ISSN 0030400X, Optics and Spectroscopy, 2011, Vol. 111, No. 1, pp. 100106. Pleiades Publishing, Ltd., 2011.Original Russian Text G.A. Lenkova, 2011, published in Optika i Spektroskopiya, 2011, Vol. 111, No. 1, pp. 107114.

100

INTRODUCTION

In monochromatic light, a standard phase Fresnellens is similar to a onefocus refractive lens. With adecrease in the height of the sawtooth profile, which ischaracteristic of the Fresnel structure, to the heightcreating a half wavelength delay, the light is mainlydirected at the zeroth and +1st diffraction orders (by~40% to each) [1]. When a lens with this profile is illuminated by a convergent spherical wave, the light isfocused at two points, the first of which (the zerothorder) is formed at the curvature center of the wave,while the second one (+1st order, with allowance forthe lens power) is formed closer to the lens. A similarpicture is observed when a Fresnel lens is combinedwith a refractive lens, as in hybrid (diffractiverefractive) artificial intraocular lenses. In this case, if arefractive lens (placed in front) is illuminated by a parallel light beam, a convergent beam falls onto theFresnel lens. As a result, such hybrid lens has two fociin the zeroth and +1st diffraction orders. If the aperture of the lens (or of the beam) is fairly large, the focaldistances (or the position of the focal points) can becalculated by formulas of geometrical optics. With adecrease in the aperture, the position of the foci (or thefocal points) begins to be affected by diffraction at theaperture edges, which causes a socalled focal shift.

The focal shift has been studied in numerous works[210] in which the intensity distribution in the fieldbehind the diaphragm upon the incidence of convergent or divergent waves was analyzed [24] dependingon the diaphragm number and the Fresnel number[58], the applicability limits of the Debye and Kirchhoff integrals were determined [9, 10], and simplifiedformulas for determining the focal shift were proposed. However, a visual physical explanation of theappearance of the shift has not been presented in theavailable literature.

In addition to the focal shift, a decrease in the aperture of the Fresnel lens leads also to a decrease in thenumber of working zones and to the extension of theintensity distribution functions. As a result, the conversion of the lens into a monofocal lens, if only thecentral zone is open, and the mutual overlap of theintensity distribution functions at the foci of the zerothand +1st orders become possible. A similar situationoccurs in bifocal diffractiverefractive intraocularlenses (IOLs, artificial lenses of an eye) when the pupilis constricted [1113]. The number of the zones in theaperture can be increased by a decrease in the diameter of the central zone of the Fresnel lens if the phaseshift is introduced into its structure; however, thisresults in a redistribution of the intensity along the axisand mutual shift of the foci.

The aims of the work are a physical interpretationof the appearance of the focal shift based on theFresnel numbers and an analytical study of the lightintensity distribution in the focal regions of the phaseFresnel lens in the case when the height of the structure creates a half wavelength delay and the lens is illuminated by a convergent spherical beam.

DIFFRACTION ON DIAPHRAGM AND FOCAL SHIFT

When a parallel light beam is incident onto a diaphragm, one part of the light passes through it withoutchanging its direction and the other part diffracts in theform of convergent and divergent waves (Fig. 1a) thatcorrespond to negative (imaginary) and positive(real) diffraction foci (intensity maxima). The distances of these foci and from the diaphragm are

, , (1)

where is the diaphragm radius; is the wavelength;n is the refractive index of the surrounding medium;

NFNF+

Nf Nf+2

Nf r n N = /2

Nf r n N+ = /

r

PHYSICAL OPTICS

On Focal Shift and Phase Fresnel LensG. A. Lenkova

Institute of Automation and Electrometry, Siberian Branch of the Russian Academy of Sciences,Novosibirsk, 630090 Russiaemail: lenkova@iae.nsk.su

Received January 26, 2011

AbstractA physical interpretation of the focal shift is given based on geometrical optics. The results of ananalytical study of the focal shift depending on the Fresnel numbers are presented. The effect of the aperturediameter and the width of the central zone of the phase Fresnel lens on the distribution of the light intensityalong the axis upon illumination by a convergent light beam is analyzed.

DOI: 10.1134/S0030400X11070137

OPTICS AND SPECTROSCOPY Vol. 111 No. 1 2011

ON FOCAL SHIFT AND PHASE FRESNEL LENS 101

and N = 1, 3, 5, ... are the odd Fresnel numbers. Themost intense maxima of the 1st order with N = 1 areespecially distinctive diffraction maxima. They areformed at distances f1 from the diaphragm corresponding to the diffraction optical power as follows:

, . (2)

As an example, Table 1 presents the quantities and calculated by formula (2) for several values ofr at = 0.5461 m and n = 1.3377, which correspondto the conditions inside an eye. One can say that, in thecase of the eye, the effect of the diameter of the pupilvisibly manifests at r < 1 mm, when > 0.55 dpt( < 2449.6 mm).

