Exact Equations for the Back and Effective Focal Lengths of a Plano-Concave Thick Lens

Abstract

In this work, we present the exact equations for the back and effective focal lengths of a plano-concave thick lens. These equations are derived through a detailed ray-tracing approach, calculating the meridional ray’s path parallel to the optical axis while taking into account lens thickness, refractive index, and surface curvature. Our analysis bridges the gap between paraxial approximations and exact ray behavior, highlighting how traditional models (e.g., thin-lens equations) can be extended to thick lenses. Additionally, we review recent advancements in negative refraction and surface phase manipulation techniques to demonstrate how our equations compare to modern numerical methods and metacoating-enhanced designs. This work provides a comprehensive and exact framework for understanding the optical properties of plano-concave thick lenses, offering new insights into their application in precision optics.

1. Introduction

In the analysis of optical systems, extended objects are approximated as arrays of point sources, whose location and size of images are determined by tracing their rays through the system. This is typically achieved through ray tracing governed by Snell’s law, which relates the angles of incidence and refraction to the refractive indices of two media:

(1)${\mathit{n}}_{\mathbf{1}}\mathbf{sin}{\mathit{\theta }}_{\mathbf{1}}={\mathit{n}}_{\mathbf{2}}\mathbf{sin}{\mathit{\theta }}_{\mathbf{2}},$

where $n_1$ and $n_2$ are the refractive indices of the initial and final media, and ${\theta }_1$ and ${\mathit{\theta }}_{\mathbf{2}}$ represent the angles of incidence and refraction relative to the normal. However, by assuming small angles ($\theta \ll$, where $\sin \theta \to \theta$), one enters the domain of Gaussian or first-order optics, where the positions and sizes of optical images can be calculated with reasonable accuracy. This approximation holds true in the paraxial region, an infinitesimal area near the optical axis [1]. Paraxial approximations simplify ray tracing but can become limiting for thick lenses, where complexities arise in accurately predicting focal properties [2,3].

In optical ray tracing, rays are classified into paraxial rays, meridional rays, and skew rays. Meridional rays pass through the lenses’ meridional plane, while skew rays do not. For thick lenses, this distinction is critical, as meridional skew rays behave differently outside the paraxial region, affecting focal length and aberration properties [4,5]. Additionally, as examined in earlier works, lens thickness significantly affects the transformation of beam waist parameters [6], requiring precise calculations beyond traditional thin-lens approximations.

A thick lens is defined by a non-negligible thickness along the optical axis compared to the radius of curvature of its surfaces. In [7], the author explored the optical behavior of concentric lenses with paraxial focusing, while, in [8], the authors extended ray-tracing methods to conic surfaces, specifically analyzing plano-concave geometries. In [9], the authors investigated the first-order properties of thick lenses, contributing foundational understanding. For plano-convex lenses, exact analytical solutions for the back and front focal lengths have been developed using meridional ray tracing [10]. However, comprehensive solutions for plano-concave lenses—often used in systems requiring the divergence rather than convergence of rays—are limited [11].

This study builds on advancements in focusing properties through lens designs that accommodate unique refractive conditions. For instance, recent research has demonstrated that plano-concave lenses fabricated from materials with negative refractive indices achieve focusing properties atypical for convex systems, as explored in photonic crystal-based lenses [12]. Moreover, the design of metacoatings with gradient refractive indices has enabled high-numerical-aperture plano-concave focusing lenses with applications in compact optical devices [13].

To advance our understanding of focal properties in plano-concave thick lenses, we hereby derive exact equations for the back focal length $\left(F_B\right)$ and effective back focal length $\left(F_E\right)$, employing meridional ray path calculations. By tracing a single meridional ray parallel to the optical axis, we obtain exact expressions for $\left(F_B\right)$ and $\left(F_E\right)$, incorporating parameters like ray height, lens thickness, and refractive index. This approach contrasts with paraxial approximations and provides a greater accuracy, which is critical for applications where precision optics are required [6,14].

