1. Introduction

Reaching the smallest possible focal spot size is useful in many applications, such as in high-resolution microscopy [1] and lithography [2]. Such sharper focal spots can be generated when the incident optical beam is focused in such a way that the rays are directed toward the focal point from nearly all directions. This requires a focusing element that subtends a solid angle approaching 4π srad, such as a high numerical aperture parabolic mirror [3,4]. Nonetheless, the parabolic reflector is rarely exploited for laser focusing, probably because of the difficulty in manufacturing high-aperture paraboloidal mirrors of sufficient optical quality. In fact, a paraboloidal surface showing deviations from ideal curvature that are smaller than the laser wavelength is needed to achieve a diffraction-limited focal spot. Still, it has been verified that a confocal microscope using a high numerical aperture parabolic mirror can deliver excellent images with only small deviations from the calculated focal electric energy distribution [5,6]. Therefore, it appears that a parabolic mirror objective has a promising future as the focusing element in high-resolution confocal microscopy.

Focusing a radially polarized beam with a parabolic mirror can further reduce the focal area. In fact, it has been pointed out that smaller spot sizes can be achieved with a radially polarized beam instead of a linearly polarized beam [7–10]. On the one hand, when a linearly polarized beam is strongly focused, the longitudinal component of its electric field is not cylindrically symmetric and, as a consequence, the focal spot becomes asymmetrically deformed and elongated in the direction of the polarization. On the other hand, when a radially polarized beam is tightly focused, the radial symmetry of the electromagnetic fields yields a perfectly symmetric focal spot with a strong longitudinal electric field component centered on the optical axis, resulting in a reduction of the area of the focal region. Furthermore, it has been shown experimentally and theoretically that parabolic mirrors can focus a radially polarized laser beam to a considerably tighter focal spot than aplanatic lenses do [11,12]. Thus the use of a high numerical aperture parabolic mirror with a radially polarized beam is very well suited to achieve minimum focal spot size [13].

A collimated beam whose axis of propagation is parallel to the axis of revolution of a parabolic mirror is, by definition, perfectly focused at the focal point of the mirror without any aberration in the geometrical optics approximation. However, when the propagation axis of the incident collimated beam is slightly tilted with respect to the mirror axis, the rays are no longer focused at a single point (Fig. 1 ): rather, they form a caustic (more precisely a catacaustic), which is defined as the envelope of light rays reflected by the mirror [14,15]. When the angle of incidence of the illuminating collimated beam with respect to the axis of revolution of the parabolic mirror is very small, the electromagnetic field distribution can be regarded as a focal spot modestly affected by optical aberrations. The amplitude distribution of the fields of the focused beam is therefore deformed with respect to the amplitude distribution that would be observed if the angle of incidence were zero. It should be noted that we are considering exclusively the case of a simple reflection produced by the parabolic mirror; the case of multiple reflections is beyond the scope of this paper (for multiple reflections, see for example Ref. [16].).

 

Fig. 1 An exact ray tracing shows that, when the incident rays of a collimated beam are not perfectly parallel to the axis of revolution of the mirror, the focused rays form a catacaustic.

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Lieb and Meixner pointed out that the parabolic mirror shows strong coma if it is not illuminated exactly along the mirror axis [5]. Thus the alignment of the propagation axis of the incident beam is critical in order to use a parabolic mirror as a focusing device. To calculate the electric field of a focused beam in the focal region of a slightly tilted parabolic mirror, Lieb and Meixner included an additional phase factor into the integral representation of the focused electric field, in such a way as to take into account the tilted wavefront. Nonetheless, they did not provide the explicit form of the phase factor.

The electromagnetic field distribution in the focal region of an aberration-free high numerical aperture parabolic mirror has been investigated by a few authors [17–19]. However, to the best of our knowledge, a detailed theoretical study of the focusing properties of a parabolic mirror used with an incident beam having a nonzero angle of incidence is not available in the literature. It is of interest to obtain an explicit expression for the intensity distribution in the focal region of a slightly tilted parabolic mirror, since this may give a clearer understanding of the effect of the angle of incidence on the focusing properties of a parabolic mirror as such a focusing element may be increasingly used in microscopy systems. The aim of this work is to provide a clear insight into the focusing of the lowest-order radially polarized beam with a parabolic mirror when the angle of incidence is nonzero.

This paper is organized as follows. In section 2, we apply the Richards–Wolf theory to analyze the electric field of a focused beam in the focal region of a parabolic mirror. In section 3, we provide the expression of the electric field of a tightly focused TM₀₁ beam having a nonzero angle of incidence on the mirror. In section 4, we use the expression presented in section 3 to investigate in detail the characteristics of the focused TM₀₁ beam.

