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Paraxial waves in the far-field region
Optik 113, No. 8 (2002) 361–365 ª 2002 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
Carlos Javier Zapata-Rodrı´guez, Paraxial waves in the far-field region
361
International Journal for Light and Electron Optics
Paraxial waves in the far-field region Carlos Javier Zapata-Rodrı´guez Department of Optics, University of Valencia, E-46100 Burjassot, Spain
Abstract: By investigating the changes suffered by a paraxial beam propagating in the near-field and in the far-field regions, it has been found a set of wave equations valid for points gradually closer to the near field. A relevant expression for the validity of the far-field approximation is given from the paraxial Helmholtz equation. It is pointed out that the well-known Fresnel number associated with every transverse diffraction pattern can be interpreted as a magnitude that measures the relative standard deviation of the Fraunhofer pattern and a first-order field, thus reporting on an integral expression suitable for a general case. Finally, the Rayleigh range of the optical beam is deduced from the previously inferred Fresnel number, what has been applied for the cases of a spherical Gaussian beam and a uniformilluminated circular aperture. Key words: Diffraction – wave propagation
1. Introduction Within the last decade a great number of studies have been published on optical beam characterization [1–8]. The principal aim is the description of the diffraction behavior in terms of a low number of parameters. Most of them are particularized to low-angular spherical electromagnetic waves where the scalar paraxial regime holds. This approximation may be accomplished by means of two equivalent representations: the scalar paraxial wave equation and the Fresnel-Kirchhoff integral formula [1, 9]. One of the most important targets in optical beam characterization is the identification of the near and far fields where the diffraction behavior clearly separates. In the far field the wavefronts are concentric spheres centered on the beam waist located at the near field. Also the transverse irradiance pattern suffers a linear spreading whose magnification depends on the radius of the spherical wavefront. Conversely, the near field is featured of transverse patterns with no substantial change in size. Up to now the evaluation of such characterization is based on the calculation of the moments of the transverse irradiance distribution of the optical
Received 21 May 2002; accepted 15 August 2002. Fax: ++34-963-864715 E-mail: [email protected]
beam [2–6]. They consider exclusively the FresnelKirchhoff integral formulation. However, to the knowledge of the author this problem has not been accomplished from the paraxial wave equation. The aim of this paper is to present a new procedure for characterizing the near and far field of a paraxial optical beam deduced from the scalar wave equation. To this end it is reported on a wave equation that arises from the paraxial Helmholtz equation but is valid exclusively in the far field. The restrictions this wave equation imposes to be satisfied will give some remarks on the far field boundary. The well-known Fraunhofer integral is demonstrated to fulfill the deduced far-field wave equation and higher-order fields are obtained for completeness in order to satisfy the paraxial wave equation in closer points to the near field. Integral diffraction formulas are provided to satisfy these higher-order wave equations. Finally an integral expression is given for the evaluation of the Rayleigh range of an optical beam, which is a standard parameter that estimates the boundaries of the near and far fields.
2. Far-field wave equation Let us start by considering the paraxial wave equation 2 @ @2 @ þ 2ik uðx; y; zÞ ¼ 0 ; ð1Þ @z @x2 @y2 which is deduced from the homogeneous Helmholtz equation [1], assuming that a primary spatial dependence of the field exists in the form of exp ðikzÞ, where k ¼ 2p=l is the wavenumber. The above scalar wave equation describes accurately the transverse pattern of a beam when a slowly varying dependence of uð x; y; zÞ on z is satisfied. This means that the spectral components of the spatial wavefield distribution are plane waves propagating with a maximum tilt of 0.5 rad [1]: Some common features can be found for wavefields that satisfy eq. (1). The near field is observed when the transverse amplitude distribution does not substantially change in size. The beam waist is derived at a transverse plane in the near field satisfying that the transverse pattern size is minimum. For example, a 0030-4026/02/113/08-361 $ 15.00/0
362
Carlos Javier Zapata-Rodrı´guez, Paraxial waves in the far-field region
monochromatic Gaussian beam has a spot size practically constant in the near field within an axial distance that is commonly called the Rayleigh range [1] and its minimum value is found at a transverse plane where the wavefront is flat. However, when the beam propagates in regions considerably far from the beam waist, the wavefield spreads out as a nonuniform spherical wave whose focus is located at the waist plane and its complex amplitude remains constant in shape but with a lateral magnification that varies linearly with distance from focus. Multifocal optical beams like those arising from a zone plate [10] are out of the scope of this study. In order to obtain the spatial distribution corresponding to the far field from eq. (1), it is used the fact that in this region the beam behaves essentially like a spherical wave. It is assumed that the center of curvature of such a spherical wave is located at the origin ð x; y; zÞ ¼ ð0; 0; 0Þ. Consequently uðx; y; zÞ may be factorized as 1 k 2 2 exp i ðx þ y Þ wðx; y; zÞ ; uðx; y; zÞ ¼ z 2z ð2Þ where the factor in braces corresponds to the transverse distribution of a paraxial spherical wave whose amplitude is given by the function wðx; y; zÞ. Substitution of eq. (2) into eq. (1) gives the following equation @ @ @ x þy þz @x @y @z 2 2 z @ @ þ wðx; y; zÞ ¼ 0 : ð3Þ þi 2k @x2 @y2 The above equation is comprised of terms involving first and second-order derivatives in terms of the spatial coordinates. The combination of all of these terms cancel each other in order to satisfy eq. (3). However, it can be elucidated that in certain regions of space some of these terms give rise to low values so we may consider they vanish independently from the rest of the terms. In this case we can neglect their contributions in order to evaluate the diffracted far field. Note that the diffraction pattern in the far-field region stabilizes in shape and diverges linearly from focus. The wavefield varies slowly along the transverse direction and therefore high-order derivatives may be ignored. This reasoning is expressed in mathematical terms as 2 z @ @2 jwðx; y; zÞj : ð4Þ þ wðx; y; zÞ 2k @x2 @y2 The above inequality gives relevant information for testing which points may be considered as to belong to the far-field region. As far as the author knows this kind of conditions have been previously presented only for the spatial axial coordinate [1, 5, 6] meaning that the limits between the far-field and the near-field regions are given in terms of certain transverse planes. However, eq. (4) provides a more complete characterization of the boundaries of the far field.
