- Email: [email protected]
Optical Principles of Loupe Magnification
831
O P T I C A L P R I N C I P L E S OF L O U P E
children o f I I 8, by two husbands, became
nance o f "juvenile" glaucoma in the reported
blind from glaucoma. In turn, almost half
pedigrees
the offspring o f these affected
from selection o f unusual cases f o r publica
have
to
date manifested
early
individuals glaucoma.
of
her affected offspring from two
marriages
clearly implicate her as transmitting the dis ease.
glaucoma
results
tion. SUMMARY
Since I I 8 died at the age o f 57 years, she did not herself manifest glaucoma, although
hereditary
1. A pedigree illustrating dominant inherit ance o f chronic simple glaucoma is
pre
sented. 2. In 13 to 25 percent o f cases o f glau
difficulties
coma, careful investigation has shown other
attendant upon obtaining information on a
members o f the family to be glaucomatous.
This
pedigree illustrates the
(1)
3. It is recommended that patients with
disease;
glaucoma should be advised o f its hereditary
disease manifested only in later l i f e : M a n y die before manifesting the
( 2 ) the family history becomes very sketchy
nature. In this w a y their affected
beyond one generation
m a y receive the benefits o f earlier diagnosis
in the g r a v e ;
(3)
younger generations are not yet old enough
and therapy. The Ohio State University
to exhibit the disease. I am in complete agreement
with
such
authors as Posner and Probert ( R e f s . 2 and 3, Table 1) w h o point out that the predomi
OPTICAL
PRINCIPLES
relatives
(10).
Grateful acknowledgment is made to Dr. Madge Macklin for her help in the preparation of this paper.
OF LOUPE
MAGNIFICATION
ARTHUR LINKSZ, M . D . New York
Interest
in
telescopic
and
so-called
error a n d that, within reasonable
limits,
m i c r o s c o p i c lenses for the partially sighted
it c a n a c c o m m o d a t e for a n y desired dis
has lately increased c o n s i d e r a b l y . Little,
tance.
h o w e v e r , has been heard
about
another
B y c o m m o n understanding,
the m a g n i
t y p e o f visual aid, the m a g n i f y i n g l o u p e .
fying l o u p e is a c o n v e x lens of m o d e r a t e
Like the m i c r o s c o p i c lens, the m a g n i f y i n g
d i o p t r i c p o w e r , held at s o m e distance from
l o u p e serves for the i m p r o v e m e n t of near
the e y e . T h i s distance usually is greater
vision. It is i n v a r i a b l y less e x p e n s i v e a n d
than the distance of c o n v e n t i o n a l specta
in s o m e cases j u s t as effective.
cles, b u t
In order to p u t m a g n i f y i n g loupes t o
n o t greater than a b o u t
arm's
length. V i e w e d t h r o u g h it are rather small
best use, their o p t i c s m u s t b e u n d e r s t o o d ;
o b j e c t s w h i c h are held within
it is the p u r p o s e of this paper t o explain
length of the l o u p e . T h i s latter limitation is
the optical principles of l o u p e magnifica
essential. If an o b j e c t is o u t s i d e the
tion in an e l e m e n t a r y manner. D e t a i l s o f
f o c u s of a c o n v e x lens, b u t n o t further from
clinical a p p l i c a t i o n subsequent
will b e treated in a
paper. F o r the simplification
the
the lens t h a n t w i c e its focal length,
focal first the
o p t i c a l i m a g e will b e magnified. T h i s i m a g e
o f b o t h the d i a g r a m s a n d the f o r m u l a s it
h o w e v e r will be real and therefore i n v e r t e d .
will be assumed t h a t the magnifying l o u p e
T h e image obtained through loupe magni
is infinitely thin and that the e y e has a
fication
s i m p l e nodal p o i n t . It also will b e taken
b e virtual.
is n o t t o b e c o m e i n v e r t e d ; it must
for granted that the e y e has n o refractive
l o o k s at this erect virtual image which is
O p t i c a l l y speaking,
the
eye
832
ARTHUR LINKSZ
Fig. 1 (Linksz). Diagram to sliow relationsliips between an object and its virtual image produced by a convex lens.
the effective o b j e c t as far as the e y e is c o n c e r n e d . T h e size of this effective o b ject, or o c u l a r o b j e c t , and its distance from the e y e determine the size of the retinal image and the effective retinal magnification. T h e relationships b e t w e e n an o b j e c t and its virtual image p r o d u c e d b y a c o n v e x lens are easily established with the help of the d i a g r a m g i v e n in Figure 1. In this figure, AB represents an o b j e c t o, and A 'B' the image i; F is the first princi pal focus and 0 the optical center of a (infinitely thin) c o n v e x lens. T h e focal distance FO of the lens will be called f; the distance FA of the o b j e c t from the focus will b e called /. T h e letter ν will stand for the distance AO of the i m a g e from the lens. T h e distance AO oí the o b j e c t from the lens equals (f—l). T h e size and position of the i m a g e are f o u n d b y using the rays (F)BH and BO in the well
k n o w n manner. ( T h e first o f these rays, while a c t u a l l y issuing from B, b e h a v e s as if c o m i n g from the direction of F. T h e r e fore, hitting the lens at H, it leaves the lens parallel to the principal axis. T h e second ray passes through 0; it thus re mains undeflected. T h e b a c k w a r d p r o longations of these rays m e e t at B'; thus B' is the virtual i m a g e of Β and the posi tion a n d length of the line A'B' gives b o t h the position and the size of the vir tual i m a g e w h i c h e v e n t u a l l y is t o b e c o m e the o b j e c t for the e y e . T h i s is n o t y e t s h o w n in the d i a g r a m . ) T h e j u s t g i v e n points and their c o n n e c t i o n s outline several pairs of similar triangles, like FAB and FOH o r OAB and OA'B', and since s o m e of these linear values are g i v e n , others c a n b e calculated b y well k n o w n rules w h i c h g o v e r n similar triangles. T h e linear values A'B' and OH are o b v i o u s l y e q u a l ; i c a n therefore b e substituted for OH and, from
833
OPTICAL PRINCIPLES OF LOUPE
the first pair of similar triangles it can b e determined that 1 = 1
or 0
(1)
I
(la)
In the latter equation, l ' all three values of the right side are k n o w n and i can b e calculated. E q u a t i o n (1) is, b y the w a y , the well k n o w n N e w t o n i a n magnification formula w h i c h expresses the size relations between i m a g e and o b j e c t {i'.o) in terms of t w o other g i v e n linear values, / a n d /. T h e second pair of similar triangles presents another expression for this rela tionship, v i z .
0
(2)
J-l
and since the left sides in equations (1) and (2) are equal it is o b v i o u s that
/
fif-i) I
Figure 2 s h o w s a n o b j e c t AB, o r o, at the distance « from the nodal p o i n t Ν of an e y e . In order t o r e c e i v e a sharp reti nal image, this e y e , assumed t o b e e m m e t r o p i c , need a c c o m m o d a t e 1 / M d i o p ters.* T h e angular size a o f the retinal image is characterized b y the relationship of the t w o j u s t g i v e n linear v a l u e s : tan a = —.
J-l
T h u s , in the case of the virtual i m a g e p r o d u c e d b y a c o n v e x lens, it can b e stated that v= -
the lens than the latter. T h e e q u a t i o n s s o far q u o t e d are valid for c o n v e x lenses a n d virtual images formed b y t h e m in general. O n l y b y its being at a reasonable distance from an e y e and in c o n n e c t i o n with the optical s y s t e m of this e y e , d o e s a c o n v e x lens b e c o m e a l o u p e and influence the size of the retinal image. It w a s stated earlier that, in order to simplify matters, it will b e assumed t h a t the eye has b u t o n e nodal p o i n t and t h a t it is able t o a c c o m m o d a t e for a n y desirable distance. M o r e o v e r , it will b e assumed that a c c o m m o d a t i o n d o e s n o t measurably c h a n g e the position o f the o c u l a r n o d a l p o i n t in relation t o the retina.
f-n I
(3)
Since the essential c o n d i t i o n w h i c h m a k e s c o n v e x lenses a c t as loupes is the presence of the o b j e c t inside the principal focus, / can v a r y o n l y b e t w e e n 0 and / . In the first e x t r e m e case, b o t h i a n d ν b e c o m e infinite; in the s e c o n d e x t r e m e case, i b e c o m e s equal t o o and ν b e c o m e s z e r o w h i c h m e a n s t h a t i m a g e and o b j e c t c o i n c i d e and that there will b e n o magnifi cation. A t a n y in-between value, i is a l w a y s larger than o and ν is larger than (J—l); thus, for all practical purposes, the virtual image (which is the o b j e c t of vision in l o u p e magnification) is always larger than the actual o b j e c t and farther from
(4)
O b v i o u s l y , the larger the distance u, the smaller is the retinal i m a g e of a n y o b j e c t 0. It is the increase of the angular v a l u e a of this retinal i m a g e with w h i c h l o u p e magnification is c o n c e r n e d . Figure 3 s h o w s a c o n v e x lens inter posed b e t w e e n the o b j e c t and the e y e . T o m a k e it serve as a l o u p e , the distance o f the o b j e c t f r o m the lens m u s t be smaller than / . T h e distance u o f the o b j e c t from the e y e is n o w a c o m p o s i t e of t w o values, n a m e l y the s u m of the distance b e t w e e n l o u p e and e y e , z, and of the distance (f—l) b e t w e e n o b j e c t and loupe. N o w , the inter position of a l o u p e b e t w e e n e y e and o b j e c t throws the virtual i m a g e of the latter as far as
At-I)
v= -
I
* u being expressed in meters.