Our work is also relevant to contemporary innovations, such as bi-aspheric and freeform lens designs that address aberrations like spherical aberration and astigmatism [15,16]. Additionally, displacement sensors, which benefit from precise optical alignment and divergence control, underscore the importance of rigorous focal length calculations [17]. Advances in computational methods have significantly enhanced the precision of optical modeling, especially for complex systems involving thick lenses. Numerical modeling enables the simulation of ray behavior beyond paraxial approximations, essential for accurately predicting the focal properties of plano-concave lenses. Techniques such as finite element analysis (FEA) and numerical ray tracing facilitate the examination of meridional and skew rays, providing insights into aberrations, wavefront distortions, and precise focal length calculations [18,19]. By implementing these methods, researchers can analyze the intricate interactions between rays and lens surfaces, accounting for material heterogeneities, non-linear refractive indices, and complex geometries that cannot be adequately addressed through analytical solutions alone. By presenting a robust analytical framework, this study provides insight into the exact optical properties of plano-concave thick lenses, with applications ranging from conventional imaging to emerging photonic technologies.

Following this introduction, Section 2 details the procedure and methodology for deriving the focal lengths of a plano-concave thick lens, employing a comprehensive ray-tracing approach to capture the optical intricacies which arise due to lens thickness and curvature. This section presents the exact analytical framework developed for calculating the back focal length $\left(F_B\right)$ and the effective focal length $\left(F_E\right)$, distinguishing it from conventional thin-lens approximations. Section 3 then discusses the results obtained through this model, emphasizing the implications of these findings in both standard and advanced optical applications. Finally, Section 4 concludes by summarizing the key insights gained from this study, highlighting the relevance of our exact focal length equations within the broader context of optical lens design and potential applications in fields requiring precise beam divergence and focal control.

2. Procedure and Calculation of Focal Lengths

2.1. Focal Lengths for Meridional Rays Arriving, from the Left, at the Flat Surface

2.1.1. Back Focal Length $\left(F_B\right)$

In this section, we analyze the path of a meridional ray arriving from the left at the flat surface of a plano-concave thick lens. We begin by defining the necessary parameters: the thickness $t$, the refractive index $N$, and the radius of curvature $R$ of the concave surface. A single meridional ray, parallel to the optical axis at height $H$, refracts at the concave surface. Using the established sign convention, the back focal length $F_B$ is calculated from the intersection of the prolongation of the refracted ray with the optical axis. The resulting equations express $F_B$ as a function of $H$, $R$, $N$, and other geometric parameters.

The key result is the exact equation for $F_B$ in terms of these quantities. For rays with $H>R/N$, total internal reflection occurs, and the back focal length must be recalculated under this condition.

In accordance with the sign convention, in Figure 1, $\rho$, $H$, $\theta$, $I,$ and $R$ are positive; $F_B$ is negative. Moreover, if we consider that the lens is immersed in air ($N_{air}=1$), then, from the geometry of Figure 1, we find the following expressions:

(2)$\sin I={\displaystyle \frac{H}{R}} ,$

(3)$N\sin I=\sin \rho ,$

(4)${\displaystyle \frac{\sin \rho }{-F_B+R}}={\displaystyle \frac{\sin \rho }{R}}, and$

(5)$\sin \theta =\sin \left(\rho -1\right).$

From Equations (2) and (3), we obtain

(6)$\sin \rho ={\displaystyle \frac{NH}{R}}.$

Using Equations (4) and (6), we obtain

(7)$\sin \theta ={\displaystyle \frac{NH}{-F_B+R}} .$

Then, Equation (5) can be written as follows:

(8)$\sin \theta =\sin \rho \cos I-\cos \rho \sin I=\sin \rho \sqrt{1-{\sin }^{2}I}-\sqrt{1-{\sin }^2\rho }\left(\sin I\right).$

Substituting Equations (2), (6), and (7) into (8) yields

(9)${\displaystyle \frac{N}{-F_B+R}}={\displaystyle \frac{1}{R^2}}\left[N\sqrt{R^2-H^2}-\sqrt{R^2-N^2{H}^2}\right],$

from which

(10)$F_B=R-{\displaystyle \frac{N^2\sqrt{R^2-H^2}+N\sqrt{R^2-N^2{H}^2}}{N^2-1}} .$

Equation (9) provides the back focal length as a function of the height H of the meridional ray. By observing the two roots, it is clear that H < R/N; hence, the height H_M for the marginal meridional ray (ray passing at the maximum aperture of the lens) is given by

