Exact vectorial model for nonparaxial focusing of freeform wavefronts

Rafael G. González-Acuña; Simon Thibault

doi:10.1364/OE.459930

1. Introduction

The modeling of nonparaxial beams is necessary in several area of science, including high-resolution microscopy [1–4], optical trapping [5], electron acceleration [6], and optical vortex knots [7,8]. Generally, papers have focused on beam shapes with prescribed polarization states that are critical for mentioned applications [9–16].

In focusing systems, the scalar approach can be considered if the angles of incidence are in the paraxial regime. However, if the numerical aperture increases the scalar approach is no longer valid and a vectorial treatment to the electromagnetic fields in the focal volume is needed. A well-know approach of the mentioned to this is the Richards–Wolf theory (RWT) [17–20]. In its initial formulation RWT assumes an input plane phase collimated beam and an output a spherical beam (stigmatic). The Richards–Wolf theory has been extended to radially and azimuthally polarized electromagnetic beams by axisymmetric aberrated systems [21,22].

The goal of this paper is to further extend RWT for non-paraxial focusing for a freeform wavefront in the optical system exit pupil. Notice that the original Richards-Wolf vector diffraction only considers perfect stigmatic optical systems. This generalization freeform wavefronts are also considered. To achieve this goal, we work directly with the propagation vector and the unit vector representing the polarization direction. We put emphasis that both must always be orthogonal. Also, we take account of the Jacobian metric during a change of variables that we will see in section 2. The manuscript is divided as follows, first an introduction to RWT is presented. Then, the new formalism is developed to compute integrals for an arbitrary wavefronts. Finally, examples are presented and conclusions are given.

2. Richard-Wolf theory

Richards–Wolf theory, consists in vector diffraction integral useful to analyze tightly focused beams. Richards–Wolf theory considers a plane wave incident on a focusing system that is transformed into a converging spherical wave. Under this paradigm, this spherical wave can be expanded into an angular spectrum of plane waves. Those plane waves contribute to the field in the focus the propagation directions of which correspond to the geometric optical rays. Therefore the field in the focus can be evaluated by superposing those plane waves. Based the aforementioned considerations and using cylindrical coordinates $(r,\phi,z)$ the electric field at the focus is,

(1)$${\mathbf E}(r,\phi,z)=\frac{E_0}{2\pi}\int\int \frac{{\mathbf A}(k_x,k_y)}{k_z} \exp({-}i {\mathbf k}\cdot{\mathbf r})\text{d}k_x\text{d}k_y$$

where ${\mathbf E}(r,\phi,z)$ is the electric field in the neighborhood of the focus, $E_0$ is a constant amplitude, ${\mathbf k}$ is the wave vector oriented toward the focus and ${\mathbf r}$ is the position vector near the focus. As it can be seen from Eq. (1) is an integral over the vector field amplitude ${\mathbf A}(k_x,k_y)\equiv q(k_x,k_y)l_0(k_x,k_y)\hat {{\mathbf a}}(k_x,k_y)$. $q(k_x,k_y)$ is the apodization pupil function,which is a factor obtained from energy conservation theorem, $l_0(k_x,k_y)$ is the amplitude distribution of the beam and $\hat {{\mathbf a}}(k_x,k_y)$ is a unit vector representing the polarization direction of the electric field near the focus. Also notice that the components of the wave vector ${\mathbf k}$ are $k_x$, $k_y$, and $k_z$ in the $x$, $y$ and $z$ directions, respectively.

