Introduction

The cornea functions as a lens, similar to the crystalline lens in ocular optics, and provides approximately two-thirds of the total refractive power of the eye. However, the cornea is not a perfect spherical lens; thus, asymmetrical, parallel light rays are distorted and cannot be focused at a single point. This type of refractive error is referred to as astigmatism. Astigmatism causes symptoms such as blurry vision and double vision and reduces quality of life. While astigmatism is usually corrected with glasses or contact lenses, surgical interventions are occasionally considered for severe astigmatism. Astigmatism is classified into two types: regular and irregular astigmatism or aberration. In the case of regular astigmatism, there are two orthogonal principal meridians; however, in aberration, the principal meridians are not orthogonal, the curvatures of both meridians are asymmetric, and it includes any type of higher-order aberration. Regular astigmatism can be corrected with a spherocylindrical lens; however, aberration cannot be corrected with such lenses.

The corneal refractive power components can be separated into spherical (0’: zero-order), decentration or asymmetric (1’: first-order), regular astigmatic (2’: second-order), and higher-order astigmatic (≥ 3’) components via Fourier series analysis of anterior1 or posterior2 corneal topographic data. Moreover, the original topographic image can be reconstructed by using these separate components1,3. The Fourier series of the corneal refractive power is an analysis of the variation in corneal refractive power as a function of semi-meridian by breaking down into frequency components, using Fourier analysis. Fourier series analysis of corneal topographical data has revealed that the corneal refractive power is determined by the sum of numerous refractive power components. The topographic spherical (0’) and regular astigmatic (2’) components correspond to the keratometric spherical and astigmatic powers, respectively, whereas neither the decentration (1’) nor the higher-order irregular astigmatic (≥ 3’) components are represented by keratometry; therefore, these components are considered unable to be corrected with glasses1,2,4. However, this idea remains questionable. In the decentration (1’) component, the optical axis of the cornea is tilted with respect to the visual axis, which has a deviating effect on light rays. This effect is the same as that of a prism. Therefore, I hypothesized that the decentration component of the corneal refractive power is equivalent to a prism.

A prism is a transparent material with two plane surfaces tilted at an angle. Unlike spherical and spherocylindrical lenses, a prism is not a lens by definition; however, prisms can bend and deviate light rays according to Snell’s law, and in this process, prisms can disperse, reflect, and refract light. In ocular optics, prisms are used for the examination and treatment of strabismus5,6 and visual field expansion7. In other fields, prisms are utilized in optical devices such as cameras, microscopes, telescopes, fiberoptics, medical endoscopy equipment, and laser equipment.

The prism is conventionally represented by a prism diopter, which has the maximum prismatic power in the principal section, and a base with a specific base-apex direction, e.g., with the base directed in, out, up, or down6,8. The prism has maximum prismatic power in the principal section, and an object viewed through the prism is deviated toward the apex parallel to the base-apex direction. However, similar to a cross bar being observed through a prism, a phenomenon that cannot be explained occurs unless we assume that there is a prismatic power component along the oblique meridian9,10. Moreover, when two prisms with different base directions and powers are combined, the prism diopter and the base direction of the combined prism must be calculated considering those along the oblique meridian. Therefore, the specific prismatic power in a meridional section of the prism should be investigated.

In this study, one of the Risley prisms11, resembling a round wedge prism, which is typically used as a trial prism to prescribe prism glasses, is employed as a model prism. To improve generalizability, the surrounding media of the prism are different in front of and behind the prism, with distinct refractive indices. With the rotation axis of the prism considered the provisional optical axis, the prismatic refractive power in an arbitrary section of the prism can be expressed as a mathematical equation on the basis of geometrical optics. Then, two types of prisms, made of either glass or corneal tissue, with distinct refractive indices are considered. In this study, it is assumed that the cornea resembles a round wedge prism; that is, the cornea is considered a round wedge prism without curved anterior or posterior surfaces. Then, the discrete Fourier transform (DFT) is applied to the complex equations for the glass and corneal prisms to approximate the mathematical expressions. Next, the relationships between the approximated equations and the Fourier series of the corneal refractive power are considered. With this approach, this study aims to prove that the first-order term in the Fourier series of the corneal refractive power is identical to a prismatic refractive component.

Methods

In this study, the dispersion associated with the prism’s refractive index and the wavelength of light are not considered because of the use of geometrical optics. In addition, a thin prism with a small apex angle to which the small angle approximation or the paraxial approximation can be applied is used.

Relationship between the apex angle α and deviation angle δ in a triangular prism (Fig. 1)

Let a monochromatic incident light ray coming from a medium with refractive index n0 pass through a triangular prism with apex angle α and refractive index n1. The light ray enters the first face of the prism at incident angle i, followed by the first refraction, before exiting from the second face of the prism at emergent angle e after the second refraction toward a medium with a refractive index n2. Suppose that n0 < n1 and n1 > n2 to prevent total reflection. The derivation of the deviation angle δ, which is the angle between the incident light ray and the emergent light ray, is explained in Supplementary File 1.

$$\:\delta\:=i+{\text{sin}}^{-1}\left(\frac{{n}_{1}}{{n}_{2}}\text{sin}\left(\alpha\:-{\text{sin}}^{-1}\left(\frac{{n}_{0}}{{n}_{1}}\text{sin}i\right)\right)\right)-\alpha\:$$
(1)
Fig. 1
figure 1

Refraction of a ray of monochromatic light passing through a triangular prism with an apex angle α and refractive index n1. i, incident angle; e, emergent angle; δ, deviation angle; n0 and n2, refractive indices of the media.