(3)
ARTHUR LINKSZ
834 Β
Fig. 2 (Linl
from the lens. T h u s , the distance of the
a c c o m m o d a t i o n is needed w h e n v i e w i n g
virtual image from the e y e b e c o m e s
objects
Åf-i)
y=-
through
a loupe—an
advantage
m a n y an early p r e s b y o p e m a k e s use of, (5)
i
even if he is otherwise n o t in need of m a g nification. T h e size of this virtual i m a g e
T o see this virtual image, w h i c h is the o b j e c t of vision, the e y e need a c c o m m o d a t e 1/y diopters.* Since y is, b y the nature of things, a l w a y s larger than u, less
has been found to b e
1
a v a l u e w h i c h , o b v i o u s l y , represents the
* y being again expressed in meters.
I
FT ."Λ I
(la)
I
κ u
Fig. 3 (Linksz). Shows a convex lens interposed between the object and the eye.
O P T I C A L P R I N C I P L E S OF L O U P E
size of the ocular o b j e c t . T h e size of the retinal i m a g e of the o b j e c t seen
through
the l o u p e can be characterized b y a tangen tial relationship
similar to e q u a t i o n ( 4 ) ,
namely
I
tan a = — =
y
Γ-β+ζΙ
of
(6)
r—ß+zl
ι W h i l e this is an easy equation t o solve, it is in m a n y cases m o r e a d v a n t a g e o u s t o substitute the rather incidental v a l u e ζ b y a term containing u, the distance of the actual o b j e c t from the ocular nodal point. T h u s
y = v-{f~l) + u Ρ-β ι f-2ß+P+ul
if-iy+ui
l
(5a)
and
tan a -
oj
U-iy+ul
(f-ir+ul
(6a)
Since the angular values e n c o u n t e r e d in clinical o p t i c s usually are small a n d o n l y limited a c c u r a c y is necessary, the t a n g e n t s found in equations (4) and (6a) can j u s t as well represent the actual angles. T h e magnification of the retinal i m a g e b y a l o u p e is therefore
fn =
tana' tan a
a'
=— =
a
of {f-iy+ul o
uf (f-iy+ui
(7)
A s s u m i n g n o w that, in a g i v e n case, the actual distance u b e t w e e n e y e and
835
o b j e c t is a l w a y s k e p t c o n s t a n t , the ques t i o n arises: W h a t h a p p e n s if a l o u p e o f s o m e ( o b v i o u s l y c o n s t a n t ) focal distance / is b r o u g h t closer to the e y e or t o the o b j e c t ; in other w o r d s , if the d i s t a n c e / is v a r i e d ? A c c o r d i n g t o the definition of a l o u p e , I can, in t h e case of a l o u p e , o n l y v a r y b e t w e e n zero and / and as usually in o p t i c s o n e will gain v a l u a b l e information b y analyzing these e x t r e m e values. T o m a k e I a p p r o a c h / , the l o u p e has t o b e b r o u g h t v e r y close t o the o b j e c t . Setting / = / , e q u a t i o n (7) reads
a'
uf
— · = » ί ι η ί η = —
a
=
(7a)
1.
uf
T h u s , a l o u p e has n o (or little) m a g n i f y ing effect if it is b r o u g h t t o o close t o the o b j e c t , w h a t e v e r its p o w e r m a y be. T h e size of the retinal i m a g e d e p e n d s in this case entirely o n the actual distance u of the o b j e c t from the e y e (it b e c o m e s smaller as u increases). M u c h m o r e remarkable is the result arrived a t b y setting Ζ = zero, t h e case in which the l o u p e is held a l m o s t as far f r o m the o b j e c t as the l o u p e ' s focal length. T h o u g h b o t h i and ν tend t o b e c o m e in finitely large, n o unlimited degree of m a g nification c a n b e a c h i e v e d b y bringing t h e l o u p e t o o far f o r w a r d — n o t e v e n a n y d e sired a m o u n t . Setting / = z e r o , e q u a t i o n (7) changes t o α
max
uf =»?max = —
β
u =
-—
f
(7b)
the m a x i m u m v a l u e of m for a n y g i v e n distance u. T h e existence of such a m a x i m u m s h o u l d b e understandable. T h e virtual i m a g e s o o n gets t o o far from the e y e for its actual enlargement t o b e of a n y further a d v a n t a g e for v i s i o n . W i t h a n y g i v e n u and / , m a g n i fication s o o n reaches o p t i m a l values and t h u s there will b e n o t h i n g g a i n e d b y bringing the l o u p e a n y closer. O n e e x a m p l e will suffice t o d e m o n strate this: H o l d i n g the o b j e c t at, for
836
ARTHUR
example, 18 c m . from the eye and a + 1 2 . 5 D . loupe 6 c m . in front of it, magni fication of the retinal image is X 2 . M a g n i fication increases t o X 2 . 1 5 w h e n the loupe is held 7 c m . from the o b j e c t and m a x i m u m possible magnification—holding the loupe nearly 8 c m . from the o b j e c t — is X 2 . 2 5 . T h e main difference lies n o t so m u c h in the magnification as in the a m o u n t s of a c c o m m o d a t i o n necessary t o receive a sharp retinal image. T h e nearer o n e brings the loupe, the less o n e need accommodate.* S t u d y i n g equation ( 7 b ) , s o m e m o r e a b o u t magnification b y l o u p e will b e learned. It is o b v i o u s t h a t for a n y g i v e n o b j e c t distance u, o p t i m u m magnification increases with d i m i n i s h i n g / ; thus, stronger loupes give m o r e magnification and t h e y can b e held farther from the e y e than weaker loupes, while u remains constant. T h i s is of great p s y c h o l o g i c i m p o r t a n c e . T h e elderly partially sighted patient is m o s t u n h a p p y a b o u t the short range, rigidly t o b e enforced, w h e n e v e r he tries to use a m i c r o s c o p i c lens. H e can achieve less magnification with a loupe, b u t at a m u c h m o r e c o m f o r t a b l e range. A n d he can actually hold a stronger l o u p e ( o n e ofl^ering greater magnification) further than a weaker one, while maintaining the same distance u. ( T h e disadvantages of t o o m u c h magnification, the distortions caused b y strong lenses, the retardation of read ing speed b y undue increase of letter size, the limitations of the field, are n o t t o b e considered in the present paper.) It w o u l d b e w r o n g , h o w e v e r , to c o n c l u d e from the presence of u in the numerator of equation (7b) that there are unlimited possibilities in l o u p e magnification b y increasing the distance u between o b j e c t and e y e . Substi tuting o/u for a' into equation ( 7 b ) , the
* A presbyope will hold the loupe as far from the object as possible (nearly 8 cm. in the given example) since in this case i is infinitely far and no accommo dation or reading lens is needed. t Cf. equation (4).