By dropping the second partial derivatives in x and y it is reduced the paraxial wave equation in eq. (3) to the far-field wave equation given in the following closed form rrwðrÞ ¼ 0 ;
ð5Þ
where r ¼ ðx; y; zÞ is a three-dimensional vector point in the far-field region measured from the origin and r ¼ ð@=@x; @=@y; @=@zÞ corresponds to the nabla operator. Now it should be confirmed that the spherical wave generated at the far field has an amplitude wðrÞ that is invariant in shape but with linear lateral magnification. It is important to mention that eq. (5) is an approximation of the paraxial wave equation given in eq. (3), in such a way that the amplitude wave field satisfies both expressions in the far-field region. The derivation of such a wave equation is relevant not for field evaluation purposes, since both Fresnel diffraction integral and Helmholtz wave equation are the appropriate tools for this aim. However, the relevance in its deduction is not if we can substitute eq. (5) by eq. (3) but which are the points where such a wave field is satisfied. This evaluation should be performed point by point and give the significant information for the boundaries of the far-field region.
3. Wavefields in the far-field region In this section it is confirmed that the spherical wave generated at the far field has an amplitude wðrÞ that is invariant in shape but with linear lateral magnification. The linear differential equation given in eq. (5) may be solved by factorizing the function wðrÞ into one-dimensional functions, i.e. wðx; y; zÞ ¼ wx ðxÞ wy ðyÞ wz ðzÞ :
ð6Þ
By introducing eq. (6) into eq. (5) it is obtained the following differential equation x @wx ð xÞ y @wy ðyÞ z @wz ðzÞ þ þ ¼ 0: wx ð xÞ @x wy ðyÞ @y wz ðzÞ @z ð7Þ It is observed that the above equation is comprised of three terms, each one depending on a different spatial coordinate. Therefore, all of them are imposed to give a constant quantity in order to satisfy eq. (7), that is a @wa ðaÞ ¼ ma ; wa ðaÞ @a
ð8Þ
where a ¼ x; y; z and ma is a constant. Additionally, by substitution of eq. (8) into eq. (7) it is obtained the following constraint mx þ my þ mz ¼ 0 :
ð9Þ
Any linear combination of functions that are a solution of the one-dimensional differential equation in
Carlos Javier Zapata-Rodrı´guez, Paraxial waves in the far-field region
eq. (8) may be substituted into eq. (6) to generate the wavefield wðrÞ after imposing the restriction in eq. (9). It is not difficult to demonstrate that the family of functions satisfying eq. (8) are in the form wa ðaÞ ¼ wa ð1Þ ama :
ð10Þ
After substitution of the above solution into eq. (6) it is found that the amplitude distribution in the farfield region may be expressed as a linear combination of this set of polynomial functions as mx my 1 P x y wðx; y; zÞ ¼ Cmx ; my ; ð11Þ z z mx ; my ¼ 0 where Cmx ; my is a constant coefficient and it is imposed the constraint in eq. (9). In principle, both constants mx and my are not restricted to non-negative integers, but in physical problems it is found that wðrÞ is an analytic function finite-valued for large values of z implying that the previous assumption is valid. In fact, eq. (11) describes a Taylor series of a three-dimensional function with a spatial dependence in the form wðx; y; zÞ ¼ f ð x=z; y=zÞ. Then, the amplitude function wðrÞ has a transverse distribution whose sole dependence on the distance to focus z is the magnification of the pattern but not its shape, as it is expected. Eqs. (2) and (11) provide the spatial dependence of a wavefield propagating at the far field. By simply inspection it is observed that uðx; y; zÞ has a singularity at the beam waist plane, z ¼ 0. This fact is not surprising since the points in the neighborhood of the origin clearly belongs to the near-field region where eqs. (4) and (5) fail. However, it is of practical interest to derive the wavefield in the far field when it is known a priori the transverse amplitude distribution at the beam waist plane. To this end it is considered the Fresnel-Kirchhoff formula [9] ðð 1 k 2 2 exp i ðx þ y Þ uo ðxo ; yo Þ uðx; y; zÞ ¼ ilz 2z k ðx2 þ y2o Þ exp i 2z o 2p exp i ðxxo þ yyo Þ dxo dyo ; ð12Þ lz which is found to be the exact solution for the paraxial wave equation given in eq. (1). uo ðxo ; yo Þ stands for the amplitude distribution at the beam waist plane z ¼ 0. As commonly assumed, the far field is obtained when the quadratic phase factor existing in the Fresnel-Kirchhoff integral may be approximated to unity. This approximation for the field amplitude is referred to as the Fraunhofer approximation [9], and after the factorization given in eq. (2) is applied yields ðð 1 uo ðxo ; yo Þ wðx; y; zÞ ¼ il 2p ðxxo þ yyo Þ dxo dyo ; exp i lz (13)
363
which, except for a factor 1=il, is simply a two-dimensional Fourier transform of the field amplitude at the beam waist plane z ¼ 0. The corresponding spatial frequencies are evaluated at ð x=lz; y=lzÞ. Now it is verified that eq. (13) satisfies the far-field wave equation in eq. (5) since it may be expressed in the form of eq. (11).
