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Theory of the halftone process. II. The diffraction theory-calculation of the light distribution
T H E O R Y O F T H E H A L F T O N E P R O C E S S . II. T H E DIFFRACTION THEORY--CALCULATION OF THE L I G H T DISTRIBUTION. BY J. A. C. Y U L E . Communication No. 916 from the K o d a k Research Laboratories, Rochester,
N,Y. SUMMARY.
An a p p r o x i m a t e m e t h o d of c a l c u l a t i n g the l i g h t d i s t r i b u tion in the dots f o r m e d b e h i n d a h a l f t o n e s c r e e n from the F r e s n e l d i f f r a c t i o n t h e o r y is d e s c r i b e d , a n d in a t y p i c a l c a s e , the d i s t r i b u t i o n is f o u n d t o c h e c k c l o s e l y with t h a t d e t e r m i n e d e x p e r i m e n t a l l y , b e i n g v e r y d i f f e r e n t from t h a t c a l c u l a t e d from g e o m e t r i c a l o p t i c s . In a p r e v i o u s p a p e r 1 the o b s e r v e d l i g h t d i s t r i b u t i o n in the dots f o r m e d b e h i n d a h a l f t o n e s c r e e n , u n d e r t y p i c a l c o n d i t i o n s u s e d in m a k i n g h a l f t o n e n e g a t i v e s , was f o u n d t o be very d i f f e r e n t from the d i s t r i b u t i o n c a l c u l a t e d from the p e n u m b r a t h e o r y ( g e o m e t r i c a l o p t i c s ) . It r e m a i n s t o c a l c u late the l i g h t d i s t r i b u t i o n from the t h e o r y of diffraction. T h i s c a l c u l a t i o n , if c a r r i e d out in a n e x a c t a n d g e n e r a l i z e d manner, appears to involve a laborious and complicated m a t h e m a t i c a l a n a l y s i s a n d m a y even be i m p o s s i b l e . Howe v e r , a simplified m e t h o d is s a t i s f a c t o r y for the c a l c u l a t i o n in a s p e c i a l c a s e , a n d i t is f o u n d e x p e r i m e n t a l l y t h a t d e v i a t i o n s from the c o n d i t i o n s a p p l y i n g t o this s p e c i a l case do not lead t o d i s c r e p a n c i e s of a n y i m p o r t a n c e in h a l f t o n e w o r k . P r o b a b l y , the r e a s o n for this is t h a t the lens a p e r t u r e used in p r a c t i c a l h a l f t o n e w o r k is n o t a p o i n t s o u r c e of l i g h t , b u t is r e l a t i v e l y l a r g e so t h a t all line d e t a i l s of the diffraction p a t t e r n are c o m p l e t e l y lost. T h e c a l c u l a t i o n is c a r r i e d out in t h r e e s t e p s : I. T h e l i g h t d i s t r i b u t i o n b e h i n d a parallel-line s c r e e n w i t h a v e r y s m a l l lens a p e r t u r e , i.e., a p o i n t s o u r c e of light, is c a l c u l a t e d . 483
484
J . A . C . YULE.
[J.
F. I.
T h e r e s u l t of c r o s s i n g two parallel-line s c r e e n s a t r i g h t a n g l e s is f o u n d , a p o i n t s o u r c e of l i g h t a g a i n b e i n g u s e d . 3. R e g a r d i n g the n o r m a l lens a p e r t u r e a s c o n s i s t i n g of a l a r g e n u m b e r of i n c o h e r e n t p o i n t s o u r c e s of l i g h t , each of w h i c h i n d e p e n d e n t l y p r o d u c e s a diffraction p a t t e r n s l i g h t l y d i s p l a c e d r e l a t i v e t o the o t h e r s , all t h e s e d i f f r a c t i o n p a t t e r n s are a d d e d up t o p r o d u c e the t o t a l l i g h t d i s t r i b u t i o n . This is the m e t h o d of a p p r o a c h u s e d by F r u w i r t h a n d M e r t l e 3 2.
The Theory of Diffraction: A b r i e f d e s c r i p t i o n of F r e s n e l ' s t h e o r y of diffraction will f i r s t be g i v e n . F o r f u r t h e r d e t a i l s , see C. F. M e y e r ' s t e x t b o o k on d i f f r a c t i o n or a n y a d v a n c e d t r e a t i s e on light. 3 In Fig. I, L is a p o i n t s o u r c e of l i g h t , S is a s c r e e n with
sI L
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Diffraction by a s m a l l opening in a screen S.
a n a p e r t u r e in it, a n d P is the so*called " f i e l d p o i n t " a t w h i c h the i n t e n s i t y of l i g h t is t o be f o u n d . E a c h p o i n t in the a p e r t u r e , for i n s t a n c e , the p o i n t A, is r e g a r d e d a s a s e c o n d a r y l i g h t s o u r c e , from w h i c h s e c o n d a r y w a v e s o r i g i n a t e . T h e a m p l i t u d e of l i g h t r e a c h i n g P is the r e s u l t a n t of the w a v e s from all p o i n t s in t h e s c r e e n a p e r t u r e . H o w e v e r , the w a v e s w h i c h do n o t t r a v e l in a s t r a i g h t line from L t o P go a l o n g a s o m e w h a t longer path, and hence arrive at P a little later. T h o s e w h i c h a r r i v e j u s t half a w a v e a p a r t , so t h a t the c r e s t of one w a v e a r r i v e s a t the s a m e m o m e n t a s the t r o u g h of a n o t h e r , e x a c t l y n e u t r a l i z e each o t h e r . W h e n the effects of all t h e s e w a v e s a r e a d d e d , it is f o u n d t h a t the l i g h t a t the p o i n t P is b r i g h t if the w a v e s m o s t l y r e i n f o r c e each o t h e r .
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n e u t r a l i z e each other, t h e l i g h t at P is r e l a t i v e l y w e a k . W a v e s w h i c h a r r i v e s i m u l t a n e o u s l y , or e x a c t l y one, two, t h r e e , e t c . , w a v e l e n g t h s a p a r t , reinforce each o t h e r . T h o s e w h i c h a r r i v e o.5, 1.5, 2.5, e t c . , wave l e n g t h s a p a r t n e u t r a l i z e each o t h e r . In most positions in the g e o m e t r i c a l s h a d o w b e h i n d t h e screen, t h e y tend t o neutralize each o t h e r , and b e h i n d t h e clear portions of t h e screen, t h e y reinforce each other.
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m e a n s that l i g h t a p p e a r s to travel in approximately straight l i n e s . W a v e s arriving at intermediate t i m e s m a y partly reinforce or partly n e u t r a l i z e each other, and t h e t o t a l effect may be f o u n d by calculation or by t h e u s e of a so-called vibration curve. T h e Cornu spiral, Fig. 2, may be used as a vibration c u r v e for determining t h e l i g h t distribution b e h i n d a straight e d g e , a s l i t , or a s e r i e s of parallel s l i t s , such as a p a r a l l e l - l i n e screen. It does not a p p l y , however, to openings a t a considerable distance from t h e direct ray, w h i c h are
486
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f o u n d t o have a n e g l i g i b l e e f f e c t in c o m p a r i s o n with the n e a r e s t o p e n i n g s in the t y p i c a l case u n d e r c o n s i d e r a t i o n . T h e c u r v e is u s e d a s f o l l o w s : In a n o p t i c a l s y s t e m , s u c h a s Fig. 3, a s t r a i g h t line d r a w n from l i g h t s o u r c e t o field p o i n t p a s s e s t h r o u g h the s c r e e n a t O. A line is d r a w n a c r o s s the s c r e e n t h r o u g h 0 a t r i g h t a n g l e s t o the e d g e s of the slit ( g r e a t l y e n l a r g e d in the d i a g r a m ) and is d i v i d e d i n t o e q u a l x/ abX " p a r t s , the l e n g t h of e a c h u n i t b e i n g e q u a l t o ~ 2 ( a ~ b ) ' w h e r e a is the d i s t a n c e L O from l i g h t s o u r c e t o s c r e e n , b is the d i s t a n c e O P from s c r e e n t o field p o i n t , a n d X is the w a v e l e n g t h of the l i g h t . T h e d i s t a n c e a c r o s s the s c r e e n , m e a s u r e d
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in t e r m s of t h e s e g r a d u a t i o n s , is r e p r e s e n t e d by v. T h e C o r n u s p i r a l , Fig. 2, is d i v i d e d i n t o s m a l l p a r t s of e q u a l l e n g t h a n d m a r k e d with n u m b e r s c o r r e s p o n d i n g t o the g r a d u a t i o n s on the slit. I n this p a r t i c u l a r case the slit e x t e n d s from v = 0.2 t o 1.8, P b e i n g a p o i n t in the g e o m e t r i c a l s h a d o w of the s c r e e n n e a r the edge of the slit. T h e p o i n t s of the C o r n u s p i r a l m a r k e d t h u s are j o i n e d by a s t r a i g h t line A B . T h e l e n g t h of this line r e p r e s e n t s the a m p l i t u d e of the l i g h t r e a c h i n g the p o i n t P . T h e a m p l i t u d e with no s c r e e n p r e s e n t , w h i c h is the s a m e a s t h a t w i t h a n infinitely wide slit, is r e p r e s e n t e d by the l e n g t h of the line M N . T h e c h a n g e of p h a s e of the l i g h t due t o the s c r e e n is g i v e n by the a n g l e 0 b e t w e e n t h e s e l i n e s . T h e a m p l i t u d e a n d p h a s e of the l i g h t r e a c h i n g the p o i n t P from o t h e r p a r a l l e l s l i t s can be f o u n d in the s a m e w a y . U n d e r
May, i943.]
