Precision Modulation and the Shadow Blister Phenomenon in Optical Diffraction Using Straight-Edge Apertures

The shadow blister effect, depicting the distortion of shadows when two objects overlap, is an optical phenomenon observable in sunlight without requiring specialized lab equipment. Despite its seemingly straightforward nature, this effect challenges explanation through ray theory and the Fresnel diffraction equation in certain regions. Conversely, the shadow blister effect exhibits both linear and nonlinear behavior corresponding to the steady variation of the transverse distance between the two unplanar straight edges along the optical axis. This article explores the shadow blister effect alongside the diffraction of a straight edge, revealing fundamental aspects of the diffraction phenomenon.

The experimental study introduces a diffraction model adept at elucidating the shadow blister effect. This model relies on an inhomogeneous fractal space, potentially generated by objects near their surfaces, including the edges of barriers.

Introduction

The shadow blister effect, commonly witnessed on sunny days, depicts the distortion of shadows when two objects overlap. Initially, it may seem that the shadow cast by the object closer to the observation screen extends toward the other shadow. However, upon closer inspection, the simultaneous inward movement of the other shadow becomes apparent.

Furthermore, the shadow blister effect is frequently overlooked when the transverse distance between the barriers is narrow. Moreover, the interaction between overlapping shadows has often been oversimplified, with static considerations failing to capture its true complexity. Yet, experimental findings suggest that such oversight is unwarranted.

For a better understanding of the behavior of the shadow blister phenomenon we need a dynamic consideration of the shadow’s deformation correlates with the constant velocity of the barrier closer to the observation screen, while the other barrier remains fixed.

The research reveals distinct zones of deformation in the affected parts of the shadows, each corresponding to the transverse distance between the barriers. When this distance falls below a certain threshold, the deformation exhibits non-linear behavior. This dynamic interplay can be further subdivided into two subsections characterized by positive and negative acceleration.

Conversely, when two objects overlap and the transverse distance between the barriers remains negative, the affected parts of the shadows maintain linear behavior, mirroring the constant velocity of the barrier closer to the observation screen. In this scenario, the other barrier also remains fixed. Fig. 1, titled ‘The shadow blister effect,’ illustrates the phenomena discussed in the text regarding the inadequacy of traditional ray theory, as described in Lock's work [1] and Fresnel diffraction theory to capture the intricate dynamics observed when two objects overlap.

Fig. 1. The shadow blister effect.

In light of these complexities, this study aims to provide a more accurate understanding of the shadow blister effect. Through a combination of theoretical analyses and experimental investigations, the aim is to unravel the nuanced behavior of shadows in the presence of multiple obstacles.

Experimental Setup

The measurements were conducted utilizing a standard diffraction by a straight-edge experimental setup, denoted as the primary barrier (B₁) in this context. This setup encompasses essential components, including a coherent light source, an additional movable barrier closer to the observation screen (B₂) known as the secondary barrier, a Galilean beam expander, lasers employed as the light source, and an observation screen (S).

In Fig. 2, the experimental setup is depicted, underscoring the role of B₂ in obstructing a significant intensity transmitted by B₁.

Fig. 2. Experimental setup for the shadow blister effect using a Galilean beam expander and a laser as the light source.

• ‘d’: Distance between the concave lens of the Galilean beam expander to the primary barrier = 100 mm.
• ‘p’: Distance from B₁ (fixed position barrier) to B₂ (movable barrier) = 30 mm.
• ‘q’: Distance from B₂ to ‘S’, the observation screen = 880 mm.

Equipment

The following equipment and materials were used in this study:

Diode Laser

635 nm (Polarization Extinction Ratio: 20 dB)

532 nm (Polarization Extinction Ratio: 4 dB)

450 nm (Polarization Extinction Ratio: 25 dB)

405 nm (Polarization Extinction Ratio: Not Clarified)

Lenses

A convex and a concave lens combined to create a Galilean expander, resulting in a beam with a diameter of 11 mm.

Opaque Obstacles

Two rectangular pieces (25 mm in length) and one circular cut (20 mm in diameter) made from opaque rubberized camera shutter fabric, each with a thickness of 0.07 mm.

Biased Si Detector

200–1100 nm wavelength range, 1 ns rise time, 0.8 mm² active area.

Picoscope

100 MHz. The measurements are reported in millivolts (mV).

Darkroom

Provides controlled conditions for light-sensitive measurements.

Pinhole

Diameter of 0.5 mm.

Slider in Z-axis

A 60 cm slider repurposed from a CNC machine spare tool was used for precise adjustment and alignment of the optical axis during the experiment. This slider played a crucial role in maintaining the accuracy of the experimental setup.

Micrometer Slider in X-axis

A micrometer slider was employed to finely adjust and control the position of the secondary barrier along the X-axis. This device facilitated precise adjustments, ensuring accurate measurements and enabling a systematic exploration of the diffraction pattern behavior by moving the secondary barrier.

Photography

A camera equipped with a macro lens (150 mm, f/2.8) capable of long-exposure photography up to 3 minutes and with a shutter speed of up to 1/8000 of a second was utilized for capturing detailed images of diffraction patterns.

Notably, barrier holders were manually crafted and assembled on sliders for the experiment. While efforts were made to ensure precision, the manual nature of the assembly process may have introduced a slight degree of imprecision in the experimental setup.

Experiments

Shadow Blister Phenomena

Considering the shadow blister phenomenon outdoors using sunlight, incoherent light, and an oversized object with rough edges can be quite confusing.

To ensure clarity and precision in observations, a meticulous investigation was conducted using laser beams with wavelengths of 635 nm, 532 nm, 450 nm, and 405 nm, as illustrated in Fig. 2. Moreover, a Galilean beam expander was employed to enhance the experimental setup.

Similar to the conventional outdoor shadow blister experiment conducted on a sunny day (refer to Fig. 2), Fig. 3 visually represents the blister effect. In this illustration, B2 (a circular barrier) positioned 30 mm away from B1 effectively blocks the illuminated region of B1 (a rectangular barrier) with an overlap size of 0.3 mm, integral to the overall diffraction pattern.

Fig. 3. The shadow blister effect using a Galilean beam expander with a circular and a rectangular barrier: (a) 532 nm, (b) 635 nm, (c) 450 nm, (d) 405 nm.

As depicted in Fig. 3, the shadow of B2 extends further into the shadow of B1, accompanied by several bright and dark bands. Additionally, the fringes of the diffraction pattern on the top and bottom of the overlap area transition into the shadow area of both barriers.