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Derivation of the apex angle αφ along the oblique meridian of a round wedge prism

As shown in Fig. 2a, a round wedge prism is created by carving out a wedge prism with a unit circle centered at O that corresponds to an intersection point of a rotation axis normal to the emergent surface. The shape of the prism is identical to that of the Risley prism11. Let a light ray enter the prism from above. The prism has an apex angle α0 where the elliptical incident surface and the circular emergent surface meet. The prism’s emergent surface is placed in the XY plane, and the base-apex direction is aligned with the X axis such that the base and the apex are in the + X and –X directions, respectively. The Z axis passes vertically through the XY plane at O. Therefore, the principal section of the prism is in the XZ plane.

Fig. 2
figure 2

Illustrations of round wedge prisms. (a) A round wedge prism (one of the Risley prisms) placed in the XY plane, with the principal section in the XZ plane. The base and the apex are aligned in the + X axis direction and the − X axis direction, respectively. A light ray enters the prism from above. The Z axis passing through the origin O of the emergent surface and the center point P of the incident surface corresponds to the rotation axis of the prism. (b) The half cylinder sectioned by the YZ plane and the plane at P parallel to the XY plane is removed, and the remaining parts are reconstructed as a Fresnel-type prism, as illustrated in Fig. 3. α0, apex angle.

Full size image

Since the deviation angle does not depend on the thickness of the prism, the deviation angles of a conventional prism and a Fresnel prism are equivalent8,12. Similarly, the deviation angle of a round wedge prism is equivalent. A half cylinder is outlined by the YZ plane, the prism’s emergent surface, and the plane parallel to the XY plane at point P, where the rotation axis intersects the incident surface (Fig. 2b). The half cylinder is removed from the round wedge prism to reconstruct a Fresnel type of round wedge prism (Fig. 3).

Figure 3 shows that the section with an angle φ (0 ≤ φ < 2π) with respect to the principal section normal to the XY plane is a prism with apex O or Q and apex angle αφ along the oblique meridian. Triangles OAB and OPG are the prism’s sections at φ = 0 and φ = π, respectively. When 0 ≤  φ < π/2 or 3π/2 ≤ φ < 2π, triangle OCD is a prism with apex O and apex angle αφ; on the other hand, when π/2 ≤ φ <3π/2, quadrangle OPFE is a prism with apex Q, which represents the intersection point of lines OE and PF in the XY plane, and apex angle αφ. Points C’, D', E', and F’ are the projections of points C, D, E, and F, respectively, perpendicularly in the XZ plane.

Fig. 3
figure 3

A perspective view of the reconstructed round wedge prism. The section passing through the origin O perpendicular to the XY plane is a prism with apex angle αφ, forming an angle φ with the principal section. The triangles OAB (φ = 0) and OPG (φ = π) are the principal sections of the original round wedge prism. When 0 ≤ φ < π/2 and 3π/2 ≤ φ < 2π, the triangle OCD is a prism with apex O and apex angle αφ, whereas when π/2 ≤ φ < 3π/2, the quadrangle OPFE is a prism with apex Q and apex angle αφ. C, D, E, and F are vertically projected to the XZ plane, at points C’, D', E', and F’, respectively. Q, intersection point of OE and PF in the XY plane.

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Figure 4a shows a top view of the reconstructed round wedge prism illustrated in Fig. 3. Figure 4b shows a side view of the prism along the direction of the Y axis, showing triangles OAB and OPG, which are the principal sections. Similarly, Fig. 4c shows a side view of triangle OCD and quadrangle OPFE, which represent oblique sections at angles of 0 ≤ φ < π/2 or 3π/2 ≤ φ < 2π for triangle OCD and π/2 ≤ φ < 3π/2 for quadrangle OPFE. The derivation of the following Eqs. (2) and (3) is explained in Supplementary File 1.

  1. (i)

    When \(\:0\le\:{\upphi\:}<\frac{\pi\:}{2}\:\text{o}\text{r}\:\frac{3\pi\:}{2}\le\:\phi\:<2\pi\::\)

    $$\:{\alpha\:}_{\phi\:}={\text{tan}}^{-1}\left(\text{tan}{\alpha\:}_{0}\text{cos}\phi\:\right)$$
    (2)
  2. (ii)

    When \(\:\frac{\pi\:}{2}\le\:\phi\:<\frac{3\pi\:}{2}\):

    $$\:{\alpha\:}_{\phi\:}={-\text{tan}}^{-1}\left(\text{tan}{\alpha\:}_{0}\text{cos}\phi\:\right)$$
    (3)
Fig. 4
figure 4

