Modeling Several Optical Components Using Scalar Diffraction Theory

Abstract

Compound systems are generally treated by geometrical optics, for instance, through the Gauss’ formalism. The objective is to simplify the process of image formation. However, this formalism does not include the wave characteristics of light and boundary effects. The treatment of diffraction is not straightforward. Thus, the extension of this formalism towards the scalar theory of diffraction is very desired. This work offers this extension and emphasizes its importance. Compound systems, including the human eye, are then modeled by Fresnel theory. For illustration, a lens-based model of the Fresnel transform is used to treat the human eye system.

1. Introduction

During the running wheel of history, several theories have been used to model the behavior of light, with respect to obstacles [1,2,3]. First, geometrical optics was used to analyze the behavior of light. Light is seen as rays traveling in straight lines [2]. Geometrical optics is the least rigorous theory modeling the behavior of light (Figure 1). Wave optics, or undulatory optics, is a more rigorous theory (Figure 1) [4]. This theory considers light as a propagating wave. It allows us to give an interpretation of the interference phenomenon [5,6]. The scalar theory of diffraction is more rigorous (Figure 1) [7,8,9]. It allows us to handle diffraction and the effects of the boundaries. The electromagnetic theory is more rigorous (Figure 1) [7,10]. Propagation is seen as interaction between the interaction of the electric and magnetic fields in the medium as vector, and not scalar, entities. Quantum physics is more rigorous (Figure 1) [11].

In the present work, we intend to use the scalar theory of diffraction, with the aim of proposing a powerful tool to easily treat and design optical components and systems. Such analysis and design become easier and more illustrative if all elements of the system, the human eye, for example, are combined into one compound system and handled as one entity. To give an example of the usefulness of such a demarch, the design of diffractive optical systems, fulfilling preestablished constraints, becomes straightforward and doable through such a compound system, within a much rigorous theory than geometrical optics, namely the scalar theory of diffraction (Figure 1). Another interesting system is the second order dispersive medium, such as the single-mode fiber [12,13]. The compound system presents a powerful tool in the design of fiber components compensating for chromatic dispersion. It is worth noting that these components are key elements in broadband communication.

The contributions of this paper may be summarized as follows:

• With the proposed formalism, we can analyze higher diffraction orders, which is not straightforwardly feasible with conventional geometrical optics.
• The new formalism allows for easily calculating the (1) PSF (point spread function), (2) DTF (optical transfer function), and (3) MTF (modulation transfer function). These three metrics are very meaningful in measuring the optical behavior of various systems. Special interest goes to the optical system of the human eye.
• The system of the human eye is easily covered by our approach.
• Optical performance criteria, as well as image quality criteria, including the optical transfer (OTE), are easily treated by our diffractive approach.
• We can easily treat optical information processing and invariance issues [14].

The remainder of the paper is organized as follows. Section 2 provides an overview on related works. Section 3 presents the mathematical formulation of the proposed compound system. Three systems are handled as examples in Section 4. Section 5 is devoted to the discussion and concluding remarks.

2. Related Works

Ray tracing is the practical tool of geometrical optics [10]. It is based on the following three postulates:

• The wave direction is given by the normal, to what is referred to as the equiphase planes.
• Rays are assumed to progress in straight lines in a homogeneous medium.
• Power is conserved inside a bundle of rays.
• Reflection and refraction obey the refraction rule.

It is obvious that geometric optics cannot offer a comprehensive interpretation of image formation for an object through a given optical instrument. For example, aberrations and diffraction could not be straightforwardly treated by geometrical optics. In this paper, we intend to bypass these constraints by offering the concept of a diffraction-based compound system.

Diffraction was first observed by Grimaldi, when illuminating a rectangular aperture in an opaque medium. He used a light source at his disposal at that time. He observed the image at some distance behind the screen [15]. He did not observe sharp borders. He observed a gradual transition from light to shadow. Huygens built up Grimaldi’s observations and advanced wave optics [16]. Young, who discovered interference [5], supported wave theory [6]. Fresnel was convinced that the postulation of Huygens does not offer an explanation for the non-existence of propagating backwards waves and advanced his theory. He proposed a combination of the Huygens’ principle, with the “envelope” building [7]. G. Kirchhoff gave a full mathematical formulation to the scalar diffraction theory, based on the works of Fresnel and Huygens [7]. Sommerfeld and Rayleigh further developed Kirchhoff’s work and advanced what was later referred to as the “Rayleigh-Sommerfeld diffraction theory” [7].

