but are 

 complicated. We shall, therefore, consider only the case in which 

 the axial ray of the incident beam is perpendicular to the rulings. 



§ 2. Diffraction of an astigmatic hea^n. In Fig. 1 let OX, OY , OZ 

 coincide with ON , OT, OL, the axes of the grating, as defined in 

 Part I, § 2. For convenience, OZ will be taken as 

 vertical. 



Let a beam, which started from a luminous 

 point and therefore has a wave front, fall upon 

 the grating near 0. Let OP^ be the continuation 

 of the ray through 0, which has been restricted to 

 lie in the plane OXY, and let OP^ be taken as the 

 axial ray of the beam. Let P^OX = d^. Take OP^ 

 as the axis of r^ in a new set of axes Or^, Os^, Ot^, 

 of which Osi is in the plane OXY and Ot^ coincides with OZ. Let 

 the equation to the incident wave front passing through be 



r, = i>SiV + W,s,t, + iT,t,^ (1) 



Let R he a point on the grating and let its coordinates referred 

 to the grating axes be 0, qd, z, where d is the grating interval and 

 q is an integer. Then the coordinates of R referred to the axes of 

 the incident beam are 



ri = qd sindi, Si = qd cos dj^, t^ = z (2) 



If a line through R parallel to OP^ cuts the wave front O-^^ in F^, 

 the second and third coordinates of F^ are qd cos ^j and z. By 

 (1) and (2), the distance of F^ from the plane r^ = 0, which touches 

 the wave front at 0, is ^S-^qH^ cos^ 6-^ + W^qdz cos ^^ + ^TjZ^, and 

 the distance of R from the same plane is qd sin ^j. Hence 



F^^R - qd sin 6^ - ^S^qU^ cos^ d^ - W^qdz cos 6^ - \T^z^....{^) 



7—2 



