- Email: [email protected]
Diffraction criterion for a slit under spherical illumination
Optics Communications 274 (2007) 32–36 www.elsevier.com/locate/optcom
Diffraction criterion for a slit under spherical illumination Edgar A. Rueda *, Francisco F. Medina, John F. Barrera ´ ptica y Foto´nica, Instituto de Fı´sica, Universidad de Antioquia, A.A. 1226 Medellı´n, Colombia Grupo de O Received 14 July 2006; received in revised form 29 January 2007; accepted 2 February 2007
Abstract A quantitative criterion to determine the type of diffraction produce by a slit, which is an aperture that cannot be circumscribed, is defined. The slit is illuminated with a spherical wave that eliminates the alignment problems produced by filamentary light sources. The determination of the first minimum of intensity in the diffraction pattern was used to define the criterion that differentiates between Fraunhofer and Fresnel diffraction. The criterion is defined in terms of a parameter c, which is connected to the number of Fresnel zones. Simulations are used to define the criterion and experimental results are presented to corroborate it. 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Fx Keywords: Fresnel diffraction; Fraunhofer diffraction; Fresnel zones
1. Introduction Christian Huygens, in 1678, formulated the principle of the wave theory of light, which explains in a qualitative form the interference phenomena between two or more point light sources. Later, in 1818, Fresnel presented a mathematical formulation for the Huygens principle, now called the Huygens–Fresnel principle, which takes into account interference and diffraction phenomena. Finally, in 1882, Kirchhoff enunciated some solid mathematical bases for the Huygens–Fresnel principle, and in his theory of diffraction presented the diffraction integral of Fresnel– Kirchhoff for the amplitude of the luminous field along the optical axis for a wave diffracted by an aperture. This integral, after considering the distances between the source and the observation screen to the aperture as greater than the dimensions of the aperture, and observing the value of the amplitude only for points close to the optical axis, can be rewritten with an exponential integrand that is proportional to the term f(n, g), where n and g are the aperture
*
Corresponding author. Tel.: +57 4 2105630; fax: +57 4 2105666. E-mail address: [email protected] (E.A. Rueda).
0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.02.009
coordinates [1], and where f(n, g) is a power series. When second and higher orders terms of f(n, g) can be neglected the approximation is called the Fraunhofer diffraction, and when the terms of second-order cannot be neglected it is called the Fresnel diffraction. The Fraunhofer diffraction, less rigorous than Fresnel, when valid simplifies the calculations enormously by presenting the value of the amplitude of the field, except for one factor, as the Fourier transform of the amplitude distribution in the aperture [2], which makes it a favorite in those applications where it is valid, for example, the location of the maximum of intensity in a diffraction slit, useful in spectroscopy [1], the location of minimums of intensity produced in image forming systems to calculate its power of resolution [1], to overcome resolution limits, because of the pixel size of two dimensional detectors, moving the detector to the far field where the pattern can be observed [3]. Nevertheless, in certain applications it is necessary to work with a better approximation like Fresnel, for example, in the optical image formation for proximity printing in X-ray lithography [4], in the description of the diffraction of an off-resonant probe beam to observe weakly absorbing cold atoms in phase contrast [5], in the observation of minute details in simple structures without any kind
E.A. Rueda et al. / Optics Communications 274 (2007) 32–36
of mechanical contact between the diffracting structure and the recording media [6]. All this generates the necessity of establishing some useful criterion that limits the region of applicability of both approximations for every particular case, and that preferably depends on the optical setup parameters: the aperture geometry, the light wavelength, the source-aperture and aperture-screen distance, etc. Criteria to validate the Fresnel approximation has been treated in text books like Goodman’s [2] and in articles like Mezouari and Harvey [7] and Steane and Rutt [8] for all kinds of aperture geometries able to be circumscribed and which diffract plane and spherical wavefronts. To distinguish the limit between Fresnel and Fraunhofer domains several criteria have been proposed by different textbooks authors, some restrict the validity of the Fraunhofer diffraction when there is a great distance between the aperture and the screen [9–11], others present qualitative criteria where they basically restrict the geometry of the optical setup [12,13], and a third group defines quantitative criteria about the optical setup geometry [1,2,14]. Among the articles found for the determination of the validity of the Fraunhofer diffraction is Mezouari and Harvey [7], where they obtain a criterion based on the electric field amplitude error for apertures which can be circumscribed; Medina [15] presents a criterion based on the number of Fresnel zones observed through the aperture and limited to situations where the spherical wavefront illuminates an aperture with constant pupil function and with comparable maximal and minimal dimensions, i.e., for apertures which can be circumscribed; a recent work proposed a criterion based on the number of Fresnel zones observed through an aperture with an irregular geometry if those irregularities can be encircled by two circles not differing by more than 10% [16]. Nevertheless, the apertures that have some infinite dimension, for example a slit, cannot be circumscribed and make the application of the criterion unreliable. There is a qualitative criterion that speaks of the validity of the Fraunhofer diffraction for a slit illuminated with a plane wave whenever the slit is very narrow and the screen is very far from the slit [10], because it is qualitative its use is very limited. The objective of this work is to define a new criterion for slits illuminated with a spherical wave, based on the identification of the first minimum of diffraction. The criterion will be given in terms of a parameter c, based on the solution given by the Fresnel approximation to the diffraction produced by a slit on a spherical wave, and then connected to the equivalent number of Fresnel zones observed through the slit. 2. Diffraction by a slit Kirchhoff’s theory of diffraction says that the amplitude at point P of a spherical wave centered on P0, vertex on O and radius q (Fig. 1), diffracted by a slit of width b, under paraxial approximation, is [14]
33
y y0 Q
x
P0
r P O
x0
r0’ r0 z
b
Fig. 1. Spherical wave centered on P0, vertex on O and radius q, diffracted by a slit of width b, in plane R.