Let us go back to geometrical optics. There are recent researches using geometrical optics to handle new issues, such as paraxial-based optical predesign of systems, polarization, inverse problems, refractive index distributions, surface geometries from ray-tracing, and aberrations. A recent work highlighted the topological space of 2 × 2 matrices. The authors used the classical versus quantum, as well as the wave models. They proposed 2 × 2 matrices to treat the quantization process [17]. Based on geometrical optics, Rakich proposed a new type of atmospheric dispersion corrector, suitable for wide-field high-resolution imaging and spectroscopy on large telescopes [18]. Chee et al. proposed a new optical design method of a floating type collimator for microscopic camera inspection [19]. An improved analytical theory of ophthalmic lens design, based on geometrical optics, was proposed recently [20]. To control the linear astigmatism aberration in a perturbed axially symmetric optical system and tolerancing was performed by using geometrical optics [21]. A focus-adjustable, head-mounted display with off-axis system was proposed [22]. The objective of this paper is not to be limited to a geometrical theory of diffraction as suggested by Keller [1]. What it is proposed in this paper is a powerful tool in a similar way as done with the compound system in geometrical theory.

In summary, the wave aspect is intrinsic to scalar theory of diffraction but not to geometrical theory. The addition of wave aspects, such as wavefront aberration, polarization, PSF, OTF, and MTF, into geometrical optics are neither natural nor straightforward in this theory. What is generally done is an addition of wave aspects to ray-tracing. In other words, the wave character is artificially added in modified geometrical optical approaches.

3. Analysis

The Rayleigh–Sommerfeld diffraction theory uses some approximations, which are valid if [7] the diffraction features are large, compared to the wavelength:

• The observation distance is significantly larger than the diffracting objection;
• The angles involved in the diffraction process are small.

In this paper, we limit attention to the framework of these conditions.

3.1. Fresnel Transform

For brevity of notation, we will consider only one lateral dimension, x. Extension to (x,y) analysis is straightforward. The diffraction field g(x,z), seen at a distance, z (Figure 2a), is described by the Fresnel transform [7,23]:

$$g\left(x,z\right)=F{R}_z\left\{g\left(x,0\right)\right\}=\frac{\exp \left(i2\pi z/\lambda \right)\exp \left(-i\pi /4\right)}{\sqrt{\lambda z}}g\left(x\right)\ast f_k\left(x,z\right)$$

(1)

where, g(x) = g(x,0) means the initial field, and λ stands for the wavelength in vacuum (or air), whereas ∗ is the simple of convolution; g(x,0) is also called the object. In general, this object has a complex expression. The Fresnel kernel is then [4]:

$$f_k\left(x,z\right)=\exp \left(i\pi \frac{x^2}{\lambda z}\right)$$

(2)

The constant phase term of propagation $\exp \left(i2\pi z/\lambda \right)$, as well as the complex factor $\exp \left(-i\pi /4\right)/\sqrt{\lambda z}$, are not considered, since they do not intervene in the analysis. In the Fourier plane, Equation (1) becomes [24,25]:

$$G\left(u,z\right)=G\left(u\right)\exp \left(-i\pi \lambda z{u}^2\right)$$

(3)

The distribution G(u,z) is a spectrum, namely the Fourier transform (FT) of g(x,z), whereas G(u) is the FT of the object.

The diffraction field at z is expressed by an FT [24]:

$$g\left(x,z\right)=\exp \left(i\pi \frac{x^2}{\lambda z}\right){\left.G’\left(u^{\prime }\right)\right|}_{u’=x/\left(\lambda z\right)}$$

(4)

$$\mathrm{with} G’\left(u’\right)=FT\left\{\exp \left(i\pi \frac{u^2}{\lambda z}\right)g\left(x\right)\right\}$$

Once the FT is performed, the spatial frequency u’ will be replaced by the scaled term x/(λz). The two phase-terms, in relation to (4), namely $\exp \left(i\pi \frac{x^2}{\lambda z}\right)$ and $\exp \left(i\pi \frac{u^2}{\lambda z}\right)$, describe two divergent waves (spherical) having a radius that is equal to z. To obtain an exact FT of the object, we should omit two waves (spherical) by using two identical spherical lenses. The focal length of each lens should be z (Figure 2b). The FT of the object (initial field) g(x) is realized physically by using a spherical lens. This object should be placed in its front (anterior) focal plane [7]. The FT of the object is then observed in the back (posterior) focal plane of this lens, as depicted in Figure 3b. To implement the Fresnel transform, FR_z, of the initial field, we use two divergent spherical lenses, performing the two phase-terms in relation (4), as shown in Figure 3a. This lenses-based formalism enables handing the compound systems.

