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The diffractive optical power of a diaphragm
15 January 2000
Optics Communications 174 Ž2000. 1–5 www.elsevier.comrlocateroptcom
The diffractive optical power of a diaphragm I.G. Palchikova a
a,b,)
, S.G. Rautian
a,b
Institute of Automation and Electrometry, Sib. Br. RAS. Prosp. Koptyug, 1, 630090, NoÕosibirsk, Russia b NoÕosibirsk State UniÕersity, PirogoÕa Street, 2, 630090, NoÕosibirsk, Russia Received 28 May 1999; received in revised form 21 October 1999; accepted 22 October 1999
Abstract The systematic analysis of the diffraction image shift enabled us to extend the notion of the optical power of a pinhole to the case of an aperture of an arbitrary form and any dimension. The mathematical expression for the diffractive optical power of an aperture has been obtained by the linear hyperbolic interpolation with three parameters. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Diffraction; Focal shift; Pinhole camera; Optical power
1. Introduction The focusing action of the diaphragm is connected with the well-known peculiarities of Fresnel diffraction pattern. According to the theory of the wave diffraction at the semi-plane, the light field diffuses into the area of geometrical shade. Its intensity increases as it approaches the shade boundary, but at the boundary the intensity reaches only a quarter of the geometrical optics value. The light stripes are observed in the lighting area, the brightest stripe is the nearest to the boundary, and is 0.86'l z Ž l is the wavelength, z denotes the distance from the aperture plane. distant from it. The maximum intensity is 1.37 in the same units. Qualitatively, these
) Corresponding author. Tel.: q7-3832-399-333; fax: q73832-354-851; e-mail: [email protected]
peculiarities are retained also under the diffraction at an aperture of any form and dimension, i.e. the light stripes fringe the shade boundary only if the distance to the first light stripe does not exceed half the diameter of the aperture, that is if the condition 'l z F a is fulfilled. Any finer structure in the transversal intensity distribution cannot exist because the condition l z ; a 2 means that there can be only one Fresnel zone on the aperture. Hence, when the first light stripe of Fresnel diffraction pattern comes onto the geometrical optics axis of the diffracted wave, intensity distribution at transversal section of the beam is unstructured. The intensity at the beam axis appears greater than its geometrical optics values, and the intensity distribution at the beam transversal section at the distance z s a 2rl becomes narrower than when the diffraction is not allowed for. This fact is interpreted as a focusing by the aperture and the definite optical power is attributed to the aperture. In its turn, the focusing apparently must result in the shift of the point source image.
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 4 5 - 8
I.G. PalchikoÕa, S.G. Rautianr Optics Communications 174 (2000) 1–5
2
Thus the focusing and the diffraction shift of image are inherent in the apertures of arbitrary forms. The phenomenon of diffraction pulling or diffraction shift of image Žfocus. had been studied in details for gaussian beams w1,2x and is at present being under the active investigation for spherical and cylindrical waves w3–8x. But up to now there has not been found simple approximate equation which would permit to calculate the diffraction focal shift for converging light wave with different Fresnel numbers N and any form of aperture. The use of the analytical properties of the exact solutions permitted us to propose the interpolation equation for the image shift. The precision of these equations proves to be as over the precision of the equations offered in Refs. w4–6x as ten times. In our work we discuss the concept of diffractive optical power of the aperture and offer sufficiently exact equations for the diffraction shift.
2. Fresnel diffraction at apertures of arbitrary forms For the quantitative analysis we shall consider the modulus of the field amplitude < EŽ0, z .< at the axis of a spherical wave with the radius of curvature R after its diffraction at the aperture. Unlike previous papers devoted to the focal shift, we refuse to characterise the effect by the relation Ž z y R .rR, because this quantity is sufficient if N 4 1 and is inadequate for our problem if N ; 1 and N < 1, and hence, is inconsistent with the physical statement of the general problem. We shall use the focal tolerance D z of spherical wave as a parameter with respect to which we shall estimate the value of the focal shift. In addition we put the co-ordinate origin at the centre of aperture and on analysing a Fresnel–Kirchhoff integral in parabolic approximation we use the effective Fresnel numbers n e and n e z , which we define with the help of the radius ae of circle of the same area S as that of the aperture: p ae2 s S,
n e s ae2rl R ,
n e z s ae2rl z ,
Ž 1.
the difference ne z y ne s z will serve as a non-dimensional variable.
