An Elementary Treatise on Optics/Chapter 11

by Henry Coddington

CHAP. XI.

CAUSTICS PRODUCED BY REFRACTION.

106. These caustics are exactly analogous to those before treated of, being formed by the successive intersections of refracted rays, as those were by reflected ones.

Prop. Required the caustic to a plane refracting surface.

Let QR (Fig. 104.) represent an incident ray, qRS the refracted one; AM, MP are the rectangular co-ordinates of a point P on that line; PN is parallel to AM.

LetAQ	=∆,
∠ AQR	=θ, ∠ AqR=θ′,
Am	=x,
mP	=y.

Then tan θ′=NR/NP=AR−MP/AM=∆tanθ−y/x,

xtanθ′−∆tanθ+y=0.

This is the equation to any point P on the refracted ray. If this point be on the caustic, it must be common to two successive refracted rays infinitely near each other, that is, x and y must be the same for the refracted rays answering to θ and θ+dθ. We may therefore equate to nothing the differential of our equation with respect to θ and θ′, considering x and y as invariable. This gives us

xdθ′/cosθ′²−∆dθ/cosθ²=0.

We have, moreover, between θ and θ′ the equation

sinθ=m·sinθ′.

These three equations must, by the elimination of the functions of θ and θ′, give the one containing only x, y, and ∆, which will be the equation to the caustic.

107. Prop. Required the focus of a thin pencil of rays after being refracted obliquely at a curved surface.

Let QR, QR′, (Fig. 105.) represent two rays inclined to each other at an infinitely small angle, incident obliquely on a curved surface at R, R′; Rq, R′q the refracted rays; RE, R′E, normals.

LetEQ	=q,
Eq	=t,
ER	=r,
QR	=u,
Rq	=v,
∠ QRZ	=φ,
∠ ERq	=φ′.

Then we have these equations:

sinφ=m·sinφ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(1)

du+mdv=0 (see last Chapter) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(2)

q2=u2+r2+2urcosφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(3)

t2=v2+r2−2vrcosφ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(4)

and the following, derived by differentiation,

from	(1)	dφ/mdφ′=cosφ′/cosφ,
	(3)	0=(u+rcosφ)du−ursinφdφ,
	(4)	0=(v−rcosφ′)dv+vrsinφ′dφ′,

that is, u	=u+rcosφ/rsinφdu/dφ,
v	=v−rcosφ′/rsinφ′·−mdv/mdφ′

Dividing the latter of these by the former, and putting 1 for −mdv/du and cosφ′/cosφ for dφ/mdφ′, we obtain

v/u=	v−rcosφ′/u+rcosφ·sinφ/sinφ′·cosφ′/cosφ;
∴ v=	urcosφ′·tanφ/utanφ−(u+rcosφ)tanφ′.

Particular cases.

(1) When u is infinite, or the incident rays are parallel

v=rcosφ′·tanφ/tanφ−tanφ′=rcosφ′²·sinφ/sin(φ−φ′).

This is easily constructed:

Draw Em (Fig. 106.) perpendicular to Rq, mn to ER; nq parallel to QR, determines the point q.

It is easy to see that by this construction we have

Rn=RE·(cosERq)²=rcosφ′²,

Rq=RnsinRnq/sinRqn=Rn·sinSRE/sinSRq=rcosφ′²·sinφ/sin(φ−φ′).

When φ is a right angle, or QR a tangent to the surface, v=rcosφ′.

In this case we have only to draw Em perpendicular to the refracted ray.

(3) When v is to be infinite, or the refracted rays parallel utanφ−tanφ′−rcosφtanφ′=0;

∴ u=rcosφtanφ′/tanφ′−tanφ=rcosφ²·sinφ′/sin(φ′−φ).

(4) When φ′ is a right angle, v=0.

108. It has been shown that an infinite number of different surfaces may reflect rays proceeding from the same point, so as to produce the same caustic: the same thing is true of refracting surfaces, for the equation

du+mdv=0,

will have for its integral an arbitrary constant, as well when Rq the line represented by v is drawn every where to one point (as in last Chapter,) as when Rq is always a tangent to a certain curve, or in short whatever law it is guided by.

We may now easily see what will be the form of the caustic in particular instances.