3.2. Compound Systems Using Geometrical Optics

Let us consider a concrete example: two lenses, separated by a distance (d). The generalization towards more complex systems, including various types of optical components, is straightforward.

Two steps are needed to determine the position of the image formed by a two-lenses system. We determine the position of the image that is formed by the first lens. Then, we take this first image as an object for the second lens. We then obtain the image position for the second lens, which is also the position of the final image of the entire system.

It is worth noting that “paraxial ray propagator matrix” could be applied, which is more general than geometric optics [26]. Two lenses, separated by a distance (d), are combined (Figure 4). Given that the thickness of both thin lenses does not enter into play, the matrixes of lenses 1 and 2 becomes:

$$A_1=\left[\begin{array}{cc}1& 1/f_1\\ 0& 1\end{array}\right] \mathrm{and} A_2=\left[\begin{array}{cc}1& 1/f_2\\ 0& 1\end{array}\right]$$

(5)

The transfer matrix from the first lens towards the second lens becomes:

$${\mathrm{T}}_{21}=\left[\begin{array}{cc}1& 0\\ -d& 1\end{array}\right]$$

(6)

We then obtain the matrix of the two lens-based compound system:

$$\mathrm{A}=A_2{T}_{21}{A}_1=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]=\left[\begin{array}{cc}1-d/f_2& 1/f_1+1/f_2-d/\left(f_1{f}_2\right)\\ -d& 1-d/f_1\end{array}\right]$$

(7)

The resulting anterior (front) and posterior (back) focal lengths, f and f’, of the compound system become:

$$\left\{\begin{array}{c}\frac{1}{f’}=A_{12}=\frac{1}{f_1}+\frac{1}{f_2}-\frac{d}{f_1{f}_2}\\ f=-f’\end{array}\right.$$

(8)

The principal planes are located as follows:

$$h=\frac{1-A_{11}}{A_{12}}=d\frac{f’}{f_2} \mathrm{and} h’=\frac{A_{22}-1}{A_{12}}=d\frac{f}{f_1}$$

(9)

The following Gaussian formula gives us an image, described by the field g₂(x,z₃):

$$z_3=\frac{A_{21}-A_{22}{z}_1}{A_{11}-A_{12}{z}_1}$$

(10)

We then obtain the linear magnification or transverse magnification (m):

$$m=\frac{A_{22}-z_3{A}_{12}}{A_{11}{A}_{22}-A_{12}{A}_{21}}=\frac{1}{A_{11}-A_{12}{z}_1}=\frac{f_1{f}_2}{f_1\left(f_2-d\right)-z_1\left(f_2+f_1-d\right)}$$

(11)

The transverse magnification is defined as the image size, divided by the object size.

3.3. Difraction-Based Compound System

Let us again consider the compound system that is composed of two lenses (Figure 4). We intend to determine the output, means of the field g₂(x,z₃) as a function of the object, and means of g(x) and the system parameters: d, f₁, and f₂. For simplicity, we decompose the calculation of the diffraction field in four successive steps (Figure 4). We go through the intermediate fields: $g\left(x,z_1\right), g_1\left(x,0\right), g_1\left(x,z_2\right), g_1\left(x,d\right), g_2\left(x,0\right)$. To bypass heavy mathematical algebra by using the Fresnel transform (relation (1)), we suggest using the lens-based model of the diffraction system (Figure 3a).