Ž 2.
With this specific notation the modulus of the field amplitude < EŽ0, z .< can be written as < E Ž 0, z . < s Ž 1 q n erz . f Ž z . , f Ž z . s z < H exp Ž i pz P r 2 . d r < .
Ž 3.
Here r is the radius-vector at the aperture plane, normalized on ae . The co-ordinate z is measured from the centre of the aperture. The integral is evaluated over the surface of the aperture, using conventional normalization the geometrical optics quantity is < E < s 1. It can be seen the function < EŽ0, z .< explicitly depends on two parameters — n e and z , and unexplicitly — on the aperture form, which defines the domain of integration. Substantially, the function f Ž z . depends only on z and the form of the aperture, and explicitly it does not depend on the radius of curvature R for the incident wave front. In a sense f Ž z . is universal, it equally describes the diffraction of plane, converging and diverging waves. The quantity R begins to influence f Ž z . only after the transition from z to variable z by Eqs. Ž1. and Ž2.. With variables n e , z the field amplitude depends on R only over the factor 1 q n erz . The phenomenon of the diffraction shift for image is connected just with the factor 1 q n erz .
3. The circular aperture In the case of the circular aperture with the radius a the intensity distribution < EŽ0, zrR .< 2 s I Ž0, zrR . along the axis for the diffracted spherical wave is well known w3x and it can be written as Eq. Ž3., if the traditional Fresnel numbers for the centre of the wave front curvature Ž N . and for the current point z Ž Nz . are introduced instead of the effective ones: < E Ž 0, z . < s Ž 1 q Nrz . <2 sin Ž pzr2 . < , f Ž z . s <2 sin Ž pzr2 . < , N s a 2rl R ,
Nz s a 2rl z ,
Ž 4. z s Nz y N.
The conclusion on focusing action of the aperture immediately follows from the ratio Ž4.. In the case of a plane incident wave Ž N s 0. the main maximum for amplitude is determined by the maximum for the function f Ž z ., which appears first as z increases up
I.G. PalchikoÕa, S.G. Rautianr Optics Communications 174 (2000) 1–5
to the value z ' z 0 s Nz s 1 Žor z decreases from the infinity to the value z s z 0 s a2rl.. At this point < EŽ0, z .< s 2, i.e. it is twice the value obtained from the geometrical optics, and the intensity is four times, and the transversal distribution has approximately half the width determined under the geometrical optics. This fact means only diffraction focusing. The maximum intensity position is determined by roots for the transcendental equation w3x:
ž
tan Ž u . s u 1 q
2u pN
/
,
u s pzr2.
Ž 5.
Due to the periodicity of tanŽ u. Eq. Ž5. has the infinite number of solutions, answering the maxima for the amplitude modulus. We are interested in the main maximum, which corresponds to the root, placed within the interval 0 - z F 1. We shall denote this root zm . Roots for Eq. Ž5. depend on N and the main question discussed in the paper consists in clearing up the nature of dependence zm s zm Ž N .. The explicit form for the inverse function N Ž zm . follows immediately from Eq. Ž5.: N Ž zm . s pzm2r 2 tan Ž pzm r2 . y pzm r2 4 .
Ž 6.
The calculation N Ž zm . with sufficiently small step allows easily to find zm Ž N . by the linear interpolation. Let single out two limiting cases i.e. the small diffraction shift when zm < 1Ž N 4 1., and the case of the great diffraction shift when zm is placed nearby z 0 and the inequality 1 y zm < 1 Ž N < 1. is fulfilled. The behaviour of N Ž zm . under the limiting cases can be easily deduced from the ratio Ž6., if tanŽ pzm r2. is replaced by the corresponding power expansion: N s 12r Ž p 2zm . ,
zm < 1;
N s p 2 Ž 1 y zm . r4,
1 y zm < 1.
3
which, instead of the trigonometric functions included in the ratio Ž6., the power series are used: N s yd zm q c q
b
.
zm
Ž 8.
The coefficients b, c, d are defined according to the limiting values Ž7.: b s 12rp 2 s 1.21585, d s p 2r4 y b s 1.25155, c s d y b s 0.03570.
Ž 9.