U/
Z Z
0
f ðx; yÞ
R
eikðq þrÞ dxdy; q0 r
ð1Þ
where k = 2p/k and f(x, y) is the aperture function, and it is separable in the spatial coordinates of the aperture. Thus, f(x, y) = h(x)g(y) and ( 1; b 6 x 6 b2 ; 2 hðxÞ ¼ 0; jxj > b2 ; ð2Þ gðyÞ ¼ 1: Because the limits of integration are very small, the amplitude of the spherical wave over the slit does not change appreciably, in the denominator of the integral q 0 ! q and r ! r00 , thus Z 1 Z b2 0 U/ eikðq þrÞ dxdy: ð3Þ 1
b 2
From Fig. 1, q02 ¼ x2 þ y 2 þ q2 ; 2
2
r2 ¼ ðx x0 Þ þ ðy y 0 Þ þ r20 ;
ð4Þ
using the binomial expansion for the Fresnel approximation q0 ¼ q þ
x2 þ y 2 ; 2q 2
2
ðx x0 Þ þ ðy y 0 Þ r ¼ r0 þ ; 2r0 and replacing Eq. (5) in Eq. (3) Z 1 Z b2 x2 þ y 2 exp ik qþ U/ 2q 1 b 2 !)# 2 2 ðx x0 Þ þ ðy y 0 Þ þ r0 þ dxdy: 2r0
ð5Þ
ð6Þ
Reorganizing Eq. (6) with regard to the separable coordinates
34
E.A. Rueda et al. / Optics Communications 274 (2007) 32–36
ik 2 2yy 0 z y 20 z y þ dy 2z r0 r0 1 ik 2 2xx0 z x20 z x exp þ dx; ð7Þ 2z r0 r0
U / exp½ikðq þ r0 Þ
Z
b 2
b 2
Z
1
exp
qr0 . The first integral is a constant due to its where z ¼ qþr 0 limits of integration. The second integral can be approximated to hthe icase of Fraunhofer diffraction if the parabolic 2 term exp ikx can be omitted. Knowing that the maximal 2z value the parabolic term can reach is when rffiffiffiffiffiffiffiffiffiffiffiffiffi2 kx2 pb2 p q þ r0 b ¼ ¼ ; ð8Þ 2z 4zk 4 qr0 k
and defining a parameter rffiffiffiffiffiffiffiffiffiffiffiffiffi q þ r0 c¼b ; qr0 k
counterpart was carried out with the setup shown in Fig. 3. A laser beam was expanded using a convergent lens to create a spherical wavefront that eliminates any dependence of the diffraction pattern with regard to the aperture-screen distance when there is a relative rotation with regard to the optical axis between the source and the slit, a problem that appears when using filament sources [17]. This wavefront was then diffracted by a slit and the corresponding diffraction pattern was captured using a CCD camera and Matrox inspector software; Matlab was used to process the images. The results of the simulation and the experimental counterpart are presented in Fig. 4. The experimental results were obtained by using the same parameters as those employed in the computer simulations.
ð9Þ
for a c value small enough the parabolic term can be omitted. But, for practical purposes it is important to be more specific and it becomes necessary to find experimentally a threshold where the parabolic term can be omitted. Then, a criterion in terms of the parameter c will be defined to distinguish between the Fraunhofer and Fresnel diffraction. This parameter depends on the geometry of the optical setup (q, r0), wavelength k and slit width b. 3. Simulation, experiment and criterion
Laser
Lens Slit CCD
Fig. 3. Experimental setup.
The definition of the criterion is closely related to the determination of the first minimum of the diffraction pattern [16]. In Fraunhofer diffraction the first minimum of the pattern corresponds to a null intensity (Fig. 2), thus, to define the criterion some simulations for different values of c have to be carried out until the first null minimum is found. Simulations of the diffraction produced on a spherical wave by a slit were carried out for different values of c (Eq. (9)). The slit’s width is b = 0.3 mm and the wavelength is k = 632.8 nm. The numerical matrix has 2048 · 2048 pixels with a sampling frequency of 9 lm · 9 lm. The experimental
Fig. 2. Fraunhofer diffraction pattern of a slit (Fourier transform of a slit). b is the slit’s width, k ¼ 2p , k is the wavelength, and h is the angle k made by the optical axis and a line from the slit to the observation point.
