The classical wave theory can trace its historical origins to the seminal works of Christian Huygens, Thomas Young, and Augustin Fresnel. To explain some of light’s observed properties, such as rectilinear propagation, reflection, and refraction, Huygens proposed a simple geometrical construction of secondary spherical wavelets with centers of disturbance located on a primary wavefront. More than a century later, Young formulated the law of interference to both predict the formation of fringes in his now famous double slit experiment and also to estimate the wavelengths associated with different colors. A decade after that, Fresnel combined Huygens’ construction with Young’s interference law to qualitatively and quantitatively describe diffraction, which is the bending of light upon encountering an obstacle or an aperture. This grand synthesis, called the Huygens–Fresnel principle, acts as a powerful pictorial aid and conceptual tool that can describe a wide variety of complicated optical phenomena. However, the applications of the principle and its later developments, such as the Kirchhoff–Fresnel integral, are strewn with several simplifying assumptions and approximations that are aimed at minimizing the mathematical challenges involved. Consequently, two distinct formalisms are necessary to account for diffraction effects when the source of light or observation screen is placed nearby and far away from the aperture or obstacle. Recently, a hyperbola framework for analyzing wave interference at a multi-slit barrier was shown to successfully circumvent all conventionally imposed ad hoc conditions. The method commences directly from the Huygens–Fresnel principle and the ensuing predictions pertaining to the distribution of fringe characteristics, namely, positions, widths, and intensities on a detection screen can, therefore, justifiably claim accuracy in both the near field (Fresnel regime) and the far field (Fraunhofer regime). In this paper, the analysis that was previously carried out for the special case of slits of negligible widths is further extended to encompass slits of finite widths as well.

The hypothesis that light has a wave nature was qualitatively articulated in two independent publications that are dated back to the same year 1665—Physico de Luminie by Grimaldi and Micrographia by Hooke.1 In fact, the term “diffraction” was coined by Grimaldi.2,3 The first semiquantitative account of the wave model was presented by Christian Huygens in his Traité de la Lumière in 1690. In it, he proposed a geometrical construction known today as the Huygens principle that could elegantly explain some of the observed properties of light, namely, the laws of propagation, reflection, and refraction (see Fig. 1). A modern restatement of the principle is as follows: “Every point on the leading surface of a wave of disturbance (primary wavefront) at a given instant, acts as a source of secondary spherical wavelets. The common tangential envelope determines the position of the new wavefront at a subsequent time.”4 

FIG. 1.

Huygens’ construction of wavefronts from a set of point sources acting as centers of disturbance can effectively account for: (a) rectilinear propagation of a plane wavefront (left) and radial expansion of a spherical wavefront from a point-source P (right) and (b) the laws of reflection (left) and the laws of refraction (right).

FIG. 1.

Huygens’ construction of wavefronts from a set of point sources acting as centers of disturbance can effectively account for: (a) rectilinear propagation of a plane wavefront (left) and radial expansion of a spherical wavefront from a point-source P (right) and (b) the laws of reflection (left) and the laws of refraction (right).

Close modal

Between 1800 and 1803, Thomas Young delivered a series of lectures in the Royal Institution on an astonishingly wide range of topics, including the wave nature of light. While most were compiled together in his two-volume magnum opus A Course of Lectures on Natural Philosophy and the Mechanical Arts (1807), the rest appeared in his later three-volume Miscellaneous Works (1855).5 For a list of his salient papers on physical optics, alongside the principal assertions made in each, see Table I. He is credited for both the coinage of the term “interference” and the formulation of the mathematical law that is associated with the wave phenomenon.6 

TABLE I.

List of Thomas Young’s important papers on the theory of wave interference.