As mention earlier RWT considers that the output waterfront is spherical and it is the superposition of several plane waves. Therefore the components of the wave vector ${\mathbf k}$ turn to,

(2)$$k_x={-}k\sin\alpha\cos\beta,\;\;k_y={-}k\sin\alpha\sin\beta,\;\;k_z=k\cos\alpha,$$

where $k\equiv |{\mathbf k}|=2\pi /\lambda$ is the wave number, $\lambda$ is the wavelength, $\alpha$ and $\beta$ are the polar and the azimuthal angles of the incident plane wave component, respectively, the domain of the angles are $\alpha \in [\alpha _{\text {min}},\alpha _{\text {max}}]$ and $\beta \in [0,2\pi ]$. See Fig. 1. Tacking into account Eq. (2) and a radially polarized beam, the polarization vector becomes,

(3)$$\hat{{\mathbf a}}(\alpha,\beta)=\cos\alpha\cos\beta\;\hat{{\mathbf x}}+\cos\alpha\sin\beta\;\hat{{\mathbf y}}+\sin\alpha\;\hat{{\mathbf z}},$$

$\hat {{\mathbf a}}$ is the unit vector oriented along the Cartesian axes $x$, $y$, $z$. Finally the differentials considering the change of $(k_x,k_y)$ coordinates from to $(\alpha,\beta )$ and their respective Jacobian,

(4)$$\frac{\text{d}k_x\text{d}k_y}{k_z}=\sin\alpha\text{d}\alpha\text{d}\beta,$$

Replacing Eqs. (2) (3) and (4) in Eq. (1), ${\mathbf E}(r,\phi,z)$ becomes,

(5)$$\begin{aligned}{\mathbf E}(r,\phi,z)=\frac{E_0}{2\pi}\int_0^{2\pi}\int_{0}^{\alpha_\text{max}} q(\alpha,\beta)& l_0(\alpha,\beta) [\hat{{\mathbf x}}\cos\alpha\cos\beta+\hat{{\mathbf y}}\cos\alpha\sin\beta+\hat{{\mathbf z}}\sin\alpha]\\&\times \exp[{-}ik(z\cos\alpha-r\sin\alpha\cos(\phi-\beta))] \sin\alpha \text{d}\alpha\text{d}\beta,\end{aligned}$$

Eq. (5) expresses the electric field near the focus when the system is stigmatic, which in other means, it has a single-point image. This means that the wavefront of the focusing beam is a perfect sphere. In the next section, we are going to consider the case when the wavefront is not necessary as a sphere.

Fig. 1. Diagram of the RWT. The dashed line is the wavefront. $\alpha$ and $\beta$ are the polar and the azimuthal angles of the incident plane wave component, respectively and $\hat {{\mathbf a}}$ is the polarization unit vector oriented along the Cartesian axes $x$, $y$, $z$.

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3. Generalisation of the Richards-Wolf theory

The fundamental background of the RWT is that the focused spherical wave can be expanded into an angular spectrum of plane waves. Without loosing generalization we can also express the any other focused wavefront as sum of plane waves with the appropriate wave vector ${\mathbf k}$ and its Cartesian components $k_x$, $k_y$, and $k_z$. Therefore, given a wavefront $w(x,y)$ and according to the Malus-Dupin theorem, the wave vector ${\mathbf k}$ of the wavefront $w(x,y)$ has the direction of normal to $w(x,y)$ [23,24]. Here, $w(x, y)$ is an aberrated wavefront generated by an imperfect system. Then, the cosine directors of the wavefront wave vector ${\mathbf k}$ are,

(6)$${\mathbf k}=[k_x,k_y,k_z]=k\frac{-\partial_xw}{\sqrt{(\partial_xw)^{2}+(\partial_yw)^{2}+1}}\hat{{\mathbf x}}+k\frac{-\partial_yw}{\sqrt{(\partial_xw)^{2}+(\partial_yw)^{2}+1}}\hat{{\mathbf y}}+k\frac{1}{\sqrt{(\partial_xw)^{2}+(\partial_yw)^{2}+1}}\hat{{\mathbf z}},$$

where $\partial _x \equiv \partial w/\partial x$ is the derivative of $w$ respect to $x$ and $\partial _yw\equiv \partial w/\partial y$ is the derivative of $w$ respect to $y$. $k_{x}$, $k_{y}$ and $k_{z}$ are the cosine directors in $x$, $y$ and $z$ directions, respectively, of the wave vector ${\mathbf k}$. Since, we are assuming that the electrical field is rotationally polarised the unit vector representing the polarisation direction $\mathbf {\hat {a}}$ for the an arbitrary wavefront is given by

