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Optics Communications 284 (2011) 2455–2459
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Finite-area centroid propagation in homogeneous media and range of validity of the Optical Ehrenfest's Theorem Jorge Ares a, Justo Arines a,b, Salvador Bará b,⁎ a b
Departamento de Física Aplicada (Área de Óptica), Universidad de Zaragoza, C\ Pedro Cerbuna 12, 50009, Zaragoza, Spain Área de Óptica, Dept. de Física Aplicada, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain
a r t i c l e
i n f o
Article history: Received 17 November 2010 Received in revised form 14 January 2011 Accepted 19 January 2011 Available online 4 February 2011 Keywords: Ehrenfest's theorem Wavefront Sensing Hartmann–Shack Aberrometry Centroid
a b s t r a c t The estimation of unknown wavefronts using Hartmann–Shack and other wavefront sensing devices relies on the fact that the irradiance centroids in homogeneous media propagate along straight lines whose slopes are given by the irradiance-weighted spatial average of the local wavefront slopes, a result which is a particular case of the Optical Ehrenfest's Theorem. The strict analytic validity of this theorem, however, heavily depends on the use of an infinitely extended integration region to compute the irradiance centroid. In this paper we describe the equation governing the centroid propagation in homogeneous media when it is computed within a finite region of the detection plane and determine the minimum size of this region sufficient to ensure an approximate fulfillment of the Optical Ehrenfest's Theorem. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The Ehrenfest's theorem regarding the expectation value of the position vector of a particle in non-relativistic quantum mechanics [1] can be applied to study the propagation of light centroids within the parabolic approximation to the wave equation. This result (henceforth referred to as the Optical Ehrenfest's Theorem) has its roots in the formal analogy between the Schrödinger equation in a two-dimensional space and the tridimensional paraxial wave equation for harmonic fields. The Optical Ehrenfest's Theorem (OET) states that the irradiance centroid of any light beam propagates in a weakly inhomogeneous medium according to the ray equations of geometrical optics, with an effective refractive index given by the irradiance-weighted spatial average of the actual index distribution [2]. This light propagation feature has enabled the development of a wide range of metrological applications, particularly in the areas of deflectometry, triangulation and wavefront sensing. Hartmann–Shack wavefront sensors are probably one of the most versatile and widely used devices whose performance is based on this theorem, constituting nowadays the key sensing element of many aberrometric and adaptive optical systems deployed in advanced research, industrial or clinical settings. In its simplest version the OET describes the centroid propagation in a homogeneous medium. In order to write it in its explicit form let's ⁎ Corresponding author. Tel.: + 34 881813525; fax: + 34 981590485. E-mail addresses:
[email protected] (J. Ares),
[email protected] (J. Arines),
[email protected] (S. Bará). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.01.054
consider an arbitrary harmonic optical field u(r) = a(r)exp{ikW(r)} at the plane z = 0, where a(r) and kW(r) are the field amplitude and phase, respectively, and k = 2π/λ, being λ the wavelength, with associated irradiance I(r) = |u(r)|2. The irradiance centroid at this plane, rc(0) = (xc, yc), is given by: 2
rc ð0Þ = ð1 = EÞ ∫ IðrÞrd r;
ð1Þ
∞
2
where E = ∫ IðrÞd r is the total radiant flux and d2r stands for the ∞
differential surface element at this plane. The centroid position ρc(z) = (ρcx, ρcy) in a parallel plane located at an arbitrary distance z away is similarly defined from the corresponding irradiance distribution I(ρ) as 2
ρc ðzÞ = ð1 = Ez Þ ∫ IðρÞρd ρ;
ð2Þ
∞
where Ez is the total flux at this plane (Ez = E for nonabsorbing media). The irradiance I(ρ), in turn, is given by the squared modulus of the diffracted field u(ρ) obtained by applying the Fresnel integral to the initial field u(r). Performing these operations one gets, after some algebra, the final OET result [3–5]: 2
ρc ðzÞ = rc ð0Þ + z ∫ iðrÞ∇W ðrÞd r;
ð3Þ
∞
where ∇ ≡ (∂/∂ x, ∂/∂ y) is the transversal gradient operator acting on the r coordinates and iðrÞ = IðrÞ = E is the irradiance normalized to the total flux.
