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APPLIED OPTICS AND OPTICAL DESIGN
By A. E. CONRADY, Rudolf Kingslake
Dover Publications, Inc.Copyright © 1985 Dover Publications, Inc.
All rights reserved.
SPHERICAL surfaces (including the plane as part of a spherical surface of infinite radius) are the only ones which can be produced by the optical grinding and polishing process with a sufficient approach to the necessary accuracy. Very slight departures from the spherical form, amounting at most to a few wavelengths in depth, can indeed be secured by the process of 'figuring'; such small amounts can be taken care of in our computations by allowing for the slight departure from strictly spherical form by approximate corrections. We therefore limit ourselves in our general formulae to exact spherical surfaces.
A spherical surface is the simplest of all possible, curved surfaces, for it is perfectly defined by its radius and by the location of its centre. A radius drawn from the centre through any point of a spherical surface stands at right angles to the tangent-plane in that point and is therefore the normal of the surface. As by the laws of reflection and refraction both the reflected and the refracted ray lie in the plane defined by the incident ray and the normal at the point of incidence ('incidence-normal') we can at once conclude that the plane of reflection and refraction in the case of spherical surfaces always contains the centre of the sphere and can thus be determined with the greatest ease.
In all ordinary optical instruments we have another vast simplification by reason of the centring of all the surfaces. It is intended, and usually achieved with sufficient accuracy, that the centres of curvature of all the component spherical surfaces shall lie on one and the same straight line, the optical axis of the instrument. That evidently means that any ray which originally cut the optical axis and therefore entered the system in a plane containing the optical axis will permanently remain in the same plane, and can therefore be traced right through the whole system by plane geometry or trigonometry. As this saves us the decidedly considerable trouble of determining a new incidence plane separately for each successive surface it will be one of the chief aims of our theoretical discussions to develop computing methods which avoid as far as possible the complication of tracing 'skew-rays'.
LAWS OF REFLECTION AND REFRACTION
By the law of reflection the reflected ray lies in the plane defined by the incident ray and the incidence-normal and forms the same angle with the latter as the incident ray; but as the two rays lie on opposite sides of the normal, the angles have the opposite clock-sense and we express this by giving them the opposite sign: we therefore state the law of reflection as
I' = -I.
By the law of refraction the refracted ray also lies in the plane defined by the incident ray and the normal of the refracting surface, but on the other side of both the surface and the normal: hence the angle of refraction has the same clock-sense or sign as the angle of incidence, and if N is the refractive index of the medium containing the incident ray, N' that of the medium containing the refracted ray, the law of refraction states that
Small corresponding changes of I and I' are frequently of interest, and are found with sufficient accuracy by differentiating (I)* with the result or
N' cos I' . dI' = N cos I . dI dI' = dI . N cos I/N' cos I',
and this equation will be included under (I)*.
When angles become very small their sines become equal to the angles themselves expressed in radians, hence the law of refraction for 'paraxial' rays which enter a refracting surface at very small angles with the incidence-normal becomes, using small letters for 'paraxial' angles,
N'i' = Ni; di' = di . N/N.
Comparing the two fundamental laws, we at once see that we may treat the law of reflection mathematically as a particular case of the law of refraction, resulting from putting N = - N', for this gives
N' sin I' = -N' sin I, or sin I' = -sin I,
which with the necessarily acute angles can only be if I' = -I. This is a very important deduction because it enables us to apply practically all our refraction-formulae to problems of reflection by simply putting N'= N or N'/N = -1.
THE FUNDAMENTAL FORMULAE AND SIGN-CONVENTIONS
It has been shown above that any problem of refraction at a spherical surface can be reduced to one of plane trigonometry by first finding the plane containing the ray to be traced and the centre of curvature.
In the diagram, Fig. 2, let the paper represent this incidence-plane, OP the ray to be traced through the refracting surface, and let the latter cut the incidence-plane in the circle AP with C as centre. Let ACB be a straight line passing through the centre C to which it is convenient to refer the ray in the particular problem: very often it will be the optical axis of a complete system of lenses, but we do not restrict ourselves at all to that assumption. A reference-axis ACB which does not coincide with the principal optical axis of a centred system will be referred to in future as an auxiliary optical axis.
We then define the position of the ray by the distance AB = L from the pole or vertex of the surface to the point where the ray (produced if necessary) intersects the adopted axis ACB, and by the angle of obliquity or convergence U under which the ray meets that axis. It will be seen at once that both these quantities leave an ambiguity. We can find a point to the left of the pole A which is also at the given distance L from it, and we may in either of these intersection-points apply the angle U either above the axis AB as shown in Fig. 2 or below that axis. To remove these ambiguities we stipulate that the intersection-length AB shall be given the positive sign when it falls to the right of the pole A, and the negative sign when it falls to the left of the pole. In the case of the angle of obliquity U we stipulate that it shall always be an acute angle, and that it shall have the positive sign when a clockwise turn will bring a ruler from the direction of the adopted axis into that of the ray: and the negative sign if the turn is counter-clockwise. The only additional datum which enters into the calculation is the radius of curvature r of the refracting surface. Obviously this also requires a sign-convention to distinguish concave from convex surfaces. In agreement with the convention for the intersection-length L we stipulate that r shall have the positive sign when the centre C lies to the right of the pole A, and that r shall be negative when C lies to the left of A. Fig. 2 shows all the quantities in their positive position. The formulae to be deduced are such as will invariably bring out the new data for the refracted ray in accordance with the above sign-conventions.