Diffraction by a Double Straight Edge

To investigate the robustness of the observed phenomena, the experiment was conducted under various conditions, including different setups such as the use of a Galilean beam expander, a pinhole, and variations in the projection angle of the laser beam. Remarkably, while the numerical values for fringe distances varied across these conditions, a consistent pattern emerged in the dynamic system.

In this stage, the Galilean beam expander is removed, and the experiment continues with solely a 532 nm diode laser.

As depicted in Fig. 4, B₂ is located at a distance of p = 30 mm from B₁ while $x_B$ = 3.0 mm, and then it is shifted to the geometrical shadow region of B₁, 0.1 mm at a time along the negative X-axis and continues to $x_B$ = −6.0 mm.

Fig. 4. The position of the nth dark or bright fringe.

Fig. 5 illustrates an asymmetric diffraction pattern on the observation screen ‘S’ at a distance of 880 mm from B₂. Fig. 5a illustrates the diffraction pattern for $x_B$ = 1.0 mm to $x_B$ = −0.3 mm, and Fig. 5b is about $x_B$ = 3.0 mm to $x_B$ = 1.0 mm. Fig. 5c shows the diffraction pattern at $x_B$ = 3.0 mm.

Fig. 5. Diffraction pattern by a double straight edge due to the displacement of the secondary barrier, B₂: (a) −0.3 mm ≤ $x_B$ ≤ 1.0 mm; (b) 1.0 mm ≤ $x_B$ ≤ 3.0 mm; (c) $x_B$ = 3.0 mm.

As illustrated in Fig. 5, the diffraction pattern displays a configuration reminiscent of the blister effect for $x_B$ ≤ 0, as indicated by the region between the leading lines in Fig. 3b. Interestingly, it also stabilizes its position, particularly for the band with the maximum intensity at $x_B$ around 1 mm. This stabilization may be attributed to the prevalence of diffraction from B₁ (a rectangular barrier) when $x_B$ is relatively large. This dominance essentially overshadows the contribution from the second barrier, as if B₂ is absent, and gradually shifts as $x_B$ decreases. Additionally, fringes are visible on either side of the diffraction pattern for $x_B$ > 0.2 mm.

Furthermore, the diffraction pattern appears similar to the diffraction by a straight edge, denoted as B₁, because there is no interaction with B₂ when $x_B$ > 1.0 mm, and notably when $x_B$ ≥ 2.0 mm.

Table I presents the results of the intensity at the maxima with maximum intensity (n = 0) for displacements in the domain of −1.0 ≤ $x_B$ ≤ 0.

Table I. Variation of Maximum Intensity (Data from the Oscilloscope in mV) with Respect to the Displacement of B₂ along the X-Axis (mm)

Displacement ( ${\mathit{x}}_{\mathit{B}}$ ) of secondary barrier in the X-axis (mm)
0.0	−0.1	−0.2	−0.3	−0.4	−0.5	−0.6	−0.7	−0.8	−0.9	−1.0
Intensity (a.u.)
160	37	11.6	4.8	2.55	1.61	1.15	0.88	0.7	0.58	0.5
Intensity ratio, I [n + 1]/I [n]
0.231	0.313	0.413	0.531	0.631	0.714	0.765	0.795	0.828	0.862	–
Wavelength: 532 nm

As shown in Fig. 6, the results from Table I reveal a non-linear decay graph for the measured intensity.

Fig. 6. Maximum intensity of the maxima for n = 0 with respect to the displacement of B2 along the X-axis (mm).

Given that the displacement domain of B₂ corresponds to the geometrical shadow produced by B₁, we anticipate that the behavior of the intensity decay ratio aligns with the Fresnel integrals of diffraction by a straight edge by the equation [2]:

(1) $$\begin{array}{r}I=\frac{1}{2}{I}_0\{{(\frac{1}{2}+{\int }_0^w\cos \left(\frac{\pi }{2}{t}^2\right)dt)}^2+{(\frac{1}{2}+{\int }_0^w\sin \left(\frac{\pi }{2}{t}^2\right)dt)}^2\}\end{array}$$

Substituting the terms of the Maclaurin series, we obtain:

(2) $$\begin{array}{r}I=\frac{1}{2}{I}_0\{{(\frac{1}{2}+\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\pi }^{2k}}{2^{2k}\left(2k\right)!}\frac{w^{4k+1}}{(4k+1)})}^2+{(\frac{1}{2}+\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\pi }^{2k+1}}{2^{2k+1}(2k+1)!}\frac{w^{4k+3}}{(4k+3)})}^2\}\end{array}$$

This formulation leads to the Euler spiral, also known as the Cornu spiral, expressed by:

(3) $$\begin{array}{r}C\left(w\right)={\int }_0^w\cos \left(\frac{\pi }{2}{t}^2\right)dt,S\left(s\right)={\int }_0^w\sin \left(\frac{\pi }{2}{t}^2\right)dt\end{array}$$

Fig. 7 illustrates the graph of the intensity of Fresnel diffraction at a straight edge. Moreover, the data presented in Table II corresponds to the Cornu spiral [3],[4].

Fig. 7. Graphical representation of the fresnel integrals.

Table II. Fresnel Integrals Data

w	C(w)	S(w)	I/I0	[I/I0](n+1)/[I/I0](n)
0	0	0	0.25	0.37909076
0.5	0.4923	0.0647	0.094773	0.433410194
1.0	0.7799	0.4383	0.041075	0.511234083
1.5	0.4453	0.6975	0.020999	0.587232733
2.0	0.4882	0.3434	0.012331	0.649699142
2.5	0.4574	0.6192	0.008012	0.699435201
3.0	0.6058	0.4963	0.005604	0.736464439
3.5	0.5326	0.4152	0.004127	0.767975963
4.0	0.4984	0.4204	0.003169	0.790514489
4.5	0.5261	0.4342	0.002505	0.809908499
5.0	0.5637	0.4992	0.002029	0.825524292
5.5	0.4784	0.5537	0.001675	0.838519513
6.0	0.4995	0.447	0.001405	–

As illustrated in Fig. 8, the data presented in Table II exhibits a non-linear decay trend for the intensity of Fresnel integrals similar to Fig. 6.

Fig. 8. The decay trend for the intensity in the shadow region of B₁ on the plane at the position of B₂: insights from Fresnel integrals.

Fig. 9 visually highlights the positive decay similarities between the intensity ratio ${\eta }_{Fr}={\left[I/I_0\right]}_{(n+1)}/{\left[I/I_0\right]}_{\left(n\right)}$ relevant to the Fresnel integrals and the measured intensity ratio denoted as $\eta =I_{(x_B+0.1)}/I_{\left(x_B\right)}$ for the comparison of the maximum intensity at $x_B$ + 0.1 mm with the maximum intensity at $x_B$ (refer to Table I).