By using the model indicated in Figure 3a, Fresnel diffraction is performed in four steps. This leads to the setup of Figure 5. While the initial (original) lenses of Figure 4 are presented in dark gray color in Figure 5, the lenses of the lenses-based model are presented in light gray color. In step 1, two divergent lenses (in light gray), with focal length –z₁, as well as a convergent spherical lens, with focal length z₁, are involved in the diffraction process. These three lenses implement the Fresnel transform of g(x,0), at distance z₁, means g(x,z₁) = FR_z1{g(x,0)}, according to the scheme of Figure 3a. Then, light goes through the first original lens (dark gray), with focal length f₁. Step 2 puts three lenses into play, which physically implement the Fresnel transform of g₁(x,0), at distance z₂ and means g₁(x,z₂) = FR_z2{g₁(x,0)}, according to the scheme of Figure 3a. Just behind the first initial lens (dark gray), we find the first divergent lens (light gray). Three side-by-side lenses are shown on the left-hand side of Figure 5. The three lenses cancel each other if:

$$\frac{1}{f_1}=\frac{1}{z_1}+\frac{1}{z_2}$$

(12)

For z₂ satisfying Equation (12), the three lenses are without effect (see cross in Figure 5).

Step 3 puts three lenses into play, which implement the Fresnel transform of g₁(x,z₂), at distance d-z₂ and means g₁(x,z₂+d-z₂) = g₁(x,d) = FR_d-z2{g₁(x,z₂)}, according to the scheme of Figure 3a.

As depicted in Figure 5, we obtain two side-by-side lenses, with respective focal lengths -z₂ and z₂-d, which could be replaced by one lens with power: F_c = −1/z₂ −1/(d-z₂). The focal length is its inverse:

$$f_c=\frac{1}{-{1/z}_2-{1 /(d-z}_2)}=z_2\frac{z_2-d}{d}$$

(13)

Thus, light propagating until just behind this compound lens, with focal length f_c, comes across four lenses, with respective focal lengths (from left to right in Figure 5): −z₁, z₁, z₂, and f_c, bearing in mind that three lenses on the left-hand side of Figure 5 are canceled out, and two other lenses are compounded together into one lens, with focal length f_c. Among these four lenses, the first and fourth lens are divergent (focal lengths: −z₁ and f_c), whereas the two lenses in the middle, namely the lenses with focal lengths z₁ and z₂, are convergent. Each of these two convergent lenses physically implement the Fourier transform (FT) of the wavefront in its front focal plane. The two successive FTs give us the zoomed version of the input wavefront of the first lens by the factor: z₂/z₁. The first and fourth lens, among these four lenses, respectively introduce the two quadratic phase distributions $\exp \left(j\pi \frac{x_1^2}{\lambda z_1}\right)$ and $\exp \left(-j\pi \frac{x^2}{\lambda f_c}\right)$, which are the transmittances of two divergent lenses.

Therefore, behind the compound lens, with focal length f_c and the diffraction field g’(x), is:

$$g’\left(x\right)=\exp \left(j\pi \frac{x_1^2}{\lambda z_1}\right)g\left(-x_1,0\right)\exp \left(-j\pi \frac{x^2}{\lambda f_c}\right)$$

(14)

$$\mathrm{with}\hspace{1em}{x}_1=\frac{z_1}{z_2}x$$

(15)

We obtain g(−x₁,0), instead of g(x₁,0), because of the application of two successive FTs, which is the concept of the telescope setup. Indeed, FT{FT{g(x)}} = g(−x).The scale (zoom) factor of Equation (15) is because the two FTs are performed, with respect to two respective factors: z₁ and z₂. For the special case of z₁ = z₂, we obtain a 4f-setup.

By combining relations (14) and (15), we obtain:

$$g’\left(x\right)=\exp \left(j\frac{\pi }{\lambda }\left[\frac{z_1}{z_2^2}-\frac{1}{f_c}\right]x^2\right)g\left(-\frac{z_1}{z_2}x,0\right)$$

(16)

For steps 3 and 4, we suppose that the three side-by-side lenses on the right (Figure 5) satisfy:

$$\frac{1}{f_2}=\frac{1}{d-z_2}+\frac{1}{z_3}=\frac{z_3+d-z_2}{z_3\left(d-z_2\right)}$$

(17)

If we solve Equation (17) for z₃, we obtain relation (9). The output diffraction field becomes:

$$g_2\left(x,z_3\right)=g’\left(-x_2\right)\exp \left(j\pi \frac{x^2}{\lambda z_3}\right)$$

(18)

$$\mathrm{where}\hspace{1em}{x}_2=\frac{d-z_2}{z_3}x$$

(19)

Leading to:

$$g_2\left(x,z_3\right)=g’\left(-\frac{d-z_2}{z_3}x,0\right)\exp \left(j\pi \frac{x^2}{\lambda z_3}\right)$$