(11)$H_M={\displaystyle \frac{R}{N}} .$

It is easy to show that any meridional ray arriving at the lens with a height $H_M>R/N$ is subject to total internal reflection at the concave surface. So, if $H_M=R/N$, $\rho =90°$ (the refracted marginal ray is tangent to the concave surface) and $I=I_C$ (critical angle). The back focal length ${{(F}_B)}_M$ for the marginal ray is obtained by substituting Equation (11) in Equation (10), as follows:

(12)${\left(F_B\right)}_M=R\left(1-{\displaystyle \frac{N\sqrt{N^2-1}}{N^2-1}}\right).$

2.1.2. Effective Focal Length $\left(F_E\right)$

In this section, the effective focal length $F_E$ is determined by calculating the position of the principal plane, which is defined by the intersection of incident and refracted ray prolongations. This results in an equation that relates $F_E$ to the height $H$, refractive index $N$, and curvature $R$, incorporating corrections for higher-order effects. For meridional rays at the maximum aperture (marginal rays), the effective focal length is obtained.

The principal plane of a lens is an imaginary plane defined by the intersections of the extensions of the incident rays and the final refracted rays, as shown in Figure 2. The effective focal length $F_E$ is the distance from the second focal point to this imaginary plane [20].

It is evident from Figure 2 that the effective focal length $F_E$ is given by the following:

(13)$-F_E=-F_B+\left(R-\sqrt{R^2-H^2}\right).$

Substituting Equation (10) into (13) gives the following:

(14)$F_E=-{\displaystyle \frac{N\sqrt{R^2-N^2{H}^2}+\sqrt{R^2-H^2}}{N^2-1}}.$

The effective focal length ${\left(F_E\right)}_M$ for the marginal meridional ray is obtained by substituting Equation (11) in Equation (14), as follows:

(15)${\left(F_E\right)}_M=-{\displaystyle \frac{R\sqrt{N^2-1}}{N\left(N^2-1\right)}}.$

2.1.3. Paraxial Focal Lengths

By considering the limit as $H\to 0$, the paraxial focal lengths are derived. These approximations yield simplified equations for both $F_B$ and $F_E$, consistent with known paraxial optics results for thick lenses. The derived paraxial back focal length and effective focal length align with standard thin-lens models, providing a useful verification of the exact equations.

The paraxial focal lengths (focal lengths for a meridional ray that is infinitesimally close to the optical axis) for a thick lens are the following [20]:

(16)${\displaystyle \frac{1}{F_B}}=\left(N-1\right)\left[{\displaystyle \frac{1}{R_1\left(1-\frac{\left(N-1\right)t}{R_{1}N}\right)}}-{\displaystyle \frac{1}{R_2}}\right] \left(\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k} \mathrm{f}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l} \mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\right)$

(17)${\displaystyle \frac{1}{F_E}}=\left(N-1\right)\left({\displaystyle \frac{1}{R_1}}-{\displaystyle \frac{1}{R_2}}\right)+{\displaystyle \frac{t{\left(N-1\right)}^2}{N{R}_1{R}_2}} \left(\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{f}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l} \mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}\right).$

In Equations (16) and (17), $R_1$ is the radius of curvature of the frontal surface and $R_2$ of the rear surface. Since the frontal surface of the plano-concave lens in Figure 1 is the flat surface ($R_1=\infty$), and the rear surface is the concave surface ($R_2=R$), then, from Equations (16) and (17), we obtain

(18)${\displaystyle \frac{1}{F_B}}=-{\displaystyle \frac{\left(N-1\right)}{R}},$

which is the paraxial back focal length for the plano-concave thick lens, and

(19)${\displaystyle \frac{1}{F_E}}=-{\displaystyle \frac{\left(N-1\right)}{R}},$

which is the paraxial effective focal length for the plano-concave thick lens.

From Equations (10) and (14), the paraxial focal lengths of Equations (18) and (19) can be obtained. If $H\to 0$ (paraxial ray) into Equation (10), we obtain

(20)$\underset{H\to 0}{\lim }{F}_B=R-{\displaystyle \frac{N^2\sqrt{R^2-{\left(0\right)}^2}+N\sqrt{R^2-N^2{\left(0\right)}^2}}{N^2-1}}=R-{\displaystyle \frac{N^{2}R+NR}{N^2-1}}={\displaystyle \frac{R{N}^2-R-N^{2}R-NR}{N^2-1}}=-{\displaystyle \frac{R\left(N+1\right)}{\left(N+1\right)\left(N-1\right)}}=-{\displaystyle \frac{R}{N-1}},$

from which

(21)${\displaystyle \frac{1}{F_B}}=-{\displaystyle \frac{N-1}{R}},$

where $R$ is positive.