4. Higher-order fields The previous approach to the far-field wave equation assumes that the optical beam propagates in a region infinitely distant from focus. In practical cases it should be considered the finite distance to the observation point and higher-order corrections to eq. (5) must be found. Following Davis [11] and Cao and Deng [12] it may be examined a first-order approximation wð1Þ ðrÞ to the wave function by simply introducing a nonhomogeneous term in the far-field wave equation in the form z r2 wð0Þ ðrÞ ; ð14Þ rrwð1Þ ðrÞ ¼ i 2k t where r2t ¼ @ 2 =@x2 þ @ 2 =@y2 is the Laplacian operator over the transverse coordinates, and wð0Þ ðrÞ is the exact solution to the far-field wave equation given in eq. (5). It is possible to give an inequality similar to that of eq. (4), particularized to the first-order wave function wð1Þ ðrÞ, in order to assure the field amplitude fulfills eq. (14) that we write as z ð15Þ r2t ½wðrÞ wð0Þ ðrÞ jwðrÞj : 2k Obviously, this inequality yields a weaker restriction so that points where eq. (4) holds must satisfy this new constraint. Once again it results appropriate to derive an integral formula that allows us to evaluate the diffracted wavefield in terms of the amplitude distribution at the beam waist plane. It is straightforward to demonstrate that the new diffraction integral may be given as ðð 1 ð1Þ uo ðxo ; yo Þ w ðx; y; zÞ ¼ il k ðx2o þ y2o Þ 1i 2z 2p exp i ðxxo þ yyo Þ dxo dyo ; lz (18) which is a more suitable approximation compared with eq. (13) when the wave is not exactly propagating over the far-field region. Following the previous reasoning, it is proposed a recursive procedure that results useful for evaluating the amplitude distribution of the beam in regions gradually closer to focus. The n-th order field generates a non-homogeneous term in the far-field wave equation
364
Carlos Javier Zapata-Rodrı´guez, Paraxial waves in the far-field region
that the next higher-order field must satisfy, having z ð17Þ rrwðnþ1Þ ðrÞ ¼ i r2 wðnÞ ðrÞ ; 2k t for any integer n 0. Thus the diffraction integral which is found to be the exact solution to the previous wave equation can be expressed as ðð 1 uo ðxo ; yo Þ wðnþ1Þ ðx; y; zÞ ¼ il 2 m 3 k 2 2 ðx þ yo Þ 7 i 6 nþ1 2z o 5 4P m! m¼0 2p exp i ðxxo þ yyo Þ dxo dyo : lz (18) Consequently, first it should be estimated the order of the wavefield to give its amplitude distribution accurately. To this end it is necessary to ensure that the following inequality holds z ð19Þ r2t ½wðrÞ wðnÞ ðrÞ jwðrÞj ; 2k which is a particularization of eq. (15) for a higher-order field. Otherwise it should be increased the field order until eq. (19) is satisfied. As a limiting case, by using the Taylor expansion exp ðxÞ ¼
1 P
xm =m!
m¼0
it is obtained the Fresnel-Kirchhoff diffraction formula given in eq. (12) for n ¼ 1, which is valid both in the near and far-field regions within the paraxial approximation.
bution corresponding to a paraxial wave satisfying the first-order wave equation in eq. (14). Performing the proposed analytical operation it is obtained kwð1Þ wð0Þ k kwð0Þ k !1=2 Ð Ð ð1Þ jw ðx; y; zÞ wð0Þ ðx; y; zÞj2 dx dy ¼ ÐÐ jwð0Þ ðx; y; zÞj2 dx dy !1=2 ÐÐ juo ðxo ; yo Þj2 ðx2o þ y2o Þ2 dxo dyo k ¼ ; ÐÐ 2jzj juo ðxo ; yo Þj2 dxo dyo
sðzÞ ¼
(20) where it has been made use of eqs. (13) and (16) together with the Parseval’s theorem [9] in order to express s ðzÞ in terms of the amplitude distribution at the beam waist plane, uo ðxo ; yo Þ. Eq. (20) shows that the relative standard deviation of the Fraunhofer integral and the first-order wave function wð1Þ ðrÞ is inversely proportional to the distance from the beam waist, what agrees with the fact that wð0Þ ðrÞ is valid for a distance well beyond focus. Otherwise the parameter sðzÞ increases dramatically since a singularity arises at the origin. To avoid this drawback it may be considered the standard deviation of the Fraunhofer pattern respect to a higher-order field. However in the limiting case of considering such a deviation at the beam waist plane, z ¼ 0, the treatment should be performed with the paraxial wave function given in eq. (3). In this case the standard deviation s ðzÞ takes a more complex expression given by 0ðð B sðzÞ ¼ 2B @
11=2 k 2 ðxo þ y2o Þ dxo dyo juo ðxo ; yo Þj2 sin2 C 4z C : ÐÐ A 2 juo ðxo ; yo Þj dxo dyo (24)
5. Rayleigh range In this section the author wants to show that the socalled Rayleigh range may be obtained from the previously deducted wave fields. It is not the intention of the author to give a new definition of the Rayleigh range, but to provide a new interpretation of this parameter. On the other hand, such a relevant parameter for the understanding of the far-field region should be introduced for the better comprehension of the following discussion. The far-field wave equation shown in eq. (5) holds when the wave function can be described accurately by wð0Þ ðrÞ. In the case the inequality in eq. (4) is not satisfied it should be evaluated a higher-order field. By inspection of eq. (15) it is possible to assert that a paraxial beam propagating in the far-field region makes the differences between the first-order wð1Þ ðrÞ and the farfield wave functions negligible. Thus it is investigated the parameter kwð1Þ wð0Þ k=kwð0Þ k that measures at every plane the relative standard deviation given between the Fraunhofer pattern and the amplitude distri-
For high values of the axial coordinate z we obtain the result given in eq. (20). As it has been previously mentioned in Gaussian beam characterization it is commonly used the Rayleigh range, namely zR, which marks the boundary between the near-field (jzj zR ) and the far-field (jzj zR ) regions for a diffracted beam whose beam waist plane is located at the origin z ¼ 0. In accordance to the reasoning of the previous paragraph it is possible to use the parameter s ðzÞ in order to evaluate the Rayleigh range of any Gaussian-like paraxial wave. It is included spherical beams with focus in the sense of geometrical optics [13] for a given transverse amplitude profile and aberrated wavefront. Then it is proposed the following formula zR ¼ jzjs ðzÞ !1=2 ÐÐ juo ðxo ; yo Þj2 ðx2o þ y2o Þ2 dxo dyo p ¼ ; ÐÐ l juo ðxo ; yo Þj2 dxo dyo
ð25Þ
Carlos Javier Zapata-Rodrı´guez, Paraxial waves in the far-field region