Title Highlights
Outlines of experiments and inquiries respecting sound and light (Misc. Works vol. 1: Read in 1800)  Draws an analogy between different colored rings in the air gap between thin plates to the various tones produced in an organ pipe. Observes that the effect of changing the thickness of the air gap on the colors of light is akin to that of changing the length of an organ pipe on the frequencies of sound 
On the theory of light and colors (Misc. Works vol. 1: Read in 1801)  Inspired by his acoustical studies, Young provides a description of the principle of wave superposition but does not mention either of the terms, interference or superposition. Proposes that the different colors seen in thin plates, thick plates, striated plates, and insect integuments are an immediate consequence of this effect. Measures the wavelengths of different colors of light 
An account of some cases of the production of colors not hitherto described (Misc. Works vol. 1: Read in 1802)  Introduces the law of interference. Makes hint to two fixed quantities that are better understood today as the path difference and the wavelength of light, and also suggests the quantitative relationship that holds between them 
On the theory of hydraulics (Natural Philosophy vol. 1: Ripple tank experiment was performed in 1802)  Provides a practical demonstration of wave interference using water in a device of his own invention (ripple tank). Observes conspicuous hyperbolic curves in the interference pattern and speculates on its implications for acoustics and optics 
Experiments and calculations relative to physical optics (Misc. Works vol. 1: Read in 1803)  Provides another practical demonstration of wave interference using sunlight, a slip of card 1/30th in. in breath, and a small pin hole on a window shutter. Notes the hyperbolic projection of fringes to a distant wall and the hyperbolic shaped fringes 
Reply to animadversions of the Edinburgh Reviewers (Misc. Works vol. 1: Published in 1804)  Illustrates the general law of interference by means of a thought experiment with water waves. The same idea is stated to hold true for the case of light waves as well 
On the nature of light and colors (Natural Philosophy vol. 1: The double slit experiment was likely performed in the period between 1803 and 1807, according to Young’s biographer Andrew Robinson (2006) The Last Man Who Knew Everything Describes the modern-day textbook version of the double slit (or pin hole) interference experiment with an explicit statement of the relationship between path difference and bright/dark fringe formation (i.e., constructive/destructive interference). In the caption of one particular figure (Plate XXX, Fig. No. 442), a clear reference is made to the shape of the trajectory that an interference fringe takes toward the distant screen, namely, a hyperbola. However, no theoretical or experimental data are furnished to support this specific claim 
Title Highlights
Outlines of experiments and inquiries respecting sound and light (Misc. Works vol. 1: Read in 1800)  Draws an analogy between different colored rings in the air gap between thin plates to the various tones produced in an organ pipe. Observes that the effect of changing the thickness of the air gap on the colors of light is akin to that of changing the length of an organ pipe on the frequencies of sound 
On the theory of light and colors (Misc. Works vol. 1: Read in 1801)  Inspired by his acoustical studies, Young provides a description of the principle of wave superposition but does not mention either of the terms, interference or superposition. Proposes that the different colors seen in thin plates, thick plates, striated plates, and insect integuments are an immediate consequence of this effect. Measures the wavelengths of different colors of light 
An account of some cases of the production of colors not hitherto described (Misc. Works vol. 1: Read in 1802)  Introduces the law of interference. Makes hint to two fixed quantities that are better understood today as the path difference and the wavelength of light, and also suggests the quantitative relationship that holds between them 
On the theory of hydraulics (Natural Philosophy vol. 1: Ripple tank experiment was performed in 1802)  Provides a practical demonstration of wave interference using water in a device of his own invention (ripple tank). Observes conspicuous hyperbolic curves in the interference pattern and speculates on its implications for acoustics and optics 
Experiments and calculations relative to physical optics (Misc. Works vol. 1: Read in 1803)  Provides another practical demonstration of wave interference using sunlight, a slip of card 1/30th in. in breath, and a small pin hole on a window shutter. Notes the hyperbolic projection of fringes to a distant wall and the hyperbolic shaped fringes 
Reply to animadversions of the Edinburgh Reviewers (Misc. Works vol. 1: Published in 1804)  Illustrates the general law of interference by means of a thought experiment with water waves. The same idea is stated to hold true for the case of light waves as well 
On the nature of light and colors (Natural Philosophy vol. 1: The double slit experiment was likely performed in the period between 1803 and 1807, according to Young’s biographer Andrew Robinson (2006) The Last Man Who Knew Everything Describes the modern-day textbook version of the double slit (or pin hole) interference experiment with an explicit statement of the relationship between path difference and bright/dark fringe formation (i.e., constructive/destructive interference). In the caption of one particular figure (Plate XXX, Fig. No. 442), a clear reference is made to the shape of the trajectory that an interference fringe takes toward the distant screen, namely, a hyperbola. However, no theoretical or experimental data are furnished to support this specific claim 

A modern restatement is as follows (see Fig. 2): “When light from a single source, after being split so as to propagate along optical paths of different lengths, is made to recombine again on a distant screen, then either a bright or dark fringe may be observed depending on whether the difference of the routes taken was an even or odd integer multiple of a half-wavelength, respectively.”1 Equipped with this law, also referred to as the principle of interference, Young was able to explain Newton’s ring formation in the air gap between the surfaces of a plano-convex lens system, the appearance of colors in glass plates and insect integuments, estimate the wavelength of light, and predict the appearance of fringes in the now famous double slit experiment. On their independent standing, neither Huygens’ principle nor Young’s law can account for the phenomenon of diffraction, which is the bending of light upon encountering an obstacle or an aperture. The crucial synthesis of both these notions was achieved by Augustin Fresnel in a series of communications between 1815 and 1818 to the French Academy of Sciences, which culminated in his prize-winning essay Memoirs on the diffraction of light (crowned in 1819).7 His efforts paved the way for the eventual acceptance of the wave model of light and also for the emergence of Physics as a major scientific discipline in the 19th century.8 By carefully moving back and forth between theory and experiment, and repeatedly revising all his initial hypotheses in order to fit newer observations, Fresnel came to the foregone conclusion that the secondary sources of spherical wavelets of a wavefront that is incident on an aperture or obstacle, could combine or mutually interfere to form the transmitted wavefront in the region of the geometrical shadow, and not merely remain confined to the region of geometrical illumination, as was previously assumed by Huygens (see Fig. 3). A modern restatement of the Huygens–Fresnel principle is as follows: “Every unobstructed point of a wavefront serves as a source of secondary spherical wavelets that have the same frequency as the primary wave. The amplitude of the optical field at any point beyond the primary wavefront is the superposition of all of these wavelets, giving due consideration to their respective amplitudes and relative phases.”9 The principle per se does not account for the backward propagation of secondary waves toward the sources (more on this artifact below).