Fig. 4. Diffraction pattern for a spherical monochromatic wave of wavelength k = 632.8 nm, diffracted by a slit of width 0.3 mm. For each image the left side corresponds to the experimental result and the right side to the simulated result, for a c value: (a) 1.7, (b) 1.5, (c) 1.4, (d) 1.0, (e) 0.8 and (f) 0.7.
E.A. Rueda et al. / Optics Communications 274 (2007) 32–36
Fig. 4a–c, with c values: 1.7, 1.5 and 1.4, present a not null first minimum which indicates a Fresnel diffraction pattern; Fig. 4d, with c value: 1.0, although very close to null intensity present a not null first minimum; Fig. 4e and f, with c values: 0.8 and 0.7, have a null first minimum belonging to Fraunhofer diffraction, where 0.8 is the first one with a null minimum. From these results, a criterion based on parameter c is proposed to determine the validity of the Fraunhofer approximation when a spherical wave is diffracted by a slit. The criterion in terms of parameter c is (Eq. (9)) c 6 0:8 Fraunhofer domain; c > 0:8 Fresnel domain: 4. Criterion in terms of the number of Fresnel zones The criteria to distinguish between Fraunhofer and Fresnel diffraction, which appears in the literature, are directly or easily expressed in terms of the number of Fresnel zones seen through the aperture. Medina et al. [16] present many of these criteria in terms of the number of Fresnel zones, for comparison purposes. To allow the comparison with other criteria, a proportionality between the parameter c and the equivalent number of Fresnel zones seen through the slit is found. For a circular aperture as the one in Fig. 5a or with apertures of different geometries that satisfy the condition of circumscription as in Fig. 5b, where the concentric circles represent the different Fresnel zones over the aperture and where the white region is the aperture, illuminated with a spherical monochromatic wave, usually only the Fresnel zone that limits with the aperture edge is observed partially, making possible to neglect it or to completely consider it depending on the criterion used for the calculation of the number of Fresnel zones being observed. Moreover, in the majority of cases of interest for the determination of the domain of diffraction only the first zone is observed partially, eliminating any contribution from the other zones. Nevertheless, in the case of a slit, Fig. 5c, the problem is very different because now a great quantity of Fresnel zones segments are seen. Thus, it is important to calculate
35
the contribution of each segment to the equivalent number of Fresnel zones observed through the slit. In Fig. 6a a spherical wave of radius q is observed; generated by a point source in P and vertex in O, diffracted by a slit of width b and whose pattern of diffraction is centered on P 0 . The Fresnel zones are defined as the ringshaped regions in which the spherical wavefront centered on P is divided and whose frontiers correspond to the intersections of the wavefront with a series of spheres centered on P 0 and of radii r0 + lk/2, where l = 0, 1, 2, . . . and k is the wavelength. For representing the zones in the plane of the slit q b and r0 b are defined, this is known as the paraxial approximation. In the plane of the slit the l + 1 Fresnel zone is drawn with the segment observed through the slit lined. From Fig. 6a and the parabolic approximation [15] we know that the radius hl of the zone l is: sffiffiffiffiffiffiffiffiffiffiffiffiffi qr0 kl hl : ð10Þ q þ r0 Defining ARl 2hlb and AR(l+1) 2hl+1b as the rectangular areas of the Fig. 6b, the difference between these areas is A0Rðlþ1Þ 2bðhlþ1 hl Þ:
ð11Þ
Approximating the area of the zone segment observed through the slit to the area calculated in Eq. (11), a valid supposition under paraxial approximation, and taking into account the p difference in phase between consecutive Fresnel zones, it is possible to say that the total effective area of the m zone segments observed through the slit that contributes to the diffraction pattern in the central point of the observation plane, is equal to the sum of the areas of all zone segments but with a change in sign between consecutive zone segments, thus the total effective area is m X
A0Rl ffi 2bh1 2bðh2 h1 Þ þ 2bðh3 h2 Þ þ ð1Þmþ1 2bðhm hm1 Þ " ! # mþ1 m1 X ð1Þ lþ1 hm ; ffi 4b ð1Þ hl þ 2 l¼1
ð12Þ
b b ρ
P ρ
hl+1
hl
r0+(l+1)λ /2 r0+lλ /2
2hl
O
2hl+1
r0 P’
Fig. 5. Fresnel zones seen through: (a) circular aperture, (b) complex geometry aperture and (c) slit. The concentric circles represent the different Fresnel zones over the aperture and the white region represents the aperture.
Fig. 6. (a) l + 1 Fresnel zone of the spherical wave centered on P, vertex on O and of radius q, diffracted by a slit of width b. The shaded segment is the part of the zone observed through the slit from point P 0 at a distance r0. (b) Comparison of the area of the lined segment zone with the area generated by two rectangles.