Fig. 9. Similarities between the Fresnel integrals and the measured intensity ratio.

Furthermore, the graphical representation of the Fresnel integrals is presented in Fig. 7. As depicted in this figure, the intensity decreases non-linearly from $0.25I/I_0$ within the geometrical shadow zone. This observed behavior aligns with the data discussed earlier and is visually represented in Fig. 6. It is worth noting that the diffraction pattern in the illuminated region, as illustrated in this figure, are obstructed by B₂, as previously described for $x_B$ ≤ 0 (refer to Fig. 2).

Besides the aforementioned similarities, there are additional observations within the shadow blister experiment that diverge from the characteristics of the ray theory and the Kirchhoff-Fresnel diffraction equation.

As illustrated in Fig. 5, the diffraction pattern’s band with maximum intensity does not align with the optical axis across the entire $x_B$ range. Notably, except for some instances near $x_B$ ≈ 1, the bands shift towards the right side (negative X-axis) within the illuminated region of B₂.

The region 0 < $x_B$ ≤ 1 mm displays a non-linear behavior in the variation of dark and bright fringe positions corresponding to the constant velocity of B₂, indicating sensitivity to changes in experimental parameters. In contrast, the region $x_B$ ≤ 0 consistently exhibits a linear response, unaffected by the steady movement of B₂, suggesting stability and predictability in the observed diffraction pattern.

As $x_B$ decreases, the band with maximum intensity undergoes deformation, transitioning from a vertical fringe to a drop shape horizontally, with the tail stretching to the positive X-axis (geometrical shadow region of B₂). This effect suggests that future calculations may benefit from referring to the first dark band rather than the central band. It is crucial to note that specific values, such as ‘1 mm’ for $x_B$, may vary depending on the experimental setup.

Moreover, when $x_B$ = 0, where the edges of both barriers meet the optical axis, a noticeable change occurs in the diffraction pattern’s behavior. Shifting B₂ to $x_B$ < 0 causes the first dark band to move linearly in the same direction. This movement becomes non-linear for displacements within 0 < $x_B$ ≤ 1 mm.

Table III presents the results of the displacement ($x_n^{\prime min},n=1$) of the first dark band in the negative X-axis on the observation screen ‘S’, corresponding to the displacement of B₂ along the X-axis for 0.1 ≤ $x_B$ ≤ 1 mm.

Table III. The Position of the First Dark Band ($x_1^{min}$) Corresponding to Displacement of B₂ within 0.1 ≤ $x_B$ ≤ 1 along the X-Axis

Displacement ( ${\mathit{x}}_{\mathit{B}}$ ) of the secondary barrier in the X-axis (mm)
1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1
Position of the first dark band in the negative X-axis, ${\mathit{x}}_{\mathit{n}}^{\mathbf{\prime }}$ (mm)
−0.084	−0.188	−0.31	−0.462	−0.642	−0.867	−1.153	−1.552	−2.134	−3.152
Wavelength: 532 nm

As mentioned, in the negative X-axis, where B₂ is in the shadow region of B₁, this behavior becomes entirely linear. During this experiment, we may consider the displacement of the first dark band in the negative X-axis up to −6.0 mm and even more. However, observing this diffraction pattern requires a long exposure time of about 3 minutes or more.

Table IV presents the results of the first dark band displacement in the negative X-axis corresponding to B₂ displacement along the X-axis for −1 ≤ $x_B$ ≤ 0.

Table IV. The Position of the First Dark Band ($x_1^{min}$) Corresponding to Displacement of B₂ within −1 ≤ $x_B$ ≤ 0 along the X-Axis

Displacement ( ${\mathit{x}}_{\mathit{B}}$ ) of the secondary barrier in the X-axis (mm)
0	−0.1	−0.2	−0.3	−0.4	−0.5	−0.6	−0.7	−0.8	−0.9	−1
Position of the first dark band in the negative X-axis, ${\mathit{x}}_{\mathit{n}}$ (mm)
−5.328	−8.328	−11.328	−14.328	−17.328	−20.328	−23.328	−26.328	−29.328	−32.328	−35.328
Wavelength: 532 nm

Fig. 10 illustrates the non-linear behavior of the first dark band displacement ($x_n^{\prime }$) in the X-axis for 0 < $x_B$ ≤ 1, based on the data presented in Table III, and a straight line representing the linear displacement of the first dark band (${\mathit{x}}_{\mathit{n}}$) for $x_B$ ≤ 0, derived from the data in Table IV.

Fig. 10. Displacement of the first dark band in the x-axis relative to the displacement of B₂ for −0.3 ≤ $x_B$ ≤ 1.

Fig. 11 illustrates the displacement ratio of the first dark band for −1 ≤ $x_B$ ≤ 1, relevant to the comparison of the position for $x_B$ + 0.1 mm with the position for $x_B$, denoted as ($x_{n:(x_B+0.1)}^{min}/x_{n:(x_B)}^{min}$) for every 0.1 mm displacement of B₂.

Fig. 11. Displacement ratio of the first dark band for −1 ≤ $x_B$ ≤ 1: 0.1 mm increment of $x_B$ ($x_{n:(x_B+0.1)}^{min}/x_{n:(x_B)}^{min}$).

Considering the displacement ratio of the first dark band ($x_n^{\prime min},and{x}_n^{min},n=1$) relative to the displacement of B₂ within −1 ≤ $x_B$ ≤ 1 referenced to the position of band with the maximum intensity at $x_B$ = 1 (refer to Fig. 5), introduces more definitions of the behavior of the diffraction fringe displacement in the X-axis. As shown in Fig. 11, the transition between behaviors (refer to Fig. 10) occurs at $x_B$ = 0 and $x_B$ ≈ 0.20 mm, considered transition points $x_Z$ and $x_T$.

Analysis of the Diffraction Pattern by Unplanar Straight Edges

As previously discussed, exploring the system’s dynamics based on the constant velocity of B₂ reveals a linear movement of the diffraction pattern for $x_B$ ≤ 0 and a non-linear behavior for 0 < $x_B$ ≤ 1, indicating positive and negative acceleration in the fringes’ motion. However, this is not the only dissimilarity in the appearance of the behavior of the fringes corresponding to the movement of B₂ within −6 ≤ $x_B$ ≤ 3 mm.

Here, we are going to analyze the diffraction patterns in four regions: $x_B$ ≤ 0, 0 < $x_B$ ≤ $x_T$, $x_T$ < $x_B$ ≤ 1 mm and $x_B$ > 1 mm.