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J. Ares et al. / Optics Communications 284 (2011) 2455–2459
Eq. (3) shows that within a homogeneous medium the light centroid propagates along a straight line, whose z-slope is the irradiance-weighted spatial average of the local wavefront slope ∇W ðrÞ at the input plane. This key result allows, under certain additional simplifications [4] and an adequate sampling of the input plane by using a Hartmann screen or a Hartmann–Shack microlens array [6,7] to extract information about the local wavefront slopes ∇W ðrÞ by measuring physically observable magnitudes (e.g. the centroid positions at two separate planes, or the in-plane centroid displacements between a reference and an unknown wavefront). From a basic standpoint, Eq. (3) also allows to describe the performance of gradient-based wavefront sensors using the language of geometrical optics, with the proviso that instead of the conventional geometrical rays, the description shall refer to the centroid propagation paths. The analytical validity of Eq. (3) is strongly tied to the definition of the centroid as an infinite-limit integral. Said otherwise, the irradiance has to be measured in unbounded regions at planes z = 0 and z to ensure that the OET version given in Eq. (3) strictly describes the centroid propagation. Obviously this situation cannot be met in practice, being the centroid computation region mainly limited by the detector size, the space allocated in the detector for the measurement of the irradiance spots of each individual sampling pupil or microlens and by further restrictions in region size resulting from the thresholding levels used to counteract the effects of detector noise [8]. In this paper we analyze from a basic standpoint the effects of computing the irradiance centroid on a finite region of the detector, and establish the minimum size of the centroid integration area in order to ensure that Eq. (3) reasonably holds. 2. Finite-area centroid propagation in homogeneous media If the irradiance centroid is computed within a finite region instead of across the whole plane, its value ρ'c(z) will in general differ from that given by Eq. (2). Defining the integration region by the binary function g(ρ), valued 1 for points within it and 0 otherwise, the new finite-area centroid is given by 0 0 2 ρc ðzÞ = 1 = Ez ∫ g ðρÞI ðρÞρd ρ;
ð4Þ
∞
0
where I(ρ) = u*(ρ)u(ρ), ‘*’ stands for complex conjugate, and Ez = 2
∫ g ðρÞIðρÞd ρ is the radiant flux through g(ρ).
∞
Within the framework of the Fresnel approximation, the field u(ρ) at plane z is related to the initial field u(r) by
ik 2 2 jr−ρj d r uðρÞ = ð1 = iλzÞexpðikzÞ∫uðrÞexp 2z
ð5Þ
Substituting Eq. (5) for u(ρ) in I(ρ) = u*(ρ)u(ρ) and introducing the result into Eq. (4) we get for the numerator 2
∫ g ðρÞIðρÞρd ρ = ∞
1 λz
2 −z ik
0 0 −ik 0 2 2 r−r 2 2 0 r −r d rd r ; ∫ ∫ uT r uðrÞexp ∇G 2z λz ∞ ∞
ð6Þ and, for the denominator, 2
∫ g ðρÞIðρÞd ρ = ∞
1 λz
2
0 0 −ik 0 2 2 r−r 2 2 0 d rd r ; G ∫ ∫ uT r uðrÞexp r −r λz 2z ∞ ∞
is the Fourier transform of the integration region g(ρ). Recall that ∇ operates on the r coordinates. Both r and r' are coordinates in the initial plane. Integrating by parts on r in Eq. (6) and rearranging terms in Eqs. (6)–(7) we get:
2
1 λz
∫ g ðρÞIðρÞρd ρ = ∞
2
∫ g ðρÞIðρÞd ρ = ∞
1 λz
2 z k 2 2 ∫ ∇ uðrÞexp i r Bðr; zÞd r; ik ∞ 2z
2
k 2 2 ∫ uðrÞexp i r Bðr; zÞd r 2z ∞
ð9Þ
ð10Þ
where the dependence on G is accounted for through the function 0 r−r0 k 2 2 0 uT r exp −i r 0 d r : Bðr; zÞ = ∫ G 2z λz ∞
ð11Þ
Substituting Eqs. (9)–(10) into Eq. (4), we finally get the expression describing the z evolution of the finite-area centroid ρ'c(z): 0
ρc ðzÞ = ∫ bðr; zÞ ∞
i k 2 2 ∇aðrÞ + zaðrÞ∇W ðrÞ + aðrÞr exp½ikW ðrÞexp i r d r ik 2z
h z
ð12Þ where b(r,z) is a normalized version of the G-dependent function B(r,z), defined as k 2 2 bðr; zÞ = Bðr; zÞ = ∫ uðrÞexp i r Bðr; zÞd r: 2z ∞