which satisfies that in the Fraunhofer regime, jzj zR , the inequality s ðzÞ 1 is achieved. It is observed that the Rayleigh range is proportional to the root square of certain even-order moments of the irradiance distribution at the beam waist plane. Siegman [2] first proposed a similar but somewhat different treatment for evaluating the effective spot size of an optical beam to be introduced in the beam propagation factor M2 . This concept has been extended for evaluating the Rayleigh range elsewhere [3, 4]. At this point it results convenient to give a physical interpretation of the parameter s ðzÞ. In agreement with eq. (2) it may be expressed as s ðzÞ ¼ zR =jzj which is commonly denoted as the effective Fresnel number. This parameter was first introduced as a normalized axial coordinate to describe the irradiance distribution along the axis of a uniform plane wave diffracted by a circular opaque screen [1]. In this case the Fresnel number indicates the number of halfwave zones that fill the aperture as viewed from an axial point located at a distance z. Afterwards this parameter was extended to untruncated optical beams by considering the effective spot radius instead of the aperture radius [14, 13]. Then s ðzÞ may be used to characterized not only the limits of the far field but also every transverse pattern of a propagating beam. Finally, let us evaluate the proposed Rayleigh range for two practical cases such as a spherical Gaussian beam whose beam waist is located at origin, z ¼ 0, and has an effective spot radius given by w0, and a uniform, plane wave diffracted by a circular clear screen of radius a. By using eq. (22) it is found that pffiffiffi pw0 ; ð23Þ 2 zG R ¼ l and p a2 p ffiffiffi ; zC ¼ R 3 l
ð24Þ
respectively. Both are in good agreement with the results given previously [1]. It is pointed out that the simplicity of eq. (22) provides a tool for the analytical evaluation of the Rayleigh range of a great variety of either untruncated optical beams or electromagnetic waves truncated by nonuniform opaque screens.
6. Summary The paraxial wave equation is particularized to a new wave equation that is valid exclusively for the far field. A given inequality must be satisfied in order to assure the field propagates in the far-field region, which depends on the observation point and not on the transverse plane it belongs to. They would allow us to establish the boundaries between the near-field and the farfield regions. For completeness, higher-order wave equations have been presented in order to evaluate
365
the amplitude distribution of a paraxial wave at points gradually closer to the near field. Additional integral formulas are deducted from the Fresnel diffraction integral to satisfy high-order wave equations. Finally, from this formalism it is reported on an expression which evaluates the distance from the beam waist plane where the Fraunhofer approximation can be applied, what is usually named as the Rayleigh range. Actually it is related with the concept of effective Fresnel number, and both were inferred from comparison of the far-field wave function and a firstorder field. The presented expressions have been applied to the particular cases of a spherical Gaussian beam and a uniformly-illuminated circular aperture. These results are in good agreement with those reported formerly. Acknowledgements. It is a pleasure to thank Albert Ferrando for helpful conversations. This work was supported by the Plan Nacional I+D+I (grant DPI2000-0774), Ministerio de Ciencia y Tecnologı´a, Spain. The author is also grateful to a Postdoctoral Grant from the Ministerio de Educaciı´on, Cultura y Deporte, Spain.
References [1] Siegman A: Lasers. University Science Books, Mill Valley 1986 [2] Siegman AE: New developments in laser resonators. Proc. Soc. Photo-Opt. Instrum. Eng. 1224 (1990) 2–14 [3] Be´langer P-A: Beam propagation and the ABCD ray matrices. Opt. Lett. 16 (1991) 196–198 [4] Pare´ C, Be´langer P-A: Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam. Opt. Commun. 123 (1996) 679–693 [5] Dragoman D, Dragoman M: Near and far field optical beam characterization using the fractional fourier transform. Opt. Commun. 141 (1997) 5–9 [6] Saghafi S, Sheppard CJR: The beam propagation factor for higher order Gaussian beams. Opt. Commun. 153 (1998) 207–210 [7] Herman RM, Wiggins TA: Rayleigh range and the M2 factor for Bessel Gauss beams. Appl. Opt. 37 (1998) 3398–3400 [8] Roundy C: Propagation factor quantifies laser beam performance comparing M2 of an actual laser beam to a pure TEM00 Gaussian beam allows beam-propagation characteristics to be accurately predicted. Laser Focus World 35 (1999) 119–124 [9] Goodman J: Introduction to Fourier Optics. McGraw-Hill, New York 1996 [10] Zapata-Rodrı´guez CJ, Martı´nez-Corral M, Andre´s P, Pons A: Axial behavior of diffractive lenses under Gaussian illumination: complex-argument spectral analysis. J. Opt. Soc. Am. A 16 (1999) 1–7 [11] Davis LW: Theory of electromagnetic beams. Phys. Rev. A 19 (1979) 1177–1179 [12] Cao Q, Deng X: Corrections to the paraxial approximation of an arbitrary free-propagation beam. J. Opt. Soc. Am. A 15 (1998) 1144–1148 [13] Martı´nez-Corral M, Zapata-Rodrı´guez CJ, Andre´s P, Silvestre E: Effective Fresnel-number concept for evaluating the relative focal shift in focused beam. J. Opt. Soc. Am. A 15 (1998) 449–455 [14] Carter WH: Focal shift and concept of effective Fresnel number for a Gaussian laser beams. Appl. Opt. 21 (1982) 1989–1994
Carlos Javier Zapata-Rodrı´guez, Paraxial waves in the far-field region
361
International Journal for Light and Electron Optics
Paraxial waves in the far-field region Carlos Javier Zapata-Rodrı´guez Department of Optics, University of Valencia, E-46100 Burjassot, Spain
Abstract: By investigating the changes suffered by a paraxial beam propagating in the near-field and in the far-field regions, it has been found a set of wave equations valid for points gradually closer to the near field. A relevant expression for the validity of the far-field approximation is given from the paraxial Helmholtz equation. It is pointed out that the well-known Fresnel number associated with every transverse diffraction pattern can be interpreted as a magnitude that measures the relative standard deviation of the Fraunhofer pattern and a first-order field, thus reporting on an integral expression suitable for a general case. Finally, the Rayleigh range of the optical beam is deduced from the previously inferred Fresnel number, what has been applied for the cases of a spherical Gaussian beam and a uniformilluminated circular aperture. Key words: Diffraction – wave propagation