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T H E O R Y OF T H E H A L F T O N E PROCESS.
n o r m a l c o n d i t i o n s of u s e of a h a l f t o n e s c r e e n , it is f o u n d e x p e r i m e n t a l l y t h a t the n e a r e s t t h r e e s l i t s of a parallel-line s c r e e n are the only ones w h i c h h a v e an i m p o r t a n t i n f l u e n c e on the l i g h t d i s t r i b u t i o n a t the p o i n t P . In Fig. 4, A is a d i a g r a m of the o p t i c a l s y s t e m . T h e l i n e s r e p r e s e n t i n g the t h r e e s l i t s of the s c r e e n , a, b, a n d c, n e a r e s t t o the field p o i n t , are s h o w n on the C o r n u s p i r a l . I t is n o t s u f f i c i e n t m e r e l y ~8 L~
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t o add the a m p l i t u d e s of the l i g h t a r r i v i n g from d i f f e r e n t s l i t s , s i n c e , if the p h a s e s are o p p o s i t e , they will tend t o n e u t r a l i z e each o t h e r . T h e i r a m p l i t u d e s a n d p h a s e s , i.e., the l e n g t h s a n d d i r e c t i o n s of the l i n e s , m u s t be c o m b i n e d v e c t o r i a l l y by p l a c i n g the l i n e s end t o e n d , a s in Fig. 4C, in w h i c h the n a r r o w l i n e s r e p r e s e n t i n g the t h r e e s l i t s c o m b i n e t o give the r e s u l t a n t s h o w n a s a h e a v y l i n e . A f t e r the a m p l i t u d e h a s b e e n f o u n d in this w a y a t a s i n g l e p o i n t P, the p r o c e s s is r e p e a t e d u n t i l a sufficient n u m b e r of p o i n t s have b e e n f o u n d .
488
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In a c t u a l c o m p u t a t i o n s it is o f t e n m o r e c o n v e n i e n t t o s o l v e the p r o b l e m by c a l c u l a t i o n i n s t e a d of g r a p h i c a l l y , a s will be s h o w n in the m a t h e m a t i c a l p a r t . T h e r e s u l t i n g l i g h t d i s t r i b u t i o n b e h i n d a parallel-line s c r e e n with e q u a l l i n e s a n d s p a c e s , the w i d t h of each of w h i c h is 2.4 in t e r m s of v (for i n s t a n c e , with d i s t a n c e from l i g h t s o u r c e t o s c r e e n = 23.5 in., s c r e e n s e p a r a t i o n = o . 4 1 3 in., w a v e l e n g t h = 435 mtz, a n d s c r e e n r u l i n g = I2o l i n e s per i n c h ) , is s h o w n in Fig. 5.
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Fit. 5. Calculated diffraction pattern behind a x : I parallel-line screen, with a smalllight source, under conditions such that the width of one screen opening equals 2.4 in terms of v. Crossed Screens: F o r the s p e c i a l case w h i c h we are c o n s i d e r i n g , i t will be s h o w n (see m a t h e m a t i c a l p a r t ) t h a t , with a p o i n t s o u r c e of light, the i n t e n s i t y a t a g i v e n p o i n t of o b s e r v a t i o n , due t o two parallel-line s c r e e n s c r o s s e d a t r i g h t a n g l e s , is e q u a l t o t h e p r o d u c t of the i n t e n s i t i e s w h i c h w o u l d be p r o d u c e d by e a c h of the s c r e e n s t a k e n s e p a r a t e l y , t h e i n t e n s i t y in the a b s e n c e of a n y s c r e e n b e i n g t a k e n as u n i t y . T h i s h a s a s u p e r f i c i a l s i m i l a r i t y t o the f a m i l i a r law of a d d i t i v i t y of d e n s i t y of two s u p e r i m p o s e d l a y e r s of a n y lighta b s o r b i n g m a t e r i a l , a l t h o u g h it m u s t be r e m e m b e r e d t h a t for d i f f r a c t i n g s c r e e n s it is only t r u e u n d e r the r e s t r i c t e d c o n d i t i o n s u n d e r c o n s i d e r a t i o n . E a c h s c r e e n m a y be r e g a r d e d a s h a v i n g a c e r t a i n a p p a r e n t o p t i c a l d e n s i t y with r e f e r e n c e t o a p o i n t s o u r c e of l i g h t a n d p o i n t of o b s e r v a t i o n . T h e a p p a r e n t
May, '943.]
489
T H F O R Y OF THE H A L F T O N E PROCESS.
d e n s i t y of the c r o s s l i n e s c r e e n is then e q u a l t o the sum of the a p p a r e n t d e n s i t i e s of the two parallel-line s c r e e n s of w h i c h it is c o m p o s e d . T h i s does n o t r e f e r only t o the a v e r a g e int e n s i t y o b s e r v e d u n d e r c o n d i t i o n s w h e r e the i n d i v i d u a l l i n e s of the s c r e e n are i n d i s t i n g u i s h a b l e , but also t o the b r i g h t n e s s a t i n d i v i d u a l p o i n t s in the diffraction p a t t e r n . T h e d i f f r a c t i o n p a t t e r n for a c r o s s l i n e s c r e e n with a p o i n t s o u r c e of light, c a l c u l a t e d in this w a y from the r e s u l t s for a parallel-line s c r e e n s h o w n in Fig. 5, is g i v e n in T a b l e I. TABLE I. Calculated light distribution i n the diffraction pattern behind a crossline screen with equal lines and spaces, a t such a distance that vl - v2, f o r the two sides o f a single opening, equals 2.4, with a p o i n t source o f light. The two p o i n t s given i n heavy type represent the center and the corner o f one screen element. Values are expressed as a percentage o f the i n t e n s i t y with the screen removed.
212
262 '44 163-o 82.4 83.3 32.7 19.8 9.6 ,6.2 2.0 324 I77 I 77.8 IO2.O ,o3.o 40.3 24.5 I3.2 2o.0 2.5 ~o.9 97.I / 42.61 55.6 56.4 122.I 13.4 6.5 4.8 5.9 2.8 ,8.6 24.4 24.6 9-72-7 3t .9 32-3 7-7 3.7 6.3 .79 .8 b 32.7 12.8 7.8 3.8 6.4 5.0 3.0 •5 2.5 .3I 1.8 -9 t.5 .I9 .44 i .73 .O9 i t.2 .I5 .02 i
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O n l y o n e - e i g h t h of the p a t t e r n c o r r e s p o n d i n g t o a u n i t s c r e e n e l e m e n t h a s been t a b u l a t e d , s i n c e the p a t t e r n is s y m m e t r i c a l a n d the rest of the p a t t e r n m e r e l y d u p l i c a t e s the f i g u r e s in this t a b l e . Diffraction Pattern Due to a Large Light Source: T h e l i g h t distribution just g i v e n is produced by a p o i n t source of light. In halftone practice, t h e l i g h t source is a lens w h i c h may be regarded as being made of a large n u m b e r of s m a l l sources of light. D i v i d i n g t h e aperture into 49 e l e m e n t s was considered sufficient for t h i s calculation. T h e relative proportions of screen and aperture, as t h e y w o u l d be s e e n from a p o i n t i n t h e plane of observation i m m e d i a t e l y
49 °
J . A . C YULE.
[J.