Eq. Ž8. allows one to get the analytical expression for the function zmŽ N .: 2b
zm s
(
.
2
Ž 10 .
N y c q Ž N y c . q 4 bd Usually the absolute value of the image position uncertainty d z is not important, but its ratio to the focal tolerance D z Žto the length of the beam waist. is essential: D zs
lz2 a
2
z s Nz
,
dz Dz
s Nz
dz z
s ydz .
Ž 11 .
Hence, the accuracy of result zm Ž N . is of the immediate interest. The results of the numerical computation are presented in Fig. 1: curve 1 is a plot of zmŽ N ., calculated according to the Eq. Ž6.; the approximated values zmŽ N . are calculated from Eq. Ž10. and is no different from the curve 1 at scale of Fig. 1. The difference between the results of the first and second calculations have been shown at the bottom of the Fig. 1 Žcurve 2., increased 100 times.
Ž 7.
The result Ž7. coincides with that derived in Ref. w3x and Ref. w7x by another method. For the analytical description of the dependence zm on N, we propose the approximating equation which is based on the asymptotic values Ž7., and in
Fig. 1. Curve 1 is the plot of the function zm Ž N ., calculated from the exact solution Ž6. and from the approximating Eq. Ž10. for the circular aperture. Curves 2, 3, and 4 describe the differences zmexact y zm , increased 100 times, for the circular, the square Ž l s 4. and hexagonal Ž l s6. apertures accordingly.
I.G. PalchikoÕa, S.G. Rautianr Optics Communications 174 (2000) 1–5
4
The maximum error of the expression Ž10. takes place at N s 2 and equals to f 0.0019. In other words, the error for the calculating image position from Eq. Ž10. does not exceed 0.19% of the focal tolerance within the interval ` ) N G y0.5, that is more than enough in many cases Žthe values N - 0 correspond to the divergent wave.. Mark some physical conclusions following from Eq. Ž10.. Turn into the non-dimensional variables and write Eq. Ž10. as 1 1 s qF D , z R l 2b FDs 2 . Ž 12 . a N y c q Ž N y c . 2 q 4 bd
(
The relation Ž12. has the form of a lens equation, and furthermore the value F D can be considered the optical power of the aperture due to the diffraction by origin. In addition to the factor lra2 it contains the additional factor describing the dependence F D on N or on the wave front curvature 1rR. In limiting cases N 4 1 and N < 1 we find from Eqs. Ž12. and Ž7.: 2
F D s l br Ž Na2 . s 12 R Ž lr Ž p a2 . . ,
N 4 1,
Ž 13 . F D Ž 1 y 4 Nrp 2 . lra2 , N < 1. Ž 14 . In Eq. Ž13. the image diffraction shift d z s z m y R from the wave front curvature centre is a fraction of the focal tolerance D z: < d z < s R 2F D s 12D zr Ž p 2 N . , D z s l R 2ra2 , Ž 15 . for example it gives < d z < f 0.12D z if N s 10. The expression Ž14. complies with a weakly converging wave when its curvature centre is far away in Fraunhofer zone Ž R 4 a 2rl. and the focusing action due to the curvature of the initial wave front is weaker than the pure diffraction focusing. The following fact mentioned in w7x has engaged our attention: diminishing of the optical power F D as increasing 1rR seems to reduce the summary part of ‘ray focusing’: 1rz s Ž 1rR . Ž 1 y 4rp 2 . q lra2 . Ž 16 . Ž . The relationship 16 is inconsistent with the well known Rayleigh result obtained in connection with
the analysis of the lensless image, in particular, at the pinhole camera w9,10x. According to Rayleigh quantities z, < R < and a are connected by the following equation, writing in ours notation, 1rz q 1r< R < s lra2 ,
Ž 17 .
which reflects only the phase properties of waves. The addendum 4rŽ p 2 < R <. in Eq. Ž16., complementing Ž17. is caused by the diffractive shift due to the change of interference wave amplitude. 4. The polygon apertures In the case of an arbitrary form aperture the diffraction shift of the image can also be described by the linear hyperbolic interpolation with three parameters by means of bringing the difference zmexact y zm of values to zero at two limiting points and the difference of derivatives at one of them, the same as it was done in the case of the circular aperture. We have carefully examined this method in Ref. w11x. In the particular case of the polygon aperture it is convenient to define the effective Fresnel numbers NP through the radius a 0 of the circumscribing circle: NP s a 02rl R, NP z s a 02rl z, z s NP z y NP . The diffractive optical power F PD of polygon aperture has the form similar Ž12.:
l F PD s
a 20
2 bP
P
(
.