(20)

or:

$$g_2\left(x,z_3\right)=\exp \left(j\frac{\pi }{\lambda R}{x}^2\right)g\left(\frac{x}{S}\right)$$

(21)

We obtain the radius of divergence R:

$$\frac{1}{R}=\left[\frac{z_1}{z_2^2}-\frac{1}{f_c}\right]{\left(\frac{d-z_2}{z_3}\right)}^2+\frac{1}{z_3}$$

(22)

where the scaling factor S:

$$S=\frac{z_2{z}_3}{z_1\left(d-z_2\right)}$$

(23)

By having a close look at Equations (11), (12), (17) and (23) it turns out that: S = m.

Moreover, by considering Equation (22), it yields:

$$R=\frac{z_3\left(f_1+f_2-d\right)-f_2\left(f_1-d\right)}{f_1+f_2-d}$$

(24)

Equations (8), (9) and (24) lead, after little algebra, to (Figure 4):

$$R=z_3-f’-h’$$

(25)

The quadratic phase term $\exp \left(j\frac{\pi }{\lambda R}{x}^2\right)$ of relation (21) represents a spherical wave, focusing on the back focal point F’ (compound system). If the input of the compound system is a plane wave, then the output field is that of a spherical wave converging towards F’. After this point, the wave diverges. At the distance z₃, we again obtain: $\exp \left(j\frac{\pi }{\lambda R}{x}^2\right)$. The term g(x/S), in relation to (21), becomes the constant 1, which is in full agreement with the framework of geometrical optics.

z₃ = h’ presents another interesting situation. This case that the wave is an observed plane at the back principal plane H’. We observe a spherical wave that converges towards (or diverges from) the focal point F’, which is in total agreement with the framework of geometrical optics.

In Figure 5, we see 14 lenses. Six of them cancel each other out. It is possible to apply a paraxial ray propagator matrix. We then obtain eight matrixes, i.e., A₁ to A₈:

$$\begin{gathered} A_1=\left[\begin{array}{cc}1& -1/z_1\\ 0& 1\end{array}\right], A_2=\left[\begin{array}{cc}1& 1/z_1\\ 0& 1\end{array}\right],A_3=\left[\begin{array}{cc}1& 1/z_2\\ 0& 1\end{array}\right], A_4=\left[\begin{array}{cc}1& -1/z_2\\ 0& 1\end{array}\right], \\ {A}_5=\left[\begin{array}{cc}1& 1/(z_2-d)\\ 0& 1\end{array}\right], A_6=\left[\begin{array}{cc}1& 1/\left(d-z_2\right)\\ 0& 1\end{array}\right], A_7=\left[\begin{array}{cc}1& 1/z_3\\ 0& 1\end{array}\right], A_8=\left[\begin{array}{cc}1& -1/z_3\\ 0& 1\end{array}\right] \end{gathered}$$

(26)

The matrixes A₄ and A₅ could be combined into one matrix $A_{4,5}=\left[\begin{array}{cc}1& -1/\mathrm{d}\\ 0& 1\end{array}\right]$, since the two corresponding lenses are adjacent and can be replaced by a single lens, with focal length -d. We then obtain six transfer matrixes:

$$\begin{gathered} T_{21}=\left[\begin{array}{cc}1& 0\\ -z_1& 1\end{array}\right], T_{32}=\left[\begin{array}{cc}1& 0\\ -z_1-z_2& 1\end{array}\right],T_{43}=\left[\begin{array}{cc}1& 0\\ -z_2& 1\end{array}\right], \\ {T}_{54}=\left[\begin{array}{cc}1& 0\\ z_1-d& 1\end{array}\right], T_{65}=\left[\begin{array}{cc}1& 0\\ z_2-d-z_3)& 1\end{array}\right], T_{76}=\left[\begin{array}{cc}1& 0\\ -z_3& 1\end{array}\right] \end{gathered}$$

(27)

After little algebra, using these 13 (7 + 6) matrixes, we obtain the same system parameters found by using the diffraction theory. The advantage of diffraction theory is that we obtain the expression of the fields and not only the image and its size and position. Among others, aberrations become straightforwardly treated.