The Structure of the Diffraction Pattern for xB ≤ 0

Table IV demonstrates that with every 0.1 mm movement of B₂, $x_n$ shifts with a constant ratio, as referenced in Figs. 4 and 5. Additionally, the intensity distribution in this region, as shown in Fig. 12, conforms to the Fresnel integrals function. Furthermore, as depicted in Fig. 10, the diffraction pattern moves in this region with a constant velocity corresponding to the steady movement of B₂.

Fig. 12. Adaptation of the Fresnel integrals function to the image captured for $x_B$ = −0.1 mm.

The Structure of the Diffraction Pattern for 0 < xB ≤ xT

Fig. 13 illustrates an asymmetric shape for the central bright band on the optical axis and the appearance of the fringes on either side of the diffraction pattern for $x_B$ = 0.2 mm. Moreover, the number of these bands on the negative X-axis seems uncountable, whereas there is only a single bright fringe on the positive X-axis.

Fig. 13. Adaptation of the Fresnel integrals function to the position of the fringes for $x_B$ = 0.2 mm.

Interestingly, although the band with the maximum intensity appears somehow on the optical axis, it is impossible to adjust the fringes to the Fresnel integrals function if we set this band on the optical axis (see the failed adjustment with the lower graph in Fig. 13, although the bright bands for n = 1, n = 2, and n = 3 on the negative X-axis are somehow adjusted).

Conversely, the positions of the dark and bright bands (ignoring the intensity) on either side of the optical axis still seem to follow the Fresnel integrals function except for the band with the maximum intensity on the optical axis, as if the graph starts beyond the optical axis (see the upper graph in Fig. 13). This is evident since different wavefronts contribute to forming the band with maximum intensity.

Additionally, as referenced in Figs. 10 and 11, the diffraction pattern moves in this region with a negative acceleration corresponding to the steady movement of B₂.

The Structure of the Diffraction Pattern for xT < xB ≤ 1

Fig. 14a presents the diffraction pattern at $x_B$ = 0.3 mm. Similarly to the diffraction pattern for 0 < $x_B$ ≤ $x_T$, it is impossible to adjust the fringes to the Fresnel integrals function if we set the band with the maximum intensity on the optical axis (see the failed adjustment with the lower graph, although the bright bands for n = 1, n = 2, and n = 3 on the negative X-axis are somehow adjusted). Conversely, the positions of the fringes on either side of the optical axis (ignoring the intensity) still seem to follow the Fresnel integrals function from positive to the negative X-axis, except for the band on the optical axis, as if the graph starts beyond the optical axis (see the upper graph).

Fig. 14. Aligning arbitrary maxima of Fresnel integrals with fringes on either side of the optical axis: (a) Positions on the positive and negative X-axis still follow for $x_B$ = 0.3 mm, except for the central band, as if an extra bright band is involved; (b) Positions on the positive X-axis fail to coincide for $x_B$ = 0.8 mm.

Fig. 14b presents the diffraction pattern at $x_B$ = 0.8 mm. As observed, with the increasing value of $x_B$, the gradual diminishment of the adaptation of the Fresnel integrals function with the positions of the fringes on the left side of the optical axis is evident. Additionally, the diffraction pattern remains asymmetric, and the width of the bright and dark bands on the left side of the optical axis are growing.

As mentioned earlier, although the Fresnel integrals function can be somehow adjusted with the positions of the fringes on the negative X-axis, especially for the smaller value of $x_B$, except for the band with the maximum intensity, it fails to adjust with the bands on the positive X-axis for $x_T$ < $x_B$ ≤ 1.

Notably, the width of the successive fringes, both maxima and minima, on the positive X-axis is increasing in the far-field. Conversely, the width of the successive fringes on the negative X-axis decreases in the far-field. Therefore, the Fraunhofer diffraction equation and Rayleigh-Sommerfeld integral are invalid because the Fourier transform provides a constant periodicity of maxima and minima. This invalidation becomes more pronounced when analyzing the positions of the maxima and minima in the near-field.

Furthermore, as illustrated in Fig. 11, the displacement of the diffraction pattern, aligning with the constant movement of B₂, exhibits a positive acceleration in this region.

The Structure of the Diffraction Pattern for xB > 1

As $x_B$ increases, the interaction with B₂ gradually decreases, and therefore, the diffraction pattern appears similar to the diffraction by a straight edge (refer to Eq. (3) and Fig. (7)). Fig. 15 shows the diffraction caused by B₁ in the absence of B₂ when $x_B$ > 1 mm.

Fig. 15. Adaptation of the Fresnel integrals function to the image captured for $x_B$ = 3 mm.

Finally, due to the diminishing influence of B₂ as $x_B$ increases, the diffraction pattern in this region maintains a fixed position, primarily influenced by the fixed primary barrier (B₁).

Mathematical Analysis of Diffraction Patterns

Let ‘d’ represent the distance from the light source to B₁ (100 mm), and ‘p’ be the distance between the barriers (30 mm).

For the nth maxima and minima of the diffraction pattern caused by a straight edge (here is B₁) on the plane of B₂, we have [5]: where ${\chi }_n$ is the distance of the nth maxima or minima for the values of n = 0, 1, 2, 3,…, and ∆ is $(2n+1)\lambda /2$ for maxima and nλ for minima. Correspondingly, the band with the maximum intensity appears at approximately ${\chi }_0^{max}=0.1440$ mm, and the subsequent minima occur at approximately ${\chi }_1^{min}=0.2037$ mm, both of which are considered potential candidates for the transition point, $x_T$ (refer to Fig. 16).

(4) $$\begin{array}{r}{\chi }_n=\sqrt{\frac{2p(d+p)\mathrm{\Delta }}{d}}\end{array}$$

Fig. 16. The potential candidates for the transition points $x_Z$ and $x_T$.

As discussed earlier regarding the structure of the diffraction pattern for different zones, an eminent progression occurs as $x_B$ approaches a potential transition points $x_Z$ and $x_T$ (around 0.1440 mm and 0.2037 mm relevant to the specific dimensions in the experiment).

At point $x_T$, the bright bands on the left merge and produce a single stretched band, while at point $x_Z$, the central band with the maximum intensity disappears, as illustrated in Fig. 5.

As $x_B$ continues to increase beyond $x_T$ (refer to Fig. 16), multiple bright bands can pass by the edge of the secondary barrier and may contribute to producing extra diffraction, resulting in multiple bright bands visible on either side of the diffraction pattern from the positive X-axis to the negative X-axis on the observation screen ‘S’. Conversely, as $x_B$ decreases from $x_T$, these multiple bright bands are effectively blocked, leading to a gradual decrease in the number of visible fringes on the left, on the positive X-axis.

