ISSN 0030�400X, Optics and Spectroscopy, 2009, Vol. 107, No. 5, pp. 773–775. © Pleiades Publishing, Ltd., 2009.Original Russian Text © A.V. Lensky, 2009, published in Optika i Spektroskopiya, 2009, Vol. 107, No. 5, pp. 814–816.

773

In the nighttime, a cat’s pupil is round. In the day�time, depending on the illumination, it has the shapeof a more or less narrow, biangular, vertical diaphragmwhose height varies much less than its width. A char�acteristic shape of the daytime pupil, along with its ori�entation can be well explained by a highly efficientsuppression of sunlight that is diffracted to the retina,i.e., by the suppression of the perturbing illuminationof the retina.

Indeed, on a clear day, only the sun, which, in themajority of situations, illuminates cat’s eye from above(clearly, in a more or less narrow/wide range of posi�tion angles), can blind a cat. If the shape of the pupilwas horizontal or remained round, as in a human eye,the so�called active zones of the pupil contour, forwhich the optical path length sun–contour–retina isextremal, would be secondary sources of diffractionillumination. At the vertical orientation of the pupilwith the sun above it, active zones are absent, and sun�light is diffracted to the retina by angular points of thecontour. In this case, the diffraction illumination ismuch weaker than that provided by active zones.

These conclusions follow from calculations of thediffraction of light on contours of different geometriesusing the Kirchhoff integral in the Rubinovich repre�sentation [1]. This representation makes it possible toquantitatively estimate the effects of diffraction in thiscase as well. This can be done similar to the consider�ation of contours studied in [2, 3].

Let i0 and i1 be the intensities of light diffracted bya round pupil and a cat’s pupil, respectively. We willassume that an infinitely far point light source islocated at an angle ϕ to the optical axis of the eye (witheither pupil). Angular points of the cat’s pupil will beassumed to be located at equal distances from the opti�cal axis in a plane that contains this axis and the direc�tion toward the source. This arrangement correspondsto the vertical orientation of the cat’s pupil with thelight source located above the pupil. In both cases, we

will consider the intensity of light diffracted by thepupil to the axial point of the retina.

In the case of a round pupil,

(1)

where J1(γ) is the first�order Bessel function of the firstkind [4], γ = 2π(r/λ)sinϕ, λ is the wavelength of light,and r is the radius of the pupil. This is the well�knownAiry formula (see [1]) with the normalization pre�sented below.

For a cat’s pupil,

(2)

Here, α = 2π(p/λ)sinϕ, and 2p is the distance betweenthe angular points of the pupil, i.e., its height.

The areas of the round pupil and cat’s pupilassumed to be the same; in this case, the normaliza�tion ensures the equality of the principal diffractionmaxima in both cases for all values of ϕ and theabsence of identical multipliers that depend on theangle ϕ not via γ of function (1) and not via α of func�tion (2). At ϕ = 0, both functions are unity.

Expression (2) proves to be simple thanks to thesimplicity of approximation of the arcs of the biangu�lar contour of the pupil by two symmetric parts ofparabolas. This gives rise to the drawback, which,however, is insignificant in this case. Namely, evenwhen the height 2p and width 2q of cat’s pupil are thesame, and the pupil becomes round, the contour thatwe used contains two angular points. However, sincewe are interested in the ratio between the diffractionilluminations of the retina by an external source atq < p, this drawback has a formal character. In themost interesting cases, where q is severalfold smallerthan p, the choice of a particular approximation of thearcs of the pupil contour, with its characteristic shapebeing preserved, is hardly of any importance at all.

i0 i0 ϕ( )2J1 γ( )

�����������

2

,= =

i1 i1 ϕ( ) 3 αcos α/αsin–

α2������������������������������⎝ ⎠

⎛ ⎞ 2.= =

PHYSICAL OPTICS

Cat’s Pupil and ApodizationA. V. Lensky

Subbotin Institute of Geophysics, National Academy of Sciences of Ukraine, Kiev, 03680 Ukrainee�mail: alex.v.lensky@gmail.com

Received December 15, 2008; in final form, May 14, 2009

Abstract—The shape and vertical orientation of a cat’s pupil in the daytime are explained by the highly effi�cient suppression of sunlight that is diffracted by the pupil to the retina.

PACS numbers: 42.25.Fx, 42.66.Lc, 87.19.lt

DOI: 10.1134/S0030400X09110137

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OPTICS AND SPECTROSCOPY Vol. 107 No. 5 2009

LENSKY

The narrower the pupil, the closer its angular pointsto their mathematical ideal and the closer our theoret�ical estimate to the real pattern.

As a measure of suppression of the diffraction illu�mination of the retina by means variation in the shapeand orientation of the pupil, we can take the ratio g offunctions of the angle ϕ that determine the intensity ofthe secondary diffraction maxima in the two consid�ered cases. This estimation will be correct if the areasπr2 of the round pupil and 8pq/3 of cat’s pupil are equalto each other. The ratio of the corresponding function

from (1) at r = to the analogous functionfrom (2) is given by

(3)

This expression was obtained using the first terms ofthe expansions of functions (1) and (2) in terms ofnegative powers of γ and α, respectively, which quitesatisfactorily replace expressions (1) and (2) beginningfrom values of ϕ already on the order of a few degrees.