2. Focusing a collimated beam with a slightly tilted parabolic mirror

The Richards–Wolf theory consists in a vector diffraction integral that can be applied to describe the electromagnetic fields of strongly focused beams [20]. We consider an incident collimated beam, whose electric field has a given state of polarization and amplitude distribution in the entrance pupil of a parabolic mirror of focal length f. At the exit pupil of the mirror, the beam converges toward the focal region. The coordinate system (r,ϕ,z) is attached to the parabolic mirror and its origin is centered at the focal point of the mirror. The Richards–Wolf vector field equation is valid under the assumption of the Debye approximation, according to which the field can be determined as a superposition of plane electromagnetic waves. The Debye approximation is applicable for high numerical aperture systems when the focal length is much larger than the wavelength in vacuum λ of the beam, i.e. kf >> 1, where k = 2π/λ is the wave number of the illumination [21]. This condition is always satisfied in optical microscopy and astronomy. Explicitly, the electric field E(r,ϕ,z) in the neighborhood of the focus is an integral over a vector field amplitude [5,8,22,23]:

In the following, we consider the focusing with a parabolic mirror whose axis of revolution coincides with the Z-axis. For such a focusing system, the aperture angle α _max can be greater than π/2 and may approach π when the transverse dimensions of the mirror are very large and the focal length is relatively short. The apodization factor of a parabolic mirror is given by [5,11,24,25]

The height $r_{\infty }$ of a ray of the incident collimated beam in the entrance pupil plane is in general related to the angle α subtended at the focus by $r_{\infty }=fg(\alpha )$. From the definition of a paraboloid, the function g(α) in the case of a parabolic mirror is given by [25]

For comparison, the function g(α) in the case of an aplanatic lens is $g(\alpha )=\sin \alpha$ [19,21]. The Cartesian coordinates in the pupil plane are therefore $x_{\infty }=r_{\infty }\cos \beta$ and $y_{\infty }=r_{\infty }\sin \beta$ (Fig. 2 ). In some situations, it is useful to characterize the optical system by its f-number, defined by $f/\#=f/D$, where f is the focal length of the optical system and $D=2{(r_{\infty })}_{\max }$ is the diameter of the entrance pupil. Using Eq. (3), the f-number of the parabolic mirror is given by $f/\#={\left[2g({\alpha }_{\max })\right]}^{-1}=\frac{1}{4}\cot \left(\frac{1}{2}{\alpha }_{\max }\right)$.

 

Fig. 2 The height $x_{\infty }$ of a ray of the incident collimated beam is related to the polar angle α and the focal length f of the parabolic mirror.

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We now consider an incident collimated beam having an angle of incidence δ with respect to the Z-axis. In the following, we assume that the angle δ is very small so that the small-angle approximation applies, i.e. $\sin \delta \approx \delta$ and $\cos \delta \approx 1$. When the propagation axis of a collimated beam is tilted with a very small angle with respect to the Z-axis in the x-z plane (see Fig. 2), the field distribution acquires an additional phase factor given by $\exp (-jk{x}_{\infty }\delta )$. The coordinate $x_{\infty }$ can be written in terms of the angular spherical coordinates α, β as $x_{\infty }=fg(\alpha )\cos \beta$.

The function $\Phi (\alpha ,\beta )\equiv f\delta g(\alpha )\cos \beta$ can be seen as the aberration function of the optical system. Using Eq. (3), the aberration function of the parabolic mirror is then $\Phi (\alpha ,\beta )=2f\delta \tan (\frac{1}{2}\alpha )\cos \beta$. To get some insight into the form of this particular aberration function, let us expand the function $2\tan (\frac{1}{2}\alpha )$ as a power series of $\sin \alpha$: $2\tan (\frac{1}{2}\alpha )\approx \sin \alpha +\frac{1}{4}{\sin }^3\alpha +\dots$ for $0\le \alpha \le \pi /2$. Hence, the first terms of the aberration function are

The first term corresponds to the distortion, which displaces the position of the focal spot by an amount fδ along the X-axis with respect to the focus of the parabolic mirror [26,27]. The second term is associated to the primary coma whose aberration coefficient is $\frac{1}{4}f\delta$ [27,28]. Higher-order terms correspond to higher-order coma such as secondary coma, etc. Therefore, since the focal length f of the mirror is large (many wavelengths), we expect that the tilting of the propagation axis of the collimated beam incident on a parabolic mirror results in an electric energy density affected by severe coma, even for small values of the angle of incidence δ. For a rigorous analysis, one should include in the present treatment the effect of the change in the apodization factor resulting from the presence of aberrations; however, this effect can be neglected because it is assumed that q(α)δ << 1 for $0\le \alpha \le {\alpha }_{\max }$.