Particularly, for the displacement ratio of the first dark band within 0 ≤ $x_B$ ≤ 1 (refer to Fig. 11), suggesting a concentration of light in a specific region. This observation adds a layer of complexity to the diffraction pattern’s response to changes in $x_B$, revealing intriguing variations in the distribution of intensity bands within the experimentally explored range.

Therefore, the analysis is divided into four subsections (similar to the subsections as discussed earlier in the regions: $x_B$ ≤ 0, 0 < $x_B$ ≤ $x_T$, $x_T$ < $x_B$ ≤ 1 mm and $x_B$ > 1 mm) to comprehensively explore the underlying dynamics.

Analysis of Linear Behavior within xB ≤ 0

When $x_B$ = 0, as discussed earlier, the band with the maximum intensity on the screen ‘S’ may appear on the negative X-axis. As illustrated in Fig. 17, the path OP₂ in the illuminated region of B₁ is blocked by B₂, causing $Q_1$ to take on the role of the light source.

Fig. 17. The position of the nth minima in the negative X-axis for $x_B$ = 0.

Let’s consider a wavefront, $W_{Q_1}{W}_{Q_1}^{\prime }$ diverging from $Q_1$, the edge of B₁. Hence, ∆, the path difference $Q_2{M}_2-P_3{M}_2$ for minima equals nλ, and $(2n+1)\lambda /2$ for maxima. Thus, we have:

$$\begin{array}{r}{Q}_2{M}_2-(Q_1{M}_2-Q_1{P}_3)=\mathrm{\Delta }\text{ and }{Q}_1{P}_3=Q_1{Q}_2=p\end{array}$$

$$\begin{array}{r}{(q^2+x_n^2)}^{\frac{1}{2}}-[{({(p+q)}^2+x_n^2)}^{\frac{1}{2}}-p]=\mathrm{\Delta }\end{array}$$

(5) $$\begin{array}{r}\sum _{k=0}^{1/2}(\genfrac{}{}{0ex}{}{1/2}{k}){\left(q^2\right)}^{1/2-k}{\left(x_n^2\right)}^k-\sum _{k=0}^{1/2}(\genfrac{}{}{0ex}{}{1/2}{k}){\left({(p+q)}^2\right)}^{\frac{1}{2}-k}{\left(x_n^2\right)}^k+p=\mathrm{\Delta }\end{array}$$

By applying binomial expansion and neglecting higher-order terms in Eq. (5) for the values of n = 0, 1, 2, 3,…, similar to Eq. (4) we have:

$$\begin{array}{r}{x}_n=\sqrt{\frac{2q(p+q)\mathrm{\Delta }}{p}}\end{array}$$

Hence, the minima and maxima are as follows:

(6) $$\begin{array}{r}{x}_n^{min}=\sqrt{\frac{2q(p+q)n\lambda }{p}}\end{array}$$

(7) $$\begin{array}{r}{x}_n^{max}=\sqrt{\frac{q(p+q)(2n+1)\lambda }{p}}\end{array}$$

Eqs. (6) and (7) yield results akin to Eq. (4) for diffraction caused by a straight edge. Subsequently, with p representing the distance from the light source $Q_1$ to B₂ (30 mm) and q as 880 mm, we have $x_1^{min}=5.329$ mm, as determined by Eq. (6). This numerical value closely matches the experimental results exhibited in Table IV for $x_B$ = 0.

Additionally, by moving B₂ further, $Q_2$ goes completely into the shadow region of B₁.

Thus, similarly to what was discussed earlier for $x_B$ = 0, a wavefront, $W_{Q_1}{W}_{Q_1}^{\prime }$diverging from $Q_1$, as shown in Fig. 18. Hence, ∆, the path difference $Q_2{M}_3-P_3{M}_3$ for minima equals nλ, and $(2n+1)\lambda /2$ for maxima. Thus, we have:

Fig. 18. The position of the nth dark band in the negative X-axis for $x_B$ < 0.

$$\begin{array}{r}{Q}_2{M}_3-(Q_1{M}_3-Q_1{Q}_2)=\mathrm{\Delta }\end{array}$$

$$\begin{array}{r}{(q^2+{(x_n-x_B)}^2)}^{\frac{1}{2}}-{({(p+q)}^2+x_n^2)}^{\frac{1}{2}}+{(p^2+x_B^2)}^{\frac{1}{2}}=\mathrm{\Delta }\end{array}$$

(8) $$\begin{array}{r}\sum _{k=0}^{1/2}(\genfrac{}{}{0ex}{}{1/2}{k}){\left(q^2\right)}^{\frac{1}{2}-k}{\left({(x_n-x_B)}^2\right)}^k-\sum _{k=0}^{1/2}(\genfrac{}{}{0ex}{}{1/2}{k}){\left({(p+q)}^2\right)}^{\frac{1}{2}-k}{\left(x_n^2\right)}^k+\sum _{k=0}^{1/2}(\genfrac{}{}{0ex}{}{1/2}{k}){\left(p^2\right)}^{\frac{1}{2}-k}{\left({x_B}^2\right)}^k=\mathrm{\Delta }\end{array}$$

Finally, on application of binomial expansion and neglecting the higher-order terms in Eq. (8), we achieve a quadric equation: where

(9) $$\begin{array}{r}{u}_1{x}_n^2+u_2{x}_n+u_3=0\end{array}$$

$$\begin{array}{r}{u}_1=p^2\end{array}$$

$$\begin{array}{r}{u}_2=-2p(p+q)x_B\end{array}$$

$$\begin{array}{r}{u}_3=(p+q)[(p+q)x_B^2-2\mathrm{\Delta }pq]\end{array}$$

Correspondingly, in addition for $x_B$ = 0, the position of the first dark band ($x_1^{min}$) for $x_B$ < 0 appears as shown in Table V:

Table V. The Position of the First Dark Band ($x_1^{min}$) Corresponding to Displacement of B₂ for $x_B$ = 0 and $x_B$ < 0 along the X-Axis, as Calculated by Eqs. (6) and (9)

Displacement ( ${\mathit{x}}_{\mathit{B}}$ ) of the secondary barrier in the X-axis (mm)
0	−0.1	−0.2	−0.3	−0.4	−0.5	−0.6	−0.7	−0.8	−0.9	−1
Position of the first dark band in the negative X-axis, ${\mathit{x}}_{\mathit{n}}$ (mm)
−5.329	−8.362	−11.395	−14.429	−17.462	−20.495	−23.529	−26.562	−29.595	−32.629	−35.662
Wavelength: 532 nm

Comparison between experimental results (see Table IV) and mathematical solutions (see Table V) reveals discrepancies of 0.001 mm for $x_B$ = 0, 0.034 mm for $x_B$ = −0.1 mm, and 0.334 mm for $x_B$ = −1 mm. Additionally, the mathematical rate of displacement for the first dark band in the negative X-axis, with every 0.1 mm movement of B₂, is approximately 3.033 mm, while this rate in the experimental results is 3 mm. These variations may be attributed to mathematical approximations and the precision limitations of the experimental setup. Furthermore, Eq. (9) is invalid for $x_B$ = 0. This emphasizes $x_Z$ as a transition point once again.