F. I.
b e h i n d the c e n t e r of a s c r e e n o p e n i n g , are s h o w n in Fig. 6. The 49 e l e m e n t s of the a p e r t u r e p r o d u c e 49 diffraction p a t t e r n s , i d e n t i c a l but s l i g h t l y d i s p l a c e d with r e g a r d t o each o t h e r . F o r each p o i n t in the p l a n e of the e m u l s i o n , the i n t e n s i t i e s of the a p p r o p r i a t e p o i n t s in t h e s e diffraction p a t t e r n s are a d d e d t o g e t h e r t o give the t o t a l i n t e n s i t y . In this c a s e , it i s n o t n e c e s s a r y t o t a k e p h a s e relationships i n t o a c c o u n t , if the l i g h t s o u r c e s are i n d e p e n d e n t .
FIG. 6.
Illustrating the division of the lens aperture into 49 elements which may be regarded as point sources of light.
If the lens a p e r t u r e w e r e i l l u m i n a t e d by a u n i f o r m l i g h t s o u r c e p l a c e d c l o s e t o the l e n s , the 49 e l e m e n t s of the lens a p e r t u r e w o u l d o b v i o u s l y be i n d e p e n d e n t , s i n c e t h e y w o u l d a r i s e from d i f f e r e n t p o i n t s on the l i g h t s o u r c e . H o w e v e r , in p r a c t i c e , the c o p y , w h i c h m a y be r e g a r d e d a s a l i g h t s o u r c e , is i m a g e d on the p l a n e of o b s e r v a t i o n , w h i c h w o u l d m e a n t h a t the l i g h t from all p o i n t s of the lens a p e r t u r e is c o h e r e n t , i.e., in the S a m e p h a s e . In s p i t e of t h i s , it was f o u n d e x p e r i -
May, I 9 4 3 . ]
491
T H E O R Y OF THE H A L F T O N E PROCESS.
m e n t a l l y t h a t , with a l a r g e u n i f o r m l i g h t s o u r c e , the l i g h t d i s t r i b u t i o n in the p l a n e of o b s e r v a t i o n was the s a m e , w i t h i n the l i m i t s of e x p e r i m e n t a l e r r o r , w h e t h e r the l i g h t s o u r c e was i m a g e d on the film o r was i m m e d i a t e l y in f r o n t of t h e l e n s . T h i s u n e x p e c t e d r e s u l t is c o n s i d e r e d s u f f i c i e n t justification for r e g a r d i n g t h e lens a p e r t u r e a s a collection of i n d e p e n d e n t l i g h t s o u r c e s . T h e c a l c u l a t i o n h a s b e e n simplified by c h o o s i n g a n a p e r t u r e of s u c h size (o.196 in.) t h a t its p r o j e c t i o n , as seen from the p l a n e of o b s e r v a t i o n , b e a r s a s i m p l e r e l a t i o n s h i p t o the size of a s c r e e n e l e m e n t . I n this c a s e , the p r o j e c t e d w i d t h of the a p e r t u r e was 7/24 t i m e s the w i d t h of a s i n g l e screen element. T h e r e s u l t i n g l i g h t d i s t r i b u t i o n , c a l c u l a t e d u n d e r the cond i t i o n s of s c r e e n d i s t a n c e , e t c . , j u s t g i v e n , is s h o w n in T a b l e I I. TABLE II. Light distribution behind a crossline screen with a light source equal in size to the lens aperture illustrated in Fig. 6, calculated from the diffraction pattern tabulated in Table I . 96.2'
9° 8
93.5
73 9
! 24,2 15.8 33.5 18.4 12.o 82.5 65.o 38.4! 2o.8 t2.8 ' 65 4 42 8 i 22 9 13 2 8 56 43.4 25.9i I3.7 ' 8.48 ' / 27.8 t15.I 7.88 I 6"°51 15.2 / 7.25 4-83 81.5
47 2
6I.i
]
7 .21
4 I5
3 ~9
"
9.37 6.86 4.o8
3 I3 k
2.26
I i i
if. 3
2"34
1.80
1.62
2.()8
1.43 1.4o i i I.I4
I
By j o i n i n g p o i n t s of e q u a l b r i g h t n e s s , a n d c o m p l e t i n g the p a t t e r n for one or m o r e c o m p l e t e s c r e e n e l e m e n t s , the l i g h t d i s t r i b u t i o n m a y be m a p p e d a s s h o w n in Fig. 7. T h i s m a y be c o n t r a s t e d with the r e s u l t s of the p e n u m b r a t h e o r y c a l c u l a t i o n g i v e n in Fig. 8. T h e s h a p e of the i s o l u x l i n e s is very d i f f e r e n t in the two c a s e s . T h e m e t h o d of d e t e r m i n i n g the l i g h t d i s t r i b u t i o n e x p e r i m e n t a l l y was g i v e n in the p r e v i o u s p a p e r . T h e o p t i c a l c o n d i t i o n s are s l i g h t l y d i f f e r e n t from t h o s e c h o s e n in the f o r m e r c a s e , so a s t o s i m p l i f y the diffraction c a l c u l a t i o n .
492
J . A . C . YULE.
[J. F. 1.
T h e l i g h t d i s t r i b u t i o n in the h a l f t o n e dot, f o u n d by e x p e r i m e n t in this w a y , is s h o w n in Fig. 9. It is seen t h a t the s h a p e s of the i s o l u x l i n e s s h o w n in Fig. 9 c o r r e s p o n d well w i t h t h o s e c a l c u l a t e d by the d i f f r a c t i o n t h e o r y , Fig. 7, a n d not a t all with the p e n u m b r a c a l c u l a t i o n s , Fig. 8.
PR~.,~...,
,~,.
~
~,....
~
~,-. ,
~
RELATIVE Fie;. 7.
INTENSITY
Light distribution behind halftone screen, calculated from diffraction theory.
T h e v a l u e s for l i g h t i n t e n s i t y are more e a s i l y c o m p a r e d by p l o t t i n g the a r e a of the d o t a g a i n s t the r e l a t i v e i n t e n s i t y a t its b o u n d a r y (see Fig. io). H e r e a g a i n , the diffraction, b u t n o t the p e n u m b r a c a l c u l a t i o n , a g r e e s well w i t h e x p e r i m e n t a l r e s u l t s , e x c e p t t h a t a little too m u c h l i g h t h a s p e n e t r a t e d t o the d a r k r e g i o n s . T h i s m a y be a t t r i b u t e d t o the s c a t t e r i n g of l i g h t by i m p e r f e c t i o n s in the s c r e e n a n d o p t i c a l s y s t e m , w h i c h it is v e r y d i f f i c u l t t o m e a s u r e , t o the low
May, t943. J
T H E O R Y OF T H E t-IALFTONE PROCESS,
493
r e s o l v i n g p o w e r of the e m u l s i o n , a n d p a r t l y t o the a p p r o x i m a t i o n s used in m a k i n g the d i f f r a c t i o n c a l c u l a t i o n s . Mathematical
Part:
Vor t h o s e w h o wish t o follow the a n a l y s i s in g r e a t e r d e t a i l , the f o l l o w i n g more c o m p l e t e d e s c r i p t i o n is g i v e n .
PROJEUTION OF
LF-N~
AP'P-KIUKP-.
. . . .
REL~T|VF.. Fro. 8.
INTEkl,5 | T Y
Light distribution behind halftone screen, calculated from geometrical optics (penumbra theory/. ASSUMPTIONS.
T h e c a l c u l a t i o n s g i v e n in this p a p e r are b a s e d on F'resnel's i n t e g r a l s , from w h i c h the C o r n u s p i r a l is p l o t t e d , a n d w h i c h are a c c u r a t e only w h e n rays c l o s e t o the s t r a i g h t line L P f r o n l l i g h t s o u r c e t o p o i n t of o b s e r v a t i o n a r e c o n s i d e r e d . T h e h a l f t o n e s c r e e n , on the o t h e r h a n d , e x t e n d s over a l a r g e a r e a . H o w e v e r , it has been f o u n d e x p e r i m e n t a l l y t h a t u n d e r
494
J . A . C . YULE.