3.4. A Scalar Theory-Based Compound System

We have seen that the scalar system of theory enables encompassing the concept of the compound system just like geometrical optics. Figure 6 shows how the compound system works. For g(x,0), propagating and crossing two lenses spaced longitudinally (deviation of d). We start with the focal length (f), and then calculate the positions h and h’. To do so, we use the relations (8) and (9). To calculate the field g’(x,z’), the Fresnel transform is applied. Then, the result is multiplied by the transmittance of the first lens. Then, the Fresnel transform is applied to cover the distance between the two lenses. The diffraction field obtained will be multiplied by the transmittance of the second lens. In the last step, the Fresnel transform is applied behind the second system. For avoiding all this complexity, the equivalent system is introduced. It replaces the two spherical lenses by a single lens, having the back focal length f’, which means having transmittance: $\exp \left(-j\pi \frac{x^2}{\lambda f’}\right)$.

All what we need is to calculate the field g(x,z) by using the Fresnel transform. This brings us to the system, the transmittance of which we know. We multiply this transmittance by g(x,z) to get g’(x,0). Then, we apply the Fresnel transform on this product, which will result in g’(x,z’). Note that the distance between H and H‘(Figure 6) has no optical effect.

The return path, i.e., from the diffraction field g’(x,z’) to the object g(x), is feasible in the same way, which gives us an inverse transformation. As a result, iterative optimization methods, such as the Gerchberg–Saxton method [27,28], become easily feasible. This enables going back and forth between the input and output of Figure 6, to satisfy certain constraints imposed by the application.

In addition, our approach covers aberrations elegantly. Let us take the case where the first lens is affected by optical aberrations (Figure 6). Our approach remains the same. The wavefront aberration w(x) is included in the analysis. The diffraction field g(x,z) will be calculated in three steps: the Fresnel transform for z-h, a multiplication by a (x), and the Fresnel transform for h (Figure 6). The field a(x) is expressed as a function of w(x):

$$a\left(x\right)=\exp \left(j\frac{2\pi }{\lambda }w\left(x\right)\right)$$

(28)

Our previous analysis limited attention to a system composed of two lenses. This analysis can be generalized straightforwardly.

Aberration is treated as a diffractive element, which has a transmission function. This transmission function is considered a wavefront field $a\left(x\right)= \exp \left(j\frac{2\pi }{\lambda }w\left(x\right)\right)$. We do not need any additional element. This field is treated as g(x) is in Equation (1).

We also handle the effect of the finite extent of refractive and diffractive elements easily. An airy spot is observed when light goes across any size aperture, and diffraction occurs. Every lens has a finite aperture; therefore, diffraction occurs because of the borders. We consider the finite extent of the lens as a circular aperture, placed just behind an infinitely extended lens. This circular aperture acts as a diffractive amplitude element, which has a binary transmission function. We treat it as we treat a(x) in Figure 6.

4. Typical Examples of Systems

Let us consider the following three typical examples:

4.1. Spherical Diopter

It is very easy to provide a diffractive model of the spherical diopter by using the compound system. It turns out that, as shown in Figure 7, the nodal points coincide with the center of curvature C, and the two principal planes, H and H’, are identical and include C. The field propagates in a medium other than the air, of refractive index n₂. Then, the transmittance of the system will be $\exp \left(-j\pi \frac{n_2{x}^2}{\lambda f’}\right)$.

For a given g(x,0), g(x,z) (of Figure 7) is expressed by the Fresnel transform (1). However, λ is replaced by λ/n₁. The field g’(x,0) is the algebraic product of g(x,z) by $\exp \left(-j\pi \frac{n_2{x}^2}{\lambda f’}\right)$. The distribution g’(x,z’) is the Fresnel transform of g’(x,0), where λ is replaced by λ/n₂ in Equation (1).

If the incident wave is spherical and originates from the focal point F, as depicted in Figure 7, then we expect a plane wave, which is an image at infinity. In our diffraction-based approach, the field g(x,z) in medium n₁ is expressed by: $\exp \left(-j\pi \frac{n_1{x}^2}{\lambda f}\right)$. In medium n₂, g’(x,0) is obtained by multiplying this exponential complex term by the transmittance of the system, therefore, resulting in: g’(x,0) = $\exp \left(-j\pi \frac{x^2}{\lambda }\left[\frac{n_1}{f}+\frac{n_2}{f’}\right]\right)$.