109. Let the refracting body be a cylinder of glass terminated by a hemisphere. (Fig. 107.)

1. Let the incident rays be parallel to the axis. Taking m=3/2, we have F=mr/m−1=3r, so that if AF be taken equal to three times AE, F is the principal focus, and it will easily be seen that there must be a cusp at that point.
   In the next place, take the extreme ray QC, which touches the surface at C, making the angle of incidence a right angle: the angle of refraction is, of course, that whose sine is 2/3. Let $ECD$ be this angle. Applying the construction discovered above, we find that if $Em$ be drawn perpendicular to $CD,\ m$ is at the extremity or edge of the caustic, which must be of a form something like $mnFn'm$.
   If the refracting surface be only part of a hemisphere, as GAg^{[errata 1]}, the caustic is of course reduced to $nFn',$ if $n,\ n'$ be the points where the rays refracted from $G,\ g$ meet the whole caustic.
2. As $Q$ advances towards the surface, (Fig. 108.) the caustic diminishes in breadth, and increases in length, till when $AQ=2AE,\ AQ$ becomes infinite, and then the caustic has two infinite branches asymptotic to the axis, (Fig. 109.).
3. Past that point, $q$ comes on the same side with $Q$, and the caustic breaks as it were into two parts, (Fig. 110.) one of which proceeding from $q$ is imaginary, the other is real, and both have infinite branches extending along the asymptotes $Oo,\ Oo'.$
4. When $Q$ comes to $A$, both parts of the caustic, of course, disappear altogether, being entirely merged in that point.

110. When the refracting surface is a concave hemisphere, the caustic lies on the same side with the radiant point.

1. If the incident rays be parallel to the axis, the form of the caustic is that represented in Fig. 111, where $F$ is the principal focus, $QC$ the extreme incident ray, $Cm$ the extreme refracted ray touching the caustic at its lip $m,\ Em$, is perpendicular to $Cm$, and the point $m$ is found by drawing $Cm$ at the proper angle, (that whose sine is 1/m) intersecting a semi-circle on $CE$.
2. As $Q$ advances towards $E$, the caustic contracts (Fig. 112.) till when $AQ=(m+1).AE,$, as there is no aberration, it vanishes altogether.
3. It then turns its arms the contrary way, as in Fig. 113, the rays refracted on one side of the axis intersecting on the other, till $Q$ gets to the centre, when the caustic again vanishes.

1. While Q is between E and A, we have a similar kind of curve, and when Q is at A, the caustic consists of a curve with two branches very much bent back, and of the point A itself, (Fig. 114.).

111. Let now the refraction take place out of the denser into the rarer medium, and as a first instance, let the refracting body be a glass hemisphere, and the rays enter perpendicularly at the flat surface, (Fig. 115.).

In the first place we may remark, that as no refraction can take place at an angle of incidence greater than that whose sine is 1/m, that is, in this case 2/3, if En be taken two-thirds of EC, and nm be drawn parallel to EA, Qm will be the extreme ray that can be refracted.

Since v=0, when φ′=π/2, or cosφ′=0, it is plain that the curve must begin at m, m′ touching the circle, and extend to F, the principal focus.

As to the rays that are without the limits of refraction, they are of course reflected at the concave surface, and their caustic consists of parts of two epicycloids, CV, cv.

A plano-convex lens represented by mAm′ would give the whole of the caustic mFm′.

112. Let now the radiant point be in the axis of a cylinder of glass terminated by a convex hemisphere, (Fig. 116.)

1. Suppose AQ>3·AE.
   The caustic here extends further both in length and breadth than in the last case. It begins of course at the point m, EmQ being the angle whose sine is 2/3.
2. When AQ=3AE, Aq is infinite, so that the branches of the caustic become asymptotic to the axis, as in Fig. 117.
3. When AQ is lest than three times AE, the curve opens, a form something similar to that in Fig. 30.
3. For instance, when Q is at the extremity of the diameter of a sphere, (Fig. 118.)

   AQ=	2AE,
   Aq=	4AE,
   EmQ=	41° 49′,
   QEm=	96° 22′;

   
   

   Rv, R′v are the asymptotes;^[1]

   Ev	=3.949AE,
   EvR	=11° 25′.

   
   

4. Let now Q come within the sphere, (Fig. 119.).
   Provided EQ be greater than 2/3AE, a segment of a circle on EQ capable of containing an angle of 41° 49′, will cut the section of the sphere in two points m, n, at which rays incident from Q will be refracted parallel to the surface. Between the points m, n, there will be no refraction: those rays which fall on Am will, after refraction, form a caustic of the same kind as that of the last case: those which fall on an will form another caustic nq′, q′ being the focus for rays refracted at α.
5. When EQ=2/3AE, (Fig. 120.) the segment of a circle just touches the sphere; all rays incident on Am are refracted accurately, so as to meet in q·(Aq=5/2AE). The other rays falling on am form a caustic mq′·(aq′=1/4aE).
6. From this last place of Q to the centre, the caustic takes a form of the kind shown in Fig. 121., in which EQ is half the radius, q is on the surface qq′=2/5qE.
7. When Q is at the center, of course there is no caustic other than that point itself: afterwards we find the figures described above, only that their places are inverted.