Projection views of the reconstructed round wedge prism in Fig. 3. (a) Top view. A, C, Cʹ, E, Eʹ, G, and Q are in the XY plane. φ is an angle formed between the principal section of the original round wedge prism and a section along an oblique meridian. The range of angle φ formed by OA and OC is \(\:0\le\:\phi\:<\frac{\pi\:}{2}\) and \(\:\frac{3\pi\:}{2}\le\:\phi\:<2\pi\:\). The range of angle φ formed by OA and OE is \(\:\frac{\pi\:}{2}\le\:\phi\:<\frac{3\pi\:}{2}\). (b) Side view of the reconstructed round wedge prism along the Y axis. O, A, B, Cʹ, Dʹ, Eʹ, Fʹ, G, and P are in the XZ plane. α0, apex angle in the principal section of the prism. (c) Side view of the oblique sections of the reconstructed round wedge prism. The range of angle φ formed by the principal section and the oblique section OCD is \(\:0\le\:\phi\:<\frac{\pi\:}{2}\) and \(\:\frac{3\pi\:}{2}\le\:\phi\:<2\pi\:\). The range of angle φ formed by the principal section and the oblique section OPFE is \(\:\frac{\pi\:}{2}\le\:\phi\:<\frac{3\pi\:}{2}\). αφ, apex angle of the prism in the oblique section.

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Derivation of the deviation angle δφ in an oblique section of the reconstructed round wedge prism (Fig. 5)

Let an incident light ray enter the reconstructed round wedge prism from above parallel to the Z axis or the rotation axis. The prism is in an oblique section at an angle φ with respect to the principal section and has an apex angle αφ. The light ray enters the prism at an incident angle iφ and exits at a deviation angle δφ. Since iφ = αφ, Eq. (1) can be rewritten as:

$$\:{\delta\:}_{\phi\:}={\text{sin}}^{-1}\left(\frac{{n}_{1}}{{n}_{2}}\text{sin}\left({\alpha\:}_{\phi\:}-{\text{sin}}^{-1}\left(\frac{{n}_{0}}{{n}_{1}}\text{sin}{\alpha\:}_{\phi\:}\right)\right)\right)$$
(4)
  1. (i)

    In triangular prism OCD \(\:\left(0\le\:{\upphi\:}<\frac{\pi\:}{2}\:\text{o}\text{r}\:\frac{3\pi\:}{2}\le\:\phi\:<2\pi\:\right)\) (Fig. 5a), the following relationships can be obtained.

    According to Eqs. (2) and (4), we have:

    $$\:{\delta\:}_{\phi\:}={\text{sin}}^{-1}\left(\frac{{n}_{1}}{{n}_{2}}\text{sin}\left({\text{tan}}^{-1}\left(\text{tan}{\alpha\:}_{0}\text{cos}\phi\:\right)-{\text{sin}}^{-1}\left(\frac{{n}_{0}}{{n}_{1}}\text{sin}\left({\text{tan}}^{-1}\left(\text{tan}{\alpha\:}_{0}\text{cos}\phi\:\right)\right)\right)\right)\right)$$
    (5)
  2. (ii)

    In triangular prism OPQ \(\:\left(\frac{\pi\:}{2}\le\:\phi\:<\frac{3\pi\:}{2}\right)\) (Fig. 5b), the following relationships can be obtained.

    According to Eqs. (3) and (4), we have:

    $$\:{\delta\:}_{\phi\:}={-\text{sin}}^{-1}\left(\frac{{n}_{1}}{{n}_{2}}\text{sin}\left({\text{tan}}^{-1}\left(\text{tan}{\alpha\:}_{0}\text{cos}\phi\:\right)-{\text{sin}}^{-1}\left(\frac{{n}_{0}}{{n}_{1}}\text{sin}\left({\text{tan}}^{-1}\left(\text{tan}{\alpha\:}_{0}\text{cos}\phi\:\right)\right)\right)\right)\right)$$
    (6)

    Equations (5) and (6) are the same except for the signs.

Fig. 5
figure 5

Deviation angle δφ in an oblique section of the reconstructed round wedge prism. A light ray enters the prism at an incident angle iφ from above parallel to the Z axis, which corresponds to the rotation axis of the original round wedge prism. The refracted light is emitted from the prism at an emergent angle eφ, with deviation angle δφ with respect to the incident light. The light is deviated by PDφ cm at a distance of 100 cm. The prism diopter PDφ represents the prismatic power in an oblique section and is determined as \(\:{\text{P}\text{D}}_{\phi\:}=100\text{tan}{\delta\:}_{\phi\:}\). αφ, apex angle of the prism in an oblique section of angle φ; n0, n1, and n2, refractive indices of the medium in front of the prism, in the prism, and behind the prism, respectively. The drawings do not represent the real lengths. (a) \(\:0\le\:\phi\:<\frac{\pi\:}{2}\) and \(\:\frac{3\pi\:}{2}\le\:\phi\:<2\pi\:\). (b) \(\:\frac{\pi\:}{2}\le\:\phi\:<\frac{3\pi\:}{2}\).