Notably, the wavefront $W_{Q_1}{W}_{Q_1}^{\prime }$ diverging from $Q_1$, the edge of B₁ (refer to Figs. 17 and 18), singularly governs the diffraction process. Consequently, the phase difference (refer to Eq. (10)) remains constant.

(10) $$\begin{array}{r}\mathrm{\Delta }\varnothing =\frac{2\pi }{\lambda }\mathrm{\Delta }r\end{array}$$

This constancy in phase difference ensures a consistent ratio of fringe displacement, and correspondingly, the movement of fringes relative to the displacement of B₂ maintains a steady velocity with zero acceleration (a = 0). Similarly, the results of Tables IV and V as well as Fig. 10 within $x_B$ ≤ 0 underpin this linear behavior, providing a clear understanding of the relationship between fringe dynamics and the movement of the secondary barrier.

Analysis of Non-Linear Behavior for 0 < xB ≤ xT

As shown in Fig. 13, an asymmetric central band appears, positioned close to the optical axis, with a variable width corresponding to the displacement of B₂ within 0 < $x_B$ ≤ $x_T$.

It is notable that here we are going to consider the displacement behavior of the diffraction pattern within a small movement of B₂, which is less than 0.2037 mm. Therefore, the light source, even if it is a narrow slit or pinhole, may not be considered a point light source, especially when contemplating the shadow blister with the source plane of sunlight. However, for simplicity, Fig. 19 assumes the light source as a point.

Fig. 19. The position of the band with the maximum intensity and nth dark band in the negative X-axis for 0 < $x_B$ ≤ $x_T$.

Thus, we need to consider the wavefront $W_O{W}_O^{\prime }$ at $Q_1$, at the edge of B₁, and the diffracted light at this point causing wavefront $W_{Q_1}{W}_{Q_1}^{\prime }$, which arrives at the edge of B₂ in the points $Q_2,Q_2^{\prime },and{Q}_2^{"}$ through the paths $Q_1{Q}_2,Q_1{Q}_2^{\prime },and{Q}_1{Q}_2^{"}$. Additionally, we should consider the wavefront $W_{O_1}{W}_{O_1}^{\prime }$ at the points $Q_2,Q_2^{\prime },and{Q}_2^{"}$ through the paths $O{Q}_2,O{Q}_2^{\prime },andO{Q}_2^{"}$.

Eq. (2) reveals a non-linear intensity decay pattern for Fresnel integrals when the edge of B₂ aligns with the points $Q_2,Q_2^{\prime },and{Q}_2^{"}$, within 0 < $x_B$ ≤ $x_T$ as $Q_2{Q}_2^{\prime }=Q_2^{\prime }{Q}_2^{"}=x_Q$ (refer to Fig. 19). Consequently, Table I anticipates a similar non-linear decay trend for the intensity of relevant bright fringes.

On the other hand, if we connect points $Q_1$ and $Q_2$ and extend this line to intersect with the screen (refer to Fig. 16), the distance from M₀, the intersection point, to the optical axis will be 6.066 mm when $x_B$ = 0.2 mm. This value closely matches the experimental result for the solitary bright band on the positive X-axis (refer to Fig. 13).

Furthermore, as depicted in Fig. 11, experimental results illustrate that the diffraction pattern displaces with a negative acceleration (a < 0) corresponding to the steady movement of B₂ within the zone between two transition points, $x_Z$ and $x_T$. Refer to Fig. 19, at point $Q_2$ we have:

$$\begin{array}{r}\mathrm{\Delta }r=O{Q}_2-Q_1{Q}_2\end{array}$$

Thus:

$$\begin{array}{r}\mathrm{\Delta }r={[{(d+p)}^2+{x_B}^2]}^{\frac{1}{2}}-{(p^2+{x_B}^2)}^{\frac{1}{2}}\end{array}$$

$$\begin{array}{r}\mathrm{\Delta }r=\sum _{k=0}^{1/2}(\genfrac{}{}{0ex}{}{1/2}{k}){\left({(d+p)}^2\right)}^{\frac{1}{2}-k}{\left({x_B}^2\right)}^k-\sum _{k=0}^{1/2}(\genfrac{}{}{0ex}{}{1/2}{k}){\left(p^2\right)}^{\frac{1}{2}-k}{\left({x_B}^2\right)}^k\end{array}$$

Applying binomial expansion and neglecting higher-order terms, we get:

(11) $$\begin{array}{r}\mathrm{\Delta }r=d-\beta x_B^{2}and\beta =\frac{d}{2p(d+p)}\end{array}$$

As depicted in Fig. 19, the wavefront from light source ‘O’ ($W_{O_1}{W}_{O_1}^{\prime }$) is obstructed at the optical axis due to the shadow region produced by B₁. Consequently, the fringes at points M₄, M₅, and M₆ result from the paths originating from $Q_1$ with phase $\varnothing$, and those from $Q_2,Q_2^{\prime },and{Q}_2^{"}$ with phases ${\varnothing }_1$, ${\varnothing }_2$, and ${\varnothing }_3$. Therefore, for the maxima x_M4, x_M5, and x_M6 at points M₄, M₅, and M₆, respectively:

1. If $\mathrm{\Delta }{\varnothing }_{2,3}=\mathrm{\Delta }{\varnothing }_{1,2}then\mathrm{\Delta }{x}_{M_5,M_6}=\mathrm{\Delta }{x}_{M_4,M_5}$, resulting in constant fringe displacement (a = 0).
2. If $\mathrm{\Delta }{\varnothing }_{2,3}>\mathrm{\Delta }{\varnothing }_{1,2}then\mathrm{\Delta }{x}_{M_5,M_6}>\mathrm{\Delta }{x}_{M_4,M_5}$, resulting in positive acceleration fringe displacement (a > 0).
3. If $\mathrm{\Delta }{\varnothing }_{2,3}<\mathrm{\Delta }{\varnothing }_{1,2}then\mathrm{\Delta }{x}_{M_5,M_6}<\mathrm{\Delta }{x}_{M_4,M_5}$, resulting in negative acceleration fringe displacement (a < 0).

