Impact of spherical aberration on optical imaging - UV

IOP PUBLISHING

EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 29 (2008) 619–627

doi:10.1088/0143-0807/29/3/021

Simple demonstration of the impact of spherical aberration on optical imaging Isabel Escobar, Genaro Saavedra, Amparo Pons and Manuel Mart´ınez-Corral Department of Optics. University of Valencia, E46100 Burjassot, Spain E-mail: [email protected]

Received 22 November 2007, in final form 6 March 2008 Published 29 April 2008 Online at stacks.iop.org/EJP/29/619 Abstract

We present an experiment, well adapted for students of introductory optics courses, for the visualization of the impact of spherical aberration in the point spread function of imaging systems. The demonstrations are based on the analogy between the point-spread function of spherically aberrated systems, and the defocused patterns of 1D slit-like screens. (Some figures in this article are in colour only in the electronic version)

1. Introduction

The subject of monochromatic spherical aberration has attracted the interest of many scientists and physics teachers through the last century. In this sense, many papers have been published in journals addressed to physics teachers, in which the nature of spherical aberration and the equations to evaluate its magnitude are explained [1–6]. Besides, some experiments for the easy demonstration of spherical aberration addressed to students of introductory optics courses have been reported [6–8]. Note that these papers deal with the spherical aberration inherent to thin or thick lenses or to spherical mirrors. In recent years, and mainly due to the development of modern scanning optical instruments, such as scanning microscopes [9] or multilayer optical data storage systems [10], the other source of spherical aberration has achieved increasing importance. We refer to the spherical aberration that is induced when the beam emerging from a high numericalaperture (NA) converging lens objective is focused deeply on an interface between two media of different refractive indices. The study of this effect, which, to our knowledge, has never been addressed in a physics-teachers’ journal, is of great interest for students of optics. This is because the study has to be performed in a wave-optics formalism, and it has to take into account non-paraxial effects. The aim of this paper is then three-fold. First to show how to evaluate the spherical aberration in high-NA scanning optical instruments. The measurement of the focal spot of the c 2008 IOP Publishing Ltd Printed in the UK 0143-0807/08/030619+09$30.00

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Figure 1. When a high-NA objective is illuminated by a plane wave, the aperture stop is projected

onto the spherical principal surface.

instruments is not an easy task. A proof of it is the fact that such measurement is currently a hot topic in microscopy research (see for example [11]). For this reason the second aim of this paper is to demonstrate that the equation that accounts for the intensity along the optical axis of the focal spot in the presence of spherical aberration is, formally, the same as the one that describes a very simple 1D diffraction phenomenon. Our third objective is the carrying out of an experiment to illustrate, by means of the 1D diffraction phenomenon, the impact of spherical aberration in scanning optical instruments. The experiment has been carried out by the students of the fourth year of physics at the University of Valencia. Previous to the realization of the experiment, they studied the theoretical concepts connected with the focalization through a stratified dielectric medium. The realization of the experiment permitted the students to understand, easily, the importance of the spherical aberration.

2. Basic theory

When one deals with the description of modern optical scanning instruments in which light beams are focused and/or collected through high-NA focusing elements, such as microscope objectives, the calculation of the focal-spot intensity distribution cannot be accurately done by using paraxial tools and representing the objective as a thin lens. Instead, a microscope objective must be represented through its principal surfaces as shown in figure 1. The back principal surface is, as in the paraxial case, a plane surface. The front principal surface, S1, is a sphere of radius f centred at the focal point. As is well known, in most high-NA objectives the aperture stop is inserted at the back-focal plane. Then, the microscope objective transforms a monochromatic plane wave into a truncated spherical wavefront1 . The amplitude transmittance of the aperture stop is mapped onto S1. To calculate the amplitude distribution in the neighbourhood of the focus one can use the first equation of Rayleigh–Sommerfeld [13], which reconstructs the amplitude distribution in the vicinity of the focus as the superposition of secondary spherical wavelets proceeding from 1 In the paraxial regime a focusing system can be characterized through its principal planes [12]. In a non-paraxial context the principal elements are a plane and a spherical shell.

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all points on S1, namely

i eiks 2 d S, U (P1 ) (1) λ0 S1 s where s represents the distance from a typical point P1 of S1 to an arbitrary point P, which belongs to the focal volume. The function U (P1 ) represents the amplitude at P1, that is e−ikf , (2) U (P1 ) = p(P1 ) f with p(P1 ) being the pupil function. After straightforward calculations, which can be found for example in [14], one finds α√ 2π h(r, z) = −i cos θ p(θ )J0 (kr sin θ ) eikz cos θ sin θ dθ , (3) λ0 0 where λ0 is the vacuum wavelength, α is the maximum value of the aperture angle, and θ is related to the pupil plane radial coordinate, rp , through h(P ) = −

rp = f sin θ.