When a collimated beam is incident on an aplanatic lens with a nonzero angle of incidence, the field distribution in the focal plane does not suffer from coma. In fact, the aberration function of the aplanatic lens is $\Phi (\alpha ,\beta )\equiv f\delta \sin \alpha \cos \beta$, because $g(\alpha )=\sin \alpha$ in this case and it corresponds only to distortion. Thus, the shape of the field distribution at the focus of an aplanatic lens remains unchanged by a nonzero angle of incidence, but the field distribution is shifted along the X-axis in the focal plane.

The specific form of the aberration function defined by Eq. (4) can be taken into account in such a way that it does not formally alter the aberration-free field. Substituting the aberration function $\Phi (\alpha ,\beta )=f\delta g(\alpha )\cos \beta$ (where g(α) is given by Eq. (3) for the parabolic mirror) in the argument of the exponential function defined by Eq. (1c) yields

The trigonometric identity $a\cos \beta +b\sin \beta ={(a^2+b^2)}^{1/2}\cos (\beta -\theta )$ where $\tan \theta =b/a$ can be used to rewrite the expression between the brackets in Eq. (5):

The function $\Gamma (\alpha ,\beta )$ contains all the dependence over the azimuthal angle β. In light of Eq. (6a), the function $\Gamma (\alpha ,\beta )$ has the same form as in the case of an aberration-free electric field. In other words, Eq. (1c) with $\Phi =0$ is formally identical to Eq. (6a), provided that the substitutions $r\to \tilde{r}(\alpha )$ and $\varphi \to \tilde{\varphi }(\alpha )$ are done.

3. Focusing a TM₀₁ beam with a slightly tilted parabolic mirror

We now apply the Richards–Wolf theory to the special case of a slightly tilted TM₀₁ beam focused with a parabolic mirror. The TM₀₁ beam is a transverse magnetic (TM) beam and it is the lowest-order member of the family of the radially polarized beams. The TM₀₁ beam in particular is analyzed in detail because of its practical importance compared to other transverse magnetic, transverse electric or linearly polarized beams. As mentioned before, Quabis et al. showed that strong focusing of radially polarized light leads to tighter spot sizes than is possible with linearly polarized light [7]. Because of their remarkable focusing properties, TM₀₁ beams are of substantial interest, for example, in high-resolution microscopy. Furthermore, the electric field of a strongly focused TM₀₁ beam has a significant longitudinal component that can be used in particle trapping and electron acceleration [29].

Many methods have been demonstrated to generate TM₀₁ beams in the laboratory. To name a few, a TM₀₁ beam can be produced interferometrically, outside the laser cavity, with a Mach–Zehnder interferometer which allows the coherent superposition of two orthogonally polarized Laguerre–Gaussian beams of order (0,1) of different parity with the same beam waist [30]. A technique to produce pseudo-radially polarized beams requires a polarization converter consisting in four half-wave plates, one in each quadrant [9]. Besides, radially polarized beams may be generated directly from a laser by inserting axially-symmetric optical elements with suitable polarization selectivity in the laser cavity; such elements include a conical reflector used as a resonator mirror, a conical Brewster window and a birefringent c-cut laser crystal [31].

Lieb and Meixner presented electric energy densities in the focal region of a parabolic mirror in the case of a TM₀₁ beam focused with an angle of incidence whose value approximately lies between 0° and 0.016° [5]. Their calculated electric energy density distributions exhibit asymmetry that is characteristic of coma. In this section, we show a detailed approach to obtain such results. Since the maximum acceptable value for the angle of incidence δ is close to 0.02° for practical applications requiring high resolution (as it will be apparent in Section 4), it is reasonable to consider that the amplitude l ₀(α) of the field distribution and the unit vector $\widehat{a}(\alpha ,\beta )$ defining the state of polarization of the incident beam are not significantly affected by the angle of incidence, i.e. they do not depend on δ. In fact, the vector amplitude distribution of the collimated beam at the entrance pupil is not as sensitive to a nonzero angle of incidence as its phase is. If the effect of the angle of incidence δ on the vector amplitude of the collimated input beam has to be taken into account, one may follow the approach given for example in Ref. [32].