Thus, based on the Eqs. (10) and (11) we have: where

$$\begin{array}{r}\mathrm{\Delta }\varnothing =\frac{2\pi }{\lambda }[((d-\beta x_{B\left(Q_2^{"}\right)}^2)-\left((d-\beta x_{B\left(Q_2^{\prime }\right)}^2)\right))-((d-\beta x_{B\left(Q_2^{\prime }\right)}^2)-(d-\beta x_{B\left(Q_2\right)}^2))]\end{array}$$

$$\begin{array}{r}{Q}_2^{\prime }{Q}_2^{"}=Q_2{Q}_2^{\prime }=x_Q\end{array}$$

Consequently:

$$\begin{array}{r}{x}_{B\left(Q_2^{\prime }\right)}=x_B-x_{Q}and{x}_{B\left(Q_2^{"}\right)}=x_B-2x_Q\end{array}$$

Hence:

$$\begin{array}{r}\mathrm{\Delta }\varnothing =\frac{2\pi \beta }{\lambda }[-{(x_B-2x_Q)}^2+2{(x_B-x_Q)}^2-x_B^2]\end{array}$$

And finally we have:

(12) $$\begin{array}{r}\mathrm{\Delta }\varnothing =-\frac{4\pi \beta x_Q^2}{\lambda }\end{array}$$

Eq. (12) aligns with the negative acceleration (a < 0) of the position of the diffraction pattern displacement, as depicted in Figs. 5 and 11, within 0 < $x_B$ ≤ $x_T$. However, it’s worth noting that the measurements of the dark and bright fringes do not entirely align with Eqs. (6)–(9).

Furthermore, a finite incident angle to B₂ results in an additional phase difference from B₁. Therefore, the positions of the maxima and minima displace corresponding to the normal incidence, and the interference pattern shifts by $q\sin {\theta }_2$. Consequently, the superposition of the diffraction pattern with the shifted diffraction pattern causes a reduction in the interference effect, which may result in a stretched bright fringe on the positive X-axis [6].

On the other hand, the path difference $Q_2{M}_1-P_2{M}_1$ for the maxima at M₁ and the path difference $Q_2{M}_4-P_3{M}_4$ for the maxima at M₄ result in an asymmetrically widened central band with maximum intensity (refer to Figs. 13 and 19).

Analysis of Non-Linear Behavior within xT < xB ≤ 1

As discussed earlier, complexities arise within the range of 0 < $x_B$ ≤ 1, particularly concerning the distance between the two barriers. Additionally, an examination of the diffraction behavior in the narrow zone 0 < $x_B$ ≤ $x_T$ revealed that one contributing factor to this complexity is the obstruction of the wavefront from the light source ‘O’ at the optical axis. Therefore, we anticipate encountering even more complexities within $x_T$ < $x_B$ ≤ 1, especially as several bright fringes produced by B₁ come into play at point $Q_2$ (refer to Fig. 21) when $x_B$ > $x_T$.

Fig. 21. The position of the nth fringe for $x_T$ < $x_B$ ≤ 1.

One significant observation in Fig. 5 is an asymmetric central band in the range $x_T$ < $x_B$ ≤ 1, which becomes less widened as $x_B$ increases. On the other hand, if we connect points $Q_1$ and $Q_2$ and extend this line to intersect with the screen (refer to Fig. 20), the distance from M₀, the intersection point to the optical axis will be 30.33 mm when $x_B$ = 1 mm. This allows several fringes to appear on the positive X-axis, produced by the primary barrier.

Fig. 20. Asymmetrically widened central band.

Furthermore, according to Eq. (4), the distance from the band with the maximum intensity ($x_n^{\prime }$) for $n=0$ produced by B₁ on the positive X-axis to the optical axis is 2.21 mm. Similarly, the distance from the band with the maximum intensity ($x_n$) for $n=0$ produced by B₂ on the negative X-axis to the optical axis is 1.9 − 1 = 0.9 mm when $x_B$ = 1 mm.

As illustrated in Fig. 20, the edges of the primary and secondary barriers produce diffraction adapted to the Fresnel integrals along both the positive and negative X-axes. The distance from B₁ to the observation screen is larger, and therefore, the graph is wider. Thus, by overlaying them, we observe an asymmetric central band stretched slightly to the left. Additionally, the first dark band on the right side appears on the negative X-axis, as presented in Table III.

As depicted in Fig. 21, for point M₃ on the positive X-axis, different sources contribute. Therefore, the path OM₃ may arrive at M₃ without contributing to B₁ and B₂, especially when $x_B$ is closer to 1 mm. In addition, $Q_1$ takes the role of the light source with the wavefront $W_O{W}_O^{\prime }$, and the consequence arrives at $Q_2$ as a diffraction pattern following Eq. (1) in the form of the wavefront $W_{Q_1}{W}_{Q_1}^{\prime }$. Finally, the path $O{Q}_2$ and, following that, the path $Q_2{M}_3$ is the third contribution.

For the point M₅ on the negative X-axis, the wavefront from ‘O’ ($W_{O_1}{W}_{O_1}^{\prime }$) is obstructed at the optical axis. Therefore, $Q_1$ and $Q_2$ take the role of the light source.

As B₂ moves from $Q_2$ to $Q_2^{\prime }$ and then to $Q_2^{"}$ at a constant velocity (since $Q_2{Q}_2^{\prime }=Q_2^{\prime }{Q}_2^{"}=x_Q$, refer to Fig. 21), the number of fringes on the positive X-axis reduces rapidly, and the fringes on the negative X-axis move further away from the optical axis, as discussed earlier, considering the first dark band, with positive acceleration. When the edge of B₂ reaches the transition point $x_T$, the acceleration changes to negative.

On the other hand, as B₂ moves from $Q_2$ to $Q_2^{\prime }$ and then to $Q_2^{"}$ at a constant velocity, the rate at which the edge of B₂ encounters the maxima and minima of the diffraction caused by B₁ on the plane of B₂, as depicted in Fig. 21 based on Eq. (1), decreases non-linearly. This suggests an increase in the time of contribution for successive fringes compared to the dark band until the edge of B₂ reaches the transition point.

In other words, according to Eq. (4), the width of each successive fringe at the edge of B₂ equals the difference in the position of two consecutive minima. Therefore, we have:

(13) $$\begin{array}{r}{\chi }_{n+1}^{min}-{\chi }_n^{min}=\sqrt{\frac{2p(d+p)\lambda }{d}}(\sqrt{n+1}-\sqrt{n})\end{array}$$

Since d, p, and λ are constants, Eq. (13) converges to zero as $n\to \mathrm{\infty }$. Consequently, in the opposite direction, when B₂ moves toward $x_T$ at a constant velocity, the time of contribution for successive fringes increases. Therefore, it appears that the diffraction pattern as discussed earlier, displace with positive acceleration (a > 0) corresponding to the constant velocity of B₂ (refer to Table III and Fig. 11).