(4)

The focal-spot amplitude distribution is usually expressed in terms of the normalized coordinates nr sin α 2nz sin2 α/2 and zN = . (5) rN = λ0 λ0 In this case, α√ sin θ h(rN , zN ) = cos θ p(θ ) exp(−i2π W (θ )) J0 2π rN sin α 0 2 sin (θ/2) × exp −i2π zN 2 sin θ dθ, (6) sin (α/2) where some irrelevant factors, external to the integral, have been omitted. Note that in the above integral we have included a new phase term exp(−i2π W (θ )), which in the forthcoming analysis will account for phase distortion occurring during the focusing. Let us suppose now that the beam emerging from the objective is focused deeply into a medium of different refractive indices. To evaluate the phase distortions induced by the refractive-index mismatch we use a para-geometrical approach and operate over an arbitrary ray that is normal to the spherical wavefront and refracts at the interface obeying the Snell law n1 sin θ = n2 sin θ (see figure 2). The phase delay suffered by the ray is proportional to the optical path difference, namely 1 W (θ ) = (n1 l1 (θ ) − n2 l2 (θ )). (7) λ0 Following the classical approach by Sheppard and Cogswell [15] we expand the above expression into a power series of sin (θ/2), up to the fourth order. We obtain 2n1 n2 d (8) sin2 (θ/2) + 2(n2 + n1 ) 13 sin4 (θ/2) . W (θ ) = (n1 − n2 ) 1 + λ0 n2 n2 Thus, the amplitude distribution in the neighbourhood of the focus of a high-NA beam that is focused deep through an interface is given by α√ sin θ h(rN , zN ; w40 ) = cos θ p(θ )J0 2π rN sin α 0 4 2 sin (θ/2) sin (θ/2) × exp i2π w40 4 sin θ dθ. (9) − zN 2 sin (α/2) sin (α/2)

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Figure 2. Scheme for the evaluation of the phase distortions occurred in the focusing. The scheme

has been drawn for the case n1 > n2 .

Figure 3. Numerically evaluated contour plots of the intensity distribution in the focal spot corresponding to w40 = 0, −0.8, −1.6, −2.4 and −3.2.

In the above equation we have omitted an irrelevant constant phase factor, and have = zN − w20 , where defined the reduced axial coordinate as zN w20 =

2d n1 (n2 − n1 ) sin2 (α/2) λ0 n2

(10)

is the defocus coefficient that accounts for the focal shift suffered by the beam (in figure 2 the focus is shifted from B to B ). The coefficient of the spherical aberration is w40 =

n2 2d 2 n2 − n21 13 sin4 (α/2) . λ0 n2

(11)

As an example of the impact of the spherical aberration induced by the refractive-index mismatch, next in figure 3 we have represented the square modulus of equation (9). In our numerical simulation we have considered the following system parameters: N A = 1.4, α = 67◦ , λ0 = 632.8 nm, n1 = 1.52, n2 = 1.33, and d = 0, 5.1 µm, 10.2 µm, 15.3 µm and 20.4 µm, which give rise to the following values for the spherical-aberration coefficient: w40 = 0, −0.8, −1.6, −2.4 and −3.2. This figure clearly illustrates the strong influence of spherical aberration on the focal-spot shape. The experimental measurement of the 3D intensity distribution is not an easy task. In fact, the optimization of the experimental techniques for the measurement of such intensity


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Figure 4. Set of 1D diffraction patterns obtained when the light proceeding from a clear slit is focused by a cylindrical lens.

distribution is still a hot research topic. Thus, the realization of such an experiment is, nowadays, beyond the student’s reach. For this reason, in the next section we show that there exists a very simple diffraction phenomenon that is governed by equations that are similar to the above formulae. 3. Axial behaviour of spherically aberrated imaging systems

To simplify the problem under study, let us concentrate our attention on the axial points of the focal spot. We find that 0.5 h(rN = 0, zN ; w40 ) = q(µ) exp(i2π w40 µ2 ) exp[−i2π zN µ] dµ, (12) −0.5

which has been obtained after the nonlinear mapping µ=

sin2 (θ/2) − 0.5; sin2 (α/2)

√ q(µ) = p(θ ) cos θ .