FIG. 2.

Young’s demonstration of interference. A plane wavefront is incident on a first barrier that bears an aperture (slit or hole), emerging out as a curvilinear wavefront (cylindrical or spherical), which is then incident on a second barrier, again bearing two similar apertures. The emergent wavefronts consequently interfere with each other and form a series of bright and dark fringes on a distant detection screen.

FIG. 2.

Young’s demonstration of interference. A plane wavefront is incident on a first barrier that bears an aperture (slit or hole), emerging out as a curvilinear wavefront (cylindrical or spherical), which is then incident on a second barrier, again bearing two similar apertures. The emergent wavefronts consequently interfere with each other and form a series of bright and dark fringes on a distant detection screen.

Close modal
FIG. 3.

(a) Fresnel’s explanation for diffraction by an aperture, (b) the incident wavefront within the opening of the aperture behaves as a continuum of Huygens’ sources of secondary spherical wavelets, (c) region of geometrical shadow and illumination for an obstacle, and (d) region of geometrical shadow and illumination for an aperture.

FIG. 3.

(a) Fresnel’s explanation for diffraction by an aperture, (b) the incident wavefront within the opening of the aperture behaves as a continuum of Huygens’ sources of secondary spherical wavelets, (c) region of geometrical shadow and illumination for an obstacle, and (d) region of geometrical shadow and illumination for an aperture.

Close modal

To the credit of his rigorous mathematical training at the Ecole Polytechnique, Fresnel produced an empirically successful theory consisting of a diffraction integral formula that could predict fringe characteristics (position and intensity distributions) with a fair degree of accuracy. It is important to note that this intellectual feat was accomplished only after invoking certain approximations that simplified the integration process (Fresnel approximation).10 Diffraction patterns tend to evolve in complicated ways with increasing screen distance from an aperture of a fixed size, gradually acquiring a stable form in the far field limit, where still further simplifications in the formalism can be made (Fraunhofer approximation). Alternatively, the size of the aperture may be decreased, and the screen distance kept fixed to produce the same result. These observations are encapsulated in the Fresnel number criteria, which are used to parameterize three distinctive optical regimes, namely, geometric (FN ≫ 1), Fresnel (FN ≈ 1), and Fraunhofer (FN ≪ 1).11,12 A detailed account of each domain and the accompanying approximations may be found elsewhere (see Fig. 4).13,14 In 1882, Kirchhoff demonstrated that a slight modification of Fresnel’s diffraction integral to include an obliquity factor was derivable from a scalar differential wave equation, which effectively resolved the problem of backward propagation. In honor of his contribution, it has since been rechristened as the Kirchhoff–Fresnel diffraction integral.15 However, Poincaré observed in 1892 that his formulation invokes a set of inconsistent boundary conditions that are not even approximately true (Poincaré paradox).16 Its basis in reality remains a topic of lively debate and contention among historians and philosophers of science.16,17 Yet, Kirchhoff’s theory is deemed a good enough working model for physicists and engineers due to the high yield of predictive successes within the visible light and microwave domain.16 Silver offers a classic discussion, in the latter context, on its reformulation as a saltus problem.18 

FIG. 4.

A succession of different diffraction patterns obtained with increasing distance L from a single slit. A similar variation may also be obtained by either changing the wavelength λ or the slit width a, keeping the screen distance fixed. The Fraunhofer, Fresnel, and Geometric regimes are also referred to as the far, near, and ultra-near fields, respectively.

FIG. 4.

A succession of different diffraction patterns obtained with increasing distance L from a single slit. A similar variation may also be obtained by either changing the wavelength λ or the slit width a, keeping the screen distance fixed. The Fraunhofer, Fresnel, and Geometric regimes are also referred to as the far, near, and ultra-near fields, respectively.

Close modal

A hyperbola framework for analyzing wave interference at a slit barrier (multiple slits, double slits, and single slit) was recently shown to circumvent all conventionally imposed ad hoc conditions.19–24 The analysis proceeds directly from the Huygens–Fresnel principle ab initio and treats each narrow slit as equivalent to a (Huygens) point source, without recourse to any further simplifying assumptions and approximations. The ensuing predictions regarding the distribution of fringe characteristics (positions, widths, and intensities) on a detection screen can, therefore, justifiably claim accuracy in both the near field (Fresnel regime) and the far field (Fraunhofer regime). Also, since the mathematics employed breaks from tradition by leaning more heavily on geometry than calculus, the formulation may rightly be considered a non-integral approach. In this paper, the new analysis that was previously carried out for the special case of slits of negligible widths is now extended to encompass slits of finite widths as well. In order to achieve this generalization, each slit is treated herein as equivalent to a linear array of coherent and in-phase point sources.