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Conversion of the deviation angle δφ for the prism diopter

One prism diopter (PD) is a prismatic power to deviate a light ray by 1 centimeter (cm) at a distance of 100 cm and can be expressed as shown in Eq. (7)6,8,13:

$$\:\text{P}\text{D}=100\text{tan}\delta\:$$
(7)

In Fig. 5, the light ray shifts by PDφ cm at a distance of 100 cm. Therefore, the prism diopter (PDφ) of the reconstructed round wedge prism in an oblique section at angle φ with respect to the principal section is given as follows:

$$\:{\text{P}\text{D}}_{\phi\:}=100\text{tan}{\delta\:}_{\phi\:}$$
(8)

Although PDφ>0 because δφ > 0, it is necessary to distinguish between two PDφ values when the magnitudes of PDφ are the same but the directions of the deviation angle are opposite. In particular, when 0 ≤ φ < π/2 or 3π/2 ≤ φ < 2π, the light ray moves away from the rotation axis; conversely, when π/2 ≤ φ < 3π/2, the light ray approaches the axis. The convex lens is considered to be composed of two prisms combined base to base, whereas the concave lens is composed of two prisms combined apex to apex10. The sign of PDφ can be determined in a similar manner to that of a lens for which the refractive power of the convex lens is positive and that of the concave lens is negative. Thus, the sign of PDφ is positive when an emergent light ray converges toward the Z axis or the rotation axis of the prism; on the other hand, the sign of PDφ is negative when the emergent light ray diverges from the axis.

Thus, when \(\:0\le\:{\upphi\:}<\frac{\pi\:}{2}\:\text{o}\text{r}\:\frac{3\pi\:}{2}\le\:\phi\:<2\pi\:\), the emergent light ray diverges, and PDφ is written as:

$$\:{\text{P}\text{D}}_{\phi\:}=-100\text{tan}{\delta\:}_{\phi\:}$$
(9)

Conversely, when \(\:\frac{\pi\:}{2}\le\:\phi\:<\frac{3\pi\:}{2}\), the emergent light ray converges, and PDφ is written as:

$$\:{\text{P}\text{D}}_{\phi\:}=100\text{tan}{\delta\:}_{\phi\:}$$
(10)

By substituting Eq. (5) for δφ into Eq. (9) or substituting Eq. (6) for δφ into Eq. (10), Eq. (11) can be obtained as follows:

$$\begin{aligned} \: & {\text{PD}}_{{\phi \:}} \\ & = - 100{\text{tan}}\left( {{\text{sin}}^{{ - 1}} \left( {\frac{{n_{1} }}{{n_{2} }}{\text{sin}}\left( {{\text{tan}}^{{ - 1}} \left( {{\text{tan}}\alpha \:_{0} {\text{cos}}\phi \:} \right)} \right.} \right.} \right. \\ & \quad \left. {\left. {\left. { - {\text{sin}}^{{ - 1}} \left( {\frac{{n_{0} }}{{n_{1} }}{\text{sin}}\left( {{\text{tan}}^{{ - 1}} \left( {{\text{tan}}\alpha \:_{0} {\text{cos}}\phi \:} \right)} \right)} \right)} \right)} \right)} \right) \\ \end{aligned}$$
(11)

Equation (11) represents the prismatic power profile along the oblique meridian at an arbitrary angle φ for a reconstructed round wedge prism with apex angle α0 and refractive index n1.

α0 is a constant apex angle, which is specific to the prism and determines the prism diopter, which represents the nominal prismatic power of the prism. In Eq. (11), without the α0 value, the PDφ value cannot be calculated. The nominal or face value of the prismatic power of the prism represented by Eq. (11) is |PD0|, the absolute value of PD0 when φ = 0 or the value is calculated in the principal section of the prism, which is expressed in Eq. (12):

$$\:\left|{\text{P}\text{D}}_{0}\right|=\left|-100\text{tan}\left({\text{sin}}^{-1}\left(\frac{{n}_{1}}{{n}_{2}}\text{sin}\left({\alpha\:}_{0}-{\text{sin}}^{-1}\left(\frac{{n}_{0}}{{n}_{1}}\text{sin}{\alpha\:}_{0}\right)\right)\right)\right)\right|$$
(12)

Equation (12) describes the relationship between the nominal value of the prism diopter |PD0| and the apex angle α0 when a prism with refractive index n1 is placed between media with refractive indices n0 and n2. Although α0 can be calculated by solving Eq. (12), this equation remains difficult to solve. According to the small angle approximation14, \(\:\text{sin}{\alpha\:}_{0}\approx\:{\alpha\:}_{0}\) and \(\:{\text{sin}}^{-1}{\alpha\:}_{0}\approx\:{\alpha\:}_{0}\). Therefore, Eq. (12) can be approximated as follows:

$$\:\left|{\text{P}\text{D}}_{0}\right|\approx\:100\text{tan}\left(\frac{{n}_{1}-{n}_{0}}{{n}_{2}}{\alpha\:}_{0}\right)$$
(13)

Solving Eq. (13) for α0 gives:

$$\:{\alpha\:}_{0}\approx\:\frac{{n}_{2}}{{n}_{1}-{n}_{0}}{\text{tan}}^{-1}\frac{\left|{\text{P}\text{D}}_{0}\right|}{100}$$
(14)

Waveform analysis of PDφ with the discrete Fourier transform

The reconstructed round wedge prism with prismatic power |PD0| is made of glass or corneal tissue.