36
E.A. Rueda et al. / Optics Communications 274 (2007) 32–36
and because the area of the first Fresnel zone is A1 ¼ ph21 , dividing Eq. (12) by this area, the equivalent number of Fresnel zones due to the presence of the m zone segments is # ! rffiffiffiffiffiffiffiffiffiffiffiffiffi" m1 pffiffi 4b q þ r0 X ð1Þmþ1 pffiffiffiffi lþ1 m m : l þ Nf ð1Þ p qr0 k 2 l¼1 ð13Þ This equivalent number of zones depends on the geometry of the optical setup, wavelength k and slit width b. It is proportional to the slit width and inversely proportional to the radius of the first Fresnel zone, where the constant of proportionality depends explicitly on the number of Fresnel zones observed through the slit. To maintain a small error in the approximation and because in Eq. (13) it is considered that only a segment of the first zone is seen, Eq. (13) is valid only for values less than one. This is not a serious restriction because seeing the first zone completely means that the diffraction pattern should be described with the Fresnel approximation [15]. If now a infinite number of zones are observed through the slit (m ! 1), a condition that is approximately satisfied in the vast majority of problems of interest, the term " # ! mþ1 m1 X pffiffi pffiffiffiffi ð1Þ lþ1 4 m l þ ð1Þ ð14Þ 2 l¼1 tends to 1.52, and the equivalent number of Fresnel zones can be written as rffiffiffiffiffiffiffiffiffiffiffiffiffi 1:52b q þ r0 : ð15Þ Nf p qr0 k The equivalent number of Fresnel zones is then proportional to parameter c (Eq. (9), N f 0:5c;
ð16Þ
and the criterion in terms of the equivalent number of Fresnel zones is N f 6 0:4
Fraunhofer domain;
N f > 0:4
Fresnel domain:
5. Conclusions In all the optical applications where the diffraction pattern is crucial in order to achieve an accurate result, knowing the type of diffraction approximation that must be used is of extreme importance. Thus, a set of criteria for different types of apertures in terms of a variable that depends on the optical setup parameters is of great utility.
To distinguish between Fraunhofer and Fresnel diffraction in the diffraction pattern produced by a slit under spherical illumination, a criterion for optical setups, in terms of a parameter c defined in Eq. (9), considers c 6 0.8 for Fraunhofer diffraction and c > 0.8 for Fresnel diffraction, and in terms of the equivalent number of Fresnel zones defined in Eq. (15), considers Nf 6 0.40 for Fraunhofer diffraction and Nf > 0.40 for Fresnel diffraction. To define a criterion for the validity of the Fraunhofer and Fresnel approximations based on the number of Fresnel zones seen through an aperture that cannot be circumscribed, i.e., that has one infinite dimension, one must take into account the contribution of all the zone segments that are observed through the aperture. Acknowledgement The grants from COLCIENCIAS (Colombia) and CODI-Universidad de Antioquia (Colombia) are gratefully acknowledged. The authors thanks the reviewers for their constructive comments that helped to improve this work. References [1] M. Born, E. Wolf, Principles of Optics, seventh ed., Cambrigde University Press, 2002, p. 446. [2] J. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1996, p. 73. [3] Q. Shen, D.H. Bilderback, K.V. Finkelstein, I.V. Basarov, S.M. Gruner, J. Phys. IV France 104 (2003) 21. [4] Y. Vladimirsky, Lithography vacuum ultraviolet spectroscopy II, in: J.A. Samson, D.L. Ederer (Eds.), Experimental methods in the physical sciences, vol. 32, ch. 10, New York, 1998, p. 205. [5] L. Turner, K. Weber, D. Paganin, R. Scholten, Opt. Lett. 29 (2004) 232. [6] J. Garcia-Sucerquia, F.F. Medina, J.F. Barrera, Opt. Commun. 253 (2005) 250. [7] S. Mezouari, A. Harvey, J. Opt. A: Pure Appl. Opt. 5 (2003) 86. [8] A. Steane, H. Rutt, J. Opt. Soc. Am. A 6 (1989) 1809. [9] M. Alonso, E. Finn, University Physics, Addison-Wesley, Palo Alto, 1967, p. 902. [10] D. Halliday, R. Resnick, Physics for Students of Science and Engineering, Part II, John Wiley and Sons, New York, 1960, p. 925. [11] F.A. Jenkins, H.E. White, Fundamentals of Optics, International Ed, McGraw-Hill, Auckland, 1981, p. 315. [12] B. Rossi, Optics, Addison-Wesley, Palo Alto, 1965, p. 177. [13] F.W. Sears, Optics, third ed., Addison-Wesley, London, 1962. [14] R.D. Guenther, Modern Optics, John Wiley and Sons, New York, 1990, p. 364. [15] F. F Medina, Revista Mexicana de Fı´sica 31 (1985) 311. [16] F.F. Medina, J. Garcia-Sucerquia, R. Castan˜eda, G. Matteucci, Optik 115 (2004) 547. [17] M.E. Mancen˜ido, G. Pozzi, L. Zunino, M. Garavaglia, J. Opt. Soc. Am. A 11 (1999) 2767.