Now, if $H\to 0$ into Equation (14), we obtain

(22)$\underset{H\to 0}{\lim }{F}_E=-{\displaystyle \frac{N\sqrt{R^2-N^2{\left(0\right)}^2}+\sqrt{R^2-{\left(0\right)}^2}}{N^2-1}}=-{\displaystyle \frac{NR+R}{N^2-1}}=-{\displaystyle \frac{R\left(N+1\right)}{\left(N+1\right)\left(N-1\right)}}=-{\displaystyle \frac{R}{N-1}} ,$

from which

(23)${\displaystyle \frac{1}{F_E}}=-{\displaystyle \frac{N-1}{R}} ,$

where $R$ is positive.

2.2. Focal Lengths for Meridional Rays Arriving, from the Left, at the Concave Surface

2.2.1. Back Focal Length $\left(F_B\right)$

For rays arriving at the concave surface, the analysis proceeds similarly, but with a reversed geometry. Using the same sign convention, the angles of incidence and refraction are now negative, as is the radius $R$. The back focal length is again determined from the geometry of the refracted ray’s intersection with the optical axis. The exact equation for $F_B$ as a function of $H$ is derived, showing how the focal point shifts depending on the height at which the ray strikes the lens.

Figure 3 shows the meridional ray arriving, from the left, at the concave surface (front surface) of the plano-concave thick lens. In Figure 3, $P$ is the length of the prolongation of the meridional ray after being refracted by the concave surface, $\rho -I$ is the incidence angle on the flat surface, and $x$ is the distance from the center of curvature to the intersection of the prolongation of the meridional ray with the optical axis. Moreover, in accordance with the sign convention, in Figure 3, the radius $R$, the back focal length $F_B$, and the angles of incidence and refraction $I$ and $\rho$, respectively, are negative; the angle of refraction $\theta$ on the concave surface and the height $H$ are positive.

From Figure 3, we find the following expressions:

(24)$\sin I={\displaystyle \frac{H}{R}},$

(25)$\sin I=N\sin \rho ,$

(26)$N\sin \left(\rho -I\right)=\sin \theta ,$

(27)${\displaystyle \frac{\sin \rho }{x}}={\displaystyle \frac{\sin \left(\rho -I\right)}{R}}={\displaystyle \frac{\sin I}{P}},$

(28)$\mathrm{a}\mathrm{n}\mathrm{d}, \tan \left(\rho -I\right)={\displaystyle \frac{H}{\sqrt{{\left(-R\right)}^2-H^2}+x}}={\displaystyle \frac{H}{\sqrt{R^2-H^2}+x}}.$

From Equations (24) and (25), we obtain

(29)$\sin \rho ={\displaystyle \frac{H}{NR}}.$

From Equations (24), (27), and (29), we find that

(30)$P={\displaystyle \frac{x\sin I}{\sin \rho }}=Nx.$

From Equations (24), (27), and (30), we obtain

(31)$\sin \left(\rho -I\right)={\displaystyle \frac{R\sin I}{P}}={\displaystyle \frac{H}{Nx}}.$

Substituting Equation (26) into Equation (21), we obtain

(32)$\sin \theta ={\displaystyle \frac{H}{x}}.$

From Figure 3, we can observe that

(33)${\left(\sqrt{R^2-H^2}+x\right)}^2+H^2=P^2=N^2{x}^2,$

from which

(34)$\left(N^2-1\right)x^2-2x\sqrt{R^2-H^2}-R^2=0.$

From Equation (34), we isolate $x$ to obtain

(35)$x={\displaystyle \frac{\sqrt{R^2-H^2}+\sqrt{R^2{N}^2-H^2}}{N^2-1}}.$

On the other hand, from Figure 3, it is clear that

(36)$\left(-F_B\right)\tan \theta -\left[\left(-R+t\right)-\sqrt{R^2-H^2}\right]\tan \left(\rho -I\right)=H,$

which can be rewritten as

(37)$\left(-F_B\right){\displaystyle \frac{\sin \theta }{\sqrt{1-{\sin }^2\theta }}}-\left[\left(-R+t\right)-\sqrt{R^2-H^2}\right]\tan \left(\rho -I\right)=H.$