[J. t,'. I.
the c h o s e n c o n d i t i o n s , with a parallel-line s c r e e n , only the t h r e e s c r e e n o p e n i n g s c l o s e t o the d i r e c t r a y h a v e a m e a s u r a b l e effect. N e a r the c e n t e r of the s c r e e n , the g r e a t e s t e r r o r c o n s i s t s in n e g l e c t i n g t h i r d - o r d e r t e r m s . T h i s i n v o l v e s p h a s e e r r o r s of c o n s i d e r a b l y less t h a n one h u n d r e d t h of a w a v e l e n g t h . A m p l i t u d e e r r o r s , due t o the a s s u m p t i o n s t h a t the
Z.B 4.O S.6
9.0 1~..4-
PROJE,.,,v,.,=
FIG. 9.
v,-,.=,~,.,
,-,,-~-,,,. v ~ . = ,
K k L ~ IIVlr-
I N I l ~ - N b l l ~"
Light distribution behind halftone screen, determined experimentally.
inclination f a c t o r for all the rays e q u a l s u n i t y a n d that loss of a m p l i t u d e a c c o r d i n g t o the i n v e r s e s q u a r e law b e c a u s e of the g r e a t e r p a t h l e n g t h of the less d i r e c t rays is n e g l i g i b l e , are e v e n less i m p o r t a n t . A t the e d g e s of the s c r e e n , p h a s e e r r o r s due t o the o b l i q u i t y of the s c r e e n m i g h t be t h o u g h t t o be a p p r e c i a b l e , b u t in p h o t o g r a p h i c e x p e r i m e n t s u n d e r the c o n d i t i o n s of u s e of a h a l f t o n e s c r e e n , it h a s n o t b e e n p o s s i b l e t o o b s e r v e a n y effects t h a t m i g h t be a s c r i b e d t o a d i f f e r e n c e
M a y , 1943.l
T H E O R Y OF
495
THE H A L F T O N E PROCESS.
in d i f f r a c t i o n p a t t e r n w h e n i m a g e s f o r m e d near t h e e d g e s of the s c r e e n are c o m p a r e d w i t h i m a g e s p r o d u c e d on the a x i s . If t h e s e a s s u m p t i o n s are m a d e , it f o l l o w s t h a t i n d i r e c t rays of l i g h t from e q u a l e l e m e n t a l a r e a s of the s c r e e n are e q u a l in a m p l i t u d e a n d a r e r e t a r d e d in p h a s e in p r o p o r t i o n t o the s q u a r e of t h e i r d i s t a n c e , in the p l a n e of the s c r e e n , from the line j o i n i n g l i g h t s o u r c e a n d p o i n t of o b s e r v a t i o n . IO0
JOIN
lal u
80
ul Z "~
GO
UP (PENUMBRA THEORY)
.DOTS JOIN UP (DIFFRACTION THEORY) -DOTS JOIN UP (EXPERIMENTAL)
O
40 O u Z ua
20
.l ha
O FIG.
Io.
20 40 GO B0 INTENSITY AT ISOt.UK LINE
100
R e l a t i o n s h i p between the i n t e n s i t y at an i s o l u x line and the area enclosed w i t h i n it. PARALLEL-LINE SCREEN.
I n s t e a d of c a r r y i n g out the c o m p u t a t i o n g r a p h i c a l l y with the help of the C o r n u s p i r a l , the v a l u e s w e r e c a l c u l a t e d from F r e s n e l ' s i n t e g r a l s , from w h i c h the C o r n u s p i r a l is p l o t t e d . T h e s e a r e a v a i l a b l e 4 up t o a v a l u e of v = 8.5. In the p a r t i c u l a r e x a m p l e we are s t u d y i n g , the s c r e e n d i s t a n c e h a s been c h o s e n with r e s p e c t t o s c r e e n r u l i n g a n d w a v e l e n g t h , so t h a t the w i d t h of the slit (vl -- v,.,) e q u a l s 2.4. T a k i n g as a n e x a m p l e a field p o i n t o n e - f o u r t h of a s c r e e n
496
J . A . C . YuLe:.
[J. F. 1.
element away from t h e c e n t e r of a screen o p e n i n g , slits will extend from v = - 1.8 t o + o.6, from 3.0 t o 5.4, a n d o n t h e o t h e r side from v = - 6.6 t o - 4.2. Openings corresponding to values of v further t h a n 7.2 o r - 7.2 are neglected. This is t h e problem w h i c h was s o l v e d graphically in Fig. 4. If x a n d y are t h e values of Fresnel's integrals c o r r e s p o n d ing t o a given value of v, we s u b t r a c t values of x a t t h e left-hand side of a slit from t h o s e a t t h e right, a n d doing t h e same for y, we g e t x l - x ~ - - o . 5 8 I + o . 3 3 4 - - o . 9 1 5 a n d y l - y~ = o . I I I + o . 4 5 I = 0 . 5 6 2 for t h e o p e n i n g from v = - 1.8 t o + 0.6. V a l u e s for t h e o t h e r two o p e n i n g s are X t - - X 2 = .072 a n d - .048, a n d Yl - y : - - .IOI a n d .o19, respectively. A d d i n g t h e values of x l - x2 for t h e three o p e n i n g s gives x = 0 . 7 9 5 a n d , similarly, a value of 0 . 4 8 0 is obtained for y. The amplitude a t this field point is t h e resultant of t h e s e two components a t r i g h t angles t o each o t h e r a n d is proportional t o
,/-
x ~ + y2 2
The intensity, being proportional t o
x2 + v2 ~ " , o r for this 2 point o.432 t i m e s t h e intensity in t h e a b s e n c e of a s c r e e n . The intensity is s i m i l a r l y determined for as many points as desired, thirteen having b e e n used in this case with t h e results shown in Fig. 5t h e square of t h e
amplitude,
CROSSED
equals
SCREENS.
The Fresnel-Kirchhoff equation for t h e disturbance a t P, when t h e wave length is very s h o r t compared with t h e distances involved, reduces t o t h e following form when L P is n o r m a l t o t h e s c r e e n :
ike<~(°+b) f f e-~k~dxdy' Up _ 4~r ab w h e r e ~ is t h e i n c r e a s e in p a t h length for t h e ray going t h r o u g h (x, y) compared with t h e d i r e c t ray L P , so t h a t &
(x'~ + 3'2) \
2ab
"
May, t943.1
" I ' t t E O R Y OF T H E
HALFTONE
I)ROCESS.
497
T h e d o u b l e i n t e g r a l t e r m then b e c o m e s
f f e-ik(~:+~)(,,+b)~2,,hdxd3, or
f
f
e - ikx"~(a+b). 2( I,C-- kv2(a ~b) / 2"bd.x:d.v.
T h e s c r e e n o p e n i n g s of a 9 o - d e g r e e c r o s s l i n e s c r e e n are r e c t a n g u l a r , so t h a t if x a n d y are m e a s u r e d in d i r e c t i o n s p a r a l l e l t o the r u l i n g s , the v a l u e s of x a n d y a n d the l i m i t s of i n t e g r a t i o n with r e s p e c t t o x a n d y are i n d e p e n d e n t of each o t h e r . T h e i n t e g r a l may, therefore, be r e s o l v e d i n t o the p r o d u c t of two i n t e g r a l s , g i v i n g the e q u a t i o n
ike-ik (a+b) f 4~ab
d : v e - i#.~'-'(, +b); 2,,~, . ~ d y e - i ~-?/-'/, 4 b). 2,d,.
E a c h of t h e s e i n t e g r a l s r e p r e s e n t s the e f f e c t of one of parallel-line s c r e e n s of w h i c h the c r o s s l i n e s c r e e n m a y c o n s i d e r e d t o be c o m p o s e d . T h e d i s t u r b a n c e a t P due t o c r o s s l i n e s c r e e n is t h e r e f o r e e q u a l t o the p r o d u c t of d i s t u r b a n c e s due t o the two parallel-line s c r e e n s .
the be the the
CONCLUSIONS.
U n d e r t y p i c a l c o n d i t i o n s , the l i g h t d i s t r i b u t i o n , w h i c h c o n t r o l s d o t s h a p e a n d tone r e p r o d u c t i o n c u r v e , c o r r e s p o n d s well with t h a t c a l c u l a t e d from the diffraction t h e o r y a n d d i s a g r e e s with c o m p u t a t i o n s from g e o m e t r i c a l o p t i c s . In the n e x t s e c t i o n of this p a p e r , f u r t h e r q u a l i t a t i v e e v i d e n c e of the effects of diffraction a n d a d i s c u s s i o n of the t y p e s of diffraction p a t t e r n s f o r m e d a t d i f f e r e n t s c r e e n d i s t a n c e s , a n d t h e i r effects on h a l f t o n e n e g a t i v e s , will be given. ACKNOWLEDGMENT.
My t h a n k s a r e due t o Dr. F. K o t t l e r of t h e s e L a b o r a t o r i e s , for his h e l p f u l criticisms" a n d s u g g e s t i o n s . ROCHESTER, N. Y. February 24, I 9 4 3 . VOL. 235, NO. 1409---I9
498
J . A . C . YULE.