(13)

The mapping transforms the angular interval [0, α] into the non-dimensional interval [−0.5, 0.5]. The same result as in equation (12) can be encountered in the analysis of 1D focusing systems as we show next. Consider the focusing system schematized in figure 4, in which a cylindrical lens is illuminated by a monochromatic plane wave. The aperture stop (a 1D slit in this case) is placed as usual at the front focal plane (FFP). Next we can make use of the well-known fact that the amplitude distribution at the back focal plane (BFP) of a lens, u(x; z = 0), is related to the amplitude distribution at its front focal plane through a Fourier transformation [13], namely /2 2π x , (14) p(x0 ) exp −i xx0 dx0 =p˜ u(x; z = 0) = λ0 fx λ0 fx −/2 where is the width of the slit, x0 is the transverse coordinate at the pupil plane and x is the transverse coordinate in the focal volume.

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Figure 5. Axial-intensity distributions corresponding to different amounts of spherical aberration.

The patterns were calculated according to equation (12).

The amplitude distribution in out-of-focus planes is obtained by convolving u(x; 0) with the free-space propagation amplitude impulse x k 2 u(x; z) = p˜ ⊗ exp i x . (15) λ0 fx 2z By expressing the convolution in its integral form and after straightforward maths, we find that 0.5 2π x 2 p(x¯ 0 ) exp i2π w20 x¯ 0 exp −i x x¯ 0 dx¯ 0 , (16) u(x; z) = λ0 fx −0.5 being x¯ 0 = x0 /. Also, we have introduced the well-known 1D defocus coefficient, defined as z2 x w20 =− . (17) 2λ0 fx2 x plays in equation (17) the same role as It is apparent that the 1D defocus coefficient w20 coefficient w40 in equation (12). Thus, we can conclude that the axial intensity distribution produced by a high-NA scanning optical instrument in which a certain amount of spherical aberration w40 is induced, is the same as the transverse intensity distribution obtained at a distance z = −2λ0 fx2 w40 /2 in the 1D focusing experiment. To illustrate this property in figure 5 we have plotted the on-axis intensity profiles corresponding to a high-NA (NA = 1.4, oil) microscope objective with a circular aperture stop and for increasing values of w40 . The similitude between these patterns and those shown in figure 4 is apparent2 .

4. Experimental procedure

Next we propose a very simple experiment which is well adapted to the level of undergraduate physics courses, and by which the students can visualize the influence of the spherical aberration in the performance of scanning optical instruments. In figure 6 we show the scheme of the experimental setup. For our experiment we used a 5 mW He–Ne laser (λ0 = 632.8 nm), an optical bench, a beam expander composed by a single-mode optical fibre and a collimating lens of focal length fc = 200 mm, a cylindrical lens of focal length fx = 100 mm, and a slit of width = 1 mm. A black and white CCD camera (Pulnix TM-765E) and a computer equipped with a frame grabber and the necessary software were used to capture the beam profile at different axial distances. 2

The only difference is that the axial intensity profile is not symmetric due to the factor fulfilment of the sine condition.

√ cos θ inherent to the


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Figure 6. Scheme of the experimental setup.

Figure 7. 1D intensity patterns obtained with the experimental setup.

The beam proceeding from the laser was expanded and illuminated the slit normally. The light-beam emerging from the slit was focused by the cylindrical lens, which was placed so that its FFP coincided with the slit. The CCD camera was mounted on a micrometric translation stage over the optical rail. This allowed the positioning, with high precision, of the CCD at different z positions. With this setup we recorded a set of 2D images corresponding to transverse patterns at different axial positions. Specifically we selected z = 0, −1 cm, −2 cm, −3 cm and −4 cm x = 0, 0.8, 1.6, which, according to equation (17), correspond to the defocus coefficients w20 2.4 and 3.2. Due to the 1D nature of the aperture and the focusing lens, the recorded diffraction patterns should not show any variation along the ordinate axis. However, in a real optical experiment this does not occur because: (a) both the slit and the cylindrical lens have a finite size; (b) the illumination beam is not a plane wavefront, but has rotational symmetry; (c) the noise inherent to an electronic detector has 2D nature. Thus, to maximize the signal-to-noise ratio we performed the average of recorded 2D patterns along the ordinate coordinate, which gives us a 1D function. In figure 7 we show the 1D intensity patterns. It is apparent from the figure that the degradation suffered by the 1D patterns increases with defocus. At the same time these experimental results illustrate the importance of the degradation suffered by the focal spot in high-NA optical instruments when scanning deep into a medium of different refractive indices.

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(a)

(b)

Figure 8. Actual transmittance of the filters: (a) high focal-depth filter, (b) axially super-resolving

filter.

Figure 9. 1D intensity patterns obtained with the high focal-depth filter (upper row) and with the

axially super-resolving filter (bottom row).