Glass prism

Let a glass round wedge prism with refractive index n1 = 1.525 be placed in air (n0 = n2 = 1). When the base of the prism is aligned in the + X direction, as shown in Fig. 2, according to Eq. (11), the following can be obtained:

$$\begin{aligned} & \:{\text{PD}}_{{\phi \:}} \\ & = - 100{\text{tan}}\left( {{\text{sin}}^{{ - 1}} \left( {1.525{\text{sin}}} \right.\left( {{\text{tan}}^{{ - 1}} \left( {{\text{tan}}\alpha \:_{0} {\text{cos}}\phi \:} \right)} \right.} \right. \\ & \quad \left. {\left. {\left. { - {\text{sin}}^{{ - 1}} \left( {\frac{1}{{1.525}}{\text{sin}}\left( {{\text{tan}}^{{ - 1}} \left( {{\text{tan}}\alpha \:_{0} {\text{cos}}\phi \:} \right)} \right)} \right)} \right)} \right)} \right) \\ \end{aligned}$$
(15)

According to Eq. (14), the following can be obtained:

$$\:{\alpha\:}_{0}\approx\:\frac{1}{0.525}{\text{tan}}^{-1}\frac{\left|{\text{P}\text{D}}_{0}\right|}{100}$$
(16)

To sample discrete PDφ values, sampled φ values (0 ≤ φ < 2π) are determined by dividing 2π by 256, yielding a function φ(t) mediated by parameter t, as shown in Eq. (17):

$$\:\phi\:\left(t\right)=\frac{2{\uppi\:}}{256}t\:\:\:\left(radians\right)\:\:\:\left(t=0,\:\text{1,2},\dots\:255\right)$$
(17)

For individual glass prisms with |PD0| values of 1, 5, 10, 15, and 20 prism diopters, the α0 given by Eq. (16) and the 256 angle φ data points given by Eq. (17) are substituted into Eq. (15), thereby producing 256 discrete PDφ values, that is, PDφ(t). Then, using the DFT, an Excel add-in (Microsoft) which represents the algorithm for calculating the fast Fourier transform15, the Fourier coefficients F(k) in Eq. (18) can be calculated:

$$\:F\left(k\right)=\sum\:_{t=0}^{255}{\text{P}\text{D}}_{\phi\:}\left(t\right){e}^{-i\left(2\pi\:tk/256\right)}\:\:\:(k=0,\:1,\:2,\:\cdots\:255)$$
(18)

F(k) is in the form of a complex number, x + iy, where x and y are real numbers and i is an imaginary number. The amplitude is given by the absolute value of F(k) divided by 128 (the number of data points divided by 2), and the phase is calculated with the atan2 function16. The frequency f is determined on the basis of the angular frequency 2πf. Although the dimension of f is hertz (Hz) or cycles/second by convention, the frequency here has no dimension of time. Therefore, the frequency f is presented in the unit cycles/circle.

Corneal prism

Suppose that the cornea (n1 = 1.376) has the form of a round wedge prism and lies between air (n0 = 1) and an aqueous medium (n2 = 1.336), similar to a human eye. The refractive indices are derived from the Gullstrand’s exact schematic eye17. The base of the corneal prism is aligned in the + X axis direction, similar to the setup for the glass prism. According to Eq. (11), the following can be derived:

$$\begin{aligned} \:{\text{PD}}_{{\phi \:}} & = - 100{\text{tan}}\left( {{\text{sin}}^{{ - 1}} \left( {\frac{{1.376}}{{1.336}}{\text{sin}}\left( {{\text{tan}}^{{ - 1}} \left( {{\text{tan}}\alpha \:_{0} {\text{cos}}\phi \:} \right)} \right.} \right.} \right. \\ & \quad \left. {\left. {\left. { - {\text{sin}}^{{ - 1}} \left( {\frac{1}{{1.376}}{\text{sin}}\left( {{\text{tan}}^{{ - 1}} \left( {{\text{tan}}\alpha \:_{0} {\text{cos}}\phi \:} \right)} \right)} \right)} \right)} \right)} \right) \\ \end{aligned}$$
(19)

From Eq. (14), the following can be derived:

$$\:{\alpha\:}_{0}\approx\:\frac{1.336}{0.376}{\text{tan}}^{-1}\frac{\left|{\text{P}\text{D}}_{0}\right|}{100}$$
(20)

Similar to the glass prism, for individual corneal prisms with |PD0| values of 1, 5, 10, 15, and 20 prism diopters, 256 discrete PDφ(t) values are calculated using Eqs. (17), (19), and (20). The Fourier coefficients are computed using the DFT. The amplitude and phase are calculated as described above.

Relationship between a Fourier series and a prismatic refractive component

A Fourier series is written as follows:

$$\:\text{F}\left(\phi\:\right)=\frac{{a}_{0}}{2}+{\sum\:}_{n=1}^{\infty\:}\left({a}_{n}\text{cos}n\phi\:+{b}_{n}\text{sin}n{\upphi\:}\right)$$
(21)

