Relationship Between Aberration Coefficients of an Optical Device and Its Focusing Property

Abstract

The best focus point of a focused Gaussian beam subject to a phase aberration is generally shifted with respect to the focal plane of the focusing lens. This focus shift is attributed to a lensing effect that belongs to the phase aberration, which mean focal length can be determined from the aberration coefficients determined in the framework of a Zernike polynomial decomposition. In this paper, we have checked the validity of this procedure, already available in literature, applied to three aberration types: a pure primary spherical aberration, the Kerr effect induced by a Gaussian beam, and an axicon illuminated by a Gaussian beam. Note that usually, the mean focal length of an aberrated lens is based on the relation between the effective radius of curvature of the wavefront before and after the lens. However, in this paper, the focal length associated with the phase aberration under study is defined from the point of the best focus, where the diffracted intensity on the axis is the maximum.

1. Introduction

Focusing a laser beam subject to aberrations is a subject having received much attention to date, whether the origin of the aberrations is inherent to the focusing lens itself [1,2,3,4,5,6,7,8,9] or to some nonlinear self-action, such as the Kerr effect [10,11,12,13,14,15,16,17,18,19] or thermal effects [20,21,22,23,24,25]. Most applications of lasers are based on their focusing, and, consequently, it is important to study all the possible causes and processes leading to the degradation of the “focusing quality”. The latter refers to the size of the focal spot and its on-axis intensity. Ideally, it is desirable that the focal spot size (on-axis intensity) should be small (high). Generally, the focusing of an aberrated wavefront of a Gaussian beam leads to a degraded focal spot: Its transverse intensity distribution is no longer Gaussian in shape, and its maximum on-axis intensity is spatially shifted with respect to the focal length of the focusing lens. The latter effect is interpreted in terms of a lensing effect, which belongs to the wavefront aberration. In the following, we will be interested in the mean focal length characterising the lensing effect associated with the aberrated wavefront, and we will ignore the beam reshaping driven by the aberrations. The wording “mean focal length” has been defined in [26], and we will retain here only its determination, as succinctly summarised in Appendix A and based on the decomposition of the aberrated wavefront on the basis of Zernike polynomials. Four focal lengths ($f_1$, $f_2$, $f_3$, and $f_4$) can be defined from the quadratic terms present in the Zernike polynomials associated with the primary, secondary, and tertiary spherical aberrations (see Appendix A). The aim of this paper is to check if one of the focal lengths ($f_1$, $f_2$, $f_3$, or $f_4$) is able to provide the right location of the best focus, where the on-axis intensity is the maximum. Note that the authors in [26] defined the mean focal length of an aberrated lens from the relation between the effective radius of curvature of the wavefront before and after the lens. However, in this paper, the focal length associated with the phase aberration under study is defined from the point of the best focus, where the diffracted intensity on the axis is the maximum.

In order to simplify the problem modelling, we will assume that we are dealing with a perfect lens of focal length $f_L$ preceded by a transparent “phase aberration profile” (PAP), $\varphi (\rho )$, representing the cumulated aberrations from the real lens and from the media through which the laser beam is propagating (Figure 1). In the following, we will consider a collimated Gaussian beam as the incident laser beam characterised by its amplitude distribution, $E_{in}(\rho )$, given by

$$E_{in}(\rho )=E_0\exp (-{\rho }^2/W^2)$$

(1)

where $\rho$ is the radial coordinate in the plane (PAP + perfect lens). The PAP is characterised by its complex transmittance, ${\tau }_{\varphi }(\rho )$, given by

$${\tau }_{\varphi }=\exp [-i\varphi (\rho )]$$

(2)

The complex transmittance, ${\tau }_L(\rho )$, of the focusing lens of focal length f_L = 50 mm is given by

$${\tau }_L=\exp \left[{\displaystyle \frac{ik{\rho }^2}{2f_L}}\right]$$

(3)

where $k=2\pi /\lambda$, and $\lambda$ = 1064 nm is the wavelength. It is well known that the focusing of a collimated Gaussian beam does not occur exactly at the position z = f_L but is shifted to the position z_max given by

$$z_{\max }={\displaystyle \frac{f_L}{1+{\left(\frac{f_L}{z_R}\right)}^2}}$$

where $z_R=\pi W^2/\lambda$ is the Rayleigh distance of the incident beam. In our case, W = 1 mm, $f_L$ = 50 mm, and $z_R$ = 3140 mm so that $z_{\max }\approx f_L$.