These results clearly illustrate how the 1D focusing pattern degrades with increasing defocus or, equivalently, how the axial point spread function (PSF) of a high-NA scanning optical instruments degrades due to an increasing amount of spherical aberration. This experimental setup can be useful, as well, for visualizing the impact of beam-shaping elements in spherically aberrated optical systems. Although easy to implement and very simple conceptually, beam shaping is a powerful tool usually aimed at improving the resolving power of imaging systems and/or their depth of field. A conscious study of properties of beamshaping elements can be found elsewhere [16, 17]. To visualize the behaviour of a given beam-shaping element of amplitude transmittance p(θ ), it is only necessary to build a slit-like filter with transmittance t (x0 ) obtained after the nonlinear mapping x0 =

sin2 (θ/2) − 0.5. sin2 (α/2)

(18)

We propose in this paper the carrying out of two experiments. One with an element designed for obtaining a high focal depth, p1 (θ ) and another with an element designed for obtaining an axial resolution, p2 (θ ) (see figure 8). Thus, we have to build two slit-like elements


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with amplitude transmittances t1 (x0 ) = 1 − 4x02 and t2 (x0 ) = 4x02 . The filters were fabricated with a high-contrast photographic film (MACO orthochromatic film). Following the same procedure as with the transparent slit, we obtained the intensity x = 0, 0.8, 1.6, 2.4 and 3.2. From patterns shown in figure 9 corresponding, again, to w20 these curves the student can easily understand that axial-resolving beam-shaping elements are very inconvenient in systems susceptible to suffering from spherical aberration. In contrast, the filters designed for obtaining a high focal depth provide the system with an important robustness against both defocus and spherical aberration. 5. Conclusions

In this paper we have shown that the impact of spherical aberration in modern high-NA scanning optical instruments can be easily visualized through the carrying out of a very simple laboratory experiment. The proposed experiment, in which different defocused patterns are generated in a 1D focusing setup, is very simple and is based on the formal analogy between two different phenomena. One is the axial intensity distribution in the focal spot generated when a monochromatic plane wave is focused, by a high-NA microscope objective, deep into a medium of different refractive indices. Another is the transverse intensity distribution obtained when the light proceeding from a transparent slit is focused by a low-NA cylindrical lens. Additionally we have used the experimental setup to visualize the impact of beamshaping elements in spherically aberrated optical systems. Acknowledgments

This work was funded by the Plan Nacional I+D+I (grant DPI2006-8309), Ministerio de Educaci´on y Ciencia. Isabel Escobar wishes also to acknowledge the same Institution for a pre-doctoral grant. References [1] Ehrenstein G 1975 On the derivation of spherical aberration Am. J. Phys. 43 745 [2] Hawkes P W 1978 On the nature of spherical aberration Am. J. Phys. 46 433 [3] Carpena P and Coronado A V 2006 On the focal point of a lens: beyond the paraxial approximation Eur. J. Phys. 27 231 [4] Trappe N, Murphy J A and Withington S 2003 The Gaussian beam mode analysis of classical phase aberrations in diffraction-limited optical systems Eur. J. Phys. 24 403 [5] Pozzi G 2001 Sommerfeld revisited: a simple introduction to imaging and spherical aberration in refracting surfaces of revolution and thin lenses Eur. J. Phys. 22 1 [6] Mak S Y 1986 Longitudinal spherical aberration of a thick lens Am. J. Phys. 55 247 [7] Hirsh F R 1945 A model to demonstrate spherical aberration of a concave spherical mirror Am. J. Phys. 13 267 [8] Lachaˆıne A R and Rochon P 1983 A simple demonstration of spherical aberration Am. J. Phys. 51 853 [9] Wilson T 1990 Confocal Microscopy (London: Academic) [10] Meinders E R, Mijiritskii A V, Pieterson L V and Wuttig M 2006 Optical Data Storage. Phase-Change Media and Recording (Berlin: Springer) [11] Dusch E, Dorval T, Vincent N, Wachsmuth M and Genovesio A 2007 Three-dimensional point spread function model for line-scanning confocal microscope with high-aperture objective J. Microsci. 228 132 [12] Longhurst R S 1973 Geometrical and Physical Optics (London: Longmans Green) [13] Goodman F W 1996 Introduction to Fourier Optics (New York: McGraw-Hill) [14] Gu M 2000 Advanced Optical Imaging Theory (Berlin: Springer) [15] Sheppard C J R and Cogswell 1991 Effects of aberrating layers and the tube length on confocal imaging properties Optik 87 34 [16] Jacquinot P and Roizen-Dossier B 1964 Apodisation in Progress in Optics vol 3 ed E Wolf (Amsterdam: North-Holland) chapter 2 [17] Dickey F M, Holswade S and Laser C 2000 Beam Shaping: Theory and Techniques (New York: Dekker)