optics in space of constant non-vanishing curvature

Optics in Space of Constant Non-Vanishing CurvatureAuthor(s): James PierpontSource: American Journal of Mathematics, Vol. 49, No. 3 (Jul., 1927), pp. 343-354Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2370667 .

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Optics in Space of Constant Non-Vanishing Curvature.

BY JAMES PIERPONT

1. Introduction. In a paper presented at the last meeting of the Society (Dec. 1926)* I showed that in such a space central optical collineation gives only an image congruent with the object. As is well known the theory of collineation enables one to establish a body of theorems of great importance in e-optics t and the fact that this theory is not available in E- and H-spaces would prove a great bar to our study of optics in such spaces if some other general method were not at hand. It is the purpose of this paper to outline a method which is partially successful, in the hope that others, more competent, may greatly improve thereon. In any case the method as here developed reaches results that can not be obtained by other means as far as known to the author.

The essential difference between this and the foregoing paper lies in this: In the former, light was regarded from the standpoint of rays, here it is regarded as a wave phenomenon.

I was led to take this view by a very remarkable memoir t by C. S. Hastings " On Certain New Methods and Results in Optics " and this present paper must be regarded as an attempt to extend Hastings' methods with more or less success to space of constant non-vanishing curvature. I shall however for the sake of clearness speak only of H-space but it will be seen that the methods hold for E-space as well.

2. Hyperbolic Geometry. As one can define the metric of H-geometry in a variety of ways we will begin by recalling the particular form adopted here and then develop a few results needed in the optical part of this paper. Let x, y, z be three real variables which we regard as definiing a point. Let R > 0 a constant

* " Optics in Hyperbolic Space." The method there used can be applied without trouble to elliptic space.

t For e-, H-, E- read euclidean, hyperbolic, elliptic. t Memoczrs of the National Academy of Sciences, Vol. 6 (1893), p. 37. Reprinted

with slight changes in his book Light, Yale Bicentennial Publications. Scribners, New York, 1901.

343



344 PIERPONT: Optics in Space of Constant Non-Vanishing Curvature.

r2 x2 + y2 + % =4R 2 r 2 > 0 ds2 dx2 + dy2 + dZ2.

The metric is then defined by

(1) d =4R2ds/A.

This is Riemann's definition. To form a concrete picture of abstract H- geometry we may regard x, y, z as ordinary cartesian coordinates of a point. Then the points of H-space are represented in the model by the points lying within the e-sphere A = 0. For points near the origin 0 we see do = ds nearly, a fact we shall employ.

H-planes are represented in the model by e-spheres cutting A = 0 orthog- onally. The intersection of two H-planes is an 1-straight. In particular H-planes through 0 are in the model also e-planes. H-spheres are e-spheres in the model whose centers are not those of the H-spheres unless at the origin. Angles in H-geometry are the same as the corresponding angles in the model. The model is thus conformal, a fact we largely employ in the sequel. Bodies in H-space may be moved about freely such that the distance do between adjacent points remains unaltered. This fact is most useful as it enables us to move a certain point to the origin 0 when H-planes and straights through it become e-planes and straights while angles remain unaltePed. From (1) we find that the 1-length of of a straight segment OP, P = (xyz) is given by

(2) s 2R tanh (oy/2R).

If s receives the increment ds

(3) ds = sech2 (a/2R) 'da.

The 1-length of an arc 4 of an 1-circle K, center A, of radius p is

(4) Ro Rsinh (pr1R).

Its curvature

(5) r dcl/da = 1/R sinh (p/R).

K regarded as an e-circle has its center A1 on the straight OA, its radius call r. Let this straight cut K in B and C. Let

OA1 = a, OB b, 06- c, in e-measure, = ~, ,B, - y, in H-measure.

Then by (2)



PIERPONT: OptTcs in Space Of Constant Non-Vanishing Curvature. 345

2r c-b 2R {tanh (y/2R) -tah (/3/2R)}, y-3 2p.

R sinh (p/R) Heiance r cosh (y/2R) cosh (/3/2R.) If K passes through 0, -O= 0, y = 2R, whence

(6) r- R tanh (p/R).