113. We have yet to consider the case of rays passing out of a denser into a rarer medium through a convex surface.

Let then E (Fig. 122.) be the centre of a surface CAc bounding a dense medium, and first let the incident rays be parallel to the axis AE.

The principal focus F is at the distance r/m−1 from A, that is, at two radii if m=3/2: and if AEm be the angle whose sine is 2/3, Qm will be the extreme ray that can be refracted, so that the caustic will touch the surface at m, and have a cusp at F.

As Q comes towards A, (Fig. 123.) the caustic contracts both in length and breadth, till on Q coinciding with A, it is reduced to that single point.

114. With respect to the forms of caustics belonging to curved surfaces not spherical, it is not worth while to say much, as the subject is difficult and of very little importance. There is, however, one case which is simple enough, namely, that in which the section of the refracting surface is a logarithmic or equiangular spiral, and the incident rays meet in its pole.

Let Q (Fig. 124.) be the pole of such a spiral, QR, QR′, QR″ three successive rays refracted into the directions RS, R′S′, R″S″, so as to meet in q, q′.

Then, since in this case the angle of incidence is every where the same, all the angles of refraction must likewise be equal, and the line $Rq$, must, like the radius of curvature, vary as $QR$; the triangle $QRq$ will be every where similar to itself, so that the angle $QqR$ will always be of a given magnitude, and since $Rq$ touches the caustic in $q$, this curve, having the angle between the radius vector and tangent constant, must be an equiangular spiral, with $Q$ for a pole.

The constant angle $QqR$ may easily be calculated by means of tables, for if $Ry$ be a tangent, we have $\cos qRy={\frac {1}{m}}\cos QRy;$ $QRq=qRy-QRy.$ Then $Rq$ being found in terms of $QR$ by the formula for $v$ (p. 82.) we have two sides and the included angle of the triangle $QRq$ to determine another angle.

115.We may now pass on to the caustics formed by rays refracted twice, such as those passing through a sphere or lens.

Let $E$ (Fig. 125.) be the centre of a sphere of glass into which rays enter parallel to the diameter $AE$.

We have seen (p. 66.) that the principal focus $F$ is at the distance of a radius and a half from the centre: the extreme ray $QC$ is refracted into the direction $Cm$ making the sine of $ECm$ two-thirds of the radius, and emerges in a tangent to the sphere at $m$. The form of the caustic is found to be such as $mkFk'm',$ having cusps at $k,\ k'.$When the sphere is rarer than the surrounding medium, the form of caustic is such as represented in Fig. 126.^{[errata 2]}

116.When parallel rays are refracted through a double-convex lens the form of the caustic is such as shown in Fig. 127 and 128, in the latter of which the caustic touches the lens, as on account of its great thickness the extreme rays do not pass through it.

A double-concave lens gives a caustic of a similar form but on the other side, (Fig. 129 and 130.).

As to other cases of the caustics given by the sphere and by lenses, it is not worth while to dwell upon them as the subject is of little or no practical utility.

It will, however, be as well to make one or two remarks connected with this subject.

117.Let $CAc$ (Fig. 131.) be a refracting surface, $mqm$ the caustic, $Cmk,\ Cm'k'$ the extreme rays touching the caustic in $m,\ m'$, and cutting it in $k,\ k'.$ It will easily be seen that all the retracted rays must pass between $k$ and $k',$ or that $kk'$ is the diameter of what in Chap. ix. we called the least circle of aberration or diffusion.

118.It is of some consequence to know how the light will be diffused over the area of this circle. It is easy to see that it will be most dense in the center and circumference; for the circumference is on the caustic where there are many rays crossing in a small space, and there is a cone of rays represented by the lines $lo,\ l'o,$ having its apex in $o$, the center.

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1. The place of the asymptote is thus calculated:
   
   Since v is to be infinite, and u=2rcosφ,

   2rcosφ=rcosφ·tanφ′/tanφ′−tanφ.

   Hence,

   sinφ′/cosφ′=2·sinφ/cosφ, or sinφ′/sinφ=2·cosφ′/cosφ, that is, m=2cosφ′/cosφ;

   and if s=sinφ,

   m√1−s²=2√1−m²s²; ∴ s²=√4−m²/3m²=√7/27, if m=3/2.

   From this we find φ, or ERQ=30° 361/2′, ERv=49° 48′.

Errata

1. Original: Gag was amended to GAg: detail
2. Original: was amended to When the sphere is rarer than the surrounding medium, the form of caustic is such as represented in Fig. 126.: detail