« PreviousContinue »
Introduce this into (1), writing in the denominator, r=R,
and develop, remembering that nt is constant all over s. Then the integral will split into two others
multiplied by sin, and by cos n(t). Thus, taking the
intensity at the sphere s as our unit, i. e. putting ≥a2=1, the intensity of light at P will be
sin .cos (r)
These formulæ are valid for any distribution of phase, n=n(0, 4), and for any form of the edge of s (diaphragm).
If, as corresponds to the subject of the present paper, n is a function of alone, and if the edge is itself a circle of latitude, 0=01 = const., then we have axial symmetry round ON, so that C, S become independent of the longitude of the point P, and we can put p=0. The most convenient integration variables being now and themselves, take ds = R2 sin 0.do .do
and integrate over 40 to 2π and over 0-0 to 01. Develop (3) and introduce the abbreviations
2πσ λ λ
(4) Then, after some easy transformations, the expression for the light intensity at the point P(p, o) will be
- (2πR)2 | 2012,
where is the absolute value of the complex integral
For the focal plane, as we shall henceforth call the plane
That is, if the "contour lines exhibited by the Lens Interferometer are circles of latitude,
ein+B cose) Jo (a sin 0). sin de.
σ=0, or B=0, for cophasal vibrations (n=0), and for small 1, the integral (6) reduces to the familiar form
giving for the intensity the well-known expression
(RO) J, (2MP 01).
Formula (5) and (6) are valid for any angular aperture 012, and for any given axially symmetrical phase distribution n=n(0). The axial displacement B (of P) from the centre enters only through the factor eis cose; the transversal displacement a enters through the zeroth Bessel function, and the phase heterogeneity through ein. Notice in passing that, by (6), a phase distribution of the type n=g.cos 0 * is equivalent to a rigid shift of the whole image (luminous
region) along the axis by
and is, therefore, unessential.
In the case of a wave issuing from a lens, with the centre O placed in its focus, and NO along its optical axis, the series development of n does not at all contain such a term, i. e., practically, no term in 02. In fact, it will be seen that, with the above choice of the reference sphere, the series for n starts, for any "uncorrected" lens (such as the simple planoconvex lens), with 04, the next term being in 0.
In all practical cases, connected with lenses, the angular semiaperture 1 hardly exceeds 4° or 5°. Under these circumstances we can write in (6), both in the factor of Jo and in Jo itself, sin 00, and in the exponential, Bcos 0B-BØ2. The first term, B, giving only the factor e outside the integral, does not influence the value ofw and can, therefore, be rejected.
Thus, introducing the new variable
u = 02,
the formula for small 1 will be
ein-Bu) J。(a √u). du,
where n is a given function of u. The corresponding intensity at P will be determined by
(5), a and
Which in the case of a small 1 becomes =-1902, the additive constant term g being irrelevant.
the coordinates of P (with X/27 as unit length). The upper limit of the integral stands for 02, the suffix having been dropped.
For certain forms of the integral (7) can be developed into more or less complicated series, as has been done by various authors, especially for the focal plane (B=0). In general, however, whether the phase distribution ʼn be given graphically or analytically, the gaol will be reached much more easily and quickly by a number of comparatively small steps Au or "du," starting from 0 and leading to the required u=01, either by mechanical quadratures or by a graphical construction analogous to that of the famous Cornu spiral. The latter method can now and then be checked by the former, which is particularly advisable for the first stages of the procedure. A curve drawn in this manner (for any fixed a, B) has also the advantage of exhibiting the local intensity as a function of the aperture; the process corresponds, in fact, to a gradual opening of (the pupil of) a lens, from no to the full required aperture.
Consider the plane of the complex variable
2w = x+iy=z,
so that, I being the distance of the point z from the origin, the corresponding intensity will be
I = (TRL/X)3.
To every fixed point P(a, B) of the luminous region belongs, in the z-plane, a curve whose element is fully given by
dz = ein-Bu) Jo(au).du.
Thus the sloping angle e at any point of the P-curve will be ε = n-Bu,
the length of an arc element
and, therefore, the curvature
By means of these formulæ any P-curve can easily be drawn step by step, much in the same way as the Cornu spiral. For any point P of the focal plane (B=0) the angle e is simply equal to the phase excess 7, and for points outside the focal plane it is smaller by Bu. The
arc elements dl corresponding to equal steps du, for axial points P(a=0), are all equal, as in the case of the Cornu spiral. Outside the optical axis, however, the steps dl become smaller and smaller as we approach the first zero of Jo, which happens the sooner the larger a. At the same time the radius of curvature dwindles to nothing
(unless =); at the apertures corresponding to the zeros
of Jo(au) the P-curves have cusps, from which they emerge with increasing steps to be lessened again when the next zero is approached, and so on. For u=0,n=0 and, therefore, e=0; thus, all P-curves start from the origin tangentially to the x-axis. Again, since, in all cases of actual interest, dn
=0 for u=0, the initial curvature of all curves belonging to the focal plane is nil; and the initial curvature for any other 2πσ point P is kß : But for any not highly corrected λ' dn lens the term in (10) soon becomes the more important one, unless σ mounts to many wave-lengths. Thus, with the exception of the first steps, the sloping angle is given
primarily by ŋ, and the curvature by Jol, the modifications due to an axial displacement being comparatively small.
In most cases, therefore, it will be found that it is sufficient to draw in detail the P-curves for the focal plane only, when the construction data become
€ = n, dl=Jo du, k =
If dn/du preserves its sign, the sense of the windings of a P-curve remains throughout the same (say, anticlockwise), even in passing through a cusp. If passes through a maximum or minimum (and Jo#0), the curve becomes flat and inflected. If a P-curve, no matter after how many windings, happens to pass again through the origin, the light at P* is extinguished, and while the curve passes on (increasing aperture), the light will reappear there, and so on. A good check at any stage of the curve construction may be to measure the whole length of the path already covered and to compare it with its correct length, which can
1 dn Jol du
* i. e., along the circle through P, centred upon and normal to the optical axis.
at once be derived from (9). Up to the first cusp, i. e., as long as the first root of Jo is not exceeded, we have, in virtue of that formula,
2√u ̧ ̧(a√/u), a√/u≤2·4048... (10′)
If au is contained between the first two roots x1, x2 of Jo(x)=0, then
i. e. 1 =
(* Jo(x) x dx
x,≤a√u≤x2, (10 a)
and so on.
The arc length (10') has, as well as the chord L, a noteworthy physical meaning. In fact, it represents the value of 2w, for any point of the focal plane, for a perfect wave (n=0), so that the normal intensity, due to such a wave, can be written, by (5),
On the other hand, the intensity due to the defective wave has been
More generally, for a ring-shaped aperture contained between a and b, the chord L in (12) is to be replaced by the chord ab joining the corresponding pair of points ua, us of the curve. Thus, up to the first cusp, the ratio of the two intensities, at the same point of the focal plane, is
(13) In words, the defective intensity is to the normal intensity as the squared chord to the squared arc of the P-curve, from the origin to the point u in question. The latter will obviously be the longer of the two, the more so the greater the value attained by n at the aperture u. More generally, for a ringshaped aperture a, b, the ratio of the two intensities will be equal to the square of the ratio of the chord Lab to the arc lab of the curve. The simple relation (13) will enable us to see at a glance on the P-curves the relative value of the intensity due to the defective wave, for various apertures. The "definition" of the image will be exhibited by the mutual position of curves corresponding to various points of the field.