Given that the front and back focal lengths of a spherical diopter are related by: f’/n₂ = -f/n₁, a plane wave. It means: g’(x,0) = g’(x,z’) = 1 is obtained. Geometrical optics leads to the same result.

4.2. The System of the Human Eye

Our approach makes it easy to treat aberrations, including aberrations of the eye (Figure 8). It makes it possible to treat extreme cases, such as the effect of the finite size of the pupil, something which is not easily treated by geometric optics. Consider the simplified eye model, what we call Gullstrand’s simplified schematic eye [29,30,31,32]. Of course, the reduced eye model, and even other more complex models, such as the schematic eye, described by Emsley [33], can be analyzed by our approach.

Given a field g(x,0) (Figure 8), g(x,z), at the anterior principal plane H, is calculated by the Fresnel transform (1). g’(x,0), observed in the plane H’, is nothing but the complex product of g(x,z), and the transmittance of the compound system, means $\exp \left(-j\pi \frac{n_2{x}^2}{\lambda f’}\right)$. The aperture function, or pupil, and wavefront aberrations can be included in the analysis. Then, g’(x,z’), which is the diffraction field observed at distance z’ inside the eye, is calculated by relation (1). Of course, λ should be replaced by λ/n₂ in Equation (1). Let us suppose that one of the surfaces of the eye crystalline lens is affected by aberrations, as depicted in Figure 8. The aberration field, a(x), as per Equation (28), is integrated in the calculation of g’(x,z’), in order to measure the effect of wavefront aberrations. It is worth noting that aberrations, related to aberration data of the outer components, such as the anterior cornea surface, are easier to treat, compared to inner components, such as the anterior lens surface.

Let us consider an emmetropic eye (F’ is located exactly in the retina), as shown in Figure 8. If the eye is myopic (nearsighted), the focal point F’_m is located before the retina (Figure 8). On the other hand, if the eye is hyperopic, the focal point F’_h is located behind the retina. Our approach remains the same for these two cases. Our diffractive approach is also easily applied to the eye corrected by an ophthalmic lens, contact lens, or any other element of correction. We can proceed according to two methods. The first method consists in calculating a compound system, including the correction itself. The second method involves an additional application of the Fresnel transform, expressed by Equation (1). In this case, we consider the corrective element and eye system separately. As a result, the Fresnel transform is applied three times, instead of two. Besides, if the correction itself is affected by aberrations, such as the case of intraocular lens, laser surgery, or corneal implant, our approach is straightforwardly applicable.

4.3. Propagation in a Dispersive Optical Fiber

The optical pulses inside a medium, such as the single-mode fibers, are generally expressed by the nonlinear Schrödinger equation [13,34,35]. For the case of pulses with a width larger than 1 ps, this Schrödinger equation is reduced to:

$$j\frac{\partial A}{\partial z}+\frac{j}{2}\alpha A-\frac{1}{2}{\beta }_2\frac{{\partial }^{2}A}{\partial T^2}+\gamma {\left|A\right|}^{2}A$$

(29)

Here:

• A: Slowly varying complex amplitude of the pulse envelope,
• α: Absorption coefficient,
• γ: Nonlinearity coefficient,
• β₂: 2nd order dispersion coefficient,
• z: Observation distance,
• T: The time measured in a reference frame running with the pulse at the group velocity, v_g (T = t-z/v_g).

Putting aside γ means γ = 0, and normalizing the complex amplitude and time scale leads to:

$$j\frac{\partial g}{\partial z}=\frac{1}{2}{\beta }_2\frac{{\partial }^{2}g}{\partial T^2}$$

(30)

here, g(z,τ) represents the normalized amplitude, where P₀ is the peak power of the incident pulse:

$$A(z,\tau )=\sqrt{P_0}\exp (-\alpha z/2)g(z,\tau )$$

(31)

and τ = T/T₀ represents the normalized time, where T₀ is the pulse width.

The FT of the solution of Equation (30) is:

$$G(\omega ,z)=G(\omega )\exp \left(\frac{i}{2}{\beta }_{2}z{\omega }^2\right)$$

(32)

Relations (32) and (3) are very similar. If we replace β₂ by −λ/2π, the two equations become identical. This inspires us to use optical elements, including diffractive elements, to reduce (or even cancel out) the effect of chromatic dispersion introduced by the propagation of the wave along the fiber. To do so, additional optical elements, such as Mach–Zender modulators and chirped fiber networks, are inserted in the telecommunication chain. It is preferable to have these additional elements, combined with the dispersive medium itself, namely the single-mode fiber, to form a compound system. This is straightforwardly doable with our approach.