2
NP y c P q Ž NP y c P . q 4 b P d P
Ž 18 . Values of parameters z 0 , b P , c P and d P was found as a result of the numerical analysis for Eq. Ž3. and are presented at the Table 1. Parameters values for the polygon apertures demonstrate the convergence
Table 1 Values of parameters for the circular aperture and for the polygon apertures l
z0
bP
cP
dP
` 8 6 5 4
1.000 1.106 1.192 1.282 1.463
12r p 2 s1.2159 1.4793 1.6965 1.9152 2.2797
0.0357 0.0393 0.0410 0.0376 y0.0230
1.2516 1.2452 1.2282 1.1940 1.0500
I.G. PalchikoÕa, S.G. Rautianr Optics Communications 174 (2000) 1–5
to the circular one as the variable l is increasing. In the case of the triangular aperture Ž l s 3. the validity of Eq. Ž18. require the additional analysis. In Fig. 1, curves 3 and 4 describe the difference zmexact y zm , multiplied by 100, for the exact value zmexact of the intensity maximum and the calculation zm with the approximate equation for l s 4 and l s 6. Note that the difference is positive Ž zmexact y zm ) 0. only for the square aperture Ž l s 4., and is negative for all other considered forms. The maximum error for the calculation of the image position by the approximate equation within the interval y0.5 - N - ` accounts for 0.19% of the focal tolerance for the circular aperture, 0.15% for the square one and 0.24% for the hexagonal one.
5
interpolation Eq. Ž18. for the optical power of the polygon aperture has been formed by the same algorithm as its analogues Ž12. for the circular aperture. The maximum calculation error for zm does not exceed f 0.2% with the proper choice for coefficients b P , c P , d P . The high precision of approximation convincingly confirms the correctness of our physical picture for the focusing action of an aperture. Calculations by equations offered at w4–6x result in significant errors. The basic disadvantage of these equations lies in the fact that their analytical properties essentially differ from the ones for exact solution in the interval of small N. The precision of our equation prove to be as over the precision of the equations offered at w4–6x as ten times.
5. Conclusion Acknowledgements Our study enabled us to extend the concept of the optical power of a pinhole, introduced by Lord Rayleigh for the investigation of the pinhole camera, to the case of the aperture of an arbitrary form and dimension. For the approximate analytical description of the diffraction shift phenomenon we used the linear hyperbolic interpolation with three parameters, reducing the difference of values zmexact y zm at two points to zero and the difference of derivatives at one of them. Main patterns of the image diffractive shift were predetermined by the factor 1 q n erz at common expression Ž3. for the field amplitude. Therefore in qualitative consideration they must be characteristic of the apertures with different forms. The analysis of the family of polygon shaped apertures clearly demonstrates this commonness of phenomenon. The
This research was carried out under Federal Directed Program ‘Integraciya’, grant N 274. References w1x H. Kogelnik, Bell Syst. Tech. J. 44 Ž1965. 465. w2x A.A. Isaev, M.A. Kazaryan, G.G. Petrash, S.G. Rautian, A.M. Shalagin, Kvantovaya Elektr. 2 Ž1975. 1125. w3x Y. Li, E. Wolf, Opt. Commun. 39 Ž1981. 211. w4x S. Szapiel, Opt. Lett. 8 Ž1983. 327. w5x Y. Li, Optik 69 Ž1984. 41. w6x Y. Li, J. Mod. Opt. 38 Ž1991. 1815. w7x X. Jiang, Q. Lin, S. Wang, Optik 97 Ž1994. 1. w8x W. Wang, E. Wolf, Opt. Commun. 119 Ž1995. 4539. w9x L. Rayleigh, On Pinhole Photography, Scientific Papers III, Cambridge University Press, 1902, p. 429. w10x R.W. Wood, Physical Optics, 3rd edn., Macmillan, New York, 1934. w11x S.G. Rautian, I.G. Palchikova, Opt. Spectrosc. 87 Ž1999. N3.