[J. F. 1.
REFERENCES.
I. YULE, J. A. C., J. FRANKLIN INST., 23I, 23--38 (1941). 2. FRUWlRTH, A., and MYRTLE, J. S., American Photo-Engraver, 27, 291-297 (1935). 3. DRUDE, P. IZ. L., " T h e o r y of Optics," Longmans, Green and Co., New York, 3rd ed., I 9 1 2 . MEYER, C. F., " T h e Diffraction of L i g h t , X-Rays, and Material Particles," p p . 1 - 1 1 1 , University of Chicago P r e s s , Chicago, I 9 3 4 . 4. JAHNKE, E., and EraDE, F., "Funktionentafeln mit F o r m e l n u n d Kurven." p. 26, B. G. Teubner, Leipzig u n d Berlin, 19o9.
N,Y. SUMMARY.
An a p p r o x i m a t e m e t h o d of c a l c u l a t i n g the l i g h t d i s t r i b u tion in the dots f o r m e d b e h i n d a h a l f t o n e s c r e e n from the F r e s n e l d i f f r a c t i o n t h e o r y is d e s c r i b e d , a n d in a t y p i c a l c a s e , the d i s t r i b u t i o n is f o u n d t o c h e c k c l o s e l y with t h a t d e t e r m i n e d e x p e r i m e n t a l l y , b e i n g v e r y d i f f e r e n t from t h a t c a l c u l a t e d from g e o m e t r i c a l o p t i c s . In a p r e v i o u s p a p e r 1 the o b s e r v e d l i g h t d i s t r i b u t i o n in the dots f o r m e d b e h i n d a h a l f t o n e s c r e e n , u n d e r t y p i c a l c o n d i t i o n s u s e d in m a k i n g h a l f t o n e n e g a t i v e s , was f o u n d t o be very d i f f e r e n t from the d i s t r i b u t i o n c a l c u l a t e d from the p e n u m b r a t h e o r y ( g e o m e t r i c a l o p t i c s ) . It r e m a i n s t o c a l c u late the l i g h t d i s t r i b u t i o n from the t h e o r y of diffraction. T h i s c a l c u l a t i o n , if c a r r i e d out in a n e x a c t a n d g e n e r a l i z e d manner, appears to involve a laborious and complicated m a t h e m a t i c a l a n a l y s i s a n d m a y even be i m p o s s i b l e . Howe v e r , a simplified m e t h o d is s a t i s f a c t o r y for the c a l c u l a t i o n in a s p e c i a l c a s e , a n d i t is f o u n d e x p e r i m e n t a l l y t h a t d e v i a t i o n s from the c o n d i t i o n s a p p l y i n g t o this s p e c i a l case do not lead t o d i s c r e p a n c i e s of a n y i m p o r t a n c e in h a l f t o n e w o r k . P r o b a b l y , the r e a s o n for this is t h a t the lens a p e r t u r e used in p r a c t i c a l h a l f t o n e w o r k is n o t a p o i n t s o u r c e of l i g h t , b u t is r e l a t i v e l y l a r g e so t h a t all line d e t a i l s of the diffraction p a t t e r n are c o m p l e t e l y lost. T h e c a l c u l a t i o n is c a r r i e d out in t h r e e s t e p s : I. T h e l i g h t d i s t r i b u t i o n b e h i n d a parallel-line s c r e e n w i t h a v e r y s m a l l lens a p e r t u r e , i.e., a p o i n t s o u r c e of light, is c a l c u l a t e d . 483
484
J . A . C . YULE.
[J.
F. I.
T h e r e s u l t of c r o s s i n g two parallel-line s c r e e n s a t r i g h t a n g l e s is f o u n d , a p o i n t s o u r c e of l i g h t a g a i n b e i n g u s e d . 3. R e g a r d i n g the n o r m a l lens a p e r t u r e a s c o n s i s t i n g of a l a r g e n u m b e r of i n c o h e r e n t p o i n t s o u r c e s of l i g h t , each of w h i c h i n d e p e n d e n t l y p r o d u c e s a diffraction p a t t e r n s l i g h t l y d i s p l a c e d r e l a t i v e t o the o t h e r s , all t h e s e d i f f r a c t i o n p a t t e r n s are a d d e d up t o p r o d u c e the t o t a l l i g h t d i s t r i b u t i o n . This is the m e t h o d of a p p r o a c h u s e d by F r u w i r t h a n d M e r t l e 3 2.
The Theory of Diffraction: A b r i e f d e s c r i p t i o n of F r e s n e l ' s t h e o r y of diffraction will f i r s t be g i v e n . F o r f u r t h e r d e t a i l s , see C. F. M e y e r ' s t e x t b o o k on d i f f r a c t i o n or a n y a d v a n c e d t r e a t i s e on light. 3 In Fig. I, L is a p o i n t s o u r c e of l i g h t , S is a s c r e e n with
sI L
F r o . I.
Diffraction by a s m a l l opening in a screen S.
a n a p e r t u r e in it, a n d P is the so*called " f i e l d p o i n t " a t w h i c h the i n t e n s i t y of l i g h t is t o be f o u n d . E a c h p o i n t in the a p e r t u r e , for i n s t a n c e , the p o i n t A, is r e g a r d e d a s a s e c o n d a r y l i g h t s o u r c e , from w h i c h s e c o n d a r y w a v e s o r i g i n a t e . T h e a m p l i t u d e of l i g h t r e a c h i n g P is the r e s u l t a n t of the w a v e s from all p o i n t s in t h e s c r e e n a p e r t u r e . H o w e v e r , the w a v e s w h i c h do n o t t r a v e l in a s t r a i g h t line from L t o P go a l o n g a s o m e w h a t longer path, and hence arrive at P a little later. T h o s e w h i c h a r r i v e j u s t half a w a v e a p a r t , so t h a t the c r e s t of one w a v e a r r i v e s a t the s a m e m o m e n t a s the t r o u g h of a n o t h e r , e x a c t l y n e u t r a l i z e each o t h e r . W h e n the effects of all t h e s e w a v e s a r e a d d e d , it is f o u n d t h a t the l i g h t a t the p o i n t P is b r i g h t if the w a v e s m o s t l y r e i n f o r c e each o t h e r .
May, m43.1
THEORY OF THE I-[ALFTONE PROCESS.
485
On tile o t h e r h a n d , if most of the w a v e s r e a c h i n g this p o i n t
n e u t r a l i z e each other, t h e l i g h t at P is r e l a t i v e l y w e a k . W a v e s w h i c h a r r i v e s i m u l t a n e o u s l y , or e x a c t l y one, two, t h r e e , e t c . , w a v e l e n g t h s a p a r t , reinforce each o t h e r . T h o s e w h i c h a r r i v e o.5, 1.5, 2.5, e t c . , wave l e n g t h s a p a r t n e u t r a l i z e each o t h e r . In most positions in the g e o m e t r i c a l s h a d o w b e h i n d t h e screen, t h e y tend t o neutralize each o t h e r , and b e h i n d t h e clear portions of t h e screen, t h e y reinforce each other.