Remark 1.

The minus sign in Equation (2) comes from the chosen notation for the propagation term of the laser beam inside the PAP material, having the form $\exp [i(\omega t-knz)]$, where z is the distance travelled inside the PAP of refractive index n. The phase shift undergone by the laser beam is equal to $\varphi =knz$, and the wave term $\exp [i(\omega t-knz)]$justifies the minus sign in Equation (2).

On the other hand, we need to determine the position, $z_{\max }$, of the best focus, where, by definition, the on-axis intensity is the maximum. This is performed by calculating, beyond the ensemble (PAP + lens), the diffracted electric field, $E_d(r,z)$, given by the Fresnel–Kirchhoff integral

$$E_d(r,z)={\displaystyle \frac{2\pi }{\lambda z}}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\tau }_L(\rho ){\tau }_{\varphi }(\rho )E_{in}(\rho )}\exp \left[{\displaystyle \frac{i\pi {\rho }^2}{\lambda z}}\right]J_0\left[{\displaystyle \frac{2\pi }{\lambda z}}r.\rho \right]\rho d\rho$$

(4)

where r (ρ) is the radial coordinate in the z-plane (PAP + Lens), and $J_0$ is the zeroth-order Bessel function of the first order. The integral given by Equation (4) is calculated using a FORTRAN routine based on the numerical integrator dqdag from the International Mathematics and Statistical Library (IMSL). The on-axis intensity beyond the ensemble (PAP + Lens) is given by $I_d(0,z)={\left|E_d(0,z)\right|}^2$. The objective of this paper is to consider a certain number of PAPs for which we will determine their aberration coefficients ($A_j$) and then check the fulfilment of the following equality:

$${\displaystyle \frac{1}{z_{\max }}}=\mathrm{or} \ne {\displaystyle \frac{1}{f_L}}+{\displaystyle \frac{1}{f_{eq}}}$$

(5)

where $f_{eq}$ stands for one of the focal lengths ($f_1$, $f_2$, $f_3$, or $f_4$). In the next sections, we will consider as the PAP the following different types of aberrations:

-

a pure primary spherical aberration;

-

the Kerr effect;

-

a refractive axicon.

2. A Pure Spherical Aberration

The phase profile, $\varphi (\rho )$, characterising a pure primary spherical aberration (SA) takes the following form:

$$\varphi (\rho )=k{W}_{40}{\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^4$$

(6)

where $W_{40}$ represents the primary spherical aberration’s coefficient. The quantity ${\rho }_0$ represents the radius of the unit circle, which, by definition, contains 99.9% of the incident beam’s power. For an incident collimated Gaussian beam of width W, the corresponding unit circle radius is ${\rho }_0=2W$. It is useful to specify, here, the meaning of “pure” when defining, above, the phase profile related to the SA. Indeed, it can be observed that the lack of a quadratic term (${\rho }^2$) in Equation (6) is not the case for the Zernike polynomials $Z_j$ (see Appendix A). However, the decomposition of the PAP given in Equation (6), on the basis of Zernike polynomials, gives rise to a mixing of Zernike aberrations, as shown below.

The plots shown in Figure 2 exhibit that the presence of a primary spherical aberration causes a shift in the best focus position, which takes place beyond (before) the focal plane $z=f_L$ when $W_{40}$ is negative (positive). This best focus shift could, therefore, be attributed to a lensing effect characterised by a focal length having the sign of $W_{40}$, which will be modelled hereafter in the framework of Zernike polynomial decomposition.

Now, it remains to express this focal length characterising the spherical aberration in the framework of Zernike polynomial decomposition. In a way similar to that used in Appendix A, we will determine the aberration coefficient ($a_j$) associated with the PAP given in Equation (6) and then apply Equation (A6) to express the equivalent focal length ($f_{40}$) associated with a pure primary spherical aberration. Taking into account Equation (6), the WAF ($S_{40}$) corresponding to the pure primary SA is written as $S_{40}=W_{40}{\overline{\rho }}^4$, and Equation (A2) can take the following form:

$$a_j=2{\displaystyle \underset{0}{\overset{1}{\int }}{S}_{40}}(\overline{\rho })Z_j(\overline{\rho })\overline{\rho }d\overline{\rho }$$

(7)

As for the numerical computing of Equation (4), we use the FORTRAN integrator dqdag for solving Equation (7). The numerical integration is performed very accurately with a global scheme based on Gauss–Konrad rules characterised by an absolute accuracy equal to 10⁻⁵ and a relative accuracy equal to 10⁻⁶.