The illumination of the diaphragm by a convergentlight beam is equivalent to placing a refractive lens in aparallel beam in front of or behind the diaphragm. Inthis case, on different sides of the lens focus (Fig. 1b), which is formed by the beam passing the diaphragm without deviation, the images and ofthe positive and negative diffractive foci and are projected at distances of and . According tothe laws of geometrical optics, these distances are

and , (3)

where f is the focal distance of the lens and andcorrespond to Eqs. (1) and (2). It follows from (3)

that, regardless of the value of , all images are

1

21f r n = /

21 1n f r = = / /

1f 1

1

1f

0F

NA+ NANF+ NF

NS+ NS

( )N N NS ff f f+ + += +/ ( )N N NS ff f f = +/

Nf+Nf

Nf+ NA+

real and located between the focus and the lens. Theimages are real and are placed behind the focus if

and are imaginary and lie in front of the dia

phragm if . The intensity distribution alongthe axis behind the lens is the total action of the intensities of all images. The position of the total maximum(focus) is most affected by the beams with N = 1,which are focused at and and are the mostintense of all the diffracting beams. However, the character of this influence depends on the aperture diameter. Here and below, we present the calculations for thefocal distance of the refractive lens f = 20 mm, whichis equal to the distance from the eye lens to the retinain the model of an eye, = 0.5461 m and n = 1.3377[11]. Figure 1b shows the formation of the images and , the relative position of which is similar to the

0F

NANf f >

Nf f

2 222

2

sin( 4)42

D DukrI U

uf

=

=

/

/

2 222 sin( '/ 4)

2 '/ 4K K

ukrI Uzf u

=

=

DI KI DU KU

2k n= /

2( ) ( )u k r f z f= / 2' ( ) ( )( )u k r f z f f z= / /

( )Dz min( )f z 2(2 )( )n f r / /

KI

maxz

1.6240

1.2123

0.844

0.45

03140

r, mmNf

24

20

S1

S+1

zmax

S, zmax, mm

16

Fig. 2. Plots of dependences of and (formula (3)) and (formula (5)) on diaphragm radius r.1S+ 1S maxz

20 78 176

OPTICS AND SPECTROSCOPY Vol. 111 No. 1 2011

ON FOCAL SHIFT AND PHASE FRESNEL LENS 103

0.5 mm (curve 3), the distribution becomes almostsymmetric, i.e., close to the Debye approximation.

The position of the maximum in the region ofthe Kirchhoff approximation can also be determinedfrom the following relation obtained by the differentiation of (5) and equating the derivative to zero:

, (7)

where u' corresponds to (6). Equation (7) has no analytical solution. The values can be determinedby the parametric method by finding the intersectionpoint of the functions from the right and lefthandsides of equality (7) or graphically (see below).

Figure 2 presented the graph of the dependence of on r calculated by formula (5) for f = 20 mm. The

difference is the focal shift. As can beseen from the plot, with an increase in r, i.e., with adecrease in the diaphragm number , the value of

tends to the value of f.

FRESNEL NUMBER AND FOCAL SHIFT

It was shown in [6] that it is more convenient tocompare different optical systems with respect to thepresence of the focal shift based on Fresnel numbers,which play a dominant role, rather than based on theratios or . If the Fresnel numbers are equal,the form of the intensity distribution is similar regardless of the relations between r, f, and . Let us represent formulas (3), (6), and (7) in terms of Fresnelnumbers

, , (8)

, (9)

, (10)

where , , , , and Nd are the

Fresnel numbers defined as for the distances , , f, , and ( is the focal dis

maxz

KdI dz/

tan / / /( ' 4) ( ' 4)( )u u f z=

maxz z=

maxz

maxf z f =

2f r/

maxz

2f r/ r /

( 1) 1S fN N+ = + ( 1) 1S fN N =

(max)' 2 ( ) 2z f du N N N= =

tan / / /( 2) ( 2)(1 )d d d fN N N N = +

( 1)SN + ( 1)SN fN (max)zN2 ( )N r n L= /

1L S+= 1S maxz df df

tance corresponding to the total action of the diaphragm). Here,

(11)

It follows from (8) that the Fresnel numbers and for light beams that focused at the points and differ by unity from the Fresnel number for the lens. As was mentioned above, for small diaphragms, the shift of the focus (position of the totalmaximum) is mainly affected by the image at a distance from the lens. Therefore, in this case, thetotal Fresnel number for the diaphragm Nd approximately equals , i.e., Nd ~ 1. With anincrease in the diaphragm, Nd tends to zero becausethe effects of the images and are opposite.Therefore, the general interval of possible values of Ndlies within the limits 1 Nd 0.

Figure 4 presents the plot of the dependence of Ndon (curve 1) constructed based on formula (10). Ascan be seen from the plot, for , and, forNf < 1, Nd ~ 1. For these two regions, the followingsimple relations are obtained from (10) in the firstapproximation:

(12)

(13)

Their plots (curves 2 and 3) are also presented inFig. 4. Formulas (12) and (13) are similar to formulasobtained in [2, 3].