Optics Communications 174 Ž2000. 1–5 www.elsevier.comrlocateroptcom
The diffractive optical power of a diaphragm I.G. Palchikova a
a,b,)
, S.G. Rautian
a,b
Institute of Automation and Electrometry, Sib. Br. RAS. Prosp. Koptyug, 1, 630090, NoÕosibirsk, Russia b NoÕosibirsk State UniÕersity, PirogoÕa Street, 2, 630090, NoÕosibirsk, Russia Received 28 May 1999; received in revised form 21 October 1999; accepted 22 October 1999
Abstract The systematic analysis of the diffraction image shift enabled us to extend the notion of the optical power of a pinhole to the case of an aperture of an arbitrary form and any dimension. The mathematical expression for the diffractive optical power of an aperture has been obtained by the linear hyperbolic interpolation with three parameters. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Diffraction; Focal shift; Pinhole camera; Optical power
1. Introduction The focusing action of the diaphragm is connected with the well-known peculiarities of Fresnel diffraction pattern. According to the theory of the wave diffraction at the semi-plane, the light field diffuses into the area of geometrical shade. Its intensity increases as it approaches the shade boundary, but at the boundary the intensity reaches only a quarter of the geometrical optics value. The light stripes are observed in the lighting area, the brightest stripe is the nearest to the boundary, and is 0.86'l z Ž l is the wavelength, z denotes the distance from the aperture plane. distant from it. The maximum intensity is 1.37 in the same units. Qualitatively, these
) Corresponding author. Tel.: q7-3832-399-333; fax: q73832-354-851; e-mail: [email protected]
peculiarities are retained also under the diffraction at an aperture of any form and dimension, i.e. the light stripes fringe the shade boundary only if the distance to the first light stripe does not exceed half the diameter of the aperture, that is if the condition 'l z F a is fulfilled. Any finer structure in the transversal intensity distribution cannot exist because the condition l z ; a 2 means that there can be only one Fresnel zone on the aperture. Hence, when the first light stripe of Fresnel diffraction pattern comes onto the geometrical optics axis of the diffracted wave, intensity distribution at transversal section of the beam is unstructured. The intensity at the beam axis appears greater than its geometrical optics values, and the intensity distribution at the beam transversal section at the distance z s a 2rl becomes narrower than when the diffraction is not allowed for. This fact is interpreted as a focusing by the aperture and the definite optical power is attributed to the aperture. In its turn, the focusing apparently must result in the shift of the point source image.
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 4 5 - 8
I.G. PalchikoÕa, S.G. Rautianr Optics Communications 174 (2000) 1–5
2
Thus the focusing and the diffraction shift of image are inherent in the apertures of arbitrary forms. The phenomenon of diffraction pulling or diffraction shift of image Žfocus. had been studied in details for gaussian beams w1,2x and is at present being under the active investigation for spherical and cylindrical waves w3–8x. But up to now there has not been found simple approximate equation which would permit to calculate the diffraction focal shift for converging light wave with different Fresnel numbers N and any form of aperture. The use of the analytical properties of the exact solutions permitted us to propose the interpolation equation for the image shift. The precision of these equations proves to be as over the precision of the equations offered in Refs. w4–6x as ten times. In our work we discuss the concept of diffractive optical power of the aperture and offer sufficiently exact equations for the diffraction shift.
2. Fresnel diffraction at apertures of arbitrary forms For the quantitative analysis we shall consider the modulus of the field amplitude < EŽ0, z .< at the axis of a spherical wave with the radius of curvature R after its diffraction at the aperture. Unlike previous papers devoted to the focal shift, we refuse to characterise the effect by the relation Ž z y R .rR, because this quantity is sufficient if N 4 1 and is inadequate for our problem if N ; 1 and N < 1, and hence, is inconsistent with the physical statement of the general problem. We shall use the focal tolerance D z of spherical wave as a parameter with respect to which we shall estimate the value of the focal shift. In addition we put the co-ordinate origin at the centre of aperture and on analysing a Fresnel–Kirchhoff integral in parabolic approximation we use the effective Fresnel numbers n e and n e z , which we define with the help of the radius ae of circle of the same area S as that of the aperture: p ae2 s S,
n e s ae2rl R ,
n e z s ae2rl z ,
Ž 1.
the difference ne z y ne s z will serve as a non-dimensional variable.