ð13Þ
Note that if the centroid computation region tends to the whole plane ρ, i.e. if g(ρ) tends to 1 for all ρ, then G½ðr−r0 Þ = ðλzÞ→ðλzÞ2 δðr−r0 Þ, and from Eqs. (11) to (13) and the fact that ρ'c(z) is a real-valued vector it is straightforward to recover the Optical Ehrenfest's Theorem in the form given by Eq. (3). 3. Size of the centroiding region and range of validity of the Optical Ehrenfest's Theorem As stated above, a sufficient condition to recover the OET given by Eq. (3) from the centroids computed within a finite region g(ρ) of the detection plane is that the Fourier transform of g(ρ), G½ðr−r0 Þ = ðλzÞ, operate as a Dirac-delta distribution within the integrand of Eq. (11). This can be achieved in an approximate way if G is a narrowly peaked – and perhaps oscillatory – function centered at the origin such that its spatial width is much smaller than the characteristic length over which the remaining term of the integrand does noticeably change. Under these circumstances the function G½ðr−r0 Þ = ðλzÞ effectively sifts out of Eq. (11) the value uTðrÞexp −iðk = 2zÞr2 required to deduce the OET (Eq. (3)). To get an estimate of the minimum size of the centroiding area required to achieve this goal, let's assume that g(ρ) can be described as a region of shape q(ρ) and typical half-width w. To get a broader scope, let's further assume that its center is displaced from the origin to any arbitrary point ρ0 in the detector plane. In this case g(ρ) = q (ρ − ρ0) and from Eq.(8) and the shifting theorem, its Fourier transform is given by
ð7Þ
r−r0 2π r−r0 0 r−r ⋅ρ0 Q = exp −i ; G λz λz λz
ð8Þ
where Q ððr−r0 Þ = ðλzÞÞis the Fourier transform of q(ρ) with the notation of Eq. (8). Let's also assume that q(ρ) is chosen in such a way that its Fourier transform approaches asymptotically a Dirac-delta as
ð14Þ
where r−r0 2π 0 2 r−r ⋅ρ d ρ = ∫ g ðρÞexp −i G λz λz ∞
J. Ares et al. / Optics Communications 284 (2011) 2455–2459
w tends to infinity. A simple choice for q(ρ) fulfilling this condition can be e.g. the two-dimensional rectangular function of half-width w qðρÞ = rect ðρx = 2wÞrect ρy = 2w ;
becomes of order 2π/N (with N≫1). This can be found by solving the equation
ð15Þ
ε2 λz = 0: + ε⋅DðrÞ− 2 N
ð16Þ
For each r the worst-case situation (in the sense of giving a smaller ε) happens when ε and D(r) are collinear, so that ε·D(r) = εD(r). Solving Eq. (20) for ε and imposing that the width (λz/w) of the peak of Q be much smaller than ε we get the following condition:
with its associated Fourier transform h i 2 Q ðνÞ = ð2wÞ sinc½2wνx sinc 2wνy ;
where sinc(α) = sin(πα)/(πα) and ν = (νx,νy) is the spatial frequency vector. Substituting Eq. (14) into Eq. (11) we get:
∞
( " #) 0 0 0 0 2 0 r−r W r r−r0 r2 a r d r: ⋅ρ0 + + exp −i2π λz 2λz λz λ
ð17Þ The narrowly peaked function Q in the integrand has its maximum at r' = r and its typical full-width is of order (λz/w). The characteristic length over which the rest of the integrand noticeably changes around r' = r is dominated by the exponential factor, which varies with r' much faster than the real amplitude factor a(r') for most cases of interest. Defining for each point r' its position vector ε relative to the location of the peak of Q as r' = r + ε, expanding W(r') around r' = r as W ðr + εÞ≈W ðrÞ + ∇W ðrÞ⋅ε and substituting ε for r' as the variable of integration in Eq. (17), we can rewrite the ε-dependent part of the exponential as ( exp −i
" #) 2 2π ε + ε⋅DðrÞ ; λz 2
ð18Þ
where ε is the modulus of ε and DðrÞ = r + z∇W ðrÞ−ρ0 :
ð19Þ
Note that D(r), as defined in Eq. (19), is just the position vector r + z∇W ðrÞ of the impact point on the detector plane of the paraxial geometrical ray passing through r at the initial plane, measured in a reference frame with origin at the center ρ0 of the centroiding region g(ρ) (Fig. 1). An estimate of the size of the region around the Q peak where the change of the exponential term (Eq. (18)) can be considered negligible can be obtained by finding the value of ε for which the argument of the exponential