The obtained results are as follows:

$$a_4={\displaystyle \frac{W_{40}}{2\sqrt{3}}}$$

(8)

$$a_{11}={\displaystyle \frac{W_{40}}{6\sqrt{5}}}$$

(9)

$$a_{22}=0$$

(10)

$$a_{37}=0$$

(11)

By applying Equation (A6), we obtain the mean focal length ($f_{40}$), in the framework of planar waves, associated with the pure primary SA

$$f_{40}={\displaystyle \frac{{\rho }_0^2}{2W_{40}}}$$

(12)

It is found for planar waves that the mean focal length ($f_{40}$) depends only on $a_4$ because the focal lengths ($f_2$, $f_3$, and $f_4$) are infinite. For an incident Gaussian beam, the numerical calculation of Equation (A3) allows for the determination of the coefficients $a_j^{\prime }$ which are

$$a_4^{\prime }=3.87\times {10}^{-2}\times W_{40}$$

(13)

$$a_{11}^{\prime }=-6.77\times {10}^{-2}\times W_{40}$$

(14)

$$a_{22}^{\prime }=9.77\times {10}^{-3}\times W_{40}$$

(15)

$$a_{37}^{\prime }=7.86\times {10}^{-3}\times W_{40}$$

(16)

Then, by applying Equations (A7)–(A10), we deduce the corresponding mean focal lengths, which have higher values than $f_{40}$. The plots in Figure 3 show that the Zernike modelling (planar wave or Gaussian beam) of the lensing effect due to the PAP, $\varphi (\rho )$, given by Equation (6) does not explain the focal shift determined by the diffraction integral modelling. Note that the differences in the values of $(z_{\max }-f_L)/f_L$ between the planar wave case and Gaussian beam cases are so small that they are practically imperceptible. As a consequence, one can conclude that the Zernike modelling, proposed in [26], of the lensing effect due to the phase aberration profile given by Equation (6), also known as a purely spherical aberration, is not able to account for the spatial shift in the best focus, where the on-axis intensity is the maximum.

3. The Kerr Effect

Before proceeding, it is useful to outline some basic elements about the optical Kerr effect (OKE) taking place in a transparent dielectric illuminated by a laser beam having an intensity profile of $I(\rho )$, where $\rho$ is the radial coordinate. The OKE causes an intensity-dependent change in the refractive index profile, $n(\rho )$, given by

$$n(\rho )=n_1+n_{2}I(\rho )$$

(17)

where $n_1$ ($n_2$) is the usual linear (nonlinear) refractive index. Note that $n_2$ can be positive or negative but is positive for a large number of optical materials, and this the case that is considered in the following. The Gaussian beam incident on a transparent dielectric of thickness d is subject to a phase shift profile (PAP), $\varphi (\rho )$, given by

$$\varphi (\rho )=kn(\rho )d=k{n}_{1}d+k{n}_{2}I(\rho )d$$

(18)

Because the term $k{n}_{1}d$ is independent of the radial coordinate and, thus, does not contribute to the lensing effect under study, it will be omitted. Finally, the PAP takes the following form:

$$\varphi (\rho )=k{n}_{2}I(\rho )d={\varphi }_0\exp (-2{\rho }^2/W^2)$$

(19)

where the on-axis phase shift (${\varphi }_0$) is given by

$${\varphi }_0={\displaystyle \frac{4n_{2}Pd}{\lambda W^2}}$$

(20)