1. Introduction Within the last decade a great number of studies have been published on optical beam characterization [1–8]. The principal aim is the description of the diffraction behavior in terms of a low number of parameters. Most of them are particularized to low-angular spherical electromagnetic waves where the scalar paraxial regime holds. This approximation may be accomplished by means of two equivalent representations: the scalar paraxial wave equation and the Fresnel-Kirchhoff integral formula [1, 9]. One of the most important targets in optical beam characterization is the identification of the near and far fields where the diffraction behavior clearly separates. In the far field the wavefronts are concentric spheres centered on the beam waist located at the near field. Also the transverse irradiance pattern suffers a linear spreading whose magnification depends on the radius of the spherical wavefront. Conversely, the near field is featured of transverse patterns with no substantial change in size. Up to now the evaluation of such characterization is based on the calculation of the moments of the transverse irradiance distribution of the optical
Received 21 May 2002; accepted 15 August 2002. Fax: ++34-963-864715 E-mail: [email protected]
beam [2–6]. They consider exclusively the FresnelKirchhoff integral formulation. However, to the knowledge of the author this problem has not been accomplished from the paraxial wave equation. The aim of this paper is to present a new procedure for characterizing the near and far field of a paraxial optical beam deduced from the scalar wave equation. To this end it is reported on a wave equation that arises from the paraxial Helmholtz equation but is valid exclusively in the far field. The restrictions this wave equation imposes to be satisfied will give some remarks on the far field boundary. The well-known Fraunhofer integral is demonstrated to fulfill the deduced far-field wave equation and higher-order fields are obtained for completeness in order to satisfy the paraxial wave equation in closer points to the near field. Integral diffraction formulas are provided to satisfy these higher-order wave equations. Finally an integral expression is given for the evaluation of the Rayleigh range of an optical beam, which is a standard parameter that estimates the boundaries of the near and far fields.
2. Far-field wave equation Let us start by considering the paraxial wave equation 2 @ @2 @ þ 2ik uðx; y; zÞ ¼ 0 ; ð1Þ @z @x2 @y2 which is deduced from the homogeneous Helmholtz equation [1], assuming that a primary spatial dependence of the field exists in the form of exp ðikzÞ, where k ¼ 2p=l is the wavenumber. The above scalar wave equation describes accurately the transverse pattern of a beam when a slowly varying dependence of uð x; y; zÞ on z is satisfied. This means that the spectral components of the spatial wavefield distribution are plane waves propagating with a maximum tilt of 0.5 rad [1]: Some common features can be found for wavefields that satisfy eq. (1). The near field is observed when the transverse amplitude distribution does not substantially change in size. The beam waist is derived at a transverse plane in the near field satisfying that the transverse pattern size is minimum. For example, a 0030-4026/02/113/08-361 $ 15.00/0
362
Carlos Javier Zapata-Rodrı´guez, Paraxial waves in the far-field region
monochromatic Gaussian beam has a spot size practically constant in the near field within an axial distance that is commonly called the Rayleigh range [1] and its minimum value is found at a transverse plane where the wavefront is flat. However, when the beam propagates in regions considerably far from the beam waist, the wavefield spreads out as a nonuniform spherical wave whose focus is located at the waist plane and its complex amplitude remains constant in shape but with a lateral magnification that varies linearly with distance from focus. Multifocal optical beams like those arising from a zone plate [10] are out of the scope of this study. In order to obtain the spatial distribution corresponding to the far field from eq. (1), it is used the fact that in this region the beam behaves essentially like a spherical wave. It is assumed that the center of curvature of such a spherical wave is located at the origin ð x; y; zÞ ¼ ð0; 0; 0Þ. Consequently uðx; y; zÞ may be factorized as 1 k 2 2 exp i ðx þ y Þ wðx; y; zÞ ; uðx; y; zÞ ¼ z 2z ð2Þ where the factor in braces corresponds to the transverse distribution of a paraxial spherical wave whose amplitude is given by the function wðx; y; zÞ. Substitution of eq. (2) into eq. (1) gives the following equation @ @ @ x þy þz @x @y @z 2 2 z @ @ þ wðx; y; zÞ ¼ 0 : ð3Þ þi 2k @x2 @y2 The above equation is comprised of terms involving first and second-order derivatives in terms of the spatial coordinates. The combination of all of these terms cancel each other in order to satisfy eq. (3). However, it can be elucidated that in certain regions of space some of these terms give rise to low values so we may consider they vanish independently from the rest of the terms. In this case we can neglect their contributions in order to evaluate the diffracted far field. Note that the diffraction pattern in the far-field region stabilizes in shape and diverges linearly from focus. The wavefield varies slowly along the transverse direction and therefore high-order derivatives may be ignored. This reasoning is expressed in mathematical terms as 2 z @ @2 jwðx; y; zÞj : ð4Þ þ wðx; y; zÞ 2k @x2 @y2 The above inequality gives relevant information for testing which points may be considered as to belong to the far-field region. As far as the author knows this kind of conditions have been previously presented only for the spatial axial coordinate [1, 5, 6] meaning that the limits between the far-field and the near-field regions are given in terms of certain transverse planes. However, eq. (4) provides a more complete characterization of the boundaries of the far field.