(7)$$\begin{aligned}\hat{{\mathbf a}}(x,y)= \frac{\partial_xw}{\sqrt{\partial_xw^{2}+\partial_yw^{2}+1} \sqrt{\partial_xw^{2}+\partial_yw^{2}}}\hat{{\mathbf x}}&+\frac{\partial_yw}{\sqrt{\partial_xw^{2}+\partial_yw^{2}+1} \sqrt{\partial_xw^{2}+\partial_yw^{2}}}\hat{{\mathbf y}}+\\&{\kern 2cm}\sqrt{1-\frac{1}{\partial_xw^{2}+\partial_yw^{2}+1}}\hat{{\mathbf z}}.\end{aligned}$$

Observe that Eq. (6) and Eq. (7) are perpendicular to each other, also notice that they are in terms of $(x,y)$, so we need to express $x$ and $y$ in terms of $\alpha$ and $\beta$. Let be $R$ the radius of the exit pupil then we can express, $(x,y)$ in terms or $(\beta,\alpha )$,

(8)$$x=R\sin\alpha\cos\beta\;\;y=R\sin\alpha\sin\beta$$

Since the output wavefront is not spherical Eq. (5) is not valid, thus to get the differentials $\text {d}\alpha \text {d}\beta$ we need to compute the Jacobian $J(\alpha,\beta )$,

(9)$$\text{d}k_x\text{d}k_y=J(\alpha,\beta)\text{d}\alpha\text{d}\beta,$$

where,

(10)$$J(\alpha,\beta)\equiv\begin{vmatrix} \frac{\partial k_x (\alpha,\beta)}{\partial \alpha} & \frac{\partial k_y (\alpha,\beta)}{\partial \beta} \\ \frac{\partial k_x (\alpha,\beta)}{\partial \beta} & \frac{\partial k_y (\alpha,\beta)}{\partial \alpha} \end{vmatrix}=\frac{\partial k_x (\alpha,\beta)}{\partial \alpha} \frac{\partial k_y (\alpha,\beta)}{\partial \beta}-\frac{\partial k_x (\alpha,\beta)}{\partial \beta} \frac{\partial k_y (\alpha,\beta)}{\partial \alpha}$$

To get Jacobian $J(\alpha,\beta )$ first we need to replace $x=R\sin \alpha \cos \beta$ and $y=R\sin \alpha \sin \beta$ in the cosine directors and then directly compute Eq. (10). Therefore, the Richard-Wolf generalized integral for arbitrary focusing wavefronts is given by,

(11)$$\begin{aligned}{\mathbf E}(r,\phi,z)=\frac{E_0}{2\pi}\int_0^{2\pi}\int_{0}^{a_\text{max}} q(\alpha,\beta) &l_0(\alpha,\beta) \hat{{\mathbf a}}(\alpha,\beta)\\&{\kern -1cm}\times \exp[{-}i(zk_z-r\cos\phi k_x-r\sin\phi k_y)] J(\alpha,\beta) \text{d}\alpha\text{d}\beta,\end{aligned}$$

Eq. (11) is the most important equation in this manuscript. It accurate describes the diffraction pattern generated by an arbitrary wavefront. Notice that all the parameters inside Eq. (11) should be expressed in terms of $(\alpha,\beta )$. Therefore, $\hat {{\mathbf a}}(\alpha,\beta )$ is the composite function between Eq. (7) and (8). The cosine directors $k_x$, $k_y$ and $k_z$ in Eq. (11) are given by the composite function between Eq. (6) and (8), respectively. Finally, $J(\alpha,\beta )$ is given by Eq. (10).

Observe that to get, azimuthally polarized components is straight forward. Minor modifications need to be added in Eq. (7). The components of Eq. (7) must be multiplied by minus one.