In the following we shall deal with H-spherical light waves and spherical lenses which pass through or near 0, the relation (6) is thus true if we neglect small quantities of order greater than 1 which we do throughout and which is also done in this kind of work in e-optics.

0~~~~~~~

FIG. 1.

In figure 2 an H-circle K of radius p (r) * and center 0 is turned through the angle 4) at P. Let the new circle have e-radius r'. We need to know how much r' differs from r where 4) is small.

Let P O A , =-- 90- 1, PA =- s. Then tan fV = cosh (pIR) tan (c/2)

r cosq/ r cos (4)+ ) cos4 + sin 4 tan f'

H]ence s- r cos eb + sin q tan (p/2) cosh (p/R)

If O is small

(7) s ( 1 + (Qp2/2) cosh (p/R)

* Means, radius is p in H-measure and r in e-measure.




Hence for small enough p we may take s = r. In figure 3, K is an H-circle whose center is 0 and radius p (r). We turn K about P through an angle O getting K' whose e-center Q has coordinates (-o , -,8) . The straight 0 Q makes the small angle 9 with 0 P and cuts K in Q.; the segment QQ., has length 8(d). The H-length of P Q is a. We wish to establish the important relation

(8) 8 =-a tan4),

neglecting small quantities of order > 1.

(

p?~~~~~~ v a~~~

FIG. 3.

To this end regard K' as an e-circle obtained by rotating K through the angle 4. Then PQ - PQ =- r. Let PC = c. The equation of K' is

(x + )2 + (y + 3)2=r2

The equation of OQ is y = - x tan 9 - xA. Thus the x coordinate of Q, in numerical value is

r =r-/30 (r/c) =r-r tani .

The projection of QQ, on the x-axis is d cos =- r - r =- r9 tanq. Thus

neglecting 02, etc.,

(9) d= r tan4).

We obtain (8) from (9) as follows. Suppose PQ -r, nearly, as it cer- tainly is if 4 is small.

By (3) d=3- sech2 (p/2R).

By (4) 9 =/Rsinh (p/R), by (2) r=- 2Rtanh (p/2R).

These in (9) give (8) at once.



PIERPONT: Optics in Space of Constant Non-Vanishing Curvature. 347

3. Relations between Curvatures. In figure 4, LML' is a cross-section of a spherical lense of H-curvature A. A spherical wave meets the lense and for small angles of incidence, is bent into another spherical wave. Let r, r' be the H-curvatures of the wave fronts LWL', LW'L'. Thus while the incident wave front travels from MI to W the refracted wave travels from M to W'. We have then the relation

(10) MW' = nMW, n index of refraction.

But MW'= MN - NW', MW = MN - NW. Hence

(11) MN-NW' =- n(MN-NW).

We may regard these H-spheres or rather H-circles as e-circles whose e-curva-

A t < r/~~~

FIG. 4.

tures are L, C, C' respectively. Then setting LN = p in e-measure

NM = jp2L, NW- jp2C, NW' = jp2C'.

These in (11) give

(12) L-C' =n(L-C) or C'=-mL+nC, (m- 1-n).

From (12) we have denoting the radii of these circles by r', 1, r respectiveIy,

1/' =- m/i + n/r.

Let us now pass back to H-space. If we take our origin 0 at or near M we have using (6)




(13) coth (p'/R) n m coth (A/R) + n coth (p/R)

where p', A, p are the radii of the above circles regarded as H-circles. By giving C, C', L in (12) appropriate signs the relation (12) is valid for all cases. For example, a convex lense with n such that C, C' have opposite curvatures we have C' = mL - nC, whence

(14) coth (p'/R) =m coth (AIR) -n coth (p/R).