1. Apercus from the past

The earliest allusion to a hyperbola in the context of interference can be found in one of Young’s lectures, recorded in his Natural Philosophy (1807)—Lecture XXXIX: On the nature of light and color. The caption of figure No. 442 Plate XXX concerning the double slit (or hole) experiment reads: “The manner in which two portions of colored light, admitted through two small apertures, produce bright and dark stripes or fringes by their interference, proceeding in the form of hyperbolas …”25 Though he offers no further theoretical or experimental reasons to validate this assertion, it is likely that his prior study of water waves in a device of his own invention, called the ripple tank, convinced him to think of light in a similar vein.26,27 The next mention appears in the context of diffraction in Fresnel’s Memoirs (1818). He observes that when the screen or source is moved away from the diffracting obstacle, the locus of a given fringe is not a straight line but a hyperbola. “But experiment shows that in the case of exterior fringes, their trajectories are hyperbolas, of which the curvature is quite sensible whenever the body which produces the shadow is sufficiently distant from the luminous point.”7 He devised a simple yet precise micrometer to establish this fact.28,29 Arthur Schuster, a student of Lord Rayleigh, made two insightful though indirect remarks on Young’s experiment in his book An Introduction to the Theory of Optics, published in 1904: (1) two point sources (either narrow slits or pin holes) act as the foci of a hyperboloid of two sheets defined by a fixed path difference; (2) the hyperboloid surfaces corresponding to the various possible path differences intersect the plane of a distant screen in hyperbolas; the non-localized fringes thus captured are hyperbolic in shape, but those in closest proximity to the screen center are almost fully straight.30 Ironically, these brilliant apercus of Young, Fresnel, and Schuster were not aptly followed by a derivation of the equation of the hyperbola/hyperboloid. It is left to the historian and philosopher of science to speculate on what course and direction the field of physical optics might have taken, as opposed to the current day integral approach, had an analytical formula been sooner sought after.

2. A welcome revival of an old idea

Over the past two decades, there has been a resurgence of interest regarding the role that hyperbolas/hyperboloids play in wave interference, more specifically in Young’s double-slit experiment.31–39 All of these efforts to derive the analytical equation are pedagogically motivated and fall in alignment with Schuster’s insights that commence from the ad hoc premise that each bright or dark fringe is associated with a particular path difference. Some authors use the derivation per se to clarify the shared misconceptions prevalent among undergraduate students on the standard path difference formula δ = d sin θ found in many textbooks, and the various approximations involved in its calculation.31–33,35,38–41 Other authors use the final equation to emphasize on the fact that the fringes are either hyperbolic or circular in shape depending on screen orientation.34,36,37 Consider two slit sources S 1 d / 2 , 0 and S 2 d / 2 , 0 separated by a distance d, with the origin O lying midway between the slits (see Fig. 5). Then, the path difference δ corresponding to a certain fringe located at a point P x , y in the XY-plane may be expressed as
δ = ± S 1 P S 2 P = ± x + d 2 2 + y 2 x d 2 2 + y 2 .
(1)
FIG. 5.

Depiction of (confocal) hyperbolas with point sources S1 and S2 as the common foci, corresponding to path differences that are an integer multiple of the wavelength λ (bright fringes). N.B. The sign convention adopted here is based on the definition δ = S1PS2P that entails that the fringe orders are positive on the right side of the origin O and negative on its left side.

FIG. 5.

Depiction of (confocal) hyperbolas with point sources S1 and S2 as the common foci, corresponding to path differences that are an integer multiple of the wavelength λ (bright fringes). N.B. The sign convention adopted here is based on the definition δ = S1PS2P that entails that the fringe orders are positive on the right side of the origin O and negative on its left side.

Close modal
Upon algebraic simplification of Eq. (1), we obtain the required hyperbola equation (see the supplementary material),
x 2 δ 2 2 y 2 d 2 2 δ 2 2 = 1 .
(2)

3. Current status and direction

While all of the preceding works cited in Sec. I C 2. have indeed deepened our understanding of wave interference between two point sources, none have actually rigorously derived the hyperbola equation from first principles. The full power and scope of applications of the new formalism in physical optics, thus, remain hitherto untapped and unexplored. In contrast, the analysis put forward by Thomas proceeds directly from the Huygens–Fresnel principle ab initio and was shown to account for multiple slit interference and single slit diffraction in the Fraunhofer regime, when the slits under consideration are of negligible widths.21 A reiteration of the central theorem employed is as follows: the locus of the points of intersections of two uniformly expanding circular wavefronts that emanate from two point sources A ( - a , 0 ) and B ( a , 0 ) separated by a finite distance 2a (where a > 0), propagating outward with a steady speed u and having a time difference of emanation ΔtAB = tB − tA, is the branch of a hyperbola (see Fig. 6),19 
x 2 u Δ t A B 2 2 y 2 a 2 u Δ t A B 2 2 = 1 .
(3)
FIG. 6.

(a) When source A emits a circular wavefront before source B (tA < tB), the locus of the intersection points is the right branch of a hyperbola, (b) when sources A and B simultaneously emit circular wavefronts (tA = tB), the locus of the intersection points is a straight line perpendicular to the line segment AB, and (c) when source A emits a circular wavefront after source B (tA > tB), the locus of the intersection points is the left branch of a hyperbola.