Substituting Equations (28) and (32) into Equation (37) gives

(38)$F_B=-\hspace{0.33em}\sqrt{x^2-H^2}\left[1+{\displaystyle \frac{-R+t-\sqrt{R^2-H^2}}{\sqrt{R^2-H^2}+x}}\right].$

Namely,

(39)$F_B=-\hspace{0.33em}\sqrt{x^2-H^2}\left[1-{\displaystyle \frac{\sqrt{R^2-H^2}+\left(R-t\right)}{\sqrt{R^2-H^2}+x}}\right],$

where $x$ is given by Equation (35).

Equation (39) gives the back focal length as a function of the height $H$ for the meridional ray. From Equation (39), $F_B=0$ for $H=x$; therefore, the height $H_M$ for the marginal meridional ray is

(40)$H_M=x.$

It is easy to show that any meridional ray at a height $H_M>x$ is not refracted (the ray is subject to total internal reflection) on the flat surface. Then, for $H_M=x$, $\theta =90°$ (the refracted marginal ray is tangent to the flat surface) and $\rho -I=(\rho -I)C$ (critical angle). The height $H_M$ is obtained by substituting Equation (39) into Equation (35), as follows:

(41)$H_M=-R\sqrt{{\displaystyle \frac{N^2+1+2\sqrt{N^2-1}}{{\left(N^2-1\right)}^2+4}}}.$

2.2.2. Effective Focal Length $\left(F_E\right)$

The principal plane for this case is also analyzed, and the effective focal length $F_E$ is computed by tracing the refracted rays’ prolongations. The result provides the exact relation between the effective focal length and the geometric and material parameters of the lens.

From Figure 4, we see that the effective focal length F is equal to the following:

(42)${-F}_E={\displaystyle \frac{H}{\tan \theta }}={\displaystyle \frac{H\cos \theta }{\mathrm{s}\mathrm{i}\mathrm{n}\theta }}={\displaystyle \frac{H\sqrt{1-{\sin }^2\theta }}{\sin \theta }}.$

Substituting Equation (32) into Equation (42) we obtain the following:

(43)$F_E=-\sqrt{x^2-H^2}.$

For the height $H_M$ given by Equation (32), the effective focal length ${\left(F_E\right)}_M$ is equal to zero.

2.2.3. Paraxial Focal Lengths

For paraxial rays (where $H\to 0$), the focal lengths reduce to simple forms, again consistent with the paraxial approximations for a thick lens. The paraxial back and effective focal lengths are shown to agree with previously known results for plano-concave lenses.

Since the frontal surface of the plano-concave lens in Figure 3 is the concave surface ($R_1=R$) and the rear surface is the flat surface ($R_2=\infty$), then, from Equations (16) and (17), we obtain

(44)${\displaystyle \frac{1}{F_B}}={\displaystyle \frac{\left(N-1\right)}{R\left(1-\frac{\left(N-1\right)t}{RN}\right)}} \left(\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} R \mathrm{i}\mathrm{s} \mathrm{n}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\right),$

which is the paraxial back focal length for the plano-concave thick lens, and

(45)${\displaystyle \frac{1}{F_E}}={\displaystyle \frac{\left(N-1\right)}{R}} \left(\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} R \mathrm{i}\mathrm{s} \mathrm{n}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\right),$

which is the paraxial effective focal length for the plano-concave thick lens.