Ž 2.
With this specific notation the modulus of the field amplitude < EŽ0, z .< can be written as < E Ž 0, z . < s Ž 1 q n erz . f Ž z . , f Ž z . s z < H exp Ž i pz P r 2 . d r < .
Ž 3.
Here r is the radius-vector at the aperture plane, normalized on ae . The co-ordinate z is measured from the centre of the aperture. The integral is evaluated over the surface of the aperture, using conventional normalization the geometrical optics quantity is < E < s 1. It can be seen the function < EŽ0, z .< explicitly depends on two parameters — n e and z , and unexplicitly — on the aperture form, which defines the domain of integration. Substantially, the function f Ž z . depends only on z and the form of the aperture, and explicitly it does not depend on the radius of curvature R for the incident wave front. In a sense f Ž z . is universal, it equally describes the diffraction of plane, converging and diverging waves. The quantity R begins to influence f Ž z . only after the transition from z to variable z by Eqs. Ž1. and Ž2.. With variables n e , z the field amplitude depends on R only over the factor 1 q n erz . The phenomenon of the diffraction shift for image is connected just with the factor 1 q n erz .
3. The circular aperture In the case of the circular aperture with the radius a the intensity distribution < EŽ0, zrR .< 2 s I Ž0, zrR . along the axis for the diffracted spherical wave is well known w3x and it can be written as Eq. Ž3., if the traditional Fresnel numbers for the centre of the wave front curvature Ž N . and for the current point z Ž Nz . are introduced instead of the effective ones: < E Ž 0, z . < s Ž 1 q Nrz . <2 sin Ž pzr2 . < , f Ž z . s <2 sin Ž pzr2 . < , N s a 2rl R ,
Nz s a 2rl z ,
Ž 4. z s Nz y N.
The conclusion on focusing action of the aperture immediately follows from the ratio Ž4.. In the case of a plane incident wave Ž N s 0. the main maximum for amplitude is determined by the maximum for the function f Ž z ., which appears first as z increases up
I.G. PalchikoÕa, S.G. Rautianr Optics Communications 174 (2000) 1–5
to the value z ' z 0 s Nz s 1 Žor z decreases from the infinity to the value z s z 0 s a2rl.. At this point < EŽ0, z .< s 2, i.e. it is twice the value obtained from the geometrical optics, and the intensity is four times, and the transversal distribution has approximately half the width determined under the geometrical optics. This fact means only diffraction focusing. The maximum intensity position is determined by roots for the transcendental equation w3x:
ž
tan Ž u . s u 1 q
2u pN
/
,
u s pzr2.
Ž 5.
Due to the periodicity of tanŽ u. Eq. Ž5. has the infinite number of solutions, answering the maxima for the amplitude modulus. We are interested in the main maximum, which corresponds to the root, placed within the interval 0 - z F 1. We shall denote this root zm . Roots for Eq. Ž5. depend on N and the main question discussed in the paper consists in clearing up the nature of dependence zm s zm Ž N .. The explicit form for the inverse function N Ž zm . follows immediately from Eq. Ž5.: N Ž zm . s pzm2r 2 tan Ž pzm r2 . y pzm r2 4 .
Ž 6.
The calculation N Ž zm . with sufficiently small step allows easily to find zm Ž N . by the linear interpolation. Let single out two limiting cases i.e. the small diffraction shift when zm < 1Ž N 4 1., and the case of the great diffraction shift when zm is placed nearby z 0 and the inequality 1 y zm < 1 Ž N < 1. is fulfilled. The behaviour of N Ž zm . under the limiting cases can be easily deduced from the ratio Ž6., if tanŽ pzm r2. is replaced by the corresponding power expansion: N s 12r Ž p 2zm . ,
zm < 1;
N s p 2 Ž 1 y zm . r4,
1 y zm < 1.
3
which, instead of the trigonometric functions included in the ratio Ž6., the power series are used: N s yd zm q c q
b
.
zm
Ž 8.
The coefficients b, c, d are defined according to the limiting values Ž7.: b s 12rp 2 s 1.21585, d s p 2r4 y b s 1.25155, c s d y b s 0.03570.
Ž 9.