Having determined Nd, one can calculate the absolute (f = ) and relative (f/f = )values of the focal shift based on relation (11) as follows:

(14)

/

/ /

2

2 2max (max)

( )

( ) ( ) .

d d

z f

N r n f

r n z r n f N N

=

= =

( 1)SN +( 1)SN 1A+1A fN

1A+1S+

( 1)S fN N+

1A+ 1A

fN1fN 1dN

/ / for212 ( ) 1.21585 1,d f f fN N N N= =

/ for21 4 1 0.405 1.d f f fN N N N= = 0.5 mm), the focal distances approach their nominalvalues (Fig. 5e). As this takes place, the Fresnel numbers increase quadratically with r (formula (11)), andthe focal shift decreases quadratically with the Fresnelnumber (formula 15)). Detailed studies of the effect ofthe phase shift and diaphragm (eye pupil) diameter onthe position of the foci for IOL (diffractiverefractiveintraocular lenses) were performed in [12], but without using Fresnel numbers.

CONCLUSIONS

Based on geometrical optics, the appearance of thefocal shift upon incidence of a convergent light beamonto the diaphragm or in the focal region of the lenswas explained. It was shown that the axial intensitydistribution behind the lens represents the total actionof all images of the diffraction foci of the diaphragm(positive and negative ones). The position of the intensity maximum is mainly affected by the foci of the +1stand 1st orders.

The intensity distributions are compared in thefocal region in the Debye and Kirchhoff approximations. It is shown that all formulas can be considerablysimplified if the focal shift is analyzed based on theFresnel numbers. It was determined that the intervalof possible values of the Fresnel numbers of the diaphragm Nd lies within the limits 1 Nd 0. Based onthe tangential dependence between Nd and the Fresnelnumber Nf for the lens, we calculated a table permitting one to determine Nd and the relative focal shifteasily and quickly for any Nf. Meanwhile, the approximate formulas are valid only for ( ) or

< 1 ( ~ 1).

The distribution of the light intensity in the focalregions of the phase Fresnel lens upon illumination bya convergent light beam was studied analytically usingthe Fresnel numbers and with allowance for the relations between and . It was shown that the focalshift is an order of magnitude less than the shifts of the+1st and zeroth foci caused by the mutual influence ofthe side maxima of the axial intensity distributions.

REFERENCES

1. G. A. Lenkova, Avtometriya, No. 5, 16 (1995) [Optoelectr., Instrum. and Data Process. No. 5, 15 (1995)].

2. Y. Li and E. Wolf, Opt. Commum. 39 (4), 211 (1981).3. Y. Li, Optik 69 (1), 41 (1984).4. X. Jiang, Q. Lin, and S. Wang, Optik 97 (1), 41 (1994).5. A. Arimoto, Optica Acta 23 (3), 245 (1976).6. W. Wang and E. Wolf, Opt. Commun. 119 (56), 453

(1995).7. Y. Li, J. Opt. Soc. Am. 72 (6), 770 (1982).8. I. G. Palchikova and S. G. Rautian, Opt. Spektrosk. 87

(3), 510 (1999) [Opt. Spectrosc. 87 (3), 472 (1999)].9. E. Wolf and Y. Li, Opt. Comm. 39 (4), 205 (1981).

10. J. J. Stamnes and B. Spjelkavik, Opt. Commun. 40 (2),81 (1981).

11. V. P. Koronkevich, G. A. Lenkova, V. P. Korolkov, andI. A. Iskakov, Opt. Zh. 74 (12), 34 (2007).

12. G. A. Lenkova, Avtometriya 46 (3), 74 (2010) [Optoelectr., Instrum. and Data Process. 46 (3), 264 (2010)].

13. M. J. Simpson and J. A. Futhey, US Patent No. 5116111.14. M. Born and E. Wolf, Principles of Optics (Pergamon,

Oxford, 1969; Nauka, Moscow, 1973).

Translated by A. Nikolskii

1fN 1dN

fN dN

dN fN

Table 3. Dependence of Nd on Nf (10) in tabular form; Nf = r2n/(f), Nd/Nf is derivative of Nd with respect to Nf, and f/f

is relative focal shift in %

Nf Nd Nd/Nf f/f Nf Nd Nd/Nf f/f

121.57 0.01 0.000 0.01 7.93 0.15 0.018 1.86

60.77 0.02 0.000 0.03 6.95 0.17 0.023 2.39

40.49 0.03 0.001 0.07 5.84 0.2 0.032 3.31

30.35 0.04 0.001 0.13 5.01 0.23 0.041 4.39

24.26 0.05 0.002 0.21 4.01 0.28 0.060 6.53

20.19 0.06 0.003 0.30 3.05 0.35 0.090 10.28

17.29 0.07 0.004 0.40 2.02 0.47 0.149 18.86

15.10 0.08 0.005 0.53 1.50 0.56 0.196 27.23

13.40 0.09 0.007 0.67 1.21 0.62 0.228 33.81

12.04 0.1 0.008 0.82 0.62 0.78 0.310 55.89

10.92 0.11 0.010 1.00 0.21 0.92 0.373 81.74

9.20 0.13 0.014 1.39 0.02 0.99 0.401 97.56

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