Equation (21) can be written in cosine form using the cosine sum formula as follows:

$$\begin{aligned} \:{\text{F}}\left( {\phi \:} \right) & = \frac{{a_{0} }}{2} + \left( {a_{1} {\text{cos}}\phi \: + b_{1} {\text{sin}}\phi \:} \right) + \left( {a_{2} {\text{cos}}2\phi \: + b_{2} {\text{sin}}2\phi \:} \right) \\ & \quad + \left( {a_{3} {\text{cos}}3\phi \: + b_{3} {\text{sin}}3\phi \:} \right) + \cdots \: + \left( {a_{n} {\text{cos}}n\phi \: + b_{n} {\text{sin}}n\upphi \:} \right) \\ & = \frac{{a_{0} }}{2} - \sqrt {a_{1}^{2} + b_{1}^{2} } \:{\text{cos}}\left( {\phi \: - \theta \:_{1} } \right) - \sqrt {a_{2}^{2} + b_{2}^{2} } \:{\text{cos}}2\left( {\phi \: - \theta \:_{2} } \right) \\ & \quad - \sqrt {a_{3}^{2} + b_{3}^{2} } \:{\text{cos}}3\left( {\phi \: - \theta \:_{3} } \right) - \cdots \: - \sqrt {a_{n}^{2} + b_{n}^{2} } \:{\text{cos}}n\left( {\phi \: - \theta \:_{n} } \right) \\ \end{aligned}$$
(22)

Here, the following relationships can be obtained:

$$\begin{array}{*{20}c} {\:{\text{cos}}\theta \:_{1} = \frac{{ - a_{1} }}{{\sqrt {a_{1}^{2} + b_{1}^{2} } }}\:,\:\:\:{\text{sin}}\theta \:_{1} = \frac{{ - b_{1} }}{{\sqrt {a_{1}^{2} + b_{1}^{2} } }}\:,\:\:\:\theta \:_{1} = {\text{atan}}2\left( {\frac{{b_{1} }}{{a_{1} }}} \right)} \hfill \\ {\:{\text{cos}}2\theta \:_{2} = \frac{{ - a_{2} }}{{\sqrt {a_{2}^{2} + b_{2}^{2} } }}\:,\:\:\:{\text{sin}}2\theta \:_{2} = \frac{{ - b_{2} }}{{\sqrt {a_{2}^{2} + b_{2}^{2} } }}\:,\:\:\:\theta \:_{2} = \frac{1}{2}{\text{atan}}2\left( {\frac{{b_{2} }}{{a_{2} }}} \right)} \hfill \\ {\: \cdots \:} \hfill \\ {\:{\text{cos}}n\theta \:_{n} = \frac{{ - a_{n} }}{{\sqrt {a_{n}^{2} + b_{n}^{2} } }}\:,\:\:\:{\text{sin}}n\theta \:_{n} = \frac{{ - b_{n} }}{{\sqrt {a_{n}^{2} + b_{n}^{2} } }}\:,\:\:\:\theta \:_{n} = \frac{1}{n}{\text{atan}}2\left( {\frac{{b_{n} }}{{a_{n} }}} \right)} \hfill \\ \end{array}$$

According to Hjortdal1, in Eq. (22), the constant term \(\:\frac{{a}_{0}}{2}\) is equal to a spherical equivalent power, the coefficient of the first-order term \(\:\sqrt{{a}_{1}^{2}+{b}_{1}^{2}}\) is equal to a decentration power component, twice the value of the coefficient of the second-order term \(\:\sqrt{{a}_{2}^{2}+{b}_{2}^{2}}\) is equal to a regular astigmatic power component, and the coefficients of the succeeding orders (n > 2) are higher-order irregular astigmatic components. By substituting \(\:2\sqrt{{a}_{2}^{2}+{b}_{2}^{2}}\:\), the magnitude of the regular astigmatic power, for C, the second-order term in Eq. (22) can be rewritten as follows:

$$\:-0.5\text{*}\text{C}\text{*}\text{cos}2\left(\phi\:-{\theta\:}_{2}\right)$$

This represents the virtual cross cylinder presented recently by Danjo18.

With this approach, I investigated whether the approximated equations obtained by applying the DFT to Eq. (11) agreed with the decentration component \(\:-\sqrt{{a}_{1}^{2}+{b}_{1}^{2}}\:\text{cos}\left(\phi\:-{\theta\:}_{1}\right)\) of the first-order term in the Fourier series. This consistency would prove that the power profile of the round wedge prism is identical to the decentration component or the first-order term in the Fourier series.

Results

Figure 6 shows plots of 256 discrete PDφ(t) values for the glass prism (Fig. 6a) and the corneal prism (Fig. 6b) when the respective prisms have nominal prismatic powers |PD0| of 1, 5, 10, 15, and 20 prism diopters. The peak value for each prism is approximately equal to the nominal prismatic power |PD0| of less than 10 prism diopters. Supplementary Figures S1-S22 and Supplementary Tables S1 and S2 show the waveform analysis of the discrete PDφ(t) data using the DFT for the glass prism and the corneal prism, respectively. Each prism has a strong peak amplitude at frequency of 1 cycles/circle and faint amplitudes at other frequencies (Table 1, Supplementary Tables S1 and S2). The amplitude at 1 cycles/circle is approximately the same as the peak value in the discrete data plots. The difference between the amplitude at 1 cycles/circle and the nominal prismatic power |PD0| tends to be greater for the corneal prism than for the glass prism. For the phase analysis, the initial phase at a frequency of 1 cycles/circle is π. Since the amplitudes at frequencies other than 1 cycles/circle are negligibly small, PDφ in Eq. (11) can be approximated as Eq. (23):

$$\:{\text{P}\text{D}}_{\phi\:}\approx\:\left|{\text{P}\text{D}}_{0}\right|*\text{cos}\left(\phi\:+\pi\:\right)=-\left|{\text{PD}}_{0}\right|\text{*cos}\phi\:$$
(23)
Fig. 6
figure 6

Plots of 256 discrete PDφ values of the glass prism (a) and the corneal prism (b), with |PD0| values of 1, 5, 10, 15, and 20 prism diopters (∆). PDφ, prismatic power of the reconstructed round wedge prism with apex angle α0 in an oblique section with angle φ; |PD0|, nominal prism power, corresponding to the absolute value of the prism power, namely, PDφ with φ = 0.