4. Examples and discussion

In this section we explore several diffraction patterns obtained directly from Eq. (11). As a first example, we are going to consider a spherical wavefront $w(x,y)=R-\sqrt {R^{2}-x^{2}-y^{2}}$, the purpose of this example is to show that the equations presented in section 3 naturally converge to the equations of section 2, which is the standard RWT. We start with the cosine directors, that for the spherical wavefront are,

(12)$${\mathbf k}=[k_x,k_y,k_z]={-}\hat{{\mathbf x}}\frac{xk}{R}-\hat{{\mathbf y}}\frac{yk}{R}+\hat{{\mathbf z}}\frac{k}{R}\sqrt{R^{2}-x^{2}-y^{2}},$$

Evaluating Eq. (12) with $x=R\sin \alpha \cos \beta$ and $y=R\sin \alpha \sin \beta$, we get Eq. (2). For the spherical wavefront $\mathbf {\hat {a}}$ is given by

(13)$$\hat{{\mathbf a}}(x,y)=\hat{{\mathbf x}}\frac{x\sqrt{R^{2}-x^{2}-y^{2}}}{R\sqrt{x^{2}+y^{2}}} +\hat{{\mathbf y}}\frac{y\sqrt{R^{2}-x^{2}-y^{2}}}{R\sqrt{x^{2}+y^{2}}}+\hat{{\mathbf z}}\frac{\sqrt{x^{2}+y^{2}}}{R}.$$

The composite function between Eqs. (13) and (8) and becomes Eq. (3). Notice that during this procedure $R$ is cancelled in Eq. (3). The Jacobian $J(\alpha,\beta )$ can directly computed using Eq. (10) and Eq. (8) and divided by $k_z(\alpha,\beta )$, that turns to be Eq. (4). If we replacing replace Eqs. (12), (13), (10) with (8) in Eq. (11) we get the Richard-Wolf Integral, Eq. (1).

In the next example, the wavefront is give by the Zernike polynomial $Z_{2}^{0}=w(x,y)=c(-1+2x^{2}+2y^{2})$ where $c\in \mathbb {R}$. The cosine directors are computed with composite function of Eq. (6) and (8),

(14)$$\begin{aligned}{\mathbf k}=[k_x,k_y,k_z]={-}\hat{{\mathbf x}}\frac{4 c R \sin (\alpha ) \cos (\beta )k}{\sqrt{-8 c^{2} R^{2} \cos (2 \alpha )+8 c^{2} R^{2}+1}}&-\hat{{\mathbf y}}\frac{4 c R \sin (\alpha ) \sin (\beta )k}{\sqrt{-8 c^{2} R^{2} \cos (2 \alpha )+8 c^{2} R^{2}+1}}\\&+\hat{{\mathbf z}}\frac{k}{\sqrt{-8 c^{2} R^{2} \cos (2 \alpha )+8 c^{2} R^{2}+1}}.\end{aligned}$$

The polarization unit vector $\hat {{\mathbf a}}(\alpha,\beta )$ is the composite function between Eq. (7) and (8), where $w(x,y)=c(-1+2x^{2}+2y^{2})$,

(15)$$\begin{aligned}\hat{{\mathbf a}}(x,y)=\hat{{\mathbf x}} \frac{\cos (\beta )}{\sqrt{-8 c^{2} R^{2} \cos (2 \alpha )+8 c^{2} R^{2}+1}}+\hat{{\mathbf y}}&\frac{\sin (\beta )}{\sqrt{-8 c^{2} R^{2} \cos (2 \alpha )+8 c^{2} R^{2}+1}}\\&+\hat{{\mathbf z}}4 \sqrt{\frac{c^{2} R^{2} \sin ^{2}(\alpha )}{-8 c^{2} R^{2} \cos (2 \alpha )+8 c^{2} R^{2}+1}}.\end{aligned}$$