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m e a n s that l i g h t a p p e a r s to travel in approximately straight l i n e s . W a v e s arriving at intermediate t i m e s m a y partly reinforce or partly n e u t r a l i z e each other, and t h e t o t a l effect may be f o u n d by calculation or by t h e u s e of a so-called vibration curve. T h e Cornu spiral, Fig. 2, may be used as a vibration c u r v e for determining t h e l i g h t distribution b e h i n d a straight e d g e , a s l i t , or a s e r i e s of parallel s l i t s , such as a p a r a l l e l - l i n e screen. It does not a p p l y , however, to openings a t a considerable distance from t h e direct ray, w h i c h are
486
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f o u n d t o have a n e g l i g i b l e e f f e c t in c o m p a r i s o n with the n e a r e s t o p e n i n g s in the t y p i c a l case u n d e r c o n s i d e r a t i o n . T h e c u r v e is u s e d a s f o l l o w s : In a n o p t i c a l s y s t e m , s u c h a s Fig. 3, a s t r a i g h t line d r a w n from l i g h t s o u r c e t o field p o i n t p a s s e s t h r o u g h the s c r e e n a t O. A line is d r a w n a c r o s s the s c r e e n t h r o u g h 0 a t r i g h t a n g l e s t o the e d g e s of the slit ( g r e a t l y e n l a r g e d in the d i a g r a m ) and is d i v i d e d i n t o e q u a l x/ abX " p a r t s , the l e n g t h of e a c h u n i t b e i n g e q u a l t o ~ 2 ( a ~ b ) ' w h e r e a is the d i s t a n c e L O from l i g h t s o u r c e t o s c r e e n , b is the d i s t a n c e O P from s c r e e n t o field p o i n t , a n d X is the w a v e l e n g t h of the l i g h t . T h e d i s t a n c e a c r o s s the s c r e e n , m e a s u r e d
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in t e r m s of t h e s e g r a d u a t i o n s , is r e p r e s e n t e d by v. T h e C o r n u s p i r a l , Fig. 2, is d i v i d e d i n t o s m a l l p a r t s of e q u a l l e n g t h a n d m a r k e d with n u m b e r s c o r r e s p o n d i n g t o the g r a d u a t i o n s on the slit. I n this p a r t i c u l a r case the slit e x t e n d s from v = 0.2 t o 1.8, P b e i n g a p o i n t in the g e o m e t r i c a l s h a d o w of the s c r e e n n e a r the edge of the slit. T h e p o i n t s of the C o r n u s p i r a l m a r k e d t h u s are j o i n e d by a s t r a i g h t line A B . T h e l e n g t h of this line r e p r e s e n t s the a m p l i t u d e of the l i g h t r e a c h i n g the p o i n t P . T h e a m p l i t u d e with no s c r e e n p r e s e n t , w h i c h is the s a m e a s t h a t w i t h a n infinitely wide slit, is r e p r e s e n t e d by the l e n g t h of the line M N . T h e c h a n g e of p h a s e of the l i g h t due t o the s c r e e n is g i v e n by the a n g l e 0 b e t w e e n t h e s e l i n e s . T h e a m p l i t u d e a n d p h a s e of the l i g h t r e a c h i n g the p o i n t P from o t h e r p a r a l l e l s l i t s can be f o u n d in the s a m e w a y . U n d e r
May, i943.]
487
T H E O R Y OF T H E H A L F T O N E PROCESS.
n o r m a l c o n d i t i o n s of u s e of a h a l f t o n e s c r e e n , it is f o u n d e x p e r i m e n t a l l y t h a t the n e a r e s t t h r e e s l i t s of a parallel-line s c r e e n are the only ones w h i c h h a v e an i m p o r t a n t i n f l u e n c e on the l i g h t d i s t r i b u t i o n a t the p o i n t P . In Fig. 4, A is a d i a g r a m of the o p t i c a l s y s t e m . T h e l i n e s r e p r e s e n t i n g the t h r e e s l i t s of the s c r e e n , a, b, a n d c, n e a r e s t t o the field p o i n t , are s h o w n on the C o r n u s p i r a l . I t is n o t s u f f i c i e n t m e r e l y ~8 L~
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t o add the a m p l i t u d e s of the l i g h t a r r i v i n g from d i f f e r e n t s l i t s , s i n c e , if the p h a s e s are o p p o s i t e , they will tend t o n e u t r a l i z e each o t h e r . T h e i r a m p l i t u d e s a n d p h a s e s , i.e., the l e n g t h s a n d d i r e c t i o n s of the l i n e s , m u s t be c o m b i n e d v e c t o r i a l l y by p l a c i n g the l i n e s end t o e n d , a s in Fig. 4C, in w h i c h the n a r r o w l i n e s r e p r e s e n t i n g the t h r e e s l i t s c o m b i n e t o give the r e s u l t a n t s h o w n a s a h e a v y l i n e . A f t e r the a m p l i t u d e h a s b e e n f o u n d in this w a y a t a s i n g l e p o i n t P, the p r o c e s s is r e p e a t e d u n t i l a sufficient n u m b e r of p o i n t s have b e e n f o u n d .
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In a c t u a l c o m p u t a t i o n s it is o f t e n m o r e c o n v e n i e n t t o s o l v e the p r o b l e m by c a l c u l a t i o n i n s t e a d of g r a p h i c a l l y , a s will be s h o w n in the m a t h e m a t i c a l p a r t . T h e r e s u l t i n g l i g h t d i s t r i b u t i o n b e h i n d a parallel-line s c r e e n with e q u a l l i n e s a n d s p a c e s , the w i d t h of each of w h i c h is 2.4 in t e r m s of v (for i n s t a n c e , with d i s t a n c e from l i g h t s o u r c e t o s c r e e n = 23.5 in., s c r e e n s e p a r a t i o n = o . 4 1 3 in., w a v e l e n g t h = 435 mtz, a n d s c r e e n r u l i n g = I2o l i n e s per i n c h ) , is s h o w n in Fig. 5.
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Fit. 5. Calculated diffraction pattern behind a x : I parallel-line screen, with a smalllight source, under conditions such that the width of one screen opening equals 2.4 in terms of v. Crossed Screens: F o r the s p e c i a l case w h i c h we are c o n s i d e r i n g , i t will be s h o w n (see m a t h e m a t i c a l p a r t ) t h a t , with a p o i n t s o u r c e of light, the i n t e n s i t y a t a g i v e n p o i n t of o b s e r v a t i o n , due t o two parallel-line s c r e e n s c r o s s e d a t r i g h t a n g l e s , is e q u a l t o t h e p r o d u c t of the i n t e n s i t i e s w h i c h w o u l d be p r o d u c e d by e a c h of the s c r e e n s t a k e n s e p a r a t e l y , t h e i n t e n s i t y in the a b s e n c e of a n y s c r e e n b e i n g t a k e n as u n i t y . T h i s h a s a s u p e r f i c i a l s i m i l a r i t y t o the f a m i l i a r law of a d d i t i v i t y of d e n s i t y of two s u p e r i m p o s e d l a y e r s of a n y lighta b s o r b i n g m a t e r i a l , a l t h o u g h it m u s t be r e m e m b e r e d t h a t for d i f f r a c t i n g s c r e e n s it is only t r u e u n d e r the r e s t r i c t e d c o n d i t i o n s u n d e r c o n s i d e r a t i o n . E a c h s c r e e n m a y be r e g a r d e d a s h a v i n g a c e r t a i n a p p a r e n t o p t i c a l d e n s i t y with r e f e r e n c e t o a p o i n t s o u r c e of l i g h t a n d p o i n t of o b s e r v a t i o n . T h e a p p a r e n t
May, '943.]
489
T H F O R Y OF THE H A L F T O N E PROCESS.
d e n s i t y of the c r o s s l i n e s c r e e n is then e q u a l t o the sum of the a p p a r e n t d e n s i t i e s of the two parallel-line s c r e e n s of w h i c h it is c o m p o s e d . T h i s does n o t r e f e r only t o the a v e r a g e int e n s i t y o b s e r v e d u n d e r c o n d i t i o n s w h e r e the i n d i v i d u a l l i n e s of the s c r e e n are i n d i s t i n g u i s h a b l e , but also t o the b r i g h t n e s s a t i n d i v i d u a l p o i n t s in the diffraction p a t t e r n . T h e d i f f r a c t i o n p a t t e r n for a c r o s s l i n e s c r e e n with a p o i n t s o u r c e of light, c a l c u l a t e d in this w a y from the r e s u l t s for a parallel-line s c r e e n s h o w n in Fig. 5, is g i v e n in T a b l e I. TABLE I. Calculated light distribution i n the diffraction pattern behind a crossline screen with equal lines and spaces, a t such a distance that vl - v2, f o r the two sides o f a single opening, equals 2.4, with a p o i n t source o f light. The two p o i n t s given i n heavy type represent the center and the corner o f one screen element. Values are expressed as a percentage o f the i n t e n s i t y with the screen removed.