Let us compare this result with that given in my December paper mentioned above for a particular case:

R =1000, a 600, r 10, n 2/3, 4 30'. We had

cos X cos (a -,8) cos 6 + sin (a -,8) sin 6 cosh (a/R)

/

FIG. 5.

or setting cosh (oaR) = 1 + C', and noting c -,B - = -,8

COSX =COS (o -f) + Ysin6 sin (a R- ) ==cos (4-1) + E,

sinh ( /R) = sin (a-/3) sinh (q/R). sin x

We find:

0=.0001479073, ax=+)+= .008874553, 83-=.0059163688, 8=-- 3=l .002810277, a-/3= .'002958184,

a 608,06, C'-.1906, E 8,3.10-8,

x = .002780641, q= = 642.37, p' = -a 34,3.

If we use (14) we get p' = 34,3 in perfect agreement.




In figure 5 the spherical wave front of curvature r strikes the interface of a lense L1 and is bent into the spherical wave front of radius pi, and curvature r1. It then strikes the interface L,. If -, is the H-distance M1M2 the radius of the wave front at L2 is pi -- -l hence its curvature is

1 1 T'Rsinh [(pi- 1)/R] sinh (pl/R) - (r1/R) cosh (pl/R)

since -q/R is small. Hence

(15) rl1'r =gr; 12 -qllcs=p1) 1 -F -qlrlcosh (pi/R)-g,P1; 1/g,=1-71T1 cos (pi/R). The lense interface L2 gives rT' the curvature r, etc.

To get relations that are easily handled we may regard these wave fronts as e-spheres whose curvatures are C, C1, C1', C,. Let M11, hi in e-measure. The radius of the first refracted wave front at M1 is 1/C1, at M, its radius is (1/C1) - h= C1/(1 - hlCl). Thus its curvature at M2 is

C1' = 1C01 /2 = 1/( -hlCl).

We have then analogous to (12) for a set of lenses

C1 = m1l1 + n1i/iC (16) C2 = m2L, + n2,X2Cl

C3- =mM3L + n3,u3C2, etc.

Here ,u = 1 is set in for symmetry, nk is the index at the kcth interface Lk Mk =1 - nk, 1/Ilk = 1 - hk-lCk-1, hk is the e-distance between the faces Lk-,Lk, Lk is the e-curvature of the length Lk. It is understood that the curvatures are taken with their proper sign. The relations (16) are the (a) of Hastings' paper. They are fundamental in his theory.

As an example, let us consider a thin convex lense in air. The lense being thin we may take h, = 0. Then at the first interface L1,

C1 = nlC -- miLl. At the second interface L2,

C2 = m2L2- n2Cl.

Set ni =-n, then n2 = 1/n, m2 = (n - 1)/n, hence

Cl + C2 =-(n-1/n (Li + L2)., 1/r, + 1/r2-[(n l/n] (/ + /)

or by (6)

(17) coth (pj/R) + coth (p2/R) = [ -(n 1)/n] [coth (A1!R) + coth (A,/R)




Here A1, A2 are the radii of the two faces of the lense; if pl is the distance of the source of light, p2 is the distance of the image, all in 11-measure. The extension of this result to a system of thin lenses in contact is obvious. It is hard to see how such results can be obtained by the trigonometric calculations of rays as in my first paper.

A, A FIEG. 6.

4. Magnification. The Sine Law. In figure 6, the small object AB of length a becomes A,B1 of length a,, after refraction. The wave front of radius p and curvature r, has radius p, and curvature r,, after refraction. We have from the right H-triangle

sinh (a/R) ==sin4 sinh (p/R) or a/R = sinh (p/R).

Similarly oa,IR - 0,, sinh (pI/R). Hence

a =/a 01/p sinh (pi/R)/sinh (p/R).

FIG. 7.

Now O1/O = n the index of refraction, hence by (15)

l/e-n1(r/r1) =n1g1(r/r1), g1 1,




g1 being introduced for symmetry. After a 2nd refraction we have similarly

a2/7l n2g2(rl/r2=

and after p refractions cp/ap- npgp=(rp1/rp).

Here g2, g3*, are as in (15). Multiplying these, gives as magnification

(18) M-ap/a - GN (r/rp) (G =a-9192 gp, N-nin2 np).