Finally, we compute the Jacobian divided by $k_z(\alpha, \beta )$,

(16)$$\frac{J(\alpha,\beta)}{k_z(\alpha, \beta)}=\frac{8 c^{2} R^{2} \sin (2 \alpha )}{\left[{-}8 c^{2} R^{2} \cos (2 \alpha )+8 c^{2} R^{2}+1\right]^{3/2}}.$$

To compute the diffraction pattern generated by $Z_{2}^{0}=w(x,y)=c(-1+2x^{2}+2y^{2})$ we replace the Eqs. (14), (15), (16) in Eq. (11) and compute the integral. In Fig. 2 we present the intensity profile for a spherical wavefront in orange. In (a) is the axial profile and (b) is the lateral profile. In the figure but in black are the axial and lateral intensity profiles of $w(x,y)=c(-1+2x^{2}+2y^{2})$, respectively. The specifications of each example can be seen in the caption of Fig. 2.

Fig. 2. Intensity profiles. (a) and (b) in orange is normalized intensity of the diffraction of the wavefront $w(x,y)=R-\sqrt {R^{2}-x^{2}-y^{2}}$, and in black is presented the normalized intensity of the diffraction of the wavefront $w(x,y)=c(-1+2x^{2}+2y^{2})$ with $c=1$.

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The orange curve corresponds to an astigmatic diffraction pattern, while black has aberrations. The full width at half maximum (FWHM) of the orange profiles are smaller than the ones in black. FWHM for the orange curves are $0.52\lambda$ and $0.23\lambda$ in the $z$ and $r$ profiles, respectively. For the black curves the FWHM is $0.57\lambda$ in the $z$ profile and $0.24\lambda$ in for $r$ profile.

From the example of Fig. 2 (a) and (b) it is clear that the method presented in this manuscript converges to RWT when the wavefront is spherical (astigmatic).

Notice that in this computation we set the apodization factor for both wavefronts as one, $q(\alpha,\beta )\to 1$, since we are only considering the wavefront at the exit pupil and the optical system that produces the aforementioned wavefront. Thus, we do not consider the distribution of ray across the optical system. Also notice that the transverse electric field at the output is radially polarized; This means that the optical system does not produces any polarization aberration. But considering polarization aberrations is straight forward by directly modifying $\hat {{\mathbf a}}(\alpha,\beta )$. The illumination pattern $l_0(\alpha,\beta )$ has been taken as uniform as well.

The method presented here also can be used to model freeform wavefronts. As an example of these freeform wavefronts we choose $w(x,y)=a \left (b y^{2}+x^{2}\right )$, where $a,b\in \mathbb {R}$. The cosine directors of the normal of the wavefront can be expressed in terms of $\alpha,\beta$ are,

(17)$$\begin{aligned}{\mathbf k}=[k_x,k_y,k_z]=&{-}\frac{2 a R \sin (\alpha ) \cos (\beta )k}{\sqrt{4 a^{2} R^{2} \sin ^{2}(\alpha ) \left[b^{2} \sin ^{2}(\beta )+\cos ^{2}(\beta )\right]+1}}\hat{{\mathbf x}}\\ &{\kern 1cm}-\frac{2 a b R \sin (\alpha ) \sin (\beta )k}{\sqrt{4 a^{2} R^{2} \sin ^{2}(\alpha ) \left[b^{2} \sin ^{2}(\beta )+\cos ^{2}(\beta )\right]+1}}\hat{{\mathbf y}}\\ &{\kern 4cm}+\frac{k}{\sqrt{4 a^{2} R^{2} \sin ^{2}(\alpha ) \left[b^{2} \sin ^{2}(\beta )+\cos ^{2}(\beta )\right]+1}}\hat{{\mathbf z}},\end{aligned}$$