LINKSZ
following i m p o r t a n t
result is arrived
α max
at:
Μ = »imax =
O
ί
J
(8)
T h i s equation, at first sight, appears strange. A c c o r d i n g t o it, the size of a retinal i m a g e w o u l d b e i n d e p e n d e n t o f o b j e c t distance (M d o e s n o t figure in the e q u a t i o n ) while, at the same time, it is clear from equation ( 7 b ) that magnifica tion ( e v e n m a x i m u m magnification) o f the retinal i m a g e increases with u. Still, the equation is c o r r e c t and the m a x i m u m a c tual size a'max of the retinal image of s o m e given o b j e c t o, seen through a l o u p e o f 1// diopters power, is in fact independent of the distance between o b j e c t and e y e ; it is o n l y the magnification t h a t changes and so does, of course, the original size a of the retinal image seen w i t h o u t the inter ference o f a loupe. T h e value a decreases with increasing M, while m increases; and since the increase of m with increasing « just m a k e s u p for the decrease of a with increasing u, the size of the magnified retinal image a ' m a x ' s in the final analysis independent of u. It is c o n s t a n t for a n y g i v e n 0 and / . If the l o u p e is held as far (or a l m o s t as far) from the o b j e c t as its focal length and if the o b j e c t and the loupe are moved together t o w a r d the e y e or a w a y from it, the size of the retinal image is unchanged. T h i s fact, while theoretically interesting, is of definite i m p o r t a n c e besides. A c c o r d ing t o the formula, a ' m a x remains un c h a n g e d w h a t e v e r the v a l u e of u. It should thus b e possible t o read print smaller than Jaeger 1 as well with the help of a l o u p e from across the street as it is from a r m ' s length. T h i s is true, at least in prin ciple, i n actuality, such m a t t e r s are l i m ited—as m u c h b y the inaccuracies of elementary optical formulas w h i c h are
OPTICAL PRINCIPLES OF L O U P E
o n l y a p p r o x i m a t i o n s , as b y the decrease, with the square of distance, of the light energy which emanates from a n y o b j e c t . Besides, the diameter of the l o u p e itself a c t s as limiting factor o f useful magnifica tion, as a kind of s t o p or entrance pupil. A s ζ increases, the angular size of this entrance pupil, as measured from t h e o b server's e y e , b e c o m e s smaller and smaller. T h e size of the largest o b j e c t w h i c h is entirely visible through a l o u p e varies, first of all, with its diameter d and, other things k e p t u n c h a n g e d , varies inversely with the distance ζ of the loupe from the eye. T h e diameter d and the distance ζ determine quasi a c o n e of lines of direction with its base at the l o u p e , its apex at N, and an apical angle of 2ω. Vision through a loupe is limited t o the lines o f d i r e c t i o n which m a k e up this c o n e . T h e d o t t e d lines a d d e d t o the diagram of the l o u p e (Fig. 3) should clarify this. L i n e PN repre sents the direction of the m o s t peripheral r a y w h i c h , modified in direction b y the l o u p e at P , still reaches the retina (assum ing t h a t the pupil of the e y e is p o i n t sized and in the plane of N). T h i s ray, while actually c o m i n g from an o b j e c t point Bj, in the o b j e c t plane, appears to c o m e from B'p in the image plane, a l o n g the b a c k w a r d prolongation of ray PN. T h u s , ABp is the m a x i m u m length of an o b j e c t , O n i a x . fully visible at a given u v a l u e ; A'B'p is the c o r r e s p o n d i n g maxi m u m image, i „ , a x . T h e angle ω s u b t e n d e d b y the c o m m o n principal axis o f the c o m bined system, loupe and eye, and the r a y {B'p)PN equals o n e half of the total angle of the entrance pupil and 5 ρ is the m o s t peripheral point seen through the loupe in the indicated setting. H o w e v e r , d and ζ determine the c o n e of lines of direction o n l y ; they are not the o n l y limiting factors to the field seen through a loupe. E v e n if they, and with them ω , are k e p t constant, and e v e n if one certain loupe of the p o w e r l / / i s c o n sidered, m o r e of a line of letters is visible through the loupe if / increases, and less if
837
it decreases. T h e b r o k e n line from BPmin to Ρ in Figure 3 indicates the possible loci of Bp and the possible lengths o f the line ABp with the changes of I, if d, z, and / are g i v e n a n d c o n s t a n t . If / a p p r o a c h e s / , in other w o r d s , if a line o f letters is b r o u g h t v e r y close to the l o u p e , m a p proaches 1* and ABp a p p r o a c h e s a maxi m u m , OP. T h u s , the greatest length of a line of letters seen through a loupe c a n n o t exceed the d i a m e t e r d of the l o u p e . If, o n the other hand, / a p p r o a c h e s zero, in o t h e r w o r d s , if a line of letters is b r o u g h t close t o the focal plane of the l o u p e , t h e m m b e c o m e s m a x i m u m , ^ b u t the length ABp of o n e half of the longest visible part of a line of letters a p p r o a c h e s a m i n i m u m v a l u e and it b e c o m e s equal t o the d i s t a n c e F5p^.„ if / b e c a m e zero. T h e actual values of the variable distance ABp are found b y the following consider ation: It is o b v i o u s from Figure 3 t h a t —' = — · = ζ y
tan ω
for
a n y g i v e n v a l u e of y. Substituting for imJ and f-fl+zl/l for y , § one gets On,ax//i
Omax/ Omax/
2
Ρ-β+21
f-ß+2l
(9)
I and ABp = o^^^ =
1 2
diP-Jl+ll)
(10)
zf
T h e distance 2omax = 2^5p, o r the maxi m u m length of a line of letters seen through a l o u p e can thus b e calculated for a n y l o u p e of k n o w n p o w e r 1// and diameter d, for a n y distance ζ b e t w e e n l o u p e and e y e
* t t §
Cf. Cf. Cf. Cf.
equation equation equation equation
(7a). (7b). (la). (5).
838
ARTHUR LINKSZ
and for a n y distance ( / — / ) between print and loupe. K e e p i n g all other values c o n stant b u t the last, the length of 2ABp m u s t v a r y with /. It is, as m e n t i o n e d , a m a x i m u m and equal to d w h e n the print is b r o u g h t close t o the loupe, and a mini m u m when the print is b r o u g h t into (or close t o ) the focal plane of the l o u p e (the c o n d i t i o n in w h i c h a' b e c o m e s c o n s t a n t ) . E q u a t i o n (9) changes in this c a s e t o hd
"max /
0 .
(9a)
Í T o i n d i c a t e the fact t h a t o^a^ = ABp varies with v a r y i n g I (cf. fig. 3 ) , its great est v a l u e OP will b e called Om^x (max)," its smallest v a l u e FB^, mm is designated as 0 max(inin).
A s s u m i n g then that the diameter d and the distance ζ of diff^erent loupes from the eye are k e p t c o n s t a n t and the print a l w a y s held in their respective focal planes, it is o b v i o u s from equation (9a) that 20max(miii). the m a x i m u m length of a line of letters visible through these different loupes, d e pends entirely o n / b e c o m i n g shorter as loupes g e t stronger. T h e actual length of these lines for loupes of a n y p o w e r 1 / / is derived b y writ ing the e q u a t i o n in the form Omax(min) = i'-Bp,miii =
¥f
= / tan ω
Ζ
or dj 2o„,ax(n,in) = — = 2 / t a n ω ζ
(10)
O b v i o u s l y , w h e n e v e r a line of letters is held in the focal plane of a g i v e n l o u p e {d a n d / a r e c o n s t a n t ) for m a x i m u m retinal image magnification ( a ' = m a x i m u m ) , then the visible length of this line d e p e n d s en tirely o n z, the distance of l o u p e (and letters) from the e y e . It increases with diminishing z. N o t magnification, b u t field is gained b y holding l o u p e and letters as close t o the e y e as possible. A few numerical e x a m p l e s will illustrate this point.
A s s u m i n g as an e x a m p l e that the length of a line of letters is 8 c m . and t h a t the focal length o f the l o u p e w h i c h is being used is also 8 c m . , tan α'max (the tangent of the largest possible retinal i m a g e of one half oí the line) is o / / = 4 / 8 = l / 2 * and the angular size of the retinal i m a g e of the total line is a b o u t 53 degrees. T h i s v a l u e will a l w a y s remain c o n s t a n t . T h e retinal i m a g e of the total line seen through the l o u p e will a l w a y s c o v e r the s a m e n u m b e r of retinal elements. If n o w the radius of the l o u p e (half of its d i a m e t e r ) is 5 c m . and the l o u p e is held 10 c m . from the e y e (the print is t o b e held another 8 c m . furth er than the l o u p e ) , the entrance pupil f o r m e d b y the l o u p e at the o b s e r v e r ' s e y e has the s a m e angular v a l u e and the w h o l e line is visible through the l o u p e . H o w e v e r , if o n e holds the l o u p e at 20 c m . from the e y e (the print is again held 8 c m . farther than the l o u p e ) , then the tangent of o n e half of the entrance pupil b e c o m e s 5 / 2 0 = 1/4 and the angular v a l u e of the entrance pupil a b o u t 28 degrees. O b v i o u s ly, o n l y a b o u t o n e half of the line of 8 c m . length will b e seen through the l o u p e at o n e time. ( T h e l o u p e w o u l d h a v e t o have a diameter of 20 c m . for a w h o l e line of 8 c m . length t o b e visible through it at this distance.) T h i s entrance pupil b e c o m e s increas ingly smaller if ζ is increased. H o l d i n g the l o u p e at the distance of 2.00 m . (the print m u s t again b e a n o t h e r 8 c m . farther), the tangent of o n e half of the entrance pupil b e c o m e s 5 / 2 0 0 = 1/40 a n d the useful magnified retinal i m a g e will have an exten sion of less than three degrees. H a r d l y a longer w o r d of newspaper print will b e seen at a time and besides, b y m o v i n g the head, the print will s o o n b e lost al together. In spite of all these o b v i o u s limitations, the present author has often d e m o n s t r a t e d t o his a u d i e n c e that even smallest lettering o n a s t a m p o r o n the face of a w a t c h c a n * Cf. equation (8J.