For an incident radially polarized beam, the unit vector $\widehat{a}(\alpha ,\beta )$ is given by [8,22]

Finally, the amplitude distribution of the collimated paraxial TM₀₁ beam incident on the parabolic mirror reads

4. Numerical simulations

The electric field of a TM₀₁ beam focused by a parabolic mirror has been computed for different values of the dimensionless parameters that characterize the optical system. The first parameter is kfδ (where δ is in radians), which quantifies the amount of aberrations in the focused beam. We recall that kf >> 1 (Debye approximation) and δ << 1 (small-angle approximation). Numerical simulations are restricted to small values of the product kfδ, corresponding to small levels of aberrations; in fact, the maximum value for fδ that will be analyzed in this paper is of the order of a few wavelengths $\lambda =2\pi /k$. The second dimensionless parameter is $a\equiv 2f^2/k{w}_o^2$, which is related to the divergence angle of the focused beam: smaller values of a lead to larger beam divergence. Large values of ka refer to the paraxial regime whereas small values of ka correspond to the nonparaxial regime for the focused beam. The frontier between the paraxial and the nonparaxial regimes can be identified as ka = 7 [33]. In fact, for the case of the TM₀₁ beam, the transverse components of the electric field are approximately of the same magnitude as its longitudinal component when ka = 7. For ka >> 1, the electric energy profile is broad and exhibits a dark core, which is unsuitable for most applications in microscopy. Hence, the focusing properties of a TM₀₁ beam are especially interesting as long as ka is less than 7, because of the existence of a sharp and relatively strong longitudinal electric field component at the focus of the parabolic mirror. The dimensionless parameter ka can be expressed in term of the f-number of the parabolic mirror. If it is assumed that ${(r_{\infty })}_{\max }$ equals w_o, then it can be shown that the f-number may be written $f/\#=f/2w_o={(ka/8)}^{1/2}$, so that $ka=8{(f/\#)}^2$.

We present in this section some profiles of the electric energy density, which is defined by $W_e\equiv \frac{1}{2}{\epsilon }_0{\left|E\right|}^2=\frac{1}{2}{\epsilon }_0\left({\left|E_x\right|}^2+{\left|E_y\right|}^2+{\left|E_z\right|}^2\right)$. Unless otherwise stated, all the results are normalized with respect to the maximum intensity of the focused beam for which ka = 1 and δ = 0. For a given value of ka, increasing the value of kfδ deforms the shape of the electric energy density profile. For ka = 1, as kfδ increases from zero to 3π, the position of the peak intensity is shifted and the rotational symmetry of the intensity distribution in the x-y plane is converted into a uniaxial symmetry with respect to the X-axis (Fig. 3 ). With a zero angle of incidence, the structure of the electric energy density is aberration free and it possesses lobes of relatively weak amplitude surrounding a central high amplitude peak [Fig. 3(a)]. When kfδ is different from zero, the rotational symmetry exhibited in the intensity distribution of a beam focused with a zero angle of incidence is destroyed by formation of unsymmetrical rings, showing a comatic image flaring in the X-direction [Figs. 3(b) and 3(c)]; these distributions, characterized by a comet-like shape, confirm the presence of coma in the focal spot when the beam is incident on the parabolic mirror with an angle of incidence, however small it is. Interestingly, the aberrated beam can be seen as a superposition of the nonparaxial fundamental TM₀₁ beam and higher-order TM beams [4]. The position of the best focus, its full-width at half-maximum (FWHM), and the value of the maximum intensity are affected by a change in the value of kfδ.

 

Fig. 3 The electric energy density distribution W_e in the focal plane (z = 0) of a focused TM₀₁ beam by a parabolic mirror for which ka = 1 with kfδ equal to (a) 0, (b) 1.5π, and (c) 3π.

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The intensity profile of a focused TM₀₁ beam in the focal plane of a parabolic mirror is strongly dependent on beam divergence angle, i.e. on parameter ka. Figure 4 shows the profiles of the electric energy density for different values of kfδ, for ka = 1, 4, and 7 (which correspond to beam divergence angles approximately equal to 66°, 40°, and 30°, respectively). Each profile is on the X-axis (y = 0), so that E_y = 0. As kfδ increases, an extended tail develops in the electric energy density profile, especially for low values of ka. The presence of comatic aberration in the system yields a positional displacement and a modification of the width of the central peak intensity accompanied with, in most cases, a reduction of the peak intensity.

 

Fig. 4 Electric energy density profiles along the X-axis (y = 0) in the focal plane (z = 0) for selected values of ka and kfδ. Note the different scale for each plot.