The ratio g(ϕ) shows how many times the intensityof light diffracted at the center of the retina upon ver�tical orientation of a cat’s pupil is smaller than that forthe round pupil with equal area if the external sourceis located in a vertical plane that contains the opticalaxis of the eye and the angular distance between thesource and this axis is ϕ.

Advantage (3) in the suppression of the diffractionillumination, in the case when this suppression isrequired, is very significant. On a bright, sunny day, acat’s pupil is narrow and, at its typical dimensions,e.g., at 2p = 5 mm and 2q = 1 mm, and at a light wave�length of λ = 550 nm, this advantage is 2.0 × 104–1.1 × 105 in the range of angles ϕ from 10° to 70°. Wecan say that, in the course of evolution of the felinefamily, the geometric shape of diaphragming of thepupil and its orientation were selected to satisfy a veryhigh level of requirements.

This result was obtained for the point source.Clearly, integration over an extended source, even if itsangular dimensions are small, such as, e.g., those ofthe sun, will change the obtained relation. However,this change is not so considerable that the result wouldno longer be as impressive.

It should also be noted that expression (2) wasobtained for the flat contour of the pupil, whereas, inreality, the pupil is nonflat. Taking into account thiscircumstance will result in the function i1(ϕ) becom�ing somewhat smoother; furthermore, its minima willno longer be zero and the secondary maxima willbecome lower. This occurs because, for the nonflatcontour of the pupil, the contributions of angularpoints to the resultant amplitude differ from eachother in magnitude.

8pq/3π

g g ϕ( ) 8α4

9πγ3��������� π3/2

p5/2 ϕsin

6q3/2λ

������������������������.= = =

To give a complete picture, we also present anexpression for the normalized intensity i2 of the lightthat is diffracted by the cat’s pupil to the axial point ofthe retina in the case where the point source is locatedupwardly and at an angle ϕ to the optical axis of theeye, while the pupil itself is horizontal as follows:

(4)

where β = , 2q is the maximal width ofthe pupil, and C(β) and S(β) are the Fresnel integrals[4]. Expression (4) was obtained using the sameapproximation of the pupil contour by symmetric partsof parabolas as that used in obtaining formula (2). Thenormalization of all the three expressions, (1), (2) and(4), is also the same (see above).

Compared to the horizontal pupil, the advantage inthe suppression of the diffraction illumination by thevertical pupil is 0.54(p/q)3/2 times greater than uponcomparison with the round pupil.

A rigorous and complete consideration of the dif�fraction of sunlight on cat’s pupil would significantlycomplicate the derivation of the result similar to (3)hardly changing it at all.

A cat’s pupil, as well as pupils of other animals,which are of the same shape but may differ in the ori�entation, provide a remarkable example of apodiza�tion [5], which was elegantly and efficiently imple�mented in wildlife long before its potential and meth�ods were realized by human. It seems that differentorientations of these pupils for animals of differentspecies are always aimed at ensuring the lowest diffrac�tion illumination of their retinas by sunlight.

ACKNOWLEDGMENTS

I am grateful to N.V. Semenenko andA.V. Rokhlenko for their help during the preparationof the manuscript.

REFERENCES

1. M. Born and E. Wolf, Principles of Optics (Pergamon,Oxford, 1969; Nauka, Moscow, 1973).

2. A. V. Lenskiі and A. V. Rokhlenko, Kinematika Fiz.Nebesnykh Tel 5 (2), 58 (1989).

3. A. V. Lenskiі, Opt. Spektrosk. 67, 1380 (1989).

4. Handbook of Mathematical Functions, Ed. by M. Ab�ramowitz and I. A. Stegun (Dover, New York, 1965;Nauka, Moscow, 1979).

5. P. Jacquinot and B. Roizen�Dossier, in Progress inOptics, Vol. 3 (North�Holland, Amsterdam, 1964),p. 29.

i2 i2 ϕ( )=

= 3π��C β( ) πβ2

/2( )sin S β( ) πβ2/2( )cos–

β3�����������������������������������������������������������������������

2

,

2 q/λ( ) ϕsin

OPTICS AND SPECTROSCOPY Vol. 107 No. 5 2009

CAT’S PUPIL AND APODIZATION 775

Comment on the Paper “Cat’s Pupil and Apodization” by A.V. Lensky

The paper expounds on the hypothesis of the scien�tific substantiation of the peculiar shape and orienta�tion of cat’s pupil upon solar illumination. It is sug�gested that the narrow and vertically elongated shapeof the pupil is determined by a decrease in the level ofsolar light incident onto the eye’s retina due to its scat�tering as a result of diffraction. Doubts about the valid�ity of the hypothesis of the author arise because of thefact that, under solar illumination, the round shape ofother animals’ pupils, as well as that in humans, andthe diffraction corresponding to it does not affect the

functioning of the visual apparatus. In addition, with aslit�like, vertically oriented shape of the pupil, the eyehas substantially different resolving powers in the ver�tical and horizontal directions, which is ignored by theauthor.

I believe that all of the above�stated can serve as asource for discussion.

A. P. Grammatin

Translated by V. Rogovoi