FIG. 6.

(a) When source A emits a circular wavefront before source B (tA < tB), the locus of the intersection points is the right branch of a hyperbola, (b) when sources A and B simultaneously emit circular wavefronts (tA = tB), the locus of the intersection points is a straight line perpendicular to the line segment AB, and (c) when source A emits a circular wavefront after source B (tA > tB), the locus of the intersection points is the left branch of a hyperbola.

Close modal
When the above hyperbola theorem (3) is applied to the double slit scenario, by treating each of the narrow slits S 1 d / 2 , 0 and S 2 d / 2 , 0 as equivalent to two coherent and in-phase point sources A a , 0 and B a , 0 , we arrive at a corollary that is identical in form to Eq. (2).20 It represents a family of confocal hyperbolas with the slit sources lying at the common foci. However, the quantity δ now bears a subtler meaning; it can take up any numerical value within the closed interval 0 , d . In other words, δ denotes the path difference to any arbitrary point P x , y in the XY-plane, not just to a particular bright or dark fringe. (N.B. Every fringe corresponds to some δ 0 , d but not every δ 0 , d corresponds to a fringe.) With this in mind, the expressions for path and phase differences may be algebraically inferred from Eq. (2),
δ 2 = d 2 + 4 x 2 + 4 y 2 d 2 + 4 x 2 + 4 y 2 2 16 x 2 d 2 2 ,
(4)
ϕ 2 = 4 π 2 λ 2 d 2 + 4 x 2 + 4 y 2 d 2 + 4 x 2 + 4 y 2 2 16 x 2 d 2 2 .
(5)
The hyperbola theorem for two slit sources S 1 , S 2 was shown to be generalizable to a linear array of N equally spaced slit sources S 1 , S 2 , S 3 , , S N using methods drawn from Discrete mathematics and Cartesian geometry (see Fig. 7).21 Note that the origin of the coordinate system X0Y0 lies midway between S 1 d / 2 , 0 and S 2 d / 2 , 0 ,
x 0 ζ i j d 2 δ i j 2 2 y 0 2 η i j d 2 2 δ i j 2 2 = 1 ,
(6)
where ζ i j = i + j 3 2 and ηij = ji are the first and second coefficients of translation, respectively, and i , j N × N i < j is any source pair combination within the array. Equation (6) represents a community of hyperbolas, i.e., a family of families of hyperbolas sharing different source pair combinations as common foci and δ i j 0 , j i d . The expressions for path and phase differences for a given source pair i , j may be algebraically inferred from Eq. (6),
δ i j 2 = 2 x 0 ζ i j d 2 + η i j d 2 2 + y 0 2 x 0 ζ i j d + η i j d 2 2 + y 0 2 x 0 ζ i j d η i j d 2 2 + y 0 2 ,
(7)
ϕ i j 2 = 8 π 2 λ 2 x 0 ζ i j d 2 + η i j d 2 2 + y 0 2 x 0 ζ i j d + η i j d 2 2 + y 0 2 x 0 ζ i j d η i j d 2 2 + y 0 2 .
(8)
FIG. 7.

(a) A linear array of N equally spaced sources S 1 , S 2 , S 3 , , S N , with a separation interval d and origin of the coordinate system X0Y0 lying midway between S 1 d / 2 , 0 and S 2 d / 2 , 0 , (b) depiction of a community of hyperbolas for adjacent source pairs in a linear array of six equally spaced sources S 1 , S 2 , S 3 , S 4 , S 5 , S 6 , (c) a family of hyperbolas for the specific non-adjacent source pair S 1 , S 4 , and (d) another family of hyperbolas for the specific non-adjacent source pair S 2 , S 5 .

FIG. 7.

(a) A linear array of N equally spaced sources S 1 , S 2 , S 3 , , S N , with a separation interval d and origin of the coordinate system X0Y0 lying midway between S 1 d / 2 , 0 and S 2 d / 2 , 0 , (b) depiction of a community of hyperbolas for adjacent source pairs in a linear array of six equally spaced sources S 1 , S 2 , S 3 , S 4 , S 5 , S 6 , (c) a family of hyperbolas for the specific non-adjacent source pair S 1 , S 4 , and (d) another family of hyperbolas for the specific non-adjacent source pair S 2 , S 5 .