Eq. Ž8. allows one to get the analytical expression for the function zmŽ N .: 2b
zm s
(
.
2
Ž 10 .
N y c q Ž N y c . q 4 bd Usually the absolute value of the image position uncertainty d z is not important, but its ratio to the focal tolerance D z Žto the length of the beam waist. is essential: D zs
lz2 a
2
z s Nz
,
dz Dz
s Nz
dz z
s ydz .
Ž 11 .
Hence, the accuracy of result zm Ž N . is of the immediate interest. The results of the numerical computation are presented in Fig. 1: curve 1 is a plot of zmŽ N ., calculated according to the Eq. Ž6.; the approximated values zmŽ N . are calculated from Eq. Ž10. and is no different from the curve 1 at scale of Fig. 1. The difference between the results of the first and second calculations have been shown at the bottom of the Fig. 1 Žcurve 2., increased 100 times.
Ž 7.
The result Ž7. coincides with that derived in Ref. w3x and Ref. w7x by another method. For the analytical description of the dependence zm on N, we propose the approximating equation which is based on the asymptotic values Ž7., and in
Fig. 1. Curve 1 is the plot of the function zm Ž N ., calculated from the exact solution Ž6. and from the approximating Eq. Ž10. for the circular aperture. Curves 2, 3, and 4 describe the differences zmexact y zm , increased 100 times, for the circular, the square Ž l s 4. and hexagonal Ž l s6. apertures accordingly.
I.G. PalchikoÕa, S.G. Rautianr Optics Communications 174 (2000) 1–5
4
The maximum error of the expression Ž10. takes place at N s 2 and equals to f 0.0019. In other words, the error for the calculating image position from Eq. Ž10. does not exceed 0.19% of the focal tolerance within the interval ` ) N G y0.5, that is more than enough in many cases Žthe values N - 0 correspond to the divergent wave.. Mark some physical conclusions following from Eq. Ž10.. Turn into the non-dimensional variables and write Eq. Ž10. as 1 1 s qF D , z R l 2b FDs 2 . Ž 12 . a N y c q Ž N y c . 2 q 4 bd
(
The relation Ž12. has the form of a lens equation, and furthermore the value F D can be considered the optical power of the aperture due to the diffraction by origin. In addition to the factor lra2 it contains the additional factor describing the dependence F D on N or on the wave front curvature 1rR. In limiting cases N 4 1 and N < 1 we find from Eqs. Ž12. and Ž7.: 2
F D s l br Ž Na2 . s 12 R Ž lr Ž p a2 . . ,
N 4 1,
Ž 13 . F D Ž 1 y 4 Nrp 2 . lra2 , N < 1. Ž 14 . In Eq. Ž13. the image diffraction shift d z s z m y R from the wave front curvature centre is a fraction of the focal tolerance D z: < d z < s R 2F D s 12D zr Ž p 2 N . , D z s l R 2ra2 , Ž 15 . for example it gives < d z < f 0.12D z if N s 10. The expression Ž14. complies with a weakly converging wave when its curvature centre is far away in Fraunhofer zone Ž R 4 a 2rl. and the focusing action due to the curvature of the initial wave front is weaker than the pure diffraction focusing. The following fact mentioned in w7x has engaged our attention: diminishing of the optical power F D as increasing 1rR seems to reduce the summary part of ‘ray focusing’: 1rz s Ž 1rR . Ž 1 y 4rp 2 . q lra2 . Ž 16 . Ž . The relationship 16 is inconsistent with the well known Rayleigh result obtained in connection with
the analysis of the lensless image, in particular, at the pinhole camera w9,10x. According to Rayleigh quantities z, < R < and a are connected by the following equation, writing in ours notation, 1rz q 1r< R < s lra2 ,
Ž 17 .
which reflects only the phase properties of waves. The addendum 4rŽ p 2 < R <. in Eq. Ž16., complementing Ž17. is caused by the diffractive shift due to the change of interference wave amplitude. 4. The polygon apertures In the case of an arbitrary form aperture the diffraction shift of the image can also be described by the linear hyperbolic interpolation with three parameters by means of bringing the difference zmexact y zm of values to zero at two limiting points and the difference of derivatives at one of them, the same as it was done in the case of the circular aperture. We have carefully examined this method in Ref. w11x. In the particular case of the polygon aperture it is convenient to define the effective Fresnel numbers NP through the radius a 0 of the circumscribing circle: NP s a 02rl R, NP z s a 02rl z, z s NP z y NP . The diffractive optical power F PD of polygon aperture has the form similar Ž12.:
l F PD s
a 20
2 bP
P
(
.