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Table 1 Results of the discrete Fourier transform of 256 discrete PDφ values for the glass prism and the corneal prism.
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With \(\:\left|{\text{P}\text{D}}_{0}\right|=\sqrt{{a}_{1}^{2}+{b}_{1}^{2}}\), Eq. (23) is identical to the first-order term in the Fourier series \(\:-\sqrt{{a}_{1}^{2}+{b}_{1}^{2}}\:\text{cos}\left(\phi\:-{\theta\:}_{1}\right)\) in Eq. (22). Thus, the first-order term in the Fourier series, defined as the decentration or asymmetric component of the corneal refractive power, is the prismatic refractive component.

Figure 7 presents a color-coded map and a graph to visually express Eq. (23). Following the terminology for the virtual cross cylinder18, I term the prism with this power profile the “virtual prism”.

Fig. 7
figure 7

A virtual prism with a prism diopter of |PD0|, depicting the power profile with a color-coded map [a] and a graphical display [b] of the power PDφ. |PD0|: nominal value of the prism diopter (∆), corresponding to the prismatic power in the principal section (φ = 0 and 180 degrees), φ: angle between the principal section and an oblique section of the prism.

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Discussion

By modeling a round wedge prism and using geometrical optics, I derived an equation approximated by applying the DFT to a relational expression of the apex angle and prism diopter in an arbitrary oblique section centered at the rotation axis of the prism. The equation indicates that the prismatic refractive power in an arbitrary section of the prism is represented by the prism’s nominal power and one cycle of a cosine function. The approximated equations derived for the glass prism and the corneal prism are similar within an applicable range of the small angle approximation, demonstrating that the form of the equation is independent of the medium. Moreover, the approximated equation is the same as the first-order term of a cosine-form of the Fourier series. Thus, I proved that the first-order term in the Fourier series, which has been called the decentration or asymmetric component of the corneal refractive power, is the prismatic refractive component, and I termed it the virtual prism.

The Risley prisms, which were presented by Risley as rotary prisms11, were first used in ophthalmology. Ophthalmologists initially utilized these prisms to measure binocular accommodations by examining the strength of the adducting or abducting muscles19. Over time, the Risley prisms continued to be used as test prisms for prescribing prism glasses. Beyond the field of ophthalmology, the Risley prisms have been applied in beam steering and laser scanning, where a pair of round wedge prisms are arranged in a tandem manner, with one surface of each prism normal to the optical axis around which each prism rotates. In this study, it was assumed that the cornea is made of a round wedge prism. In this model, one of the Risley prisms was used, with the tilted surface and the surface normal to the optical axis deemed to be the epithelium and the endothelium, respectively. Therefore, the corneal prism used in this study has a prismatic refractive power in only the anterior cornea and not in the posterior cornea. However, a Fourier series analysis of the posterior cornea in patients with keratoconus has been reported2. The study suggested that a prismatic refractive component can also be found in the posterior cornea and that the anterior and posterior corneal prisms must be combined for calculations involving the total corneal prism. This study’s finding that the prismatic refractive component involves one cycle of a cosine function suggests that calculations involving the total corneal prism could be performed by applying a method similar to the virtual cross cylinder method18. However, calculations involving the total corneal prism are beyond the scope of this study.

In prior works, attempts were made to represent the prismatic power profile in sections other than the principal section of a prism, such as oblique sections. Laurance9 derived Δ’ = Δcosr, where Δ’ is the power of a prism in an oblique section, Δ is the overall power of the prism, and r is the angle between the given meridian and the base-apex plane. In addition, Jalie10 reported that the power of a prism along an oblique meridian is given by using the difference in thickness between the oblique section and the principal section. More recently, Raasch20 states that the one-cycle component is a tilt of the cornea with respect to the keratometric axis, and optically is equivalent to a prismatic deviation. However, in contrast to this study, they did not seek to prove that the prismatic power in an oblique section can be derived from a relational expression between the prismatic power PDφ and deviation angle δφ in an oblique section using the apex angle α0 and refractive indices of the prism and surrounding media on the basis of geometrical optics.

According to the definition that a lens has at least one curved surface, prisms are not lenses. However, prisms can bend and deviate light rays according to Snell’s law. Moreover, a lens can be composed of multiple small prisms and produces a prismatic effect when its optical center is not the center of the pupil according to Prentice’s rule13. Thus, lenses and prisms are closely related. The optical center of a lens is a point where the optical axis, connecting the center of the anterior and posterior surfaces of the lens, intersects the lens. Lenses are rotationally symmetrical about the optical center, where light rays pass without deviation. However, prisms have no optical center since they are not lenses. Therefore, in this study, it was assumed that the optical center of a round wedge prism is a point where the optical axis parallel to the incident light ray and normal to the emergent surface intersects the prism. The provisional optical center lies along a line connecting the centers of the elliptical incident surface and the circular emergent surface of the prism. Accordingly, the provisional optical center of the round wedge prism used in this study is the same as the optical axis or rotation axis of the tandemly arranged Risley prisms21,22,23. One of the drawbacks of the Risley prism system is the blind spot around the optical axis22,24. In contrast, the provisional optical center of the round wedge prism used in this study was determined to evaluate the prism’s power profile but not the prismatic effect in the image space. Therefore, the blind spot was not a problem in this study.