Close modal
In addition, the light intensity or irradiance expression (time-averaged, relative, unnormalized, or normalized) for an arbitrary spatial configuration of N coherent and in-phase point sources of the same frequency and field amplitudes was shown to be a succinct matrix (triple) product,
I ̃ = 1 2 J T ϒ J , I ̃ = 1 N 2 J T ϒ J , J = 1 1 1 N × 1 , ϒ = cos ϕ 11 cos ϕ 12 cos ϕ 1 N cos ϕ 21 cos ϕ 22 cos ϕ 2 N cos ϕ N 1 cos ϕ N 2 cos ϕ N N N × N .
(9)

Intensity distribution curves were generated for a grating composed of N slits in the far field regime using Eq. (9) in conjunction with Eq. (8), after relaxing the i < j constraint imposed on the latter. In the near field regime, it becomes necessary to modify the ϒ matrix so as to accommodate the inverse square law, which mandates a fall of intensity or field amplitude with distance of propagation (see the supplementary material). The hyperbola framework is, in principle, seamlessly applicable to both the near and far fields and small and large angles relative to the screen center since no simplifying assumptions or approximations were invoked in its formulation, except for the idealization of the slits as point sources. The theory makes two salient predictions that diverge from received wisdom: (1) non-uniformity of spacings and widths of the fringes in the double-slit experiment and (2) non-fixity of the positions of the primary maxima in the multiple slit experiment. Before delving into the arguments that allow for a further extension of the new analysis to encompass slits of finite widths, a synopsis of the conventional analysis of interference and diffraction is provided below.

The main results of the calculus-based treatment of interference and diffraction in the Fraunhofer regime for slits of finite widths are summarized in this section.9 A counterpart discussion for slits of negligible widths may be found elsewhere.1 

Consider a single slit of width d and a coordinate system with origin O fixed at its center, as shown in Fig. 8(a). The intensity distribution formula for a characteristic field strength ɛS is given by
I P θ = I 0 sin β β 2 , I 0 = 1 2 ε S d R 2 , β = π d sin θ λ .
(10)
FIG. 8.

(a) Single slit, (b) double slits, and (c) multiple slits. The origin O lies at the center of the first slit. The line OP = R extends from the origin to an arbitrary point P on the distant screen and makes an angle θ with the positive X-axis direction. For each arrangement, the slits are of width d and inter-slit spacing interval b (center to center).

FIG. 8.

(a) Single slit, (b) double slits, and (c) multiple slits. The origin O lies at the center of the first slit. The line OP = R extends from the origin to an arbitrary point P on the distant screen and makes an angle θ with the positive X-axis direction. For each arrangement, the slits are of width d and inter-slit spacing interval b (center to center).

Close modal
The intensity distribution formula for double slits is given by [see Fig. 8(b)]
I P θ = I 0 sin β β 2 c o s 2 α , I 0 = 1 2 2 ε S d R 2 = 4 I 0 , β = π d sin θ λ , α = π b sin θ λ .
(11)
The intensity distribution formula for multiple slits is given by [see Fig. 8(c)]
I P θ = I 0 sin β β 2 sin N α N sin α 2 , I 0 = 1 2 N ε S d R 2 = N 2 I 0 , β = π d sin θ λ , α = π b sin θ λ .
(12)

The hyperbola-based treatment of multiple slit interference was originally carried out in the Fraunhofer regime for slits of negligible widths.21 The same method of analysis is herein extended to encompass slits of finite widths as well. For the sake of brevity, the intricate details concerning the formal structure of the complete theory have been relegated to the supplementary material. Only a superficial outline of the principal arguments and chief theorems without proofs is furnished in this section. An essential trait of the new approach is that the twin physical properties of light, namely, interference and diffraction, become mathematically indistinguishable and, thus, render a robust unitary framework to apprehend both phenomena.

The foremost mandate of the proposed program is to consider each slit of an N-slit grating as equivalent to a linear array of n coherent and in-phase point sources. Let d and b denote the uniform spacing interval between adjacent point sources within a slit and the uniform spacing interval between adjacent slits, respectively (see Fig. 9). By additionally ascribing individual coordinate systems XkYk to each of the k N point sources, in serialized fashion from the left to right direction, we can then generalize the community theorem [multiple slit interference Eq. (6)] for any source pair combination i , j N × N i < j within the grating,
x 1 ζ i j 1 d ζ i j 2 b 2 δ i j 2 2 y 1 2 η i j 1 d + η i j 2 b 2 2 δ i j 2 2 = 1 ,
(13)
where ζ i j 1 = 1 2 i + j i n j n , ζ i j 2 = 1 2 i n + j n 2 , η i j 1 = j i j n i n , and η i j 2 = j n i n are the coefficients of translation. Note that the origin of the coordinate system X1Y1 lies at the location of the point source S1 and denotes the ceiling function. We may alternatively perform an Euclidean translation operation and shift the origin to the center C of the grating. If l = N n 1 d + N 1 b is the span of the grating, then Eq. (13) can be re-written in the XcYc coordinate system,
x c + l 2 ζ i j 1 d ζ i j 2 b 2 δ i j 2 2 y c 2 η i j 1 d + η i j 2 b 2 2 δ i j 2 2 = 1 .
(14)
FIG. 9.

(a) A grating composed of a total of N slits labeled m = 1 , 2 , 3 , , N and each slit made of n point sources. The origins of the coordinate systems X1Y1 and XcYc lie at the location of the point source S1 and center C of the grating, respectively. (b) Depiction of a grating composed of a total of five slits and each slit made of three point sources. The uniform inter-source intervals and inter-slit distances (edge to edge) are denoted by d and b, respectively.

