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(1)

Letters to the Editor

Jerzy Nowak*

The Secondary Spectrum of Two-Lens

Superachromats with an Air Space

The simplet optical system that meets the super- achromatic correction is the two-lens system. Such a system may also serve as a starting point for cal­ culations of more complex systems, but with a better correction and simultaneous increase in the aperture and field angle. For example, the simplet supera- chromatic system with a corrected Petzval curve is a two-lens system with an air space; but it has to be expanded in order to be of practical value. We shall deal with this problem in the paper “An ana­ lysis of the possibility of constructing a superachroma- tic objective with a flat field”. We shall now ana­ lyse the secondary spectrum of two-lens systems with an air space. This problem was dealt with Canzek

who gave the following formula in papers [1,2]:

S l - S H P u - P x i ) '

h , i - v 2, ( 1)

where

Vi ,Pn — the Abbe’s number and the partial dis­

persion, respectively, of the first glass, v2> — the same for the second glass,

h

— hight of incidence of the aperture ray on the second lens (the hight o f incidence on the first lens is assumed to equal zero),

— the last lens surface to focus distance for wave length A,

s' — the last lens surface to focus distance for basic

colour.

The following approximation was used to derive formula (1),

h2 = s[s (2)

and the term neglected

R = ( \ - h ) [ { \ - h ) a Fak- h { , n - \ ) { a F+ a 2)\, (3)

where

aF --- n —nF, «2 = n - n 2.

It seems, however, that it is difficult to estimate the error we make when neglecting (3) without a de­ tailed analysis. It follows that in some superachro- matic systems the approximation resulting from neglec­ ting the term (3) is not sufficient. We shall, therefore deal with this problem again. Henceforth we shall use the following definition o f Abbe’s number and partial dispersion

V =

(4)

Px= »f~ » 2

«r— «2

Violet has been taken as the basic colour (deno­ ted without index). We assume that the system is corrected for violet and red. For systems with the focal length normalized to a unity we may write

h

<Pi+hcp2 = —r > (5)

s

where <pi,(p2 — focussing power of the first and se­ cond lens, respectively. The focussing power for any wavelength may be written as

<Px = » 2 -1

72 — 1 r

(6)

Formula (5) for any wavelength has the form: » n - l

« 1 -1 <Pi+hx «22-1

722— 1 (7)

*) Instytut Fizyki Technicznej Politechniki Wrocławskiej, Dividing eq. (5) by h and eq. (7) by h2, and subtra-

Wrocław, Wybrzeże S. Wyspiańskiego 27, Poland. tracting them we obtain

(2)

<P 2 = — (18)

<P i

i - 4 ) +yi f1-?Hzi)= 4 =i-.

«

1

-

1

/

\

«

2

-

1

/

SX - S

1 «U “ 1 \

h «1 — 1 /

(8)

Multiplying eq. (8) by lilt, we get

t>i(hx—h ~ x + h h x(p2 ( l ---- — r ) = ( h - s ') - A

\ « 1 - 1 / \ « 2 - 1 / sx X

X

(9) It may be shown that

7

+

m

From the conditions

h = 1— dcpi hx = l-d<pn we obtain h2 = h + d f i ( l - î ü = l \ «1 — 1 (11) ( 12)

On inserting relations (10) (11) and (12) into eq. (9) we have <Pi ^i

-+

hd(pycp

211

'1 2'

-1

« î - l « 2 A -h2<f2 1 »21-1 \

«

2 - 1

/

+

Making use o f the identity

« i - l = (« —i) | i - ^ - j · Eq. (13) may be written as

(14)

21 , , ^*22 , , ^21 r 22 < , ,.

<P 1 — +«2?>1--- = S/—S . ^2 P n P , (15)

V2 Vi v2

Since the system was assumed to have the chro matic aberration corrected for violet and red colo urs,

— + h 2 — = 0. (16)

Vi v2

From the normalization condition it follows that

<Pi+h(p2 = l . (17)

Therefore <p2, and q>2 can be found

v j i Vi h—v2

Vz

h(vih—v2)

Calcullating d from the first of eq. (11) and inser­ ting into (15) with simultaneous application o f (18), we obtain

K P n - P z z ) (1 - h ) P n Pn . ,

---= j A—^ · (19)

V i h — v 2 V i { v i h — v 2 )

The relation (19) becomes (1) if we neglect the second term. As seen from eq. (19) the approximate formula (1) may be used only when the product of the focussing power of the first lens and the distance between the lenses is relatively small. The magnitude o f error we make when using formula (1) depends also on the dispersion o f glass the first lens is made of, and also on the wavelength for which the secon­ dary spectrum is calculated. As an example we esti­ mate the secondary spectrum of an superachromatic system with the Petzval curve corrected. The first lens is made o f fluorite, the second — of LaK 11. The focussing powers and distances between the lenses are, respectively;

9?! = 6.158, <p2 = - 7 . 1 4 0 , <7=0.045.

(20)

This system is the strating system for calculating a superachromat with flat field. The magnitude of the secondary spectrum for the wavelength A = 1.014 [im obtained from formula (1) (the first term o f formula (19)) is — 0.0012, whereas from formula (19) it is — 0.002. Therefore, the relation (19) can not be used in this case. The above considerations show that, in general the foimula (1) can not used for estimating the magnitude of secondary spectrum. This formula can be used only when the second term in relation (19) is negligible.

The author expresses his thanks to Prof. M. Gaj for helpful discussions.

References

[1] Can2ek L., Optik 30, 1069, 17.

[2] Can2ek L., Private communication.

Received, June 15, 1972.

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