For a thin lense system we may take G = 1. Suppose we have a small diaphragm D D of diameter 28 at the first lense as in figure 7. Then if A1D =pi, AD =p.

sin 9 = sinh (81/R)/sinh (p/R) = 8ar

sin 01 =1rj

B

FIG. 8. Hence

sin 01/sin r/r1 =g1(PP(r/rl), g11

Similarly diaphragms 82 . . . 8p at the other interfaces give

sin 01/sin2 =g- 2(rl/r2), . ., sin Op-,/sin Op gpg(rP /rp).

Multiplying these, gives

sin 0/sin Op G(r/rp). This with (18) gives

(19) <p/a = N sin 9/sin Op - N0/0p, 0, Op small.




This is the celebrated sine law of Lagrange. Hastings by an ingenious consideration shows that (19) holds when 9, Op

are not small, provided the incident and emergent wave fronts are spherical, i. e., the instrument is free from spherical aberration. In figure 8, P Q is the incident wave front issuing from A and P Q1 that issuing from B the other extremity of a small object a.

In the other half of the figure P'Q', P'Q'1 have a similar meaning. The incident wave front is limited by a diaphragm of semi-diameter 8,. The diaphragm Sp is the optical image of 81. Then Q1, Q'1 are corresponding points and hence setting QQ, = c, Q'Q'l = p in H-measure,

(20) ep Ne

since the velocity of light in the last medium is N times that in the first. To prove (20) let us refer to figure 3 which we suppose first is euclidean. Let e QQ, in e-measure. The equation of the e-circle K' is 8s before, while

A it L/1

FIG 9.

the equation of the straight OQ is y = - x tan 9 where now O is not small. The x coordinate of Q, is now in numerical value r, = r cos - Jrcp sin 29. The projection of QQ, on the x axis is

e cos 9 r cos 9-ri= irq sin 29.

Hence e =rc sin 9.

Using as before (2), (3), (4), e = sin9.

Similarly ep =- 'p sin 9,.

These with (20) give cp/= N=- p sin Op/a sin 0.




Hence

(21) op/a=N sin 0/sin Op,

which is (19) with this difference that 0, Ov are not restricted to small values.

5. Telescopes, Microscopes. In figure 9, L and L' are object glass and ocular of a telescope or microscope. In front of L is a small diaphragm of semi-diameter 8, behind L' is another of semi-diameter 8', so placed that D' is the optical image of D. The extremities of A A1 of an object of length Or send out light waves which at B are inclined at an angle 4. We are now under the condition of figure 3. Thus by (8)

VW BWtanq = tanq.

Similarly on the right we have

V'W' - B'W' tan 4= 8' tan q/.

Now N being the ratio of the velocity of light at the two ends V'W'I N - VW. Thus

(22) tan 0'/tan4 N(8/8').

Let us now suppose figure 9 represents a telescope. Let AB , BC =

x + r =,8. Let o subtend an angle q at B, an angle t at C, and an angle o' as seen through the instrument. We define the magnification afforded by the instrument by

(23) M I '/q.

Now from figure 9 a R 1sinh (c/R) =R, sinh (ft/R).

Thus using (22)

(24) I N * sinh (x/R)

For a telescope N = 1 also r/R is small, and hence we can take a jS. Thus

M = 8/8'.

Let us take the diaphragm 8 to be the edge of the object glass, and 8' its ocular image. Hence Ramsden's rule

(25) M diameter of objective 2 diameter of its ocular image




holds for E- and H-spaces.

For a microscope in e-space the magnification is taken as

M size as seen through instrument size as seen at distance 10 inches

As we know nothing about the eyes of beings in other spaces than our own, let us suppose w is the distance of easy vision and Y is the apparent size of the object as seen through the instrument, coming from the distance w. We may take as magnification

(26) M = tanh (o,/R)

But tanh (:/R) sinh (0/R) tan 4', tanh (or/R) = sinh (o/R) tan i.

Thus M tan+' tan4/ tan4 8 tan4

tan tan tan J Wtan t Now

tan a approximately. tanq a c

Hence

(27) M=N , - ,for a microscope.

There are other results which may be obtained from Hiastings' memoir under certain restrictions, we will however stop at this point.



optics in space of constant non-vanishing curvature

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