From Equations (39) and (43), the paraxial focal lengths in Equations (44) and (45) can be obtained. If $H\to 0$ (paraxial ray) into Equations (36) and (40), we obtain the following:

(46)$$\begin{gathered} \underset{H\to 0}{\lim }x={\displaystyle \frac{\sqrt{{\left(-R\right)}^2-{\left(0\right)}^2}+\sqrt{{\left(-R\right)}^2{N}^2-{\left(0\right)}^2}}{N^2-1}}={\displaystyle \frac{-R-RN}{N^2-1}}=-{\displaystyle \frac{R}{N-1}}\underset{H\to 0}{\lim }{F}_B \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =-\hspace{0.33em}\sqrt{{\left(-{\displaystyle \frac{R}{N-1}}\right)}^2-{\left(0\right)}^2}\left[1-{\displaystyle \frac{\sqrt{{\left(-R\right)}^2-{\left(0\right)}^2}+\left(R-t\right)}{\sqrt{{\left(-R\right)}^2-{\left(0\right)}^2}-\frac{R}{N-1}}}\right] \\ =-\left(-{\displaystyle \frac{R}{N-1}}\right)\left(1-{\displaystyle \frac{\left(-R\right)+R-t}{-R-\frac{R}{N-1}}}\right) \end{gathered}$$

(47)$$\begin{gathered} =\left({\displaystyle \frac{R}{N-1}}\right)\left(1-{\displaystyle \frac{t}{R\left(1+\frac{1}{N-1}\right)}}\right)=\left({\displaystyle \frac{R}{N-1}}\right)\left(1-{\displaystyle \frac{t}{R\frac{\left(N-1\right)+1}{N-1}}}\right) \\ =\left({\displaystyle \frac{1}{N-1}}\right)\left[R\left(1-{\displaystyle \frac{t\left(N-1\right)}{RN}}\right)\right] , \end{gathered}$$

From the above equation we finally have

(48)${\displaystyle \frac{1}{F_B}}={\displaystyle \frac{N-1}{R\left(1-\frac{\left(N-1\right)t}{RN}\right)}} ,$

where $R$ is negative.

Now, if $H\to 0$ into Equation (43), we have

(49)$F_E=-\sqrt{{\left(-{\displaystyle \frac{R}{N-1}}\right)}^2-{\left(0\right)}^2}=-\left(-{\displaystyle \frac{R}{N-1}}\right),$

(50)${\displaystyle \frac{1}{F_E}}={\displaystyle \frac{N-1}{R}} ,$

where $R$ is negative.

The following flowchart, in Figure 5, outlines the step-by-step procedure for calculating the focal lengths of meridional rays in a plano-concave thick lens. This visual guide aims to clarify each part of the process discussed in this section, from defining the parameters to addressing total internal reflection and paraxial focal length approximations.

3. Discussion of Results

The exact equations derived in this work—particularly Equations (10) and (39) for the back focal length and Equations (14) and (43) for the effective focal length—represent a marked improvement over traditional approximations used in lens design. These equations transcend paraxial and thin-lens limitations by tracing meridional rays rigorously through each refractive interface, accounting for higher-order effects which simple Gaussian approximations omit. By integrating the full lens thickness and using precise sign conventions, these exact formulas reveal dependencies on factors such as curvature, refractive index, and ray height, factors which significantly impact focal length, especially in thick, plano-concave geometries.

This approach fills gaps in traditional lens analyses, where focal length calculations for plano-concave lenses are often confined to paraxial approximations, as noted in standard optics texts [1]. Furthermore, previous studies, such as Rosin’s early exploration of concentric lens systems [7] and the work by Cornejo-Rodriguez et al. [8], primarily address paraxial behavior in plano-convex designs and merely approximate diverging lens characteristics for plano-concave configurations. The novel equations derived here extend such foundational work by incorporating exact, non-paraxial conditions, which are essential in advanced applications that demand high precision, such as imaging and photonic engineering.

Comparative studies on Gaussian beam transformations, like those by Nemoto [6], show that lens thickness plays a significant role in waist parameter transformations. Our results support Nemoto’s findings and go further by detailing exact focal dependencies, aligning well with studies on wavefront manipulation in thick lenses [15]. Furthermore, Vodo et al. [12] demonstrate how a plano-concave lens with a photonic crystal can focus light using negative refraction, underscoring the relevance of specific refractive index modifications. Our equations provide an exact analytical foundation for such phenomena, enabling the practical adaptation of plano-concave lenses in negative refraction scenarios. By rigorously accounting for refraction and thickness, our equations remain accurate across applications using both positive and negative index materials, ensuring reliable performance across diverse optical designs.