OPTICAL PRINCIPLES OF LOUPE
v e r y well b e read across the r o o m , e v e n through a w e a k loupe, especially in bright d a y l i g h t illumination.* A s iar as practical application goes, t h e principle m a d e e v i d e n t b y e q u a t i o n (8) is responsible for the great versatility of loupes as visual aids. G i v e n s o m e (small) o b j e c t 0 and a loupe of the p o w e r 1//, visual performance is largely i n d e p e n d e n t of o b j e c t distance if the o b j e c t is in, o r near, the first focus of the loupe. A n o p h t h a l m i c surgeon, a philatelist, a b o t a n ist, will v a r y his distance from the o b j e c t he handles a c c o r d i n g t o the m o s t a d v a n t a g e o u s distance o f this handling w i t h o u t losing a n y effective magnification and since i is always infinitely far, i n d e p e n d e n t l y of presbyopia, if present. A s m e n t i o n e d earli er, stronger loupes can b e held farther from the e y e for a n y given c o n s t a n t v a l u e of u. It n o w turns o u t t h a t « can b e a n y dis tance. T h a t b o t h of these circumstances are w e l c o m e t o the partially sighted is o b v i o u s . All he has to d o is m a k e sure t h a t o b j e c t and loupe b e at c o n s t a n t distance f r o m e a c h o t h e r a n d t h a t this d i s t a n c e b e nearly that of the focal distance of the loupe. T h e r e are m a n y c o m m e r c i a l l y avail a b l e d e v i c e s t o k e e p a l o u p e a certain dis t a n c e from a printed page. T h e distance, of course, has t o b e well c h o s e n and rigidly kept. All this freedom of setting oneself at a desirable distance from the printed p a g e is lost w h e n e v e r a p r e s b y o p e uses a l o u p e and his reading correction a t the same time (or w h e n e v e r a m y o p e with n o c o r rection uses a l o u p e ) . In this case, y is limited. A person wearing, for e x a m p l e , a -f-4.0D. reading addition sees ( a t least theoretically) o n l y such o b j e c t s sharply which are 25 c m . from his lens. In o r d e r t o see a magnified o b j e c t sharply t h r o u g h a loupe, the distance y of the virtual image i must be 25 c m . from his reading lens. • It should be possible to put this principle to some limited practical use, for example, by making house numbers or names over entrance doors better visi ble, or in some eye-catching advertising displays.
839
E q u a t i o n (5a) has s h o w n that U-iy+ul y=-
p-2fl+P+ul I
l ρ
=
---2f+l+u.
F r o m this it follows t h a t u=
(11)
Ρ y+2f-j-l.
O b v i o u s l y , if a certain l o u p e is used ( / is c o n s t a n t ) and if there is n o c h o i c e in the v a l u e of y because of a reading a d d i t i o n (or uncorrected m y o p i a ) , u will v a r y with I. In other w o r d s , the distance between e y e and o b j e c t will h a v e t o b e c h a n g e d if the distance (/— / ) b e t w e e n o b j e c t and l o u p e is varied. A few numerical e x a m p l e s will clarify this. It will b e assumed again t h a t / = 8 c m . and t h a t y = 25 c m . and is unchangeable. If the l o u p e is held 2 c m . f r o m the o b j e c t (1 = 6 c m . ) , u m u s t b e 2 4 | c m . T h u s , o b j e c t and virtual image will a l m o s t c o i n c i d e (the latter will o n l y b e f c m . farther) a n d there will h a r d l y b e a n y useful magnifica tion. If the l o u p e is k e p t 6 c m . from the o b j e c t {1 = 2 c m . ) , u will b e c o m e 7 c m . and the l o u p e will h a v e t o b e held 1 c m . in front o f the p r e s b y o p i c reading a d d i t i o n . Magnification will again n o t b e v e r y effec t i v e . F r o m e q u a t i o n ( 7 ) it is f o u n d t h a t in this case 7X8 56 -= — = m= ( 8 _ 2 ) 2 - f 7 X 2 50
1.12.
Still, e v e n if tn is n o t v e r y great, the l o u p e will n o t b e w i t h o u t v a l u e . A l o u p e o f / = 8 c m . permits the p r e s b y o p e wearing a - | - 4 . 0 D . reading a d d i t i o n t o bring o b j e c t s as close to his e y e as 7 c m . , a distance for w h i c h he w o u l d otherwise need a rather strong m i c r o s c o p i c lens. Since 7 c m . is a b o u t o n e fifth o f t h e standard reading distance o f 14 inches ( a b o u t 35 c m . ) , the a p p r o a c h in itself will magnify the retinal i m a g e b y X 5 . 0 . T h e additional,
840
ARTHUR LINKSZ
t h o u g h small, magnifying effect increases this to X S . 6 . A person with not m o r e than 1 4 / 7 0 ( 2 0 / 1 0 0 equivalent) near vision should b e able to read the 1 4 / 1 4 print b y such arrangement—not a m i n o r a c h i e v e m e n t with such inexpensive means. A case o f especial interest is the o n e in w h i c h the o b j e c t is held halfway be tween a l o u p e and its principal focus. In this case, l = (f—l) = / / 2 and it follows from equation (3) that v=f. Thus, whenever an o b j e c t is held halfway between a loupe and its principal focus, the virtual image is in the principal focal plane. M o r e o v e r , it follows from equation ( l a ) that i = 2o, whatever the p o w e r of the l o u p e . T h e s e circumstances are of great inter est in practical w o r k . W h e n e v e r a pres b y o p e has his reading distance fixed b y a reading addition, he can be assured of X 2 . 0 magnification b y any loupe, holding the loupe in such a manner that its principal focus coincides with the reading distance. A t the same time, he is to hold the o b j e c t halfway between this distance and the loupe. In the present e x a m p l e where / is assumed t o b e 8 c m . and y is 25 c m . , it will b e necessary to hold the o b j e c t 21 c m . from the reading lens and the loupe an other 4 c m . closer. Such arrangement is ideal for patients with n o t t o o great loss of vision. If a person has sufficient visual a c u i t y (around 2 0 / 8 0 ) to read a magazine at 25 c m . with a - | - 4 . 0 D . reading addition, he will h a v e a m p l e sight t o read a news paper, or look up a telephone number, with a n y weak loupe, in case l o u p e and reading matter are placed into the posi tions just indicated. ( F o r a 4 - 1 2 . 5 D . loupe, these positions will b e 17 and 21 c m . in front of the eyeglasses, respective
l y ; for a -I-IO.OD. loupe, 14 and 20 c m . , and s o forth.) A s a m a t t e r of fact, even persons with less vision often prefer this arrangement t o the m i c r o s c o p i c lens; a patient with 2 0 / 1 6 0 visual a c u i t y will b e able t o read a magazine w i t h it. SUMM.ARY A N D
CONCLUSIONS
Starting from the well k n o w n N e w tonian formula for the relation between o (the size of the o b j e c t ) and i (the size of the virtual i m a g e ) , a s y s t e m o f e q u a t i o n s is d e v e l o p e d b y w h i c h all m a g n i t u d e s pertinent t o loupe magnification can easily b e ascertained. Such values are ν (the dis tance of the image from the l o u p e ) , y (the distance o f the image from the e y e ) , a' (the angular size of the retinal image of an o b j e c t as seen through a l o u p e ) , and m (the effective magnification of this retinal image). It is s h o w n that, while m changes with the distance of the e y e from the loupe, a' (the size of the magnified retinal image) is largely independent of this distance o n c e loupe and o b j e c t are m o v e d together. T h i s explains w h y loupes c a n b e used at a n y c o n v e n i e n t distance, also b y presby o p e s w h o have no reading c o r r e c t i o n . It is s h o w n furthermore h o w the wear ing of a p r e s b y o p i c reading addition af fects the usefulness of a l o u p e . Finally, a simple formula is d e v e l o p e d b y w h i c h X 2 magnification can b e achieved and a c o n v e n i e n t reading dis tance retained when using a l o u p e of a n y power. 6 East 76th Street {21). I wish to acknowledge the help o f Dr. Harris Ripps, Forest Hills, New York, in the preparation of the diagrams.