where P is the incident Gaussian beam power given by $P=(\pi W^2{E}_0^2/2)$. The PAP given by Equation (19) is responsible for a lensing effect that results in a shift towards the lens of the best focus position. It is important to note that this lensing effect is not pure but contains, in addition to the defocus term, some high-order spherical aberration, as will be shown below. Note that the on-axis phase shift (${\varphi }_0$) is the key parameter that controls the aberrations of the PAP. It has long been a tradition to model the Kerr lensing effect using the well-known “parabolic approximation”, which consists of the development of the exponential term in Equation (19) at the first order and then the application of ABCD formalism [27,28,29,30]. Although this method has been predominantly used for several decades, it has been shown that the Kerr focal length deduced from the parabolic approximation does not provide the best focus position, where the on-axis intensity is the maximum. It is worth noting in passing that the knowledge of the correct modelling of the Kerr lensing effect is essential, for instance, for building a performing model of KLM lasers [31]. This is why the modelling of the Kerr length has been revisited [32] in order to obtain a formulation for the Kerr focal length, allowing for an accurate prediction of the best focus position. Before recalling the results obtained in [32], let us consider the aberration coefficients ($a_j$ and $a_j^{\prime }$) determined by applying Equations (A2) and (A3) and displayed in

Table 1. Dimensionless aberration coefficients associated with phase aberration $\varphi (\rho )={\varphi }_0\exp (-2{\rho }^2/W^2)$ when the incident wave is planar or a Gaussian of width W = 1 mm, and λ = 1064 nm.

Incident wave	$A_4/{\varphi }_0$	$A_{11}/{\varphi }_0$	$A_{22}/{\varphi }_0$	$A_{37}/{\varphi }_0$
Plane wave	$+7.27\times {10}^{-2}$	$-3.38\times {10}^{-2}$	$+1.08\times {10}^{-2}$	$-2.61\times {10}^{-3}$
Gaussian beam	$+7.80\times {10}^{-2}$	$-7.04\times {10}^{-2}$	$+4.96\times {10}^{-2}$	$-2.88\times {10}^{-2}$

.

After that, we can deduce the mean focal lengths ($f_1$, $f_2$, $f_3$, and $f_4$) from Equations (A7)–(A10) and, finally, the position of the best focus ($z_{\max }^{\prime }$) on the basis of geometrical optics arguments as follows:

$${\displaystyle \frac{1}{z_{\max }^{\prime }}}={\displaystyle \frac{1}{f_L}}+{\displaystyle \frac{1}{f_j}}$$

(21)

The results are shown, for the planar wave case, in Figure 4, which displays the variations in $(z_{\max }-f_L)/f_L$ and $(z_{\max }^{\prime }-f_L)/f_L$ versus ${\varphi }_0$, the nonlinear on-axis phase shift. For ${\varphi }_0\le \pi /3$, it is seen that the Kerr lens modelling employing the focal length $f_1={\rho }_0^2/(4\lambda \sqrt{3}{A}_4)$ is satisfactory, but for higher values of ${\varphi }_0$, the value gap between $(z_{\max }-f_L)/f_L$ and $(z_{\max }^{\prime }-f_L)/f_L$ rises significantly.

Better results are obtained when the Gaussian nature of the incident beam is taken into account when calculating the aberration coefficient ($a_j^{\prime }$) (Equation (A3)), as shown in Figure 5. It is seen that the Zernike modelling of the Kerr lensing effect can be considered as a relatively good description for j = 1 if ${\varphi }_0<1 \mathrm{rad}$ and for j = 2 if $1<{\varphi }_0<4 \mathrm{rad}$. Equations (A7)–(A10) allow for the expression of the focal length ($f_K$) characterising the OKE as follows:

$$f_K={\displaystyle \frac{K{W}^2}{\lambda {\varphi }_0}}$$

(22)

where $K=7.4$ if ${\varphi }_0<1 \mathrm{rad}$, and $K=1.65$ if $1<{\varphi }_0<4 \mathrm{rad}$. Note that $f_K$, $\lambda$, and W are expressed in millimetres. For ${\varphi }_0>4 rad$, the Kerr lensing effect is well described by the Zernike modelling if j = 3 corresponds to K = 0.71. Note that in [32], a focal length of K = 4.7 was obtained using Equation (22). The discrepancy between the values of K obtained in [32] and in this paper is probably because of distinct values of ${\rho }_0$ equal to 1.5 W in [32] and to 2 W in this work.