FIG. 9.

(a) A grating composed of a total of N slits labeled m = 1 , 2 , 3 , , N and each slit made of n point sources. The origins of the coordinate systems X1Y1 and XcYc lie at the location of the point source S1 and center C of the grating, respectively. (b) Depiction of a grating composed of a total of five slits and each slit made of three point sources. The uniform inter-source intervals and inter-slit distances (edge to edge) are denoted by d and b, respectively.

Close modal
It may be readily shown that the multiple slit diffraction Eq. (14) reduces right back to the original multiple slit interference Eq. (6) upon setting n = 1, b = d, and then performing a Euclidean translation of the origin from the center of the grating to a position lying midway between the point sources S 1 d / 2 , 0 and S 2 d / 2 , 0 . The expressions for path and phase differences for a given source pair i , j may be algebraically inferred from Eq. (14),
δ i j 2 = 2 x c + l 2 ζ i j 1 d ζ i j 2 b 2 + y c 2 + η i j 1 d + η i j 2 b 2 2 x c + l 2 ζ i j 1 η i j 1 2 d ζ i j 2 η i j 2 2 b 2 + y c 2 x c + l 2 ζ i j 1 + η i j 1 2 d ζ i j 2 + η i j 2 2 b 2 + y c 2 ,
(15)
ϕ i j 2 = 8 π 2 λ 2 x c + l 2 ζ i j 1 d ζ i j 2 b 2 + y c 2 + η i j 1 d + η i j 2 b 2 2 x c + l 2 ζ i j 1 η i j 1 2 d ζ i j 2 η i j 2 2 b 2 + y c 2 x c + l 2 ζ i j 1 + η i j 1 2 d ζ i j 2 + η i j 2 2 b 2 + y c 2 ,
(16)

The prime innovation of the extended formalism is the introduction of the parameter n, denoting the number of point sources per slit. It quantitatively embodies Huygens’ notion of secondary sources of spherical wavelets that overlie a primary wavefront. Since there is no known least distance of separation between neighboring Huygens’ sources on the wavefront, the choice of the numerical value of n is rather arbitrary. However, its magnitude determines the smoothness and final form of the irradiance curve. Larger values of n yield better resolution and agreement with the conventional analysis and, therefore, act as a limiting parameter in the current model. As a rule of thumb, it is suggested that for a slit of width a and wavelength of light λ, n > a λ . This heuristic roughly implies that the separation interval between neighboring Huygens’ sources is less than a wavelength a n < λ . For the N-slit scenario, the same argument may be extended after juxtaposing all the slits together edge to edge, that is, n > N a λ .

The intensity distribution curves are generated for a slit barrier composed of N = 1 , 2 , 3 , 4 , 5 slits and n = 200 point sources per slit in the far field regime, using the irradiance theorem (9) in conjunction with the phase difference expression (16), after relaxing the i < j constraint imposed on the latter and setting the total number of coherent, in-phase point sources as N = nN.

A graphical comparison is drawn against the standard intensity formula (12) in Figs. 12 and 13. It should be noted that the grating characteristics d , b denote different measures of length in the conventional and hyperbola-based analyses. In order to make this distinction clearer, we designate appropriate subscripts for each: d con , b con represents the slit width and the inter-slit distance (center to center) in the former, and d hyp , b hyp represents the inter-point-source distance within a slit and the inter-slit distance (edge to edge) in the latter (see Fig. 10). These quantities are mutually related as follows: d con = n 1 d hyp and b con = b hyp + n 1 d hyp , or when written inversely d hyp = d con n 1 and bhyp = bcondcon. Other configuration parameters include screen distance from the grating L and the wavelength of light λ. All numerical values are expressed in Standard International units (meters) and fall within the typical range of optical experiments performed in a physics laboratory: L , λ , d con , b con = 1 m , 500 × 1 0 9 m , 1 0 5 m , 10 × 1 0 5 m for the multiple-slit apparatus and L , λ , d con , b con = 1 m , 500 × 1 0 9 m , 1 0 4 m , 0 m for the single-slit apparatus.

FIG. 10.

Double-slit apparatus illustration: (a) d hyp , b hyp denotes the inter-source spacing interval and inter-slit distance (edge to edge) in the hyperbola-based analysis and (b) d con , b con denotes the slit width and inter-slit distance (center to center) in the conventional analysis.

FIG. 10.

Double-slit apparatus illustration: (a) d hyp , b hyp denotes the inter-source spacing interval and inter-slit distance (edge to edge) in the hyperbola-based analysis and (b) d con , b con denotes the slit width and inter-slit distance (center to center) in the conventional analysis.