Diffraction criterion for a slit under spherical illumination Edgar A. Rueda *, Francisco F. Medina, John F. Barrera ´ ptica y Foto´nica, Instituto de Fı´sica, Universidad de Antioquia, A.A. 1226 Medellı´n, Colombia Grupo de O Received 14 July 2006; received in revised form 29 January 2007; accepted 2 February 2007
Abstract A quantitative criterion to determine the type of diffraction produce by a slit, which is an aperture that cannot be circumscribed, is defined. The slit is illuminated with a spherical wave that eliminates the alignment problems produced by filamentary light sources. The determination of the first minimum of intensity in the diffraction pattern was used to define the criterion that differentiates between Fraunhofer and Fresnel diffraction. The criterion is defined in terms of a parameter c, which is connected to the number of Fresnel zones. Simulations are used to define the criterion and experimental results are presented to corroborate it. 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Fx Keywords: Fresnel diffraction; Fraunhofer diffraction; Fresnel zones
1. Introduction Christian Huygens, in 1678, formulated the principle of the wave theory of light, which explains in a qualitative form the interference phenomena between two or more point light sources. Later, in 1818, Fresnel presented a mathematical formulation for the Huygens principle, now called the Huygens–Fresnel principle, which takes into account interference and diffraction phenomena. Finally, in 1882, Kirchhoff enunciated some solid mathematical bases for the Huygens–Fresnel principle, and in his theory of diffraction presented the diffraction integral of Fresnel– Kirchhoff for the amplitude of the luminous field along the optical axis for a wave diffracted by an aperture. This integral, after considering the distances between the source and the observation screen to the aperture as greater than the dimensions of the aperture, and observing the value of the amplitude only for points close to the optical axis, can be rewritten with an exponential integrand that is proportional to the term f(n, g), where n and g are the aperture
*
Corresponding author. Tel.: +57 4 2105630; fax: +57 4 2105666. E-mail address: [email protected] (E.A. Rueda).
0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.02.009
coordinates [1], and where f(n, g) is a power series. When second and higher orders terms of f(n, g) can be neglected the approximation is called the Fraunhofer diffraction, and when the terms of second-order cannot be neglected it is called the Fresnel diffraction. The Fraunhofer diffraction, less rigorous than Fresnel, when valid simplifies the calculations enormously by presenting the value of the amplitude of the field, except for one factor, as the Fourier transform of the amplitude distribution in the aperture [2], which makes it a favorite in those applications where it is valid, for example, the location of the maximum of intensity in a diffraction slit, useful in spectroscopy [1], the location of minimums of intensity produced in image forming systems to calculate its power of resolution [1], to overcome resolution limits, because of the pixel size of two dimensional detectors, moving the detector to the far field where the pattern can be observed [3]. Nevertheless, in certain applications it is necessary to work with a better approximation like Fresnel, for example, in the optical image formation for proximity printing in X-ray lithography [4], in the description of the diffraction of an off-resonant probe beam to observe weakly absorbing cold atoms in phase contrast [5], in the observation of minute details in simple structures without any kind
E.A. Rueda et al. / Optics Communications 274 (2007) 32–36
of mechanical contact between the diffracting structure and the recording media [6]. All this generates the necessity of establishing some useful criterion that limits the region of applicability of both approximations for every particular case, and that preferably depends on the optical setup parameters: the aperture geometry, the light wavelength, the source-aperture and aperture-screen distance, etc. Criteria to validate the Fresnel approximation has been treated in text books like Goodman’s [2] and in articles like Mezouari and Harvey [7] and Steane and Rutt [8] for all kinds of aperture geometries able to be circumscribed and which diffract plane and spherical wavefronts. To distinguish the limit between Fresnel and Fraunhofer domains several criteria have been proposed by different textbooks authors, some restrict the validity of the Fraunhofer diffraction when there is a great distance between the aperture and the screen [9–11], others present qualitative criteria where they basically restrict the geometry of the optical setup [12,13], and a third group defines quantitative criteria about the optical setup geometry [1,2,14]. Among the articles found for the determination of the validity of the Fraunhofer diffraction is Mezouari and Harvey [7], where they obtain a criterion based on the electric field amplitude error for apertures which can be circumscribed; Medina [15] presents a criterion based on the number of Fresnel zones observed through the aperture and limited to situations where the spherical wavefront illuminates an aperture with constant pupil function and with comparable maximal and minimal dimensions, i.e., for apertures which can be circumscribed; a recent work proposed a criterion based on the number of Fresnel zones observed through an aperture with an irregular geometry if those irregularities can be encircled by two circles not differing by more than 10% [16]. Nevertheless, the apertures that have some infinite dimension, for example a slit, cannot be circumscribed and make the application of the criterion unreliable. There is a qualitative criterion that speaks of the validity of the Fraunhofer diffraction for a slit illuminated with a plane wave whenever the slit is very narrow and the screen is very far from the slit [10], because it is qualitative its use is very limited. The objective of this work is to define a new criterion for slits illuminated with a spherical wave, based on the identification of the first minimum of diffraction. The criterion will be given in terms of a parameter c, based on the solution given by the Fresnel approximation to the diffraction produced by a slit on a spherical wave, and then connected to the equivalent number of Fresnel zones observed through the slit. 2. Diffraction by a slit Kirchhoff’s theory of diffraction says that the amplitude at point P of a spherical wave centered on P0, vertex on O and radius q (Fig. 1), diffracted by a slit of width b, under paraxial approximation, is [14]
33
y y0 Q
x
P0
r P O
x0
r0’ r0 z
b
Fig. 1. Spherical wave centered on P0, vertex on O and radius q, diffracted by a slit of width b, in plane R.