and the unit polarization vector is given by

(18)$$\begin{aligned}\hat{{\mathbf a}}(x,y)=&\frac{\cos (\beta )}{\sqrt{b^{2} \sin ^{2}(\beta )+\cos ^{2}(\beta )} \sqrt{4 a^{2} R^{2} \sin ^{2}(\alpha ) \left[b^{2} \sin ^{2}(\beta )+\cos ^{2}(\beta )\right]+1}}\hat{{\mathbf x}}\\&{\kern 1cm}+\frac{\cos (\beta )}{\sqrt{b^{2} \sin ^{2}(\beta )+\cos ^{2}(\beta )} \sqrt{4 a^{2} R^{2} \sin ^{2}(\alpha ) \left[b^{2} \sin ^{2}(\beta )+\cos ^{2}(\beta )\right]+1}}\hat{{\mathbf y}}\\&{\kern 4cm}+ \sqrt{\frac{1}{2 a^{2} R^{2} \sin ^{2}(\alpha ) \left[\left(b^{2}-1\right) \cos (2 \beta )-b^{2}-1\right]-1}+1}\;\hat{{\mathbf z}}.\end{aligned}$$

The Jacobian divided by $k_z(\alpha, \beta )$ for the $w(x,y)=a \left (b y^{2}+x^{2}\right )$, it is given by,

(19)$$\frac{J(\alpha,\beta)}{k_z(\alpha, \beta)}=\frac{2 a^{2} b R^{2} \sin (2 \alpha )}{\left[4 a^{2} R^{2} \sin ^{2}(\alpha ) \left\{b^{2} \sin ^{2}(\beta )+\cos ^{2}(\beta )\right\}+1\right]^{3/2}}$$

Inserting Eqs. (17), (18), (19) in (11) and directly computing the integral we get the diffraction patterns presented in Fig. 3. From the figure it is clear that the diffraction patterns is nor radially symmetric as is expected since $w(x,y)=a(b y^{2}+x^{2})$ is not radially symmetry. In Fig. 4, (a), (b) and (c) are the diffraction patterns in the $z$, $x$ and $y$ directions, respectively. Notice that the distribution of energy is wider than in Fig. 2, this is due the intrinsic aberration suffered by the example wavefront.

Fig. 3. Normalized intensity of the diffraction of the wavefront $w(x,y)=a(b y^{2}+x^{2})$ with uniform illumination pattern, $a=0.5$, $b=0.7$. (a) is intensity profile with respect to the $z$ axis (b) is the is intensity profile with regarding to the $y$ axis and (c) is intensity profile with respect to the $y$ axis.

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Fig. 4. Normalized intensity of the diffraction produced by an spherical mirror (a) is intensity profile with respect to the $z$ axis (b) is the is intensity profile with regarding to the $r$ axis. The radius of the mirrors is 13mm and the radius of the aperture is 10mm.

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Finally, we computed the diffraction pattern generated by a spherical mirror. The diffraction pattern is displayed in Fig. 4. In the left side is the right side of the mentioned figure correspond to the diffraction pattern in the $z$ and $r$ axis, respectively. We set the apodization factor as one, $q(\alpha )\to 1$, we use a Gaussian illumination, $l_0(\alpha )\to \alpha\,\textrm{exp}(-\alpha ^{2}/\Delta \alpha ^{2})$ and we $w(x,y)$ the following wavefront,

(20)$$w(x,y)= \sqrt{R_a^{2}-x^{2}-y^{2}}-\frac{R_a \tanh ^{{-}1}\left(\frac{\sqrt{2} \sqrt{R_a^{2}-x^{2}-y^{2}}}{R_a}\right)}{\sqrt{2}}$$

where $R_a$ is the radius of the spheric mirror. $w(x,y)$ was obtained by computing the cosine directors of the rays reflected by the spheric mirror. This cosine directors must obey Eq. (6), so they form differential equation who’s solution is $w(x,y)$. The the results are similar to the ones obtained in a previous generalization [21]. However, the method in Ref. [21] is only valid for aspherical systems and implements several approximations that do not guaranty the orthogonality between the propagation vector ${\mathbf k}$ and the polarization direction vector $\mathbf {\hat {a}}(\alpha,\beta )$.