212
262 '44 163-o 82.4 83.3 32.7 19.8 9.6 ,6.2 2.0 324 I77 I 77.8 IO2.O ,o3.o 40.3 24.5 I3.2 2o.0 2.5 ~o.9 97.I / 42.61 55.6 56.4 122.I 13.4 6.5 4.8 5.9 2.8 ,8.6 24.4 24.6 9-72-7 3t .9 32-3 7-7 3.7 6.3 .79 .8 b 32.7 12.8 7.8 3.8 6.4 5.0 3.0 •5 2.5 .3I 1.8 -9 t.5 .I9 .44 i .73 .O9 i t.2 .I5 .02 i
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O n l y o n e - e i g h t h of the p a t t e r n c o r r e s p o n d i n g t o a u n i t s c r e e n e l e m e n t h a s been t a b u l a t e d , s i n c e the p a t t e r n is s y m m e t r i c a l a n d the rest of the p a t t e r n m e r e l y d u p l i c a t e s the f i g u r e s in this t a b l e . Diffraction Pattern Due to a Large Light Source: T h e l i g h t distribution just g i v e n is produced by a p o i n t source of light. In halftone practice, t h e l i g h t source is a lens w h i c h may be regarded as being made of a large n u m b e r of s m a l l sources of light. D i v i d i n g t h e aperture into 49 e l e m e n t s was considered sufficient for t h i s calculation. T h e relative proportions of screen and aperture, as t h e y w o u l d be s e e n from a p o i n t i n t h e plane of observation i m m e d i a t e l y
49 °
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b e h i n d the c e n t e r of a s c r e e n o p e n i n g , are s h o w n in Fig. 6. The 49 e l e m e n t s of the a p e r t u r e p r o d u c e 49 diffraction p a t t e r n s , i d e n t i c a l but s l i g h t l y d i s p l a c e d with r e g a r d t o each o t h e r . F o r each p o i n t in the p l a n e of the e m u l s i o n , the i n t e n s i t i e s of the a p p r o p r i a t e p o i n t s in t h e s e diffraction p a t t e r n s are a d d e d t o g e t h e r t o give the t o t a l i n t e n s i t y . In this c a s e , it i s n o t n e c e s s a r y t o t a k e p h a s e relationships i n t o a c c o u n t , if the l i g h t s o u r c e s are i n d e p e n d e n t .
FIG. 6.
Illustrating the division of the lens aperture into 49 elements which may be regarded as point sources of light.
If the lens a p e r t u r e w e r e i l l u m i n a t e d by a u n i f o r m l i g h t s o u r c e p l a c e d c l o s e t o the l e n s , the 49 e l e m e n t s of the lens a p e r t u r e w o u l d o b v i o u s l y be i n d e p e n d e n t , s i n c e t h e y w o u l d a r i s e from d i f f e r e n t p o i n t s on the l i g h t s o u r c e . H o w e v e r , in p r a c t i c e , the c o p y , w h i c h m a y be r e g a r d e d a s a l i g h t s o u r c e , is i m a g e d on the p l a n e of o b s e r v a t i o n , w h i c h w o u l d m e a n t h a t the l i g h t from all p o i n t s of the lens a p e r t u r e is c o h e r e n t , i.e., in the S a m e p h a s e . In s p i t e of t h i s , it was f o u n d e x p e r i -
May, I 9 4 3 . ]
491
T H E O R Y OF THE H A L F T O N E PROCESS.
m e n t a l l y t h a t , with a l a r g e u n i f o r m l i g h t s o u r c e , the l i g h t d i s t r i b u t i o n in the p l a n e of o b s e r v a t i o n was the s a m e , w i t h i n the l i m i t s of e x p e r i m e n t a l e r r o r , w h e t h e r the l i g h t s o u r c e was i m a g e d on the film o r was i m m e d i a t e l y in f r o n t of t h e l e n s . T h i s u n e x p e c t e d r e s u l t is c o n s i d e r e d s u f f i c i e n t justification for r e g a r d i n g t h e lens a p e r t u r e a s a collection of i n d e p e n d e n t l i g h t s o u r c e s . T h e c a l c u l a t i o n h a s b e e n simplified by c h o o s i n g a n a p e r t u r e of s u c h size (o.196 in.) t h a t its p r o j e c t i o n , as seen from the p l a n e of o b s e r v a t i o n , b e a r s a s i m p l e r e l a t i o n s h i p t o the size of a s c r e e n e l e m e n t . I n this c a s e , the p r o j e c t e d w i d t h of the a p e r t u r e was 7/24 t i m e s the w i d t h of a s i n g l e screen element. T h e r e s u l t i n g l i g h t d i s t r i b u t i o n , c a l c u l a t e d u n d e r the cond i t i o n s of s c r e e n d i s t a n c e , e t c . , j u s t g i v e n , is s h o w n in T a b l e I I. TABLE II. Light distribution behind a crossline screen with a light source equal in size to the lens aperture illustrated in Fig. 6, calculated from the diffraction pattern tabulated in Table I . 96.2'
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By j o i n i n g p o i n t s of e q u a l b r i g h t n e s s , a n d c o m p l e t i n g the p a t t e r n for one or m o r e c o m p l e t e s c r e e n e l e m e n t s , the l i g h t d i s t r i b u t i o n m a y be m a p p e d a s s h o w n in Fig. 7. T h i s m a y be c o n t r a s t e d with the r e s u l t s of the p e n u m b r a t h e o r y c a l c u l a t i o n g i v e n in Fig. 8. T h e s h a p e of the i s o l u x l i n e s is very d i f f e r e n t in the two c a s e s . T h e m e t h o d of d e t e r m i n i n g the l i g h t d i s t r i b u t i o n e x p e r i m e n t a l l y was g i v e n in the p r e v i o u s p a p e r . T h e o p t i c a l c o n d i t i o n s are s l i g h t l y d i f f e r e n t from t h o s e c h o s e n in the f o r m e r c a s e , so a s t o s i m p l i f y the diffraction c a l c u l a t i o n .
492
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T h e l i g h t d i s t r i b u t i o n in the h a l f t o n e dot, f o u n d by e x p e r i m e n t in this w a y , is s h o w n in Fig. 9. It is seen t h a t the s h a p e s of the i s o l u x l i n e s s h o w n in Fig. 9 c o r r e s p o n d well w i t h t h o s e c a l c u l a t e d by the d i f f r a c t i o n t h e o r y , Fig. 7, a n d not a t all with the p e n u m b r a c a l c u l a t i o n s , Fig. 8.
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T h e v a l u e s for l i g h t i n t e n s i t y are more e a s i l y c o m p a r e d by p l o t t i n g the a r e a of the d o t a g a i n s t the r e l a t i v e i n t e n s i t y a t its b o u n d a r y (see Fig. io). H e r e a g a i n , the diffraction, b u t n o t the p e n u m b r a c a l c u l a t i o n , a g r e e s well w i t h e x p e r i m e n t a l r e s u l t s , e x c e p t t h a t a little too m u c h l i g h t h a s p e n e t r a t e d t o the d a r k r e g i o n s . T h i s m a y be a t t r i b u t e d t o the s c a t t e r i n g of l i g h t by i m p e r f e c t i o n s in the s c r e e n a n d o p t i c a l s y s t e m , w h i c h it is v e r y d i f f i c u l t t o m e a s u r e , t o the low
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T h e c a l c u l a t i o n s g i v e n in this p a p e r are b a s e d on F'resnel's i n t e g r a l s , from w h i c h the C o r n u s p i r a l is p l o t t e d , a n d w h i c h are a c c u r a t e only w h e n rays c l o s e t o the s t r a i g h t line L P f r o n l l i g h t s o u r c e t o p o i n t of o b s e r v a t i o n a r e c o n s i d e r e d . T h e h a l f t o n e s c r e e n , on the o t h e r h a n d , e x t e n d s over a l a r g e a r e a . H o w e v e r , it has been f o u n d e x p e r i m e n t a l l y t h a t u n d e r
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J . A . C . YULE.
[J. t,'. I.
the c h o s e n c o n d i t i o n s , with a parallel-line s c r e e n , only the t h r e e s c r e e n o p e n i n g s c l o s e t o the d i r e c t r a y h a v e a m e a s u r a b l e effect. N e a r the c e n t e r of the s c r e e n , the g r e a t e s t e r r o r c o n s i s t s in n e g l e c t i n g t h i r d - o r d e r t e r m s . T h i s i n v o l v e s p h a s e e r r o r s of c o n s i d e r a b l y less t h a n one h u n d r e d t h of a w a v e l e n g t h . A m p l i t u d e e r r o r s , due t o the a s s u m p t i o n s t h a t the
Z.B 4.O S.6
9.0 1~..4-
PROJE,.,,v,.,=
FIG. 9.
v,-,.=,~,.,
,-,,-~-,,,. v ~ . = ,
K k L ~ IIVlr-
I N I l ~ - N b l l ~"
Light distribution behind halftone screen, determined experimentally.