U/
Z Z
0
f ðx; yÞ
R
eikðq þrÞ dxdy; q0 r
ð1Þ
where k = 2p/k and f(x, y) is the aperture function, and it is separable in the spatial coordinates of the aperture. Thus, f(x, y) = h(x)g(y) and ( 1; b 6 x 6 b2 ; 2 hðxÞ ¼ 0; jxj > b2 ; ð2Þ gðyÞ ¼ 1: Because the limits of integration are very small, the amplitude of the spherical wave over the slit does not change appreciably, in the denominator of the integral q 0 ! q and r ! r00 , thus Z 1 Z b2 0 U/ eikðq þrÞ dxdy: ð3Þ 1
b 2
From Fig. 1, q02 ¼ x2 þ y 2 þ q2 ; 2
2
r2 ¼ ðx x0 Þ þ ðy y 0 Þ þ r20 ;
ð4Þ
using the binomial expansion for the Fresnel approximation q0 ¼ q þ
x2 þ y 2 ; 2q 2
2
ðx x0 Þ þ ðy y 0 Þ r ¼ r0 þ ; 2r0 and replacing Eq. (5) in Eq. (3) Z 1 Z b2 x2 þ y 2 exp ik qþ U/ 2q 1 b 2 !)# 2 2 ðx x0 Þ þ ðy y 0 Þ þ r0 þ dxdy: 2r0
ð5Þ
ð6Þ
Reorganizing Eq. (6) with regard to the separable coordinates
34
E.A. Rueda et al. / Optics Communications 274 (2007) 32–36
ik 2 2yy 0 z y 20 z y þ dy 2z r0 r0 1 ik 2 2xx0 z x20 z x exp þ dx; ð7Þ 2z r0 r0
U / exp½ikðq þ r0 Þ
Z
b 2
b 2
Z
1
exp
qr0 . The first integral is a constant due to its where z ¼ qþr 0 limits of integration. The second integral can be approximated to hthe icase of Fraunhofer diffraction if the parabolic 2 term exp ikx can be omitted. Knowing that the maximal 2z value the parabolic term can reach is when rffiffiffiffiffiffiffiffiffiffiffiffiffi2 kx2 pb2 p q þ r0 b ¼ ¼ ; ð8Þ 2z 4zk 4 qr0 k
and defining a parameter rffiffiffiffiffiffiffiffiffiffiffiffiffi q þ r0 c¼b ; qr0 k
counterpart was carried out with the setup shown in Fig. 3. A laser beam was expanded using a convergent lens to create a spherical wavefront that eliminates any dependence of the diffraction pattern with regard to the aperture-screen distance when there is a relative rotation with regard to the optical axis between the source and the slit, a problem that appears when using filament sources [17]. This wavefront was then diffracted by a slit and the corresponding diffraction pattern was captured using a CCD camera and Matrox inspector software; Matlab was used to process the images. The results of the simulation and the experimental counterpart are presented in Fig. 4. The experimental results were obtained by using the same parameters as those employed in the computer simulations.
ð9Þ
for a c value small enough the parabolic term can be omitted. But, for practical purposes it is important to be more specific and it becomes necessary to find experimentally a threshold where the parabolic term can be omitted. Then, a criterion in terms of the parameter c will be defined to distinguish between the Fraunhofer and Fresnel diffraction. This parameter depends on the geometry of the optical setup (q, r0), wavelength k and slit width b. 3. Simulation, experiment and criterion
Laser
Lens Slit CCD
Fig. 3. Experimental setup.
The definition of the criterion is closely related to the determination of the first minimum of the diffraction pattern [16]. In Fraunhofer diffraction the first minimum of the pattern corresponds to a null intensity (Fig. 2), thus, to define the criterion some simulations for different values of c have to be carried out until the first null minimum is found. Simulations of the diffraction produced on a spherical wave by a slit were carried out for different values of c (Eq. (9)). The slit’s width is b = 0.3 mm and the wavelength is k = 632.8 nm. The numerical matrix has 2048 · 2048 pixels with a sampling frequency of 9 lm · 9 lm. The experimental
Fig. 2. Fraunhofer diffraction pattern of a slit (Fourier transform of a slit). b is the slit’s width, k ¼ 2p , k is the wavelength, and h is the angle k made by the optical axis and a line from the slit to the observation point.