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Due to its high amplitude, the longitudinal component of the electric field plays a dominating role in shaping the total electric energy density for ka < 7 (in the nonparaxial regime). However, for ka = 7, the transverse components of the electric field have nearly the same magnitude as the longitudinal component. For kfδ = 0, the transverse components of the electric field are zero at the focal point, providing a dark core to the electric energy density profile associated to the transverse components, as shown in the first row of Fig. 5 . As kfδ increases, the energy in the lobe closer to the origin is gradually transferred to the lobe farther from the origin, enhancing the peak intensity of the latter, as illustrated in the second row of Fig. 5. Therefore, the peak intensity of the electric energy density distribution associated to the total field can be higher than the aberration-free case, due to this redistribution of energy in the focal plane from one peak to the adjacent. Note that this effect is observed in Fig. 4 for the case of ka = 7.

 

Fig. 5 Electric energy density distribution along the X-axis (y = 0) in the focal plane (z = 0) of a focused TM₀₁ beam for which for ka = 7, for kfδ = 0 and kfδ = 6π. (a) ${\left|E_x\right|}^2$, (b) ${\left|E_z\right|}^2$, and (c) ${\left|E_x\right|}^2+{\left|E_z\right|}^2$. The profiles are normalized by the maximum intensity of ${\left|E_x\right|}^2+{\left|E_z\right|}^2$.

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The Strehl ratio is the ratio of the observed peak intensity in the focal plane (z = 0) of the parabolic mirror compared to the maximum peak intensity of the aberration-free version of the system, i.e. when δ = 0. Specifically, we define the Strehl ratio by $S\equiv I/I_0$, where I ₀ is the maximum value of the electric energy density W_e of the unaberrated beam and I is the maximum value of the electric energy density of the aberrated beam taken at the plane z = 0. For ka = 1, the Strehl ratio decreases relatively quickly with an increase in the value of kfδ; for ka = 4, it decreases, but more slowly; and for ka = 7, it is greater than one, at least for sufficient low values of kfδ [Fig. 6(a) ]. This last observation can be explained by the fact that the amplitude of the transverse electric field components is high enough to shape the total electric energy density in such a way that the energy transferred from one lobe to the other increases the maximum value of the peak intensity (see Fig. 5).

 

Fig. 6 (a) The Strehl ratio, (b) the full-width at half-maximum (FWHM), and (c) the position x _max of the peak intensity of the electric energy density profile in the focal plane (z = 0) as a function of the parameter kfδ, for ka = 1 (red curves), for ka = 4 (green curves), and for ka = 7 (blue curves).

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The FWHM of the central lobe, calculated on the X-axis (y = 0), is affected by the angle of incidence of the beam on the parabolic mirror [Fig. 6(b)]. It increases with an increase in the value of kfδ for ka = 1; it is roughly constant for ka = 4; and it decreases with kfδ for ka = 7. The last observation can be understood by the fact that the total electric energy density profile is, for δ = 0, the superposition of three juxtaposed peaks of nearly identical height, whereas it is the superposition of practically two peaks when kfδ is larger, since the amplitude of the lobe nearer to the origin becomes relatively small.

With a nonzero value of kfδ, the electric energy density profile is shifted in one direction with respect to the origin. The position of the peak of highest intensity, denoted by x _max, depends on the amount of aberration, but it is almost independent of the beam divergence angle [Fig. 6(c)]. According to the results of Fig. 6(c), the position of the best focus in the plane z = 0 is approximately located at x _max = 1.3fδ, measured along the X-axis with respect to the focal point of the paraboloid (this is valid at least for $1\le ka\le 7$ and $0\le kf\delta \le 6\pi$). This displacement is mainly explained by the presence of distortion in the aberration function of the system; the extra amount of displacement can be attributed to the presence of coma in the aberration function of the system.

5. Conclusion

In summary, we outlined a method to analyze quantitatively the effects of the angle of incidence on the focusing properties of a high numerical aperture parabolic mirror. The aberration function of the system, for a very small tilting of the mirror, can be expanded to underline the fact that the system suffers from distortion and coma. The application of the Richards–Wolf theory to the tight focusing of a TM₀₁ beam yields the electric field of the strongly focused beam in the focal region. This analysis shows that, when a parabolic mirror is illuminated with an incident collimated beam whose axis of propagation does not exactly coincide with the axis of revolution of the mirror, a positional displacement of the focal spot, a change in the width of the central lobe and a modification of the peak intensity are observed.

The analysis presented in this paper is consistent with the results of Lieb and Meixner, which claim that, for kf = 18,000π, an angle of incidence as small as 0.006° causes a severe asymmetry of the electric energy density, resulting from the presence of coma [5]. Consequently the propagation axis of the incident beam must be properly aligned with respect to the axis of revolution of the mirror axis. The correction of the misalignment can be achieved by a steering mirror equipped with micrometer adjustment screws [5]. Other techniques to correct the aberrations in a parabolic mirror involve closed loop adaptive systems [34].