By dropping the second partial derivatives in x and y it is reduced the paraxial wave equation in eq. (3) to the far-field wave equation given in the following closed form rrwðrÞ ¼ 0 ;
ð5Þ
where r ¼ ðx; y; zÞ is a three-dimensional vector point in the far-field region measured from the origin and r ¼ ð@=@x; @=@y; @=@zÞ corresponds to the nabla operator. Now it should be confirmed that the spherical wave generated at the far field has an amplitude wðrÞ that is invariant in shape but with linear lateral magnification. It is important to mention that eq. (5) is an approximation of the paraxial wave equation given in eq. (3), in such a way that the amplitude wave field satisfies both expressions in the far-field region. The derivation of such a wave equation is relevant not for field evaluation purposes, since both Fresnel diffraction integral and Helmholtz wave equation are the appropriate tools for this aim. However, the relevance in its deduction is not if we can substitute eq. (5) by eq. (3) but which are the points where such a wave field is satisfied. This evaluation should be performed point by point and give the significant information for the boundaries of the far-field region.
3. Wavefields in the far-field region In this section it is confirmed that the spherical wave generated at the far field has an amplitude wðrÞ that is invariant in shape but with linear lateral magnification. The linear differential equation given in eq. (5) may be solved by factorizing the function wðrÞ into one-dimensional functions, i.e. wðx; y; zÞ ¼ wx ðxÞ wy ðyÞ wz ðzÞ :
ð6Þ
By introducing eq. (6) into eq. (5) it is obtained the following differential equation x @wx ð xÞ y @wy ðyÞ z @wz ðzÞ þ þ ¼ 0: wx ð xÞ @x wy ðyÞ @y wz ðzÞ @z ð7Þ It is observed that the above equation is comprised of three terms, each one depending on a different spatial coordinate. Therefore, all of them are imposed to give a constant quantity in order to satisfy eq. (7), that is a @wa ðaÞ ¼ ma ; wa ðaÞ @a
ð8Þ
where a ¼ x; y; z and ma is a constant. Additionally, by substitution of eq. (8) into eq. (7) it is obtained the following constraint mx þ my þ mz ¼ 0 :
ð9Þ
Any linear combination of functions that are a solution of the one-dimensional differential equation in
Carlos Javier Zapata-Rodrı´guez, Paraxial waves in the far-field region
eq. (8) may be substituted into eq. (6) to generate the wavefield wðrÞ after imposing the restriction in eq. (9). It is not difficult to demonstrate that the family of functions satisfying eq. (8) are in the form wa ðaÞ ¼ wa ð1Þ ama :
ð10Þ
After substitution of the above solution into eq. (6) it is found that the amplitude distribution in the farfield region may be expressed as a linear combination of this set of polynomial functions as mx my 1 P x y wðx; y; zÞ ¼ Cmx ; my ; ð11Þ z z mx ; my ¼ 0 where Cmx ; my is a constant coefficient and it is imposed the constraint in eq. (9). In principle, both constants mx and my are not restricted to non-negative integers, but in physical problems it is found that wðrÞ is an analytic function finite-valued for large values of z implying that the previous assumption is valid. In fact, eq. (11) describes a Taylor series of a three-dimensional function with a spatial dependence in the form wðx; y; zÞ ¼ f ð x=z; y=zÞ. Then, the amplitude function wðrÞ has a transverse distribution whose sole dependence on the distance to focus z is the magnification of the pattern but not its shape, as it is expected. Eqs. (2) and (11) provide the spatial dependence of a wavefield propagating at the far field. By simply inspection it is observed that uðx; y; zÞ has a singularity at the beam waist plane, z ¼ 0. This fact is not surprising since the points in the neighborhood of the origin clearly belongs to the near-field region where eqs. (4) and (5) fail. However, it is of practical interest to derive the wavefield in the far field when it is known a priori the transverse amplitude distribution at the beam waist plane. To this end it is considered the Fresnel-Kirchhoff formula [9] ðð 1 k 2 2 exp i ðx þ y Þ uo ðxo ; yo Þ uðx; y; zÞ ¼ ilz 2z k ðx2 þ y2o Þ exp i 2z o 2p exp i ðxxo þ yyo Þ dxo dyo ; ð12Þ lz which is found to be the exact solution for the paraxial wave equation given in eq. (1). uo ðxo ; yo Þ stands for the amplitude distribution at the beam waist plane z ¼ 0. As commonly assumed, the far field is obtained when the quadratic phase factor existing in the Fresnel-Kirchhoff integral may be approximated to unity. This approximation for the field amplitude is referred to as the Fraunhofer approximation [9], and after the factorization given in eq. (2) is applied yields ðð 1 uo ðxo ; yo Þ wðx; y; zÞ ¼ il 2p ðxxo þ yyo Þ dxo dyo ; exp i lz (13)
363
which, except for a factor 1=il, is simply a two-dimensional Fourier transform of the field amplitude at the beam waist plane z ¼ 0. The corresponding spatial frequencies are evaluated at ð x=lz; y=lzÞ. Now it is verified that eq. (13) satisfies the far-field wave equation in eq. (5) since it may be expressed in the form of eq. (11).