Close modal

The main goal of this paper was to perform a complete geometrization of the Huygens–Fresnel principle without invoking any of the simplifying assumptions that are generally associated with the integral approach. Specific instances include the parallel ray, the small angle, the field amplitude, and phase approximations.9 A hallmark feature of the new analysis is that its mathematical structure makes no distinction between interference and diffraction phenomena (see Fig. 11). Only the precise physical setting, like the slit size and/or slit separation and/or number of slits involved, can help ascertain the predominating influence of one or the other. On the other hand, in the conventional analysis, there are two factorized terms that disjointedly convey the weighted contribution of the interference and diffraction effects [Eq. (12)]. As noted by Richard Feynman, there is no physical difference between the two since both are an immediate consequence of wave superposition—the former when there are only a few waves involved and the latter when there are many.42 Nonetheless, college courses and textbooks treat these topics separately, owing in small part to their independent historical origins, but mostly due to the striking dissimilarity of their underlying formalisms. In contrast, the hyperbola-based analysis renders a robust unitary framework capable of concurrently apprehending both phenomena.

FIG. 11.

Summary of the hyperbola-based analysis of multiple-slit interference/diffraction.

FIG. 11.

Summary of the hyperbola-based analysis of multiple-slit interference/diffraction.

Close modal

Many of the results of the new analysis were shown to be highly versatile and information-laden. The following examples serve as a case in point: (1) the path difference formula [Eq. (4)] aided in quantifying the distribution of fringe positions on screens of varied shapes (linear, semicircular, and semielliptical) in the double-slit experiment,19,20 (2) the phase difference formula [Eq. (8)] enabled graphical plotting of the fringe intensity distributions in the double-slit, multiple-slit, and single-slit experiments,21 (3) the framework itself offers an effective scheme to count fringes and study their defining characteristics, such as shapes and sizes,24 and (4) design novel methods to measure the wavelength of light and the refractive index of a liquid medium, study 2D materials and x-ray crystallography, and potentially the detection of gravitational waves.21,22,43

The theory previously made two salient predictions that diverge from the conventional approach: (1) non-uniformity of spacings and widths of fringes in the double-slit experiment19,20 and (2) non-fixity of the positions of the primary maxima in the multiple-slit experiment.21 The basic premise of these findings is that the slits can be idealized as point sources. However, real slits have definite spatial extents that cannot be ignored. The current, more elaborate system accommodates slits of finite widths, and the results yielded are in good agreement with the conventional analysis. In Figs. 12 and 13, a numerical–graphical simulation of the classical grating Eq. (12) is compared against the irradiance theorem (9) in conjunction with the phase difference expression (16) for N = 1 , 2 , 3 , 4 , 5 slits. The intensity distributions for both analyses consist of a diffraction envelope housing an interference pattern and an equal number of subsidiary maxima between any pair of principal maxima. The degree of congruence of the curves depends directly on the magnitude of the limiting parameter n. The greater the n-value, the more precise the overlap. The author proposes a heuristic to determine the minimum of n, which essentially mandates that the separation interval between neighboring Huygens’ sources on a wavefront is less than a wavelength.

FIG. 12.

Multiple slit interference and diffraction patterns of both analyses display remarkable congruence for N = {2, 3, 4, 5} slits and n = 200 point sources per slit.

FIG. 12.

Multiple slit interference and diffraction patterns of both analyses display remarkable congruence for N = {2, 3, 4, 5} slits and n = 200 point sources per slit.

Close modal
FIG. 13.

Single-slit diffraction patterns of both analyses also display remarkable congruence for n = 200 point sources per slit.

FIG. 13.

Single-slit diffraction patterns of both analyses also display remarkable congruence for n = 200 point sources per slit.

Close modal

The new analysis may be considered a discretized counterpart of the continuous integral approach. The absence of simplifying assumptions or approximations ensures, in principle, that the ensuing predictions regarding the distribution of fringe characteristics can justifiably claim accuracy. Besides the broad scope of potential applications in areas, such as optical metrology and spectroscopy, it bears the distinctive pedagogic advantage of being visually more intuitive. It is, therefore, recommended for inclusion into the standard curriculum of an advanced undergraduate/graduate-level course in physical optics to complement the conventional analysis.

As part of a series of forthcoming projects, the hyperbola framework is used to: (i) compare against the Kirchhoff-Fresnel formulation and the Rayleigh–Sommerfeld formulation, (ii) study the Poisson spot and its associated concentric rings, (iii) explore Fresnel diffraction when a planar (or curvilinear) wavefront is incident on a wide aperture in the near field limit, (iv) determine the resolving power of different optical elements like lenses or disks or spheres, (v) explain Talbot image formation, (vi) investigate for links to Fourier optics, and (vii) revisit the Doppler effect.

In this paper, a complete geometrization of the Huygens–Fresnel principle has been achieved by means of a highly versatile hyperbola theorem. The new analysis that was previously carried out for the special case of slits of negligible widths in the Fraunhofer regime is now extended to encompass slits of finite widths as well. Its overall domain of validity has thus been enlarged to include the near and far fields, small and large angles, and narrow and wide slits (see the supplementary material for an outline of its application in the Fresnel regime).

All the essential details of the formal structure underlying the complete theory, including the principal arguments, statement of the chief theorems with proofs, additional tables enumerating special cases, etc., are provided in the supplementary material.

This research received no funding from any specific grant or agency.

The author has no conflicts to disclose.

Joseph Ivin Thomas: Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available within the article and its supplementary material.

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Supplementary Material