The sign-conventions as stated are optically the most convenient, and therefore almost universally adopted. It should be noted at once that whilst the convention with regard to L and r agrees with that of co-ordinate geometry, the convention as to the sign of angles is the reverse of that usual in co-ordinate geometry. This occasionally will call for adjustments of the sign in equations derived in the first instance by the methods of analytical geometry.
The nomenclature employed in all our computing formulae has been carefully selected so as to be easily memorized, and also to be within the capacity of the ordinary typewriting machine. It is based on the following simple principles:
1. Only English letters are employed, capitals for the data of rays at finite angles with, or at finite distances from, the optical axis, small letters for rays so close to the optical axis or to a principal ray as to allow of the use of simpler formulae.
2. Vowels are invariably used for angles; Consonants for lengths. Y is treated as a consonant.
3. Quantities which are changed in value by the refraction at a surface are distinguished by the use of 'plain' and 'dashed' letters respectively.
4. Suffixes such as I1, U4, &c, are only used when really necessary. Plain letters are usually retained for the surface actually under consideration, and the suffix (-1) then applies to the preceding, the suffix 1 to the following surface. In general formulae for a whole series of surfaces the latter are numbered successively 1, 2, 3, &c.
5. In lens-systems the surfaces will always be numbered in the order from left to right. The refractive index of the medium to the left of any surface will be denoted by plain N, that of the medium to the right by N', and plain L and U will be used for the ray in the left medium, L' and U' for the ray in the medium to the right.
It should be clearly realized and borne in mind that the use of 'plain' or 'dashed' letters is determined by the location of the actually existing ray, and not by that of the intersecting point B or B'. Thus in Fig. 2 the ray OP only exists in the medium to the left of the refracting surface, and its co-ordinates therefore are plain L and plain U, although the actual intersecting point B, found by producing the ray beyond the surface which really intercepts it and alters its subsequent course, lies to the right of the surface.
In Fig. 2(a) (which again shows all the quantities in their positive sense) we have the arriving ray OP, when produced, meeting the axis at B at the distance
AB = L from the vertex A, and making with the axis the angle U. Drawing the radius CP through the point of incidence we have the angle of incidence OPR, for which we adopt the symbol I, also appearing at CPB in the triangle CPB. In this triangle we know the side CP = r and the side CB = AB - AC = L - r, also the angle CBP = U; dropping a perpendicular CE from C upon BP, we have
sin I = CE/r and CE/L - r = sin U,
and introducing the value of CE by the second equation into the first we obtain equation
sin I = sin U · L - r/r,
by which we calculate the angle of incidence. A transposition of equation (I)* next gives
sin I' = N/N' sin I
by the law of refraction, and we thus determine the direction of the refracted ray PB' in Fig. 2(a)
We now have to determine L' and U' for the refracted ray. Referring to the diagram we see that the angle PCA at the centre of curvature is external angle to the triangle PCB and therefore = (U + I), and is also external angle to the triangle PCB', corresponding to the refracted ray and therefore = (U' + I'). Consequently we have the important relation (which should be remembered as it is very frequently employed subsequently):
U + 1 = U'+ I'
and by transposition of this we obtain the value of U', namely
U' = U + I - I'.
In the triangle PCB' we now know the side PC = r and the two angles I' and U': we therefore can determine the side CB' = L' - r. Employing the same reasoning already applied to the triangle PCB, we easily obtain
L' - r = sin I' r/sin U' and from this
L' = (L' - r) + r
which completes the work for the surface under consideration.
In accordance with a generally accepted custom the formulae have been deduced for a ray travelling from left to right. The formulae are not, however, limited in validity to this usual direction, for our definition of the meaning and of the signs of the L, U, and r is quite free from any reference to the direction in which the light is travelling. If we required to trace the same ray in the reverse direction we should have as given quantities its intersection-length AB' = L' and its inclination U' to the optical axis in the medium of index N' to the right of the surface, and we could compute its course by a simple transposition of the above formulae taken in inverse order. Equation (4) transposed would give
right-to-lefy: sin I' = sin U' L' - r/r.
By transposing equation (2) we should then have
right-to-left: sin I = N'/N sin I'.
Then by transposing (3)
right-to-left: U = U' + I' - I.
and finally by transposing (1)
right-to-left: L - r = sin I r/sin U,
right-to-left: L = (L - r) + r.
It will be noticed that these equations are mathematically identical with those for left-to-right calculations, from which they are obtainable by simply exchanging 'plain' and 'dashed' symbols.
This is very important, for many calculations of complicated lens-systems can be considerably shortened and simplified by using left-to-right and right-to-left calculations alternately.
As has already been pointed out, the fundamental computing formulae are applicable to every case of refraction or reflection at spherical surfaces as soon as the incidence-plane has been determined. In the general case of a so-called skew-ray the determination of the incidence plane calls for a calculation of some difficulty at each successive surface. To avoid this, one of our principal aims will be to avoid the tracing of skew-rays as far as possible, and to derive the required results from rays proceeding in the plane of the optical axis of a centred lens-system. For such rays the transfer of the data from surface to surface is of the simplest kind.
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