Fig. 4. Diffraction pattern for a spherical monochromatic wave of wavelength k = 632.8 nm, diffracted by a slit of width 0.3 mm. For each image the left side corresponds to the experimental result and the right side to the simulated result, for a c value: (a) 1.7, (b) 1.5, (c) 1.4, (d) 1.0, (e) 0.8 and (f) 0.7.
E.A. Rueda et al. / Optics Communications 274 (2007) 32–36
Fig. 4a–c, with c values: 1.7, 1.5 and 1.4, present a not null first minimum which indicates a Fresnel diffraction pattern; Fig. 4d, with c value: 1.0, although very close to null intensity present a not null first minimum; Fig. 4e and f, with c values: 0.8 and 0.7, have a null first minimum belonging to Fraunhofer diffraction, where 0.8 is the first one with a null minimum. From these results, a criterion based on parameter c is proposed to determine the validity of the Fraunhofer approximation when a spherical wave is diffracted by a slit. The criterion in terms of parameter c is (Eq. (9)) c 6 0:8 Fraunhofer domain; c > 0:8 Fresnel domain: 4. Criterion in terms of the number of Fresnel zones The criteria to distinguish between Fraunhofer and Fresnel diffraction, which appears in the literature, are directly or easily expressed in terms of the number of Fresnel zones seen through the aperture. Medina et al. [16] present many of these criteria in terms of the number of Fresnel zones, for comparison purposes. To allow the comparison with other criteria, a proportionality between the parameter c and the equivalent number of Fresnel zones seen through the slit is found. For a circular aperture as the one in Fig. 5a or with apertures of different geometries that satisfy the condition of circumscription as in Fig. 5b, where the concentric circles represent the different Fresnel zones over the aperture and where the white region is the aperture, illuminated with a spherical monochromatic wave, usually only the Fresnel zone that limits with the aperture edge is observed partially, making possible to neglect it or to completely consider it depending on the criterion used for the calculation of the number of Fresnel zones being observed. Moreover, in the majority of cases of interest for the determination of the domain of diffraction only the first zone is observed partially, eliminating any contribution from the other zones. Nevertheless, in the case of a slit, Fig. 5c, the problem is very different because now a great quantity of Fresnel zones segments are seen. Thus, it is important to calculate
35
the contribution of each segment to the equivalent number of Fresnel zones observed through the slit. In Fig. 6a a spherical wave of radius q is observed; generated by a point source in P and vertex in O, diffracted by a slit of width b and whose pattern of diffraction is centered on P 0 . The Fresnel zones are defined as the ringshaped regions in which the spherical wavefront centered on P is divided and whose frontiers correspond to the intersections of the wavefront with a series of spheres centered on P 0 and of radii r0 + lk/2, where l = 0, 1, 2, . . . and k is the wavelength. For representing the zones in the plane of the slit q b and r0 b are defined, this is known as the paraxial approximation. In the plane of the slit the l + 1 Fresnel zone is drawn with the segment observed through the slit lined. From Fig. 6a and the parabolic approximation [15] we know that the radius hl of the zone l is: sffiffiffiffiffiffiffiffiffiffiffiffiffi qr0 kl hl : ð10Þ q þ r0 Defining ARl 2hlb and AR(l+1) 2hl+1b as the rectangular areas of the Fig. 6b, the difference between these areas is A0Rðlþ1Þ 2bðhlþ1 hl Þ:
ð11Þ
Approximating the area of the zone segment observed through the slit to the area calculated in Eq. (11), a valid supposition under paraxial approximation, and taking into account the p difference in phase between consecutive Fresnel zones, it is possible to say that the total effective area of the m zone segments observed through the slit that contributes to the diffraction pattern in the central point of the observation plane, is equal to the sum of the areas of all zone segments but with a change in sign between consecutive zone segments, thus the total effective area is m X
A0Rl ffi 2bh1 2bðh2 h1 Þ þ 2bðh3 h2 Þ þ ð1Þmþ1 2bðhm hm1 Þ " ! # mþ1 m1 X ð1Þ lþ1 hm ; ffi 4b ð1Þ hl þ 2 l¼1
ð12Þ
b b ρ
P ρ
hl+1
hl
r0+(l+1)λ /2 r0+lλ /2
2hl
O
2hl+1
r0 P’
Fig. 5. Fresnel zones seen through: (a) circular aperture, (b) complex geometry aperture and (c) slit. The concentric circles represent the different Fresnel zones over the aperture and the white region represents the aperture.
Fig. 6. (a) l + 1 Fresnel zone of the spherical wave centered on P, vertex on O and of radius q, diffracted by a slit of width b. The shaded segment is the part of the zone observed through the slit from point P 0 at a distance r0. (b) Comparison of the area of the lined segment zone with the area generated by two rectangles.