2
NP y c P q Ž NP y c P . q 4 b P d P
Ž 18 . Values of parameters z 0 , b P , c P and d P was found as a result of the numerical analysis for Eq. Ž3. and are presented at the Table 1. Parameters values for the polygon apertures demonstrate the convergence
Table 1 Values of parameters for the circular aperture and for the polygon apertures l
z0
bP
cP
dP
` 8 6 5 4
1.000 1.106 1.192 1.282 1.463
12r p 2 s1.2159 1.4793 1.6965 1.9152 2.2797
0.0357 0.0393 0.0410 0.0376 y0.0230
1.2516 1.2452 1.2282 1.1940 1.0500
I.G. PalchikoÕa, S.G. Rautianr Optics Communications 174 (2000) 1–5
to the circular one as the variable l is increasing. In the case of the triangular aperture Ž l s 3. the validity of Eq. Ž18. require the additional analysis. In Fig. 1, curves 3 and 4 describe the difference zmexact y zm , multiplied by 100, for the exact value zmexact of the intensity maximum and the calculation zm with the approximate equation for l s 4 and l s 6. Note that the difference is positive Ž zmexact y zm ) 0. only for the square aperture Ž l s 4., and is negative for all other considered forms. The maximum error for the calculation of the image position by the approximate equation within the interval y0.5 - N - ` accounts for 0.19% of the focal tolerance for the circular aperture, 0.15% for the square one and 0.24% for the hexagonal one.
5
interpolation Eq. Ž18. for the optical power of the polygon aperture has been formed by the same algorithm as its analogues Ž12. for the circular aperture. The maximum calculation error for zm does not exceed f 0.2% with the proper choice for coefficients b P , c P , d P . The high precision of approximation convincingly confirms the correctness of our physical picture for the focusing action of an aperture. Calculations by equations offered at w4–6x result in significant errors. The basic disadvantage of these equations lies in the fact that their analytical properties essentially differ from the ones for exact solution in the interval of small N. The precision of our equation prove to be as over the precision of the equations offered at w4–6x as ten times.
5. Conclusion Acknowledgements Our study enabled us to extend the concept of the optical power of a pinhole, introduced by Lord Rayleigh for the investigation of the pinhole camera, to the case of the aperture of an arbitrary form and dimension. For the approximate analytical description of the diffraction shift phenomenon we used the linear hyperbolic interpolation with three parameters, reducing the difference of values zmexact y zm at two points to zero and the difference of derivatives at one of them. Main patterns of the image diffractive shift were predetermined by the factor 1 q n erz at common expression Ž3. for the field amplitude. Therefore in qualitative consideration they must be characteristic of the apertures with different forms. The analysis of the family of polygon shaped apertures clearly demonstrates this commonness of phenomenon. The
This research was carried out under Federal Directed Program ‘Integraciya’, grant N 274. References w1x H. Kogelnik, Bell Syst. Tech. J. 44 Ž1965. 465. w2x A.A. Isaev, M.A. Kazaryan, G.G. Petrash, S.G. Rautian, A.M. Shalagin, Kvantovaya Elektr. 2 Ž1975. 1125. w3x Y. Li, E. Wolf, Opt. Commun. 39 Ž1981. 211. w4x S. Szapiel, Opt. Lett. 8 Ž1983. 327. w5x Y. Li, Optik 69 Ž1984. 41. w6x Y. Li, J. Mod. Opt. 38 Ž1991. 1815. w7x X. Jiang, Q. Lin, S. Wang, Optik 97 Ž1994. 1. w8x W. Wang, E. Wolf, Opt. Commun. 119 Ž1995. 4539. w9x L. Rayleigh, On Pinhole Photography, Scientific Papers III, Cambridge University Press, 1902, p. 429. w10x R.W. Wood, Physical Optics, 3rd edn., Macmillan, New York, 1934. w11x S.G. Rautian, I.G. Palchikova, Opt. Spectrosc. 87 Ž1999. N3.