DðrÞ +
: 1 2λz 2 −DðrÞ N
ð21Þ
If the size of the integration area used to compute the centroid fulfills this condition, the function Q will behave like an effective Dirac-delta distribution within the integrand of Eq. (11) for that given value of r. In order to ensure that this behavior holds for all r we choose for D(r) the worst-case condition (in the sense of requiring a bigger w), which corresponds to the r for which D(r) = Dmax. By imposing this condition and rearranging terms in Eq. (21) we finally get the requirement for w, the half-width of the centroiding region g(ρ), in order to keep the approximate validity of the OET under finite centroid computation: λz = Dmax : 1 2λz 2 1+ −1 ND 2max
ð22Þ
w≫
Two particular limiting cases of Eq. (22) are of interest. One of them arises when D2max ≪ λz that is, when all geometrical rays from the initial plane are concentrated on the detector within a space around ρ0 smaller pffiffiffiffiffiffithat the size of a Fresnel zone as seen from the initial plane λz . In this case we get from Eq. (22): rffiffiffiffiffiffiffiffiffi" rffiffiffiffiffiffiffiffi Nλz N N 2 + w≫ 1 + Dmax − Dmax 2 2λz 4λz
#
⋯
:
ð23Þ
The extreme limiting situation corresponds to the case of perfect geometrical focusing precisely onto ρ0 (Dmax = 0 since D(r) = 0 for all r), when p the minimum required size of the centroiding region becomes of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi order Nλz = 2, i.e., the size of several Fresnel zones. The opposite limiting case arises when the geometrical rays from the initial plane impact on the detector verifying D2max ≫ λz. This happens for general non-focused fields whose associated geometrical rays fill a detector area much wider than a Fresnel zone, but also for geometrically perfect focused ones if the focus lies at several Fresnel zones away from the origin of the centroiding region ρ0 . Then from Eq. (22) we get: w≫NDmax :
Fig. 1. Finite centroiding region g(ρ) centered at ρ0 in the detection plane, and impact point D of an arbitrary geometrical ray coming from r in the input plane, measured with respect to ρ0.
ð20Þ
λz
w≫ 2
Bðr; zÞ = ∫ Q
2457
ð24Þ
In this case the centroiding region has to be sufficiently wide as to enclose several times the impact region of all geometrical rays coming from the initial plane. As an example to illustrate this approach, we show in Fig. 2 (left) the diffracted irradiance at an observation plane located z = 150 mm away from a clear semicircular aperture of radius R = 1 mm illuminated by an on-axis plane wave of wavelength λ = 0.5 μm, computed numerically within the Fresnel approximation. The vertical side of the figure corresponds to 3.75 mm in the observation plane. In Fig. 2 (right) we superimposed onto the former image the geometrically illuminated region (semicircle of radius R = 1 mm), a particular instance of centroid integration region (a square centered at the origin
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J. Ares et al. / Optics Communications 284 (2011) 2455–2459
Fig. 2. Left: Diffracted irradiance at an observation plane located z = 150 mm away from a clear semicircular aperture of radius R = 1 mm illuminated by an on-axis plane wave of wavelength λ = 0.5 μm. Right: Superimposed onto the former image, the geometrically illuminated region (semicircle of radius R = 1 mm), a particular instance of centroid integration region (square centered at the origin of coordinates, pffiffiffiffiffiffidrawn for the particular half-width w = R), and a circle in the upper left corner representing the size of a Fresnel zone in this plane as seen from the aperture plane (zone radius λz = 0.274 mm).