O P T I C A L P R I N C I P L E S OF L O U P E
children o f I I 8, by two husbands, became
nance o f "juvenile" glaucoma in the reported
blind from glaucoma. In turn, almost half
pedigrees
the offspring o f these affected
from selection o f unusual cases f o r publica
have
to
date manifested
early
individuals glaucoma.
of
her affected offspring from two
marriages
clearly implicate her as transmitting the dis ease.
glaucoma
results
tion. SUMMARY
Since I I 8 died at the age o f 57 years, she did not herself manifest glaucoma, although
hereditary
1. A pedigree illustrating dominant inherit ance o f chronic simple glaucoma is
pre
sented. 2. In 13 to 25 percent o f cases o f glau
difficulties
coma, careful investigation has shown other
attendant upon obtaining information on a
members o f the family to be glaucomatous.
This
pedigree illustrates the
(1)
3. It is recommended that patients with
disease;
glaucoma should be advised o f its hereditary
disease manifested only in later l i f e : M a n y die before manifesting the
( 2 ) the family history becomes very sketchy
nature. In this w a y their affected
beyond one generation
m a y receive the benefits o f earlier diagnosis
in the g r a v e ;
(3)
younger generations are not yet old enough
and therapy. The Ohio State University
to exhibit the disease. I am in complete agreement
with
such
authors as Posner and Probert ( R e f s . 2 and 3, Table 1) w h o point out that the predomi
OPTICAL
PRINCIPLES
relatives
(10).
Grateful acknowledgment is made to Dr. Madge Macklin for her help in the preparation of this paper.
OF LOUPE
MAGNIFICATION
ARTHUR LINKSZ, M . D . New York
Interest
in
telescopic
and
so-called
error a n d that, within reasonable
limits,
m i c r o s c o p i c lenses for the partially sighted
it c a n a c c o m m o d a t e for a n y desired dis
has lately increased c o n s i d e r a b l y . Little,
tance.
h o w e v e r , has been heard
about
another
B y c o m m o n understanding,
the m a g n i
t y p e o f visual aid, the m a g n i f y i n g l o u p e .
fying l o u p e is a c o n v e x lens of m o d e r a t e
Like the m i c r o s c o p i c lens, the m a g n i f y i n g
d i o p t r i c p o w e r , held at s o m e distance from
l o u p e serves for the i m p r o v e m e n t of near
the e y e . T h i s distance usually is greater
vision. It is i n v a r i a b l y less e x p e n s i v e a n d
than the distance of c o n v e n t i o n a l specta
in s o m e cases j u s t as effective.
cles, b u t
In order to p u t m a g n i f y i n g loupes t o
n o t greater than a b o u t
arm's
length. V i e w e d t h r o u g h it are rather small
best use, their o p t i c s m u s t b e u n d e r s t o o d ;
o b j e c t s w h i c h are held within
it is the p u r p o s e of this paper t o explain
length of the l o u p e . T h i s latter limitation is
the optical principles of l o u p e magnifica
essential. If an o b j e c t is o u t s i d e the
tion in an e l e m e n t a r y manner. D e t a i l s o f
f o c u s of a c o n v e x lens, b u t n o t further from
clinical a p p l i c a t i o n subsequent
will b e treated in a
paper. F o r the simplification
the
the lens t h a n t w i c e its focal length,
focal first the
o p t i c a l i m a g e will b e magnified. T h i s i m a g e
o f b o t h the d i a g r a m s a n d the f o r m u l a s it
h o w e v e r will be real and therefore i n v e r t e d .
will be assumed t h a t the magnifying l o u p e
T h e image obtained through loupe magni
is infinitely thin and that the e y e has a
fication
s i m p l e nodal p o i n t . It also will b e taken
b e virtual.
is n o t t o b e c o m e i n v e r t e d ; it must
for granted that the e y e has n o refractive
l o o k s at this erect virtual image which is
O p t i c a l l y speaking,
the
eye
832
ARTHUR LINKSZ
Fig. 1 (Linksz). Diagram to sliow relationsliips between an object and its virtual image produced by a convex lens.
the effective o b j e c t as far as the e y e is c o n c e r n e d . T h e size of this effective o b ject, or o c u l a r o b j e c t , and its distance from the e y e determine the size of the retinal image and the effective retinal magnification. T h e relationships b e t w e e n an o b j e c t and its virtual image p r o d u c e d b y a c o n v e x lens are easily established with the help of the d i a g r a m g i v e n in Figure 1. In this figure, AB represents an o b j e c t o, and A 'B' the image i; F is the first princi pal focus and 0 the optical center of a (infinitely thin) c o n v e x lens. T h e focal distance FO of the lens will be called f; the distance FA of the o b j e c t from the focus will b e called /. T h e letter ν will stand for the distance AO of the i m a g e from the lens. T h e distance AO oí the o b j e c t from the lens equals (f—l). T h e size and position of the i m a g e are f o u n d b y using the rays (F)BH and BO in the well
k n o w n manner. ( T h e first o f these rays, while a c t u a l l y issuing from B, b e h a v e s as if c o m i n g from the direction of F. T h e r e fore, hitting the lens at H, it leaves the lens parallel to the principal axis. T h e second ray passes through 0; it thus re mains undeflected. T h e b a c k w a r d p r o longations of these rays m e e t at B'; thus B' is the virtual i m a g e of Β and the posi tion a n d length of the line A'B' gives b o t h the position and the size of the vir tual i m a g e w h i c h e v e n t u a l l y is t o b e c o m e the o b j e c t for the e y e . T h i s is n o t y e t s h o w n in the d i a g r a m . ) T h e j u s t g i v e n points and their c o n n e c t i o n s outline several pairs of similar triangles, like FAB and FOH o r OAB and OA'B', and since s o m e of these linear values are g i v e n , others c a n b e calculated b y well k n o w n rules w h i c h g o v e r n similar triangles. T h e linear values A'B' and OH are o b v i o u s l y e q u a l ; i c a n therefore b e substituted for OH and, from
833
OPTICAL PRINCIPLES OF LOUPE
the first pair of similar triangles it can b e determined that 1 = 1
or 0
(1)
I
(la)
In the latter equation, l ' all three values of the right side are k n o w n and i can b e calculated. E q u a t i o n (1) is, b y the w a y , the well k n o w n N e w t o n i a n magnification formula w h i c h expresses the size relations between i m a g e and o b j e c t {i'.o) in terms of t w o other g i v e n linear values, / a n d /. T h e second pair of similar triangles presents another expression for this rela tionship, v i z .
0
(2)
J-l
and since the left sides in equations (1) and (2) are equal it is o b v i o u s that
/
fif-i) I
Figure 2 s h o w s a n o b j e c t AB, o r o, at the distance « from the nodal p o i n t Ν of an e y e . In order t o r e c e i v e a sharp reti nal image, this e y e , assumed t o b e e m m e t r o p i c , need a c c o m m o d a t e 1 / M d i o p ters.* T h e angular size a o f the retinal image is characterized b y the relationship of the t w o j u s t g i v e n linear v a l u e s : tan a = —.
J-l
T h u s , in the case of the virtual i m a g e p r o d u c e d b y a c o n v e x lens, it can b e stated that v= -
the lens than the latter. T h e e q u a t i o n s s o far q u o t e d are valid for c o n v e x lenses a n d virtual images formed b y t h e m in general. O n l y b y its being at a reasonable distance from an e y e and in c o n n e c t i o n with the optical s y s t e m of this e y e , d o e s a c o n v e x lens b e c o m e a l o u p e and influence the size of the retinal image. It w a s stated earlier that, in order to simplify matters, it will b e assumed t h a t the eye has b u t o n e nodal p o i n t and t h a t it is able t o a c c o m m o d a t e for a n y desirable distance. M o r e o v e r , it will b e assumed that a c c o m m o d a t i o n d o e s n o t measurably c h a n g e the position o f the o c u l a r n o d a l p o i n t in relation t o the retina.
f-n I
(3)
Since the essential c o n d i t i o n w h i c h m a k e s c o n v e x lenses a c t as loupes is the presence of the o b j e c t inside the principal focus, / can v a r y o n l y b e t w e e n 0 and / . In the first e x t r e m e case, b o t h i a n d ν b e c o m e infinite; in the s e c o n d e x t r e m e case, i b e c o m e s equal t o o and ν b e c o m e s z e r o w h i c h m e a n s t h a t i m a g e and o b j e c t c o i n c i d e and that there will b e n o magnifi cation. A t a n y in-between value, i is a l w a y s larger than o and ν is larger than (J—l); thus, for all practical purposes, the virtual image (which is the o b j e c t of vision in l o u p e magnification) is always larger than the actual o b j e c t and farther from
(4)
O b v i o u s l y , the larger the distance u, the smaller is the retinal i m a g e of a n y o b j e c t 0. It is the increase of the angular v a l u e a of this retinal i m a g e with w h i c h l o u p e magnification is c o n c e r n e d . Figure 3 s h o w s a c o n v e x lens inter posed b e t w e e n the o b j e c t and the e y e . T o m a k e it serve as a l o u p e , the distance o f the o b j e c t f r o m the lens m u s t be smaller than / . T h e distance u o f the o b j e c t from the e y e is n o w a c o m p o s i t e of t w o values, n a m e l y the s u m of the distance b e t w e e n l o u p e and e y e , z, and of the distance (f—l) b e t w e e n o b j e c t and loupe. N o w , the inter position of a l o u p e b e t w e e n e y e and o b j e c t throws the virtual i m a g e of the latter as far as
At-I)
v= -
I
* u being expressed in meters.