36
E.A. Rueda et al. / Optics Communications 274 (2007) 32–36
and because the area of the first Fresnel zone is A1 ¼ ph21 , dividing Eq. (12) by this area, the equivalent number of Fresnel zones due to the presence of the m zone segments is # ! rffiffiffiffiffiffiffiffiffiffiffiffiffi" m1 pffiffi 4b q þ r0 X ð1Þmþ1 pffiffiffiffi lþ1 m m : l þ Nf ð1Þ p qr0 k 2 l¼1 ð13Þ This equivalent number of zones depends on the geometry of the optical setup, wavelength k and slit width b. It is proportional to the slit width and inversely proportional to the radius of the first Fresnel zone, where the constant of proportionality depends explicitly on the number of Fresnel zones observed through the slit. To maintain a small error in the approximation and because in Eq. (13) it is considered that only a segment of the first zone is seen, Eq. (13) is valid only for values less than one. This is not a serious restriction because seeing the first zone completely means that the diffraction pattern should be described with the Fresnel approximation [15]. If now a infinite number of zones are observed through the slit (m ! 1), a condition that is approximately satisfied in the vast majority of problems of interest, the term " # ! mþ1 m1 X pffiffi pffiffiffiffi ð1Þ lþ1 4 m l þ ð1Þ ð14Þ 2 l¼1 tends to 1.52, and the equivalent number of Fresnel zones can be written as rffiffiffiffiffiffiffiffiffiffiffiffiffi 1:52b q þ r0 : ð15Þ Nf p qr0 k The equivalent number of Fresnel zones is then proportional to parameter c (Eq. (9), N f 0:5c;
ð16Þ
and the criterion in terms of the equivalent number of Fresnel zones is N f 6 0:4
Fraunhofer domain;
N f > 0:4
Fresnel domain:
5. Conclusions In all the optical applications where the diffraction pattern is crucial in order to achieve an accurate result, knowing the type of diffraction approximation that must be used is of extreme importance. Thus, a set of criteria for different types of apertures in terms of a variable that depends on the optical setup parameters is of great utility.
To distinguish between Fraunhofer and Fresnel diffraction in the diffraction pattern produced by a slit under spherical illumination, a criterion for optical setups, in terms of a parameter c defined in Eq. (9), considers c 6 0.8 for Fraunhofer diffraction and c > 0.8 for Fresnel diffraction, and in terms of the equivalent number of Fresnel zones defined in Eq. (15), considers Nf 6 0.40 for Fraunhofer diffraction and Nf > 0.40 for Fresnel diffraction. To define a criterion for the validity of the Fraunhofer and Fresnel approximations based on the number of Fresnel zones seen through an aperture that cannot be circumscribed, i.e., that has one infinite dimension, one must take into account the contribution of all the zone segments that are observed through the aperture. Acknowledgement The grants from COLCIENCIAS (Colombia) and CODI-Universidad de Antioquia (Colombia) are gratefully acknowledged. The authors thanks the reviewers for their constructive comments that helped to improve this work. References [1] M. Born, E. Wolf, Principles of Optics, seventh ed., Cambrigde University Press, 2002, p. 446. [2] J. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1996, p. 73. [3] Q. Shen, D.H. Bilderback, K.V. Finkelstein, I.V. Basarov, S.M. Gruner, J. Phys. IV France 104 (2003) 21. [4] Y. Vladimirsky, Lithography vacuum ultraviolet spectroscopy II, in: J.A. Samson, D.L. Ederer (Eds.), Experimental methods in the physical sciences, vol. 32, ch. 10, New York, 1998, p. 205. [5] L. Turner, K. Weber, D. Paganin, R. Scholten, Opt. Lett. 29 (2004) 232. [6] J. Garcia-Sucerquia, F.F. Medina, J.F. Barrera, Opt. Commun. 253 (2005) 250. [7] S. Mezouari, A. Harvey, J. Opt. A: Pure Appl. Opt. 5 (2003) 86. [8] A. Steane, H. Rutt, J. Opt. Soc. Am. A 6 (1989) 1809. [9] M. Alonso, E. Finn, University Physics, Addison-Wesley, Palo Alto, 1967, p. 902. [10] D. Halliday, R. Resnick, Physics for Students of Science and Engineering, Part II, John Wiley and Sons, New York, 1960, p. 925. [11] F.A. Jenkins, H.E. White, Fundamentals of Optics, International Ed, McGraw-Hill, Auckland, 1981, p. 315. [12] B. Rossi, Optics, Addison-Wesley, Palo Alto, 1965, p. 177. [13] F.W. Sears, Optics, third ed., Addison-Wesley, London, 1962. [14] R.D. Guenther, Modern Optics, John Wiley and Sons, New York, 1990, p. 364. [15] F. F Medina, Revista Mexicana de Fı´sica 31 (1985) 311. [16] F.F. Medina, J. Garcia-Sucerquia, R. Castan˜eda, G. Matteucci, Optik 115 (2004) 547. [17] M.E. Mancen˜ido, G. Pozzi, L. Zunino, M. Garavaglia, J. Opt. Soc. Am. A 11 (1999) 2767.