inclination f a c t o r for all the rays e q u a l s u n i t y a n d that loss of a m p l i t u d e a c c o r d i n g t o the i n v e r s e s q u a r e law b e c a u s e of the g r e a t e r p a t h l e n g t h of the less d i r e c t rays is n e g l i g i b l e , are e v e n less i m p o r t a n t . A t the e d g e s of the s c r e e n , p h a s e e r r o r s due t o the o b l i q u i t y of the s c r e e n m i g h t be t h o u g h t t o be a p p r e c i a b l e , b u t in p h o t o g r a p h i c e x p e r i m e n t s u n d e r the c o n d i t i o n s of u s e of a h a l f t o n e s c r e e n , it h a s n o t b e e n p o s s i b l e t o o b s e r v e a n y effects t h a t m i g h t be a s c r i b e d t o a d i f f e r e n c e
M a y , 1943.l
T H E O R Y OF
495
THE H A L F T O N E PROCESS.
in d i f f r a c t i o n p a t t e r n w h e n i m a g e s f o r m e d near t h e e d g e s of the s c r e e n are c o m p a r e d w i t h i m a g e s p r o d u c e d on the a x i s . If t h e s e a s s u m p t i o n s are m a d e , it f o l l o w s t h a t i n d i r e c t rays of l i g h t from e q u a l e l e m e n t a l a r e a s of the s c r e e n are e q u a l in a m p l i t u d e a n d a r e r e t a r d e d in p h a s e in p r o p o r t i o n t o the s q u a r e of t h e i r d i s t a n c e , in the p l a n e of the s c r e e n , from the line j o i n i n g l i g h t s o u r c e a n d p o i n t of o b s e r v a t i o n . IO0
JOIN
lal u
80
ul Z "~
GO
UP (PENUMBRA THEORY)
.DOTS JOIN UP (DIFFRACTION THEORY) -DOTS JOIN UP (EXPERIMENTAL)
O
40 O u Z ua
20
.l ha
O FIG.
Io.
20 40 GO B0 INTENSITY AT ISOt.UK LINE
100
R e l a t i o n s h i p between the i n t e n s i t y at an i s o l u x line and the area enclosed w i t h i n it. PARALLEL-LINE SCREEN.
I n s t e a d of c a r r y i n g out the c o m p u t a t i o n g r a p h i c a l l y with the help of the C o r n u s p i r a l , the v a l u e s w e r e c a l c u l a t e d from F r e s n e l ' s i n t e g r a l s , from w h i c h the C o r n u s p i r a l is p l o t t e d . T h e s e a r e a v a i l a b l e 4 up t o a v a l u e of v = 8.5. In the p a r t i c u l a r e x a m p l e we are s t u d y i n g , the s c r e e n d i s t a n c e h a s been c h o s e n with r e s p e c t t o s c r e e n r u l i n g a n d w a v e l e n g t h , so t h a t the w i d t h of the slit (vl -- v,.,) e q u a l s 2.4. T a k i n g as a n e x a m p l e a field p o i n t o n e - f o u r t h of a s c r e e n
496
J . A . C . YuLe:.
[J. F. 1.
element away from t h e c e n t e r of a screen o p e n i n g , slits will extend from v = - 1.8 t o + o.6, from 3.0 t o 5.4, a n d o n t h e o t h e r side from v = - 6.6 t o - 4.2. Openings corresponding to values of v further t h a n 7.2 o r - 7.2 are neglected. This is t h e problem w h i c h was s o l v e d graphically in Fig. 4. If x a n d y are t h e values of Fresnel's integrals c o r r e s p o n d ing t o a given value of v, we s u b t r a c t values of x a t t h e left-hand side of a slit from t h o s e a t t h e right, a n d doing t h e same for y, we g e t x l - x ~ - - o . 5 8 I + o . 3 3 4 - - o . 9 1 5 a n d y l - y~ = o . I I I + o . 4 5 I = 0 . 5 6 2 for t h e o p e n i n g from v = - 1.8 t o + 0.6. V a l u e s for t h e o t h e r two o p e n i n g s are X t - - X 2 = .072 a n d - .048, a n d Yl - y : - - .IOI a n d .o19, respectively. A d d i n g t h e values of x l - x2 for t h e three o p e n i n g s gives x = 0 . 7 9 5 a n d , similarly, a value of 0 . 4 8 0 is obtained for y. The amplitude a t this field point is t h e resultant of t h e s e two components a t r i g h t angles t o each o t h e r a n d is proportional t o
,/-
x ~ + y2 2
The intensity, being proportional t o
x2 + v2 ~ " , o r for this 2 point o.432 t i m e s t h e intensity in t h e a b s e n c e of a s c r e e n . The intensity is s i m i l a r l y determined for as many points as desired, thirteen having b e e n used in this case with t h e results shown in Fig. 5t h e square of t h e
amplitude,
CROSSED
equals
SCREENS.
The Fresnel-Kirchhoff equation for t h e disturbance a t P, when t h e wave length is very s h o r t compared with t h e distances involved, reduces t o t h e following form when L P is n o r m a l t o t h e s c r e e n :
ike<~(°+b) f f e-~k~dxdy' Up _ 4~r ab w h e r e ~ is t h e i n c r e a s e in p a t h length for t h e ray going t h r o u g h (x, y) compared with t h e d i r e c t ray L P , so t h a t &
(x'~ + 3'2) \
2ab
"
May, t943.1
" I ' t t E O R Y OF T H E
HALFTONE
I)ROCESS.
497
T h e d o u b l e i n t e g r a l t e r m then b e c o m e s
f f e-ik(~:+~)(,,+b)~2,,hdxd3, or
f
f
e - ikx"~(a+b). 2( I,C-- kv2(a ~b) / 2"bd.x:d.v.
T h e s c r e e n o p e n i n g s of a 9 o - d e g r e e c r o s s l i n e s c r e e n are r e c t a n g u l a r , so t h a t if x a n d y are m e a s u r e d in d i r e c t i o n s p a r a l l e l t o the r u l i n g s , the v a l u e s of x a n d y a n d the l i m i t s of i n t e g r a t i o n with r e s p e c t t o x a n d y are i n d e p e n d e n t of each o t h e r . T h e i n t e g r a l may, therefore, be r e s o l v e d i n t o the p r o d u c t of two i n t e g r a l s , g i v i n g the e q u a t i o n
ike-ik (a+b) f 4~ab
d : v e - i#.~'-'(, +b); 2,,~, . ~ d y e - i ~-?/-'/, 4 b). 2,d,.
E a c h of t h e s e i n t e g r a l s r e p r e s e n t s the e f f e c t of one of parallel-line s c r e e n s of w h i c h the c r o s s l i n e s c r e e n m a y c o n s i d e r e d t o be c o m p o s e d . T h e d i s t u r b a n c e a t P due t o c r o s s l i n e s c r e e n is t h e r e f o r e e q u a l t o the p r o d u c t of d i s t u r b a n c e s due t o the two parallel-line s c r e e n s .
the be the the
CONCLUSIONS.
U n d e r t y p i c a l c o n d i t i o n s , the l i g h t d i s t r i b u t i o n , w h i c h c o n t r o l s d o t s h a p e a n d tone r e p r o d u c t i o n c u r v e , c o r r e s p o n d s well with t h a t c a l c u l a t e d from the diffraction t h e o r y a n d d i s a g r e e s with c o m p u t a t i o n s from g e o m e t r i c a l o p t i c s . In the n e x t s e c t i o n of this p a p e r , f u r t h e r q u a l i t a t i v e e v i d e n c e of the effects of diffraction a n d a d i s c u s s i o n of the t y p e s of diffraction p a t t e r n s f o r m e d a t d i f f e r e n t s c r e e n d i s t a n c e s , a n d t h e i r effects on h a l f t o n e n e g a t i v e s , will be given. ACKNOWLEDGMENT.
My t h a n k s a r e due t o Dr. F. K o t t l e r of t h e s e L a b o r a t o r i e s , for his h e l p f u l criticisms" a n d s u g g e s t i o n s . ROCHESTER, N. Y. February 24, I 9 4 3 . VOL. 235, NO. 1409---I9
498
J . A . C . YULE.
[J. F. 1.
REFERENCES.
I. YULE, J. A. C., J. FRANKLIN INST., 23I, 23--38 (1941). 2. FRUWlRTH, A., and MYRTLE, J. S., American Photo-Engraver, 27, 291-297 (1935). 3. DRUDE, P. IZ. L., " T h e o r y of Optics," Longmans, Green and Co., New York, 3rd ed., I 9 1 2 . MEYER, C. F., " T h e Diffraction of L i g h t , X-Rays, and Material Particles," p p . 1 - 1 1 1 , University of Chicago P r e s s , Chicago, I 9 3 4 . 4. JAHNKE, E., and EraDE, F., "Funktionentafeln mit F o r m e l n u n d Kurven." p. 26, B. G. Teubner, Leipzig u n d Berlin, 19o9.