of coordinates, drawn for the particular half-width w = R), and a circle in the upper left corner representing the approximate size of a Fresnel zone p ffiffiffiffiffiffi in this plane as seen from the aperture plane (zone radius λz = 0.274 mm). The true centroid coordinates can be found analytically by taking into account that in this particular example the actual centroid propagates along the Z axis according to Eq. (3) with zero slope and hence its transverse coordinates – which do not depend on z – can be computed using the initial irradiance distribution at the semicircular aperture. This gives for the true centroid the position (0.424, 0) mm in cartesian coordinates of the observation plane. In Fig. 3 we show that integrating the diffracted irradiance distribution in a sufficiently wide area the centroid position is recovered as expected from the previous considerations. In this figure we plot the centroiding error (i.e. the distance from the computed to the true centroid, in microns) as a function of w (the half-width of the square centroiding window in mm) for this particular diffracted beam. The first vertical line, at w = 1 mm, labels the minimum half-width at which the centroid integration region fully includes the geometrically illuminated region. Using this halfwidth the centroiding error is still about 10 μm. The second vertical line is drawn at w = 2.37 mm, that is, the half-width of the former
Fig. 3. Centroiding error (microns) as a function of w (half-width of the centroiding integration region, in mm) for the example in Fig. 2. The first vertical line, w = 1 mm, labels the minimum half-width of the centroid integration region (square drawn in Fig. 2, right) that fully includes the geometrically illuminated region. The second vertical line at w = 2.37 mm corresponds to the half-width of the former region plus five Fresnel zone radii.
region plus five Fresnel zone radii. With this increased half-size the centroiding error falls already below the pixel resolution (3.75 μm) used in the calculations. 4. Conclusions We have deduced in this paper the general Eq. (12) describing the propagation of the irradiance centroid of a light field when computed within a finite integration region g(ρ). We also deduced the general condition (22) on the size of the centroiding region allowing to restore the validity of the analytical version of the Optical Ehrenfest's Theorem, which in itself assumes an infinitely extended centroid computation area. It is found that the centroid computation region has to be wide enough as to sufficiently enclose the geometrically pffiffiffiffiffiffi illuminated area with additional contributions of order λz to account for diffraction effects. As far as these conditions are satisfied, as expressed by Eq. (22), the conventional expression for the OET given by Eq. (3) can be applied to the determination of the irradianceweighted spatially averaged wavefront slopes from the measurement of observable magnitudes as the irradiance centroids produced by a given wave – or a portion thereof – in two separate parallel planes or the in-plane centroid displacements between a reference and an unknown wave. Note that these general results are valid within the framework of the Fresnel approximation, and that no particular assumptions have been made about the shape W(r) of the wavefronts under test. The use of the Fresnel approximation [9] of the Sommerfeld–Rayleigh diffraction integral is well justified in many practical wavefront sensing applications: both Hartmann-like and Hartmann–Shack sensors usually work in the low Fresnel-number regime, where the Fresnel approximation is known to provide accurate results [10]. The combination of the low numerical aperture of the microlenses used in most devices and their relatively tight focal regions overall ensure paraxial propagation. Typical arrays for Hartmann–Shack sensors have microlenses of lateral sizes about several hundred microns and focal lengths in the ten millimeters range, leading to slowly converging wavefronts (often f/20 or slower) and Fresnel numbers (R2/λz) of order 10 and below (see e.g. the system parameters in [11]). The general requirements on the size of g(ρ) can be clearly alleviated in some particular situations, for instance when the irradiance distributions at the detection plane have enough degree of symmetry. This is the case, among others, of the distributions produced by symmetric apertures illuminated by possibly tilted wavefronts with otherwise center-symmetric aberrations. In those cases, relevant information about the average tilt can be obtained just
J. Ares et al. / Optics Communications 284 (2011) 2455–2459
by finding the center of symmetry of the corresponding irradiance distribution, which coincides with the actual centroid. On the other hand, if some knowledge about the overall shape of the focal spots is available a priori, matched filters [11] can be efficiently used to determine the centroid positions using relatively small observation regions. Acknowledgments This work was funded by the Spanish Ministerio de Ciencia e Innovación (MICINN) grant FIS2008-03884. Dr. J. Arines wants to acknowledge financial support from the Isidro Parga Pondal Program 2009 of the Galician Government, Xunta de Galicia, Galicia (Spain).
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