4. Higher-order fields The previous approach to the far-field wave equation assumes that the optical beam propagates in a region infinitely distant from focus. In practical cases it should be considered the finite distance to the observation point and higher-order corrections to eq. (5) must be found. Following Davis [11] and Cao and Deng [12] it may be examined a first-order approximation wð1Þ ðrÞ to the wave function by simply introducing a nonhomogeneous term in the far-field wave equation in the form z r2 wð0Þ ðrÞ ; ð14Þ rrwð1Þ ðrÞ ¼ i 2k t where r2t ¼ @ 2 =@x2 þ @ 2 =@y2 is the Laplacian operator over the transverse coordinates, and wð0Þ ðrÞ is the exact solution to the far-field wave equation given in eq. (5). It is possible to give an inequality similar to that of eq. (4), particularized to the first-order wave function wð1Þ ðrÞ, in order to assure the field amplitude fulfills eq. (14) that we write as z ð15Þ r2t ½wðrÞ wð0Þ ðrÞ jwðrÞj : 2k Obviously, this inequality yields a weaker restriction so that points where eq. (4) holds must satisfy this new constraint. Once again it results appropriate to derive an integral formula that allows us to evaluate the diffracted wavefield in terms of the amplitude distribution at the beam waist plane. It is straightforward to demonstrate that the new diffraction integral may be given as ðð 1 ð1Þ uo ðxo ; yo Þ w ðx; y; zÞ ¼ il k ðx2o þ y2o Þ 1i 2z 2p exp i ðxxo þ yyo Þ dxo dyo ; lz (18) which is a more suitable approximation compared with eq. (13) when the wave is not exactly propagating over the far-field region. Following the previous reasoning, it is proposed a recursive procedure that results useful for evaluating the amplitude distribution of the beam in regions gradually closer to focus. The n-th order field generates a non-homogeneous term in the far-field wave equation
364
Carlos Javier Zapata-Rodrı´guez, Paraxial waves in the far-field region
that the next higher-order field must satisfy, having z ð17Þ rrwðnþ1Þ ðrÞ ¼ i r2 wðnÞ ðrÞ ; 2k t for any integer n 0. Thus the diffraction integral which is found to be the exact solution to the previous wave equation can be expressed as ðð 1 uo ðxo ; yo Þ wðnþ1Þ ðx; y; zÞ ¼ il 2 m 3 k 2 2 ðx þ yo Þ 7 i 6 nþ1 2z o 5 4P m! m¼0 2p exp i ðxxo þ yyo Þ dxo dyo : lz (18) Consequently, first it should be estimated the order of the wavefield to give its amplitude distribution accurately. To this end it is necessary to ensure that the following inequality holds z ð19Þ r2t ½wðrÞ wðnÞ ðrÞ jwðrÞj ; 2k which is a particularization of eq. (15) for a higher-order field. Otherwise it should be increased the field order until eq. (19) is satisfied. As a limiting case, by using the Taylor expansion exp ðxÞ ¼
1 P
xm =m!
m¼0
it is obtained the Fresnel-Kirchhoff diffraction formula given in eq. (12) for n ¼ 1, which is valid both in the near and far-field regions within the paraxial approximation.
bution corresponding to a paraxial wave satisfying the first-order wave equation in eq. (14). Performing the proposed analytical operation it is obtained kwð1Þ wð0Þ k kwð0Þ k !1=2 Ð Ð ð1Þ jw ðx; y; zÞ wð0Þ ðx; y; zÞj2 dx dy ¼ ÐÐ jwð0Þ ðx; y; zÞj2 dx dy !1=2 ÐÐ juo ðxo ; yo Þj2 ðx2o þ y2o Þ2 dxo dyo k ¼ ; ÐÐ 2jzj juo ðxo ; yo Þj2 dxo dyo
sðzÞ ¼
(20) where it has been made use of eqs. (13) and (16) together with the Parseval’s theorem [9] in order to express s ðzÞ in terms of the amplitude distribution at the beam waist plane, uo ðxo ; yo Þ. Eq. (20) shows that the relative standard deviation of the Fraunhofer integral and the first-order wave function wð1Þ ðrÞ is inversely proportional to the distance from the beam waist, what agrees with the fact that wð0Þ ðrÞ is valid for a distance well beyond focus. Otherwise the parameter sðzÞ increases dramatically since a singularity arises at the origin. To avoid this drawback it may be considered the standard deviation of the Fraunhofer pattern respect to a higher-order field. However in the limiting case of considering such a deviation at the beam waist plane, z ¼ 0, the treatment should be performed with the paraxial wave function given in eq. (3). In this case the standard deviation s ðzÞ takes a more complex expression given by 0ðð B sðzÞ ¼ 2B @
11=2 k 2 ðxo þ y2o Þ dxo dyo juo ðxo ; yo Þj2 sin2 C 4z C : ÐÐ A 2 juo ðxo ; yo Þj dxo dyo (24)
5. Rayleigh range In this section the author wants to show that the socalled Rayleigh range may be obtained from the previously deducted wave fields. It is not the intention of the author to give a new definition of the Rayleigh range, but to provide a new interpretation of this parameter. On the other hand, such a relevant parameter for the understanding of the far-field region should be introduced for the better comprehension of the following discussion. The far-field wave equation shown in eq. (5) holds when the wave function can be described accurately by wð0Þ ðrÞ. In the case the inequality in eq. (4) is not satisfied it should be evaluated a higher-order field. By inspection of eq. (15) it is possible to assert that a paraxial beam propagating in the far-field region makes the differences between the first-order wð1Þ ðrÞ and the farfield wave functions negligible. Thus it is investigated the parameter kwð1Þ wð0Þ k=kwð0Þ k that measures at every plane the relative standard deviation given between the Fraunhofer pattern and the amplitude distri-
For high values of the axial coordinate z we obtain the result given in eq. (20). As it has been previously mentioned in Gaussian beam characterization it is commonly used the Rayleigh range, namely zR, which marks the boundary between the near-field (jzj zR ) and the far-field (jzj zR ) regions for a diffracted beam whose beam waist plane is located at the origin z ¼ 0. In accordance to the reasoning of the previous paragraph it is possible to use the parameter s ðzÞ in order to evaluate the Rayleigh range of any Gaussian-like paraxial wave. It is included spherical beams with focus in the sense of geometrical optics [13] for a given transverse amplitude profile and aberrated wavefront. Then it is proposed the following formula zR ¼ jzjs ðzÞ !1=2 ÐÐ juo ðxo ; yo Þj2 ðx2o þ y2o Þ2 dxo dyo p ¼ ; ÐÐ l juo ðxo ; yo Þj2 dxo dyo