Studies by Malacara-Hernández and others [10,11] focused on plano-convex thick lenses illustrate how exact focal length expressions can transition seamlessly to paraxial predictions in the thin-lens limit. Our results mirror these observations for plano-concave lenses, showing a smooth reduction to paraxial results as thickness diminishes. This reduction verifies the robustness of the exact equations, making them adaptable for both high-precision optics and simplified, paraxial-based setups where approximation suffices.

In more recent contexts, our findings align with advancements in metacoating and freeform lens designs. For example, research by González-Acuña et al. [16,17] has yielded general formulas for bi-aspheric and freeform lenses, free from aberrations such as astigmatism. These studies underscore the need for exact analytical solutions in complex optical design, as our work provides here for plano-concave lenses. Additionally, metacoating applications explored by Naserpour et al. [13] further validate the importance of customized refractive index profiles, which our equations can accommodate, enabling effective plano-concave lens designs for high-numerical-aperture and gradient-index systems.

In summary, the equations developed in this study offer a comprehensive model for calculating both the back and effective focal lengths of a plano-concave thick lens. By transcending paraxial limitations, they provide a general solution applicable to both traditional and emerging optical technologies. The results underscore the relevance of exact ray-tracing methods in modern lens design, offering a robust alternative to thin-lens approximations and supporting the inclusion of plano-concave lenses in high-precision, specialized optical systems.

4. Conclusions

In this study, we derived and presented exact analytical equations for the back focal length $\left(F_B\right)$ and effective focal length $\left(F_E\right)$ of a plano-concave thick lens. These results were obtained through a rigorous application of meridional ray tracing and the use of a precise sign convention for both distances and angles along the optical axis. The equations offer a clearer and more detailed understanding of the optical properties of thick lenses, extending the analysis beyond traditional paraxial models. Our findings demonstrate how key factors—such as lens thickness, surface curvature, and refractive index—significantly influence focal lengths, making our equations critical for optical system design where such parameters cannot be neglected.

By bridging the gap between paraxial approximations and exact ray-tracing methods, our work provides optical designers with a flexible and powerful tool. The ability to transition seamlessly from exact formulations to paraxial models enhances the robustness of the analysis, allowing for accurate predictions of lens behavior in both idealized and practical optical systems. This flexibility makes the derived formulas highly versatile, applicable to scenarios requiring both high-order precision and simplified, first-order solutions.

In comparison to existing research on plano-convex lenses and modern innovations such as metacoating-enhanced optical designs, our approach offers an analytical foundation that complements numerical and experimental techniques. While previous models have focused on either specific cases or numerical approximations, our exact analytical equations provide a more generalized solution. This is particularly relevant in the context of emerging technologies like negative refraction and gradient-index materials, where precise control over optical paths is essential. As the field continues to evolve, our equations can serve as a benchmark for evaluating complex optical phenomena and optimizing the performance of plano-concave lenses in advanced applications.

Moreover, the incorporation of modern innovations, such as metacoatings and photonic crystals, offers promising future directions for applying our work. These advancements not only enhance the focusing capabilities of plano-concave lenses but also suggest potential applications in areas such as optical trapping, high-resolution imaging, and the development of high-performance optical systems. Thus, the theoretical framework presented in this paper lays the groundwork for continued exploration and innovation in the design and optimization of thick lens systems.

Author Contributions

Conceptualization, V.M.D.-R.; formal analysis, writing—original draft preparation, and writing—review and editing, J.M.-M.; formal analysis, writing—original draft preparation, and writing—review and editing, F.J.C.-R.; formal analysis, writing—original draft preparation, and writing—review and editing, M.M.-G.; and formal analysis, writing—original draft preparation, and writing—review and editing, F.G.P.-L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding authors.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures

Figure 1. Meridional ray, parallel and at a height H from the optical axis, arriving, from the left, at the flat surface (front surface) of a plano-concave thick lens.

Figure 2. Principal plane and effective focal length in a plano-concave thick lens for a meridional ray arriving, from the left, at the flat surface.

Figure 3. Meridional ray, parallel and at a height H from the optical axis, arriving, from the left, at the concave surface (front surface) of a plano-concave thick lens.

Figure 4. Principal plane and effective focal length in the plane-concave thick lens for the meridional ray arriving, from the left, at the concave surface.

Figure 5. Flowchart of the procedure and calculation of focal lengths for meridional rays in a plano-concave thick lens.

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