(3)
ARTHUR LINKSZ
834 Β
Fig. 2 (Linl
from the lens. T h u s , the distance of the
a c c o m m o d a t i o n is needed w h e n v i e w i n g
virtual image from the e y e b e c o m e s
objects
Åf-i)
y=-
through
a loupe—an
advantage
m a n y an early p r e s b y o p e m a k e s use of, (5)
i
even if he is otherwise n o t in need of m a g nification. T h e size of this virtual i m a g e
T o see this virtual image, w h i c h is the o b j e c t of vision, the e y e need a c c o m m o d a t e 1/y diopters.* Since y is, b y the nature of things, a l w a y s larger than u, less
has been found to b e
1
a v a l u e w h i c h , o b v i o u s l y , represents the
* y being again expressed in meters.
I
FT ."Λ I
(la)
I
κ u
Fig. 3 (Linksz). Shows a convex lens interposed between the object and the eye.
O P T I C A L P R I N C I P L E S OF L O U P E
size of the ocular o b j e c t . T h e size of the retinal i m a g e of the o b j e c t seen
through
the l o u p e can be characterized b y a tangen tial relationship
similar to e q u a t i o n ( 4 ) ,
namely
I
tan a = — =
y
Γ-β+ζΙ
of
(6)
r—ß+zl
ι W h i l e this is an easy equation t o solve, it is in m a n y cases m o r e a d v a n t a g e o u s t o substitute the rather incidental v a l u e ζ b y a term containing u, the distance of the actual o b j e c t from the ocular nodal point. T h u s
y = v-{f~l) + u Ρ-β ι f-2ß+P+ul
if-iy+ui
l
(5a)
and
tan a -
oj
U-iy+ul
(f-ir+ul
(6a)
Since the angular values e n c o u n t e r e d in clinical o p t i c s usually are small a n d o n l y limited a c c u r a c y is necessary, the t a n g e n t s found in equations (4) and (6a) can j u s t as well represent the actual angles. T h e magnification of the retinal i m a g e b y a l o u p e is therefore
fn =
tana' tan a
a'
=— =
a
of {f-iy+ul o
uf (f-iy+ui
(7)
A s s u m i n g n o w that, in a g i v e n case, the actual distance u b e t w e e n e y e and
835
o b j e c t is a l w a y s k e p t c o n s t a n t , the ques t i o n arises: W h a t h a p p e n s if a l o u p e o f s o m e ( o b v i o u s l y c o n s t a n t ) focal distance / is b r o u g h t closer to the e y e or t o the o b j e c t ; in other w o r d s , if the d i s t a n c e / is v a r i e d ? A c c o r d i n g t o the definition of a l o u p e , I can, in t h e case of a l o u p e , o n l y v a r y b e t w e e n zero and / and as usually in o p t i c s o n e will gain v a l u a b l e information b y analyzing these e x t r e m e values. T o m a k e I a p p r o a c h / , the l o u p e has t o b e b r o u g h t v e r y close t o the o b j e c t . Setting / = / , e q u a t i o n (7) reads
a'
uf
— · = » ί ι η ί η = —
a
=
(7a)
1.
uf
T h u s , a l o u p e has n o (or little) m a g n i f y ing effect if it is b r o u g h t t o o close t o the o b j e c t , w h a t e v e r its p o w e r m a y be. T h e size of the retinal i m a g e d e p e n d s in this case entirely o n the actual distance u of the o b j e c t from the e y e (it b e c o m e s smaller as u increases). M u c h m o r e remarkable is the result arrived a t b y setting Ζ = zero, t h e case in which the l o u p e is held a l m o s t as far f r o m the o b j e c t as the l o u p e ' s focal length. T h o u g h b o t h i and ν tend t o b e c o m e in finitely large, n o unlimited degree of m a g nification c a n b e a c h i e v e d b y bringing t h e l o u p e t o o far f o r w a r d — n o t e v e n a n y d e sired a m o u n t . Setting / = z e r o , e q u a t i o n (7) changes t o α
max
uf =»?max = —
β
u =
-—
f
(7b)
the m a x i m u m v a l u e of m for a n y g i v e n distance u. T h e existence of such a m a x i m u m s h o u l d b e understandable. T h e virtual i m a g e s o o n gets t o o far from the e y e for its actual enlargement t o b e of a n y further a d v a n t a g e for v i s i o n . W i t h a n y g i v e n u and / , m a g n i fication s o o n reaches o p t i m a l values and t h u s there will b e n o t h i n g g a i n e d b y bringing the l o u p e a n y closer. O n e e x a m p l e will suffice t o d e m o n strate this: H o l d i n g the o b j e c t at, for
836
ARTHUR
example, 18 c m . from the eye and a + 1 2 . 5 D . loupe 6 c m . in front of it, magni fication of the retinal image is X 2 . M a g n i fication increases t o X 2 . 1 5 w h e n the loupe is held 7 c m . from the o b j e c t and m a x i m u m possible magnification—holding the loupe nearly 8 c m . from the o b j e c t — is X 2 . 2 5 . T h e main difference lies n o t so m u c h in the magnification as in the a m o u n t s of a c c o m m o d a t i o n necessary t o receive a sharp retinal image. T h e nearer o n e brings the loupe, the less o n e need accommodate.* S t u d y i n g equation ( 7 b ) , s o m e m o r e a b o u t magnification b y l o u p e will b e learned. It is o b v i o u s t h a t for a n y g i v e n o b j e c t distance u, o p t i m u m magnification increases with d i m i n i s h i n g / ; thus, stronger loupes give m o r e magnification and t h e y can b e held farther from the e y e than weaker loupes, while u remains constant. T h i s is of great p s y c h o l o g i c i m p o r t a n c e . T h e elderly partially sighted patient is m o s t u n h a p p y a b o u t the short range, rigidly t o b e enforced, w h e n e v e r he tries to use a m i c r o s c o p i c lens. H e can achieve less magnification with a loupe, b u t at a m u c h m o r e c o m f o r t a b l e range. A n d he can actually hold a stronger l o u p e ( o n e ofl^ering greater magnification) further than a weaker one, while maintaining the same distance u. ( T h e disadvantages of t o o m u c h magnification, the distortions caused b y strong lenses, the retardation of read ing speed b y undue increase of letter size, the limitations of the field, are n o t t o b e considered in the present paper.) It w o u l d b e w r o n g , h o w e v e r , to c o n c l u d e from the presence of u in the numerator of equation (7b) that there are unlimited possibilities in l o u p e magnification b y increasing the distance u between o b j e c t and e y e . Substi tuting o/u for a' into equation ( 7 b ) , the
* A presbyope will hold the loupe as far from the object as possible (nearly 8 cm. in the given example) since in this case i is infinitely far and no accommo dation or reading lens is needed. t Cf. equation (4).