4. A Refractive Axicon

An axicon is an optical element described long ago by McLeod [33], sometimes called a “conical lens”, and is an optical device mainly used for generating a ring of light and nondiffracting beams [34,35]. We may note in passing that an axicon can also be used for focusing Gaussian beams to obtain an extended line segment [36,37,38]. Note that it is also possible to etch an axicon microlens on a single-mode bare optical fibre [39], thus benefiting from an axicon’s laser-beam-shaping possibilities. In a way similar to that described in the previous sections, we will be interested in the shift in the best focus because of the presence of an axicon in the focusing setup shown in Figure 6. An axicon is characterised by a wedge angle ($\alpha$) and the top angle, $\theta =(\pi -2\alpha )$, often called the “apex angle”.

The first step is to determine the on-axis intensity, $I_d(0,z)$, by applying Equation (4), where the transmittance, ${\tau }_{\varphi }(\rho )$, has to be replaced by ${\tau }_A(\rho )$, the complex transmittance of the axicon, which calculation is presented below.

$${\tau }_A(\rho )=\exp [-ik\left\{nd+\Delta z\right\}]$$

(23)

The meanings of the parameters in Equation (23) are given in Figure 7. By taking into account that $\Delta z=\rho \tan \alpha$ and that the term $\exp [-ikne]$ can be omitted because it does not play any role in the aberrations associated with the axicon, we finally obtain the complex transmission of the axicon and the WAF, $S_A(\rho )$, at the axicon’s output

$${\tau }_A(\rho )=\exp [ik\rho (n-1)\tan \alpha ]$$

(24)

$$S_A(\rho )=\rho (n-1)\tan \alpha$$

(25)

By applying Equations (A2) and (A3), we find the aberration coefficients characterising the wavefront at the axicon’s output, given as follows for an incident plane wave (Equation (A2)):

$$a_4={\displaystyle \frac{2{\rho }_0\sqrt{3}(n-1)\tan \alpha }{15}}$$

(26)

$$a_{11}=-{\displaystyle \frac{2{\rho }_0\sqrt{5}(n-1)\tan \alpha }{105}}$$

(27)

$$a_{22}={\displaystyle \frac{6{\rho }_0\sqrt{7}(n-1)\tan \alpha }{945}}$$

(28)

$$a_{37}=-{\displaystyle \frac{6{\rho }_0(n-1)\tan \alpha }{1155}}$$

(29)

and for a Gaussian incident wave (Equation (A3)):

$$a_4^{\prime }=-1.08a_4$$

(30)

$$a_{11}^{\prime }=+0.14a_{11}$$

(31)

$$a_{22}^{\prime }=+4.33a_{22}$$

(32)

$$a_{37}^{\prime }=+10.7a_{37}$$

(33)

Now, it remains to pay attention if the focal lengths $f_j$s and $f_j^{\prime }$s allow for the prediction of the position of the shifted best focal point, which is determined by the on-axis intensity calculated by applying Equation (4) provided that ${\tau }_{\varphi }(\rho )$ is replaced by ${\tau }_A(\rho )$ given by Equation (24). However, before doing that, it is important to note that the linear phase shift introduced by the refractive axicon depends on two main parameters, which are W and the angle $\alpha$. These parameters can be grouped in a single parameter denoted as $\Delta$ and defined as follows:

$$\Delta ={\rho }_{\pi }/W$$

(34)

where ${\rho }_{\pi }$ is the lateral distance at which the incident Gaussian beam is phase reversed, i.e., $k{\rho }_{\pi }(n-1)\tan \alpha =\pi$, which results in the following expression:

$${\rho }_{\pi }={\displaystyle \frac{\lambda }{2(n-1)\tan \alpha }}$$

(35)

The effect of the axicon on the on-axis intensity distribution is displayed in Figure 8, which shows a shifting of the best focus position that is particularly large as the parameter Δ is reduced. This shifting of the best focus is attributed to an axicon lensing effect, which agreement with the Zernike lens modelling, given by Equations (A7)–(A10), must be checked. For that, we reformulate the coefficients $a_j$ and $a_j^{\prime }$ by considering Equation (35).