It is important to remark the limitations of the method. The method begins with from the normal vector of the wavefront $w(x,y)$ which is in indeed, the propagation vector ${\mathbf k}$, if this vector is such that its respective Jacobian $J(\alpha,\beta )$ is not zero the computation can be done. An example of a wavefront with a zero Jacobian is a plane wave. This result, should not be surprising since the plane it is not a focusing beam. Therefore, the diffraction pattern of diverging beams can not be computed with the method presented in this paper.

Also notice that the method presented here considers the Jacobian $J(\alpha,\beta )$ as well the orthogonality of ${\mathbf k}$ and $\hat {{\mathbf a}}(\alpha,\beta )$. In the generalization presented in [11,21,22], both motioned aspects are not considered and only aberration is added in the axial axis. [11,21,22] ignore these aspects, thus as a consequence ${\mathbf k}$ and $\hat {{\mathbf a}}(\alpha,\beta )$ are no longer orthogonal and for the case $|{\mathbf k}|$ is no longer equal to $k$. Therefore, in the best knowledge of the authors, the generalization presented here is more robust than the previews ones [11,21,22]. We consider the method presented in this manuscript as a great bridge between the two paradigms, geometrical optics and diffraction theory.

5. Conclusions

In this manuscript, we presented a rigorous generalization of the RWT to arbitrary wavefronts. The standard RWT only considers spherical waterfronts (no aberration) but this new theory considers any arbitrary wavefronts. To do so, This generalization considers that ${\mathbf k}$ and $\hat {{\mathbf a}}(\alpha,\beta )$ should be always perpendicular as well considers that the Jacobian in the changes of variables for the first time. As a result, the method presented here successfully computes the diffraction patterns of freeform wavefronts, including spherical, Zernike polynomials and any freeform waterfronts. We also have discussed its limitations.

The natural step to follow in this formalism is the study of the diffraction pattern of freeforms wavefront. For example one potential application is to study the effect of optical tolerance like component decenters (coma), or off axis aberrations for non-paraxial focusing. Also the study diffraction patterns generated by on-axis and off-axis fields in microscope objectives or freeform telescopes, where phase and intensity spatial distributions of the beam should be considered.

Appendix

Here, we show how to get a wavefront generated by a spherical mirror when it reflects a plane wave aligned along with the mirror. From refraction’s law at the surface, the cosine directors of the output wavefront are,

(21)$$k_x/k={-}\frac{2 x \sqrt{R_a^{2}-x^{2}-y^{2}}}{R_a^{2}},\;\;\; k_y/k={-}\frac{2 y \sqrt{R_a^{2}-x^{2}-y^{2}}}{R_a^{2}},\;\;\;k_z/k=\frac{2 \left(x^{2}+y^{2}\right)}{R_a^{2}}-1,$$

Equaling Eq. (21) with Eq. (6) and solving for $\partial (w_x)$ and $\partial (w_y)$, we get,

(22)$$\partial_x w=\frac{2 x \sqrt{R_a^{2}-x^{2}-y^{2}}}{2 \left(x^{2}+y^{2}\right)-R_a^{2}},\;\;\;\partial_y w= \frac{2 y \sqrt{R_a^{2}-x^{2}-y^{2}}}{2 \left(x^{2}+y^{2}\right)-R_a^{2}}$$

From Eq. (22), If we integrate $\partial _xw$ respect to $x$, or we integrate $\partial _yw$ respect to $y$, we get the same result, the wavefront $w(x,y)$ expressed in Eq. (20).

Funding

Universite Laval (NSERQ RGPIN-2016-05962)).

Disclosures

The author declares no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.

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Exact vectorial model for nonparaxial focusing of freeform wavefronts

Abstract

1. Introduction

2. Richard-Wolf theory

3. Generalisation of the Richards-Wolf theory

4. Examples and discussion

5. Conclusions

Appendix

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (22)

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