The difference between the nominal prismatic power |PD0| and the peak value of the discrete PDφ(t) function becomes greater when |PD0| is larger than 10 prism diopters. This is also reflected in the DFT analysis, as the difference in amplitude at 1 cycles/circle is larger in the corneal prism than in the glass prism. Given that the small angle approximation is applicable for angles less than 0.253 radians14, the apex angle α0 corresponding to a |PD0| value of 10 prism diopters is 0.190 radians for the glass prism and 0.354 radians for the corneal prism. According to these geometrical optical relationships, the |PD0| value of the glass prism should be less than 10 prism diopters; however, that of the corneal prism should be much smaller to apply the small angle approximation.

Surprisingly, the cornea has prismatic refractive power, which has been called the decentration or asymmetric component, representing the first-order term in a Fourier series of the corneal refractive power1,3. This component is one of the irregular types of astigmatism thought to be uncorrectable with glasses. Since the component was proven to be a prismatic refractive component in this study, the component may be corrected by incorporating a prismatic power component into glasses, contact lenses, or intraocular lenses or by performing refractive surgery to neutralize the corneal prismatic refractive component. Moreover, strabismus is a condition in which the eyes are not properly aligned, and the causes of strabismus are mainly myogenic or neurogenic. However, the results of this study suggest that there is a corneogenic type of strabismus, namely, strabismus with a cornea origin, caused by a prismatic effect of the corneal prismatic refractive component. I hypothesize that corneogenic strabismus occurs due to the prismatic effect of the cornea.

An example of a potential clinical application is a laser ablation procedure on the cornea in eyes with corneogenic strabismus. Suppose a refractive surgeon intended to correct or induce some amount of corneal prism. Let’s say an ablative laser procedure was applied to the cornea over a 6 mm circular area. If the maximum ablation depth was 100 μm at the base of the principal section of the prism, the prismatic effect of that ablation would be as follows. Since the apical angle α0 would be 0.1 mm/6 mm≃0.0167 radians, according to Eq. (13), the prismatic effect in air is |PD0|≃100*tan((1.376-1)/1*0.0167)≃0.627 prism diopters. This is a small value, but it may be useful if the deviation were vertical since vertical prismatic imbalances can be more consequential than similarly small horizontal imbalances. This prismatic effect could be enhanced by decentered ablation. The amount of correction achieved in prismatic power is influenced by various factors, necessitating further studies.

Keratometric astigmatism and topographic astigmatism are inconsistent in the magnitude and axis since the measured areas on the cornea differ1. In keratometric astigmatism, regular and irregular astigmatism cannot be distinguished25. In contrast, in topographic astigmatism, the virtual prism (the first-order term) can be distinguished from the virtual cross cylinder (the second-order term) using a Fourier series analysis. Therefore, another reason for the inconsistency between keratometric and topographic astigmatism may be that in keratometric astigmatism, the magnitude of the astigmatism is overestimated or underestimated depending on the relationship between the axis locations in the virtual prism and the virtual cross cylinder.

One limitation of this study is that it was only theoretically demonstrated that the cornea has a prismatic refractive power component by modeling the cornea as a round wedge prism and comparing the approximated equation resulting from the DFT with the first-order term in a Fourier series of the corneal refractive power. However, the clinical implications of these findings for human corneas must be confirmed. If a corneal topographer is used to measure the corneal prismatic refractive component, measurement errors may occur depending on the determined center of the cornea1,26. In a model eye, the center of the cornea can be determined by tracing the optical path from the fixation point to the fovea. In a living eye, the point where the line of sight intersects the cornea is the center of the cornea; however, this point is rarely consistent with the vertex normal, which is used as the reference standard of topographers26. Since Prentice’s rule13 states that a prismatic effect is generated when a light ray passes through a point other than the optical center of the cornea, how the prismatic refractive component is affected by the location of the vertex normal of the topographer remains to be investigated. Furthermore, since a prism has two mutually inclined surfaces, investigations are needed to determine the effects of the total corneal prismatic refractive component, as determined by combining the components of the anterior and posterior corneal prisms, on the visual function of eyes with normal and pathological corneas. Finally, if corneogenic strabismus exists, the disease should be thoroughly investigated.

In conclusion, by creating a model using a round wedge prism and geometrical optics, I derived an approximated equation representing the prismatic refractive power in an arbitrary oblique section by applying the DFT to the relational expression between the apex angle and prism diopter, and this component is identical to the first-order term in a Fourier series of the corneal refractive power. The equation is represented by the prism’s nominal power and one cycle of a cosine function, corresponding to the decentration component in the Fourier series of the corneal refractive power. I term this component the virtual prism. The virtual prism is expected to be applied not only in ocular optics applications but also in optical systems using prisms, including the Risley prism system. I hope that the virtual prism and virtual cross cylinder will be considered in optics principles.