Finally, we obtain the following numerical expressions of the focal lengths ($f_j$s) in millimetres as follows:

$$f_1=2349\Delta$$

(36)

$$f_2=1375\Delta$$

(37)

$$f_3=986\Delta$$

(38)

$$f_4=848\Delta$$

(39)

The $f_j^{\prime }$ corresponding to the “Gaussian modelling” are given as follows:

$$f_1^{\prime }=-2175\Delta$$

(40)

$$f_2^{\prime }=-2409\Delta$$

(41)

$$f_3^{\prime }=+1229\Delta$$

(42)

$$f_4^{\prime }=+386\Delta$$

(43)

Before proceeding, it is interesting to display the beam reshaping achieved by the axicon. For Δ = 0.4, Figure 9 shows the radial intensity profile at two remarkable longitudinal positions: first, at position $z=z_{\max }$, where the on-axis intensity is the maximum, and second, at position $z=f_L$, where a ring-shaped pattern is generated.

In order to appreciate the accuracy of the Zernike modelling to predict the position of the best focus, we will compare, as previously in Section 2 and Section 3, the variations in $(z_{\max }-f_L)/f_L$ versus Δ with the variations in $(z_{\max }^{\prime }-f_L)/f_L$, as determined using Equation (21). The results are shown in Figure 10 (Figure 11) for the planar wave (Gaussian) modelling.

The planar wave modelling, for $j=2$, given by Equations (A2), (A7)–(A10) and (21), seems to account properly for the best focus shifting when the parameter Δ is varied. It remains now to perform the same work for the Gaussian modelling given by Equations (A3), (A7)–(A10) and (21), and the results are shown in Figure 11. It is seen that the axicon lensing effect is well described by the Gaussian–Zernike modelling for j = 3. In summary, the axicon lensing effect causing the shift in the best focal point, as illustrated in Figure 11, is well described by the focal length given by Equation (A9), which takes the following form:

$$f_{axicon}={\displaystyle \frac{0.655W}{(n-1)\tan \alpha }}$$

(44)

The axicon focal length ($f_{axicon}$) depends on the axicon parameters (wedge angle $\alpha$ and refractive index n) as well as W, the incident Gaussian beam’s width. The result given by Equation (44) is in relatively good agreement with previous results [40,41] for beam focusing by axicons.

5. Conclusions

Focusing a Gaussian beam subject to a phase aberration usually results in a longitudinal shift in the best focus. This focus shift is attributed to a lensing effect depending upon the type of aberration and characterised by a mean focal length. The latter can be deduced from the aberration coefficients determined in the framework of a Zernike polynomial decomposition, as shown in [26]. It is important to note that the authors of [26] have defined the mean focal length ($f_{eq}$) from the relation between the incoming ($R_{in}$) and outgoing ($R_{out}$) effective radii of curvature given by

$${\displaystyle \frac{1}{R_{out}}}={\displaystyle \frac{1}{R_{in}}}+{\displaystyle \frac{1}{f_{eq}}}$$

(45)

Because we are interested in the modelling of the best focus shift, we have preferred to define the mean focal length ($f_{eq}$) from the position ($z_{\max }$) of the maximum on-axis intensity, commonly known as the best focus, and the focal length ($f_L$) of the focusing lens as follows:

$${\displaystyle \frac{1}{z_{\max }}}={\displaystyle \frac{1}{f_L}}+{\displaystyle \frac{1}{f_{eq}}}$$

(46)

The procedure for defining the mean focal length from the aberration coefficients of the wavefront outgoing from the aberration has been described in [26] and is summarised in Appendix A. The theoretical results in [26] had been confirmed experimentally by considering a primary aberration (defocus, x-astigmatism, y-astigmatism, x-coma, y-coma, and spherical aberration). In this paper, we have desired to check if the concept of the mean focal length defined in [26], applied to three different aberrations (a primary spherical aberration, the Kerr effect, and an axicon), is able to predict the position of the best focus. Note that the determination of the best focus position ($z_{\max }$) from the on-axis intensity distribution calculated using Equation (4) can be considered as a “theoretical experiment” in consideration of the high degree of confidence in its proven accuracy in diffraction modelling. We have found that the concept of the mean focal length based on Zernike aberration coefficients applies very well to the Kerr effect and to an axicon but is unable to supply the best focus shift for a pure primary spherical aberration, as defined by Equation (6). For each type of aberration, it is important to find empirically the number (integer j) of high-order spherical aberrations that are required for describing accurately the shift in the best focus.