LINKSZ
following i m p o r t a n t
result is arrived
α max
at:
Μ = »imax =
O
ί
J
(8)
T h i s equation, at first sight, appears strange. A c c o r d i n g t o it, the size of a retinal i m a g e w o u l d b e i n d e p e n d e n t o f o b j e c t distance (M d o e s n o t figure in the e q u a t i o n ) while, at the same time, it is clear from equation ( 7 b ) that magnifica tion ( e v e n m a x i m u m magnification) o f the retinal i m a g e increases with u. Still, the equation is c o r r e c t and the m a x i m u m a c tual size a'max of the retinal image of s o m e given o b j e c t o, seen through a l o u p e o f 1// diopters power, is in fact independent of the distance between o b j e c t and e y e ; it is o n l y the magnification t h a t changes and so does, of course, the original size a of the retinal image seen w i t h o u t the inter ference o f a loupe. T h e value a decreases with increasing M, while m increases; and since the increase of m with increasing « just m a k e s u p for the decrease of a with increasing u, the size of the magnified retinal image a ' m a x ' s in the final analysis independent of u. It is c o n s t a n t for a n y g i v e n 0 and / . If the l o u p e is held as far (or a l m o s t as far) from the o b j e c t as its focal length and if the o b j e c t and the loupe are moved together t o w a r d the e y e or a w a y from it, the size of the retinal image is unchanged. T h i s fact, while theoretically interesting, is of definite i m p o r t a n c e besides. A c c o r d ing t o the formula, a ' m a x remains un c h a n g e d w h a t e v e r the v a l u e of u. It should thus b e possible t o read print smaller than Jaeger 1 as well with the help of a l o u p e from across the street as it is from a r m ' s length. T h i s is true, at least in prin ciple, i n actuality, such m a t t e r s are l i m ited—as m u c h b y the inaccuracies of elementary optical formulas w h i c h are
OPTICAL PRINCIPLES OF L O U P E
o n l y a p p r o x i m a t i o n s , as b y the decrease, with the square of distance, of the light energy which emanates from a n y o b j e c t . Besides, the diameter of the l o u p e itself a c t s as limiting factor o f useful magnifica tion, as a kind of s t o p or entrance pupil. A s ζ increases, the angular size of this entrance pupil, as measured from t h e o b server's e y e , b e c o m e s smaller and smaller. T h e size of the largest o b j e c t w h i c h is entirely visible through a l o u p e varies, first of all, with its diameter d and, other things k e p t u n c h a n g e d , varies inversely with the distance ζ of the loupe from the eye. T h e diameter d and the distance ζ determine quasi a c o n e of lines of direction with its base at the l o u p e , its apex at N, and an apical angle of 2ω. Vision through a loupe is limited t o the lines o f d i r e c t i o n which m a k e up this c o n e . T h e d o t t e d lines a d d e d t o the diagram of the l o u p e (Fig. 3) should clarify this. L i n e PN repre sents the direction of the m o s t peripheral r a y w h i c h , modified in direction b y the l o u p e at P , still reaches the retina (assum ing t h a t the pupil of the e y e is p o i n t sized and in the plane of N). T h i s ray, while actually c o m i n g from an o b j e c t point Bj, in the o b j e c t plane, appears to c o m e from B'p in the image plane, a l o n g the b a c k w a r d prolongation of ray PN. T h u s , ABp is the m a x i m u m length of an o b j e c t , O n i a x . fully visible at a given u v a l u e ; A'B'p is the c o r r e s p o n d i n g maxi m u m image, i „ , a x . T h e angle ω s u b t e n d e d b y the c o m m o n principal axis o f the c o m bined system, loupe and eye, and the r a y {B'p)PN equals o n e half of the total angle of the entrance pupil and 5 ρ is the m o s t peripheral point seen through the loupe in the indicated setting. H o w e v e r , d and ζ determine the c o n e of lines of direction o n l y ; they are not the o n l y limiting factors to the field seen through a loupe. E v e n if they, and with them ω , are k e p t constant, and e v e n if one certain loupe of the p o w e r l / / i s c o n sidered, m o r e of a line of letters is visible through the loupe if / increases, and less if
837
it decreases. T h e b r o k e n line from BPmin to Ρ in Figure 3 indicates the possible loci of Bp and the possible lengths o f the line ABp with the changes of I, if d, z, and / are g i v e n a n d c o n s t a n t . If / a p p r o a c h e s / , in other w o r d s , if a line o f letters is b r o u g h t v e r y close to the l o u p e , m a p proaches 1* and ABp a p p r o a c h e s a maxi m u m , OP. T h u s , the greatest length of a line of letters seen through a loupe c a n n o t exceed the d i a m e t e r d of the l o u p e . If, o n the other hand, / a p p r o a c h e s zero, in o t h e r w o r d s , if a line of letters is b r o u g h t close t o the focal plane of the l o u p e , t h e m m b e c o m e s m a x i m u m , ^ b u t the length ABp of o n e half of the longest visible part of a line of letters a p p r o a c h e s a m i n i m u m v a l u e and it b e c o m e s equal t o the d i s t a n c e F5p^.„ if / b e c a m e zero. T h e actual values of the variable distance ABp are found b y the following consider ation: It is o b v i o u s from Figure 3 t h a t —' = — · = ζ y
tan ω
for
a n y g i v e n v a l u e of y. Substituting for imJ and f-fl+zl/l for y , § one gets On,ax//i
Omax/ Omax/
2
Ρ-β+21
f-ß+2l
(9)
I and ABp = o^^^ =
1 2
diP-Jl+ll)
(10)
zf
T h e distance 2omax = 2^5p, o r the maxi m u m length of a line of letters seen through a l o u p e can thus b e calculated for a n y l o u p e of k n o w n p o w e r 1// and diameter d, for a n y distance ζ b e t w e e n l o u p e and e y e
* t t §
Cf. Cf. Cf. Cf.
equation equation equation equation
(7a). (7b). (la). (5).
838
ARTHUR LINKSZ
and for a n y distance ( / — / ) between print and loupe. K e e p i n g all other values c o n stant b u t the last, the length of 2ABp m u s t v a r y with /. It is, as m e n t i o n e d , a m a x i m u m and equal to d w h e n the print is b r o u g h t close t o the loupe, and a mini m u m when the print is b r o u g h t into (or close t o ) the focal plane of the l o u p e (the c o n d i t i o n in w h i c h a' b e c o m e s c o n s t a n t ) . E q u a t i o n (9) changes in this c a s e t o hd
"max /
0 .
(9a)
Í T o i n d i c a t e the fact t h a t o^a^ = ABp varies with v a r y i n g I (cf. fig. 3 ) , its great est v a l u e OP will b e called Om^x (max)," its smallest v a l u e FB^, mm is designated as 0 max(inin).
A s s u m i n g then that the diameter d and the distance ζ of diff^erent loupes from the eye are k e p t c o n s t a n t and the print a l w a y s held in their respective focal planes, it is o b v i o u s from equation (9a) that 20max(miii). the m a x i m u m length of a line of letters visible through these different loupes, d e pends entirely o n / b e c o m i n g shorter as loupes g e t stronger. T h e actual length of these lines for loupes of a n y p o w e r 1 / / is derived b y writ ing the e q u a t i o n in the form Omax(min) = i'-Bp,miii =
¥f
= / tan ω
Ζ
or dj 2o„,ax(n,in) = — = 2 / t a n ω ζ
(10)
O b v i o u s l y , w h e n e v e r a line of letters is held in the focal plane of a g i v e n l o u p e {d a n d / a r e c o n s t a n t ) for m a x i m u m retinal image magnification ( a ' = m a x i m u m ) , then the visible length of this line d e p e n d s en tirely o n z, the distance of l o u p e (and letters) from the e y e . It increases with diminishing z. N o t magnification, b u t field is gained b y holding l o u p e and letters as close t o the e y e as possible. A few numerical e x a m p l e s will illustrate this point.
A s s u m i n g as an e x a m p l e that the length of a line of letters is 8 c m . and t h a t the focal length o f the l o u p e w h i c h is being used is also 8 c m . , tan α'max (the tangent of the largest possible retinal i m a g e of one half oí the line) is o / / = 4 / 8 = l / 2 * and the angular size of the retinal i m a g e of the total line is a b o u t 53 degrees. T h i s v a l u e will a l w a y s remain c o n s t a n t . T h e retinal i m a g e of the total line seen through the l o u p e will a l w a y s c o v e r the s a m e n u m b e r of retinal elements. If n o w the radius of the l o u p e (half of its d i a m e t e r ) is 5 c m . and the l o u p e is held 10 c m . from the e y e (the print is t o b e held another 8 c m . furth er than the l o u p e ) , the entrance pupil f o r m e d b y the l o u p e at the o b s e r v e r ' s e y e has the s a m e angular v a l u e and the w h o l e line is visible through the l o u p e . H o w e v e r , if o n e holds the l o u p e at 20 c m . from the e y e (the print is again held 8 c m . farther than the l o u p e ) , then the tangent of o n e half of the entrance pupil b e c o m e s 5 / 2 0 = 1/4 and the angular v a l u e of the entrance pupil a b o u t 28 degrees. O b v i o u s ly, o n l y a b o u t o n e half of the line of 8 c m . length will b e seen through the l o u p e at o n e time. ( T h e l o u p e w o u l d h a v e t o have a diameter of 20 c m . for a w h o l e line of 8 c m . length t o b e visible through it at this distance.) T h i s entrance pupil b e c o m e s increas ingly smaller if ζ is increased. H o l d i n g the l o u p e at the distance of 2.00 m . (the print m u s t again b e a n o t h e r 8 c m . farther), the tangent of o n e half of the entrance pupil b e c o m e s 5 / 2 0 0 = 1/40 a n d the useful magnified retinal i m a g e will have an exten sion of less than three degrees. H a r d l y a longer w o r d of newspaper print will b e seen at a time and besides, b y m o v i n g the head, the print will s o o n b e lost al together. In spite of all these o b v i o u s limitations, the present author has often d e m o n s t r a t e d t o his a u d i e n c e that even smallest lettering o n a s t a m p o r o n the face of a w a t c h c a n * Cf. equation (8J.