ð25Þ
Carlos Javier Zapata-Rodrı´guez, Paraxial waves in the far-field region
which satisfies that in the Fraunhofer regime, jzj zR , the inequality s ðzÞ 1 is achieved. It is observed that the Rayleigh range is proportional to the root square of certain even-order moments of the irradiance distribution at the beam waist plane. Siegman [2] first proposed a similar but somewhat different treatment for evaluating the effective spot size of an optical beam to be introduced in the beam propagation factor M2 . This concept has been extended for evaluating the Rayleigh range elsewhere [3, 4]. At this point it results convenient to give a physical interpretation of the parameter s ðzÞ. In agreement with eq. (2) it may be expressed as s ðzÞ ¼ zR =jzj which is commonly denoted as the effective Fresnel number. This parameter was first introduced as a normalized axial coordinate to describe the irradiance distribution along the axis of a uniform plane wave diffracted by a circular opaque screen [1]. In this case the Fresnel number indicates the number of halfwave zones that fill the aperture as viewed from an axial point located at a distance z. Afterwards this parameter was extended to untruncated optical beams by considering the effective spot radius instead of the aperture radius [14, 13]. Then s ðzÞ may be used to characterized not only the limits of the far field but also every transverse pattern of a propagating beam. Finally, let us evaluate the proposed Rayleigh range for two practical cases such as a spherical Gaussian beam whose beam waist is located at origin, z ¼ 0, and has an effective spot radius given by w0, and a uniform, plane wave diffracted by a circular clear screen of radius a. By using eq. (22) it is found that pffiffiffi pw0 ; ð23Þ 2 zG R ¼ l and p a2 p ffiffiffi ; zC ¼ R 3 l
ð24Þ
respectively. Both are in good agreement with the results given previously [1]. It is pointed out that the simplicity of eq. (22) provides a tool for the analytical evaluation of the Rayleigh range of a great variety of either untruncated optical beams or electromagnetic waves truncated by nonuniform opaque screens.
6. Summary The paraxial wave equation is particularized to a new wave equation that is valid exclusively for the far field. A given inequality must be satisfied in order to assure the field propagates in the far-field region, which depends on the observation point and not on the transverse plane it belongs to. They would allow us to establish the boundaries between the near-field and the farfield regions. For completeness, higher-order wave equations have been presented in order to evaluate
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the amplitude distribution of a paraxial wave at points gradually closer to the near field. Additional integral formulas are deducted from the Fresnel diffraction integral to satisfy high-order wave equations. Finally, from this formalism it is reported on an expression which evaluates the distance from the beam waist plane where the Fraunhofer approximation can be applied, what is usually named as the Rayleigh range. Actually it is related with the concept of effective Fresnel number, and both were inferred from comparison of the far-field wave function and a firstorder field. The presented expressions have been applied to the particular cases of a spherical Gaussian beam and a uniformly-illuminated circular aperture. These results are in good agreement with those reported formerly. Acknowledgements. It is a pleasure to thank Albert Ferrando for helpful conversations. This work was supported by the Plan Nacional I+D+I (grant DPI2000-0774), Ministerio de Ciencia y Tecnologı´a, Spain. The author is also grateful to a Postdoctoral Grant from the Ministerio de Educaciı´on, Cultura y Deporte, Spain.
References [1] Siegman A: Lasers. University Science Books, Mill Valley 1986 [2] Siegman AE: New developments in laser resonators. Proc. Soc. Photo-Opt. Instrum. Eng. 1224 (1990) 2–14 [3] Be´langer P-A: Beam propagation and the ABCD ray matrices. Opt. Lett. 16 (1991) 196–198 [4] Pare´ C, Be´langer P-A: Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam. Opt. Commun. 123 (1996) 679–693 [5] Dragoman D, Dragoman M: Near and far field optical beam characterization using the fractional fourier transform. Opt. Commun. 141 (1997) 5–9 [6] Saghafi S, Sheppard CJR: The beam propagation factor for higher order Gaussian beams. Opt. Commun. 153 (1998) 207–210 [7] Herman RM, Wiggins TA: Rayleigh range and the M2 factor for Bessel Gauss beams. Appl. Opt. 37 (1998) 3398–3400 [8] Roundy C: Propagation factor quantifies laser beam performance comparing M2 of an actual laser beam to a pure TEM00 Gaussian beam allows beam-propagation characteristics to be accurately predicted. Laser Focus World 35 (1999) 119–124 [9] Goodman J: Introduction to Fourier Optics. McGraw-Hill, New York 1996 [10] Zapata-Rodrı´guez CJ, Martı´nez-Corral M, Andre´s P, Pons A: Axial behavior of diffractive lenses under Gaussian illumination: complex-argument spectral analysis. J. Opt. Soc. Am. A 16 (1999) 1–7 [11] Davis LW: Theory of electromagnetic beams. Phys. Rev. A 19 (1979) 1177–1179 [12] Cao Q, Deng X: Corrections to the paraxial approximation of an arbitrary free-propagation beam. J. Opt. Soc. Am. A 15 (1998) 1144–1148 [13] Martı´nez-Corral M, Zapata-Rodrı´guez CJ, Andre´s P, Silvestre E: Effective Fresnel-number concept for evaluating the relative focal shift in focused beam. J. Opt. Soc. Am. A 15 (1998) 449–455 [14] Carter WH: Focal shift and concept of effective Fresnel number for a Gaussian laser beams. Appl. Opt. 21 (1982) 1989–1994