5. Applications

The formalism advanced in this work may be applied in various application fields, including the following three typical fields.

5.1. Analysis and Design of Refractive Devices and Instruments

Refractive devices are analysed by geometrical optics in a non-rigorous way. Aberrations, especially high-order aberrations, are generally not taken into account or are very hard to be handled by geometrical optics.

Higher-order aberrations, such as the 5th to 14th Zernike terms (vertical astigmatism, vertical trefoil, vertical coma, horizontal coma, oblique trefoil, oblique quadrafoil, oblique secondary astigmatism, primary spherical aberration, and vertical secondary astigmatism vertical quadrafoil), can cause difficulty seeing at night, glare, blurring, halos, and starburst patterns, as well as diplopia. Wavefront aberration analysis presents a rigorous way to handle high-order aberrations. Our approach could easily handle Zernike terms, especially when designing refractive devices aimed at minimizing aberrations.

5.2. Analysis and Synthesis of Diffractive Optical Elements

Diffractive elements are of nature to be treated with a diffraction theory. However, when several diffractive elements are involved, the application of a diffraction theory becomes forbidding, since we do not have a mechanism at our disposal, such as the compound system in geometrical optics. Our approach offers such a mechanism. The diffractive optical elements are replaced by an equivalent system that is characterized by two principal planes, i.e., two focal and two nodal points.

For the synthesis of diffractive optical elements, we optimize the position of the principal, focal, and nodal planes. The optimization may be proceeded analytically or through an iterative process. Then, we design diffractive elements that converge the wavefront towards the focal points and redress the beam (make them parallel) at the principal planes.

5.3. Optometry and Opthalmology

The human eye and optical elements are generally analysed with geometrical optics. The design of corrective refractive elements is also generally handled with geometrical optics. Then, when it comes to wavefront aberrations, especially high-order aberrations, waves and wavefronts are used. However, when designing corrective refractive elements, it is more convenient to use, from the beginning, an approach that is more rigorous than geometrical optics, since wavefront aberrations come into play. Our approach is embedded in this spirit and presents a solution.

Moreover, when using photoablation-based (PRK, LASIK, …) corrections, especially for correcting astigmatism, including high-order astigmatism, our approach is more appropriate than the conventional method for treating this situation. Conventionally, we use a combination of geometrical optics with wavefront aberrations.

Finally, when analysing the retinal image quality and conceiving/applying metrics for image quality, our approach is appropriate [32], since it starts from diffraction theory from the beginning. For example, our approach can easily be used for measuring the influence of ametropia on the retinal image quality of the human eye [36].

Our approach also enables handling polarization; wavefront aberrations is a straightforward way, since our approach is wave-based. Very useful metrics, such as PSF, the OTF, and the MTF, are straightforwardly processed by using our formalism. For example, the results of [9,32,36] are straightforwardly reachable by using our approach without any complex calculation.

6. Conclusions

Compound systems are very used by opticians, optical component designers, optometrists, and ophthalmologists (among others). These systems are generally treated by geometrical optics—for instance, through the Gauss’ formalism. While the analysis is simplified, the wave character of light is ignored. Moreover, boundary effects are also ignored. In the present paper, the process of image formation is simplified, while taking into account the wave character of light and boundary effects. Higher diffraction orders are covered by our approach. A combination of several optical components is treated as a compound system having (1) a transmittance (or reflectance) function, (2) principal plans, (3) focal planes, and (4) nodal points.

The conception of a compound system turned out be doable, in the framework of scalar diffraction theory. Our diffraction-based approach enables calculating the output field at any observation distance, z’, by taking into account the parameters of the complex system. The complexity of the calculation is caused by two Fresnel transforms and one matrix multiplication (transmittance or reflectance of the system). It also enables handling aberrations, at the cost of some additional computational effort, because of an additional Fresnel transform. Some very useful metrics, such as PSF, the OTF, and the MTF, are straightforwardly calculated by using our approach. These functions are powerful tools, measuring the optical behavior of the optical systems [37]. The optical system of the human eye is easily covered by our approach. Optical performance criteria, as well as image quality criteria, including the OTE [32], are easily treated by our diffractive approach.