OPTICAL PRINCIPLES OF LOUPE
v e r y well b e read across the r o o m , e v e n through a w e a k loupe, especially in bright d a y l i g h t illumination.* A s iar as practical application goes, t h e principle m a d e e v i d e n t b y e q u a t i o n (8) is responsible for the great versatility of loupes as visual aids. G i v e n s o m e (small) o b j e c t 0 and a loupe of the p o w e r 1//, visual performance is largely i n d e p e n d e n t of o b j e c t distance if the o b j e c t is in, o r near, the first focus of the loupe. A n o p h t h a l m i c surgeon, a philatelist, a b o t a n ist, will v a r y his distance from the o b j e c t he handles a c c o r d i n g t o the m o s t a d v a n t a g e o u s distance o f this handling w i t h o u t losing a n y effective magnification and since i is always infinitely far, i n d e p e n d e n t l y of presbyopia, if present. A s m e n t i o n e d earli er, stronger loupes can b e held farther from the e y e for a n y given c o n s t a n t v a l u e of u. It n o w turns o u t t h a t « can b e a n y dis tance. T h a t b o t h of these circumstances are w e l c o m e t o the partially sighted is o b v i o u s . All he has to d o is m a k e sure t h a t o b j e c t and loupe b e at c o n s t a n t distance f r o m e a c h o t h e r a n d t h a t this d i s t a n c e b e nearly that of the focal distance of the loupe. T h e r e are m a n y c o m m e r c i a l l y avail a b l e d e v i c e s t o k e e p a l o u p e a certain dis t a n c e from a printed page. T h e distance, of course, has t o b e well c h o s e n and rigidly kept. All this freedom of setting oneself at a desirable distance from the printed p a g e is lost w h e n e v e r a p r e s b y o p e uses a l o u p e and his reading correction a t the same time (or w h e n e v e r a m y o p e with n o c o r rection uses a l o u p e ) . In this case, y is limited. A person wearing, for e x a m p l e , a -f-4.0D. reading addition sees ( a t least theoretically) o n l y such o b j e c t s sharply which are 25 c m . from his lens. In o r d e r t o see a magnified o b j e c t sharply t h r o u g h a loupe, the distance y of the virtual image i must be 25 c m . from his reading lens. • It should be possible to put this principle to some limited practical use, for example, by making house numbers or names over entrance doors better visi ble, or in some eye-catching advertising displays.
839
E q u a t i o n (5a) has s h o w n that U-iy+ul y=-
p-2fl+P+ul I
l ρ
=
---2f+l+u.
F r o m this it follows t h a t u=
(11)
Ρ y+2f-j-l.
O b v i o u s l y , if a certain l o u p e is used ( / is c o n s t a n t ) and if there is n o c h o i c e in the v a l u e of y because of a reading a d d i t i o n (or uncorrected m y o p i a ) , u will v a r y with I. In other w o r d s , the distance between e y e and o b j e c t will h a v e t o b e c h a n g e d if the distance (/— / ) b e t w e e n o b j e c t and l o u p e is varied. A few numerical e x a m p l e s will clarify this. It will b e assumed again t h a t / = 8 c m . and t h a t y = 25 c m . and is unchangeable. If the l o u p e is held 2 c m . f r o m the o b j e c t (1 = 6 c m . ) , u m u s t b e 2 4 | c m . T h u s , o b j e c t and virtual image will a l m o s t c o i n c i d e (the latter will o n l y b e f c m . farther) a n d there will h a r d l y b e a n y useful magnifica tion. If the l o u p e is k e p t 6 c m . from the o b j e c t {1 = 2 c m . ) , u will b e c o m e 7 c m . and the l o u p e will h a v e t o b e held 1 c m . in front o f the p r e s b y o p i c reading a d d i t i o n . Magnification will again n o t b e v e r y effec t i v e . F r o m e q u a t i o n ( 7 ) it is f o u n d t h a t in this case 7X8 56 -= — = m= ( 8 _ 2 ) 2 - f 7 X 2 50
1.12.
Still, e v e n if tn is n o t v e r y great, the l o u p e will n o t b e w i t h o u t v a l u e . A l o u p e o f / = 8 c m . permits the p r e s b y o p e wearing a - | - 4 . 0 D . reading a d d i t i o n t o bring o b j e c t s as close to his e y e as 7 c m . , a distance for w h i c h he w o u l d otherwise need a rather strong m i c r o s c o p i c lens. Since 7 c m . is a b o u t o n e fifth o f t h e standard reading distance o f 14 inches ( a b o u t 35 c m . ) , the a p p r o a c h in itself will magnify the retinal i m a g e b y X 5 . 0 . T h e additional,
840
ARTHUR LINKSZ
t h o u g h small, magnifying effect increases this to X S . 6 . A person with not m o r e than 1 4 / 7 0 ( 2 0 / 1 0 0 equivalent) near vision should b e able to read the 1 4 / 1 4 print b y such arrangement—not a m i n o r a c h i e v e m e n t with such inexpensive means. A case o f especial interest is the o n e in w h i c h the o b j e c t is held halfway be tween a l o u p e and its principal focus. In this case, l = (f—l) = / / 2 and it follows from equation (3) that v=f. Thus, whenever an o b j e c t is held halfway between a loupe and its principal focus, the virtual image is in the principal focal plane. M o r e o v e r , it follows from equation ( l a ) that i = 2o, whatever the p o w e r of the l o u p e . T h e s e circumstances are of great inter est in practical w o r k . W h e n e v e r a pres b y o p e has his reading distance fixed b y a reading addition, he can be assured of X 2 . 0 magnification b y any loupe, holding the loupe in such a manner that its principal focus coincides with the reading distance. A t the same time, he is to hold the o b j e c t halfway between this distance and the loupe. In the present e x a m p l e where / is assumed t o b e 8 c m . and y is 25 c m . , it will b e necessary to hold the o b j e c t 21 c m . from the reading lens and the loupe an other 4 c m . closer. Such arrangement is ideal for patients with n o t t o o great loss of vision. If a person has sufficient visual a c u i t y (around 2 0 / 8 0 ) to read a magazine at 25 c m . with a - | - 4 . 0 D . reading addition, he will h a v e a m p l e sight t o read a news paper, or look up a telephone number, with a n y weak loupe, in case l o u p e and reading matter are placed into the posi tions just indicated. ( F o r a 4 - 1 2 . 5 D . loupe, these positions will b e 17 and 21 c m . in front of the eyeglasses, respective
l y ; for a -I-IO.OD. loupe, 14 and 20 c m . , and s o forth.) A s a m a t t e r of fact, even persons with less vision often prefer this arrangement t o the m i c r o s c o p i c lens; a patient with 2 0 / 1 6 0 visual a c u i t y will b e able t o read a magazine w i t h it. SUMM.ARY A N D
CONCLUSIONS
Starting from the well k n o w n N e w tonian formula for the relation between o (the size of the o b j e c t ) and i (the size of the virtual i m a g e ) , a s y s t e m o f e q u a t i o n s is d e v e l o p e d b y w h i c h all m a g n i t u d e s pertinent t o loupe magnification can easily b e ascertained. Such values are ν (the dis tance of the image from the l o u p e ) , y (the distance o f the image from the e y e ) , a' (the angular size of the retinal image of an o b j e c t as seen through a l o u p e ) , and m (the effective magnification of this retinal image). It is s h o w n that, while m changes with the distance of the e y e from the loupe, a' (the size of the magnified retinal image) is largely independent of this distance o n c e loupe and o b j e c t are m o v e d together. T h i s explains w h y loupes c a n b e used at a n y c o n v e n i e n t distance, also b y presby o p e s w h o have no reading c o r r e c t i o n . It is s h o w n furthermore h o w the wear ing of a p r e s b y o p i c reading addition af fects the usefulness of a l o u p e . Finally, a simple formula is d e v e l o p e d b y w h i c h X 2 magnification can b e achieved and a c o n v e n i e n t reading dis tance retained when using a l o u p e of a n y power. 6 East 76th Street {21). I wish to acknowledge the help o f Dr. Harris Ripps, Forest Hills, New York, in the preparation of the diagrams.