Analysis of Diffraction Pattern Behavior for xB > 1

In the last stage, as discussed earlier, $x_B$ is significant enough not to contribute to the path OM₁. Therefore, the diffraction pattern remains unchanged due to the fixed position of B₁, and it appears as the diffraction caused by a straight edge (B₁) through the path difference $Q_1{M}_1-P_1{M}_1$. Thus, the intensity at M₁ (refer to Fig. 22) is measured by Eq. (1), and the nth maxima and minima of the diffraction pattern are calculated using Eq. (4).

Fig. 22. The position of the nth Fringe for $x_B$ > 1.

Discussion

As discussed earlier, although Figs. 13 and 14 illustrate how the fringes on the negative X-axis somehow conform to the principles of the Fresnel integrals, these integrals fail to account for the fringes on the positive X-axis (refer to Eq. (1)). This discrepancy is further emphasized when considering the experimental results, which reveal a decay in the distance between successive fringes on the negative X-axis and a growth trend on the other side of the diffraction pattern, while the Fresnel integral straightforwardly converges. The significant challenge arises due to the considerable distance from $Q_1$ to $Q_2$ (approximately 30 mm) and $x_B$ (approximately 1 mm), which cannot be treated as a small aperture.

Similarly, employing the Fraunhofer approximation and Rayleigh-Sommerfeld diffraction theory to calculate the departure of the transmitted wave from its geometrical optical path is also deemed inadequate. The field distribution U across the surface ‘S₀’ at the coordinates $(\xi ,\eta ,0)$—derived directly from the Fourier transform of the aperture distribution itself in the far-field—is given by Eq. (14) [7]. However, Eq. (14) fails to reproduce the fringes with the observed growth trend on the positive X-axis.

(14) $$\begin{array}{r}U(x,y,z)=\frac{e^{jkz}{e}^{jk(x^2+y^2)/2z}}{j\lambda z}{\iint }_{-\mathrm{\infty }}^{\mathrm{\infty }}U(\xi ,\eta ,0)e^{-j\frac{2\pi }{\lambda z}(x\xi +y\eta )}d\xi d\eta \end{array}$$

Conversely, analyzing the plot graph of the diffraction pattern within 0 < $x_B$ ≤ 1 (relevant to the intensity) leads to novel insights into diffraction phenomena. Fig. 23 illustrates the plot graph of the diffraction pattern when $x_B$ = 0.4 mm, reminiscent of the propagation of an Ultrashort pulse in a dense medium, where the velocity of the pulse within a medium plays a crucial role in determining the temporal characteristics of the pulse, including its duration and the degree of chirping. Furthermore, variations in the medium’s refractive index and the resulting changes in velocity contribute to the temporal evolution of the pulse and ultimately influence its chirp rate.

Fig. 23. The plot graph of the diffraction pattern for $x_B$ = 0.4 mm.

Notably, Fig. 23 is not precise since it is a plot graph over a photo, and the exposure time used, 1/8000, is not sufficient, resulting in the bright bands merging. Therefore, the minima bands do not show zero intensity. However, this figure shows how the central band appears asymmetrically widened, as discussed earlier, with the growth trend on the positive X-axis. This asymmetry is a result of the contribution of the sources O, $Q_1$, and $Q_2$ causing $x_2^{\prime }-x_1^{\prime }<x_1^{\prime }-x_0^{\prime }$.

Chirp occurs when the frequency of a signal changes with time, stemming from two principal reasons: the variation of the refractive index along the optical path length in the medium and the effects of chromatic dispersion due to the Kerr effect.

The consequences of group velocity dispersion (GVD) include the broadening of the pulse envelope and the acquisition of chirp by the carrier wave within the envelope. Specifically, when low frequencies travel faster than high frequencies, it leads to an up-chirp (positive chirp). The analogous expansion for $k(\omega )$ for the angular frequency ${\omega }_0$ is expressed in Eq. (15), where the first term represents phase velocity, the second term accounts for the propagation of the pulse envelope, and the third term relates to the changing pulse shape through the medium [8].

(15) $$\begin{array}{r}k(\omega )=k({\omega }_0)+\frac{dk}{d\omega }(\omega -{\omega }_0)+\frac{1}{2}\frac{d^{2}k}{d{\omega }^2}{(\omega -{\omega }_0)}^2+\dots \end{array}$$

Furthermore, Fig. 23 depicts an up-chirped pulse propagating into matter and experiencing positive GVD, causing the instantaneous frequency to increase with time due to the medium’s refractive index.

Moreover, the electric field of a chirped Gaussian pulse is given by Eq. (16), as depicted in Fig. 24 [9], [10].

Fig. 24. The up-chirped pulse: (a) with the larger degree of chirping; (b) with the smaller degree of chirping.

where $E_0$ is the peak amplitude, ${\omega }_0$ is the angular frequency, α is the chirp rate, and ${\tau }_0$ represents the FWHM. Additionally, for an up-chirp $\alpha >0$ and for an down-chirp $\alpha <0$.

(16) $$\begin{array}{r}E\left(t\right)=E_{0}exp(-\frac{(1+i\alpha )t^2}{2{\tau }_0^2})exp(-i{\omega }_{0}t)\end{array}$$

Comparing the up-chirped pulse illustrated in Fig. 24 with Fig. 5 reveals similarities between Fig. 24a for a larger degree of chirping with $x_B$ = 0.2 mm and Fig. 24b for a smaller degree of chirping with $x_B$ = 0.4 mm. This difference in chirping behavior can be attributed to the tighter transverse distance between B₁ and B₂ for $x_B$ = 0.2 mm. With this smaller $x_B$ value, photons have a limited opportunity to pass through the fractal space near the edges of the barriers, where the refractive index is higher. Consequently, the observed chirping effect is more pronounced due to the increased influence of the inhomogeneous refractive index in this region. These findings are further enriched by considering the contribution of the sources O, $Q_1$, and $Q_2$ to the central band.

Notably, due to the lack of chromatic dispersion effect in the experiment, we can conclude that the variation of the refractive index along the optical path length in the medium caused the chirp shape observed in Fig. 23. Thus, when comparing the experimental results for 0 < $x_B$ ≤ 1 with the mathematical solution, Fig. 24 suggests that the region close to the surface of the obstacles, along with the edges of the apertures, exhibits fractal inhomogeneity characterized by an inhomogeneous refractive index [11]. The general expression of $E\left(t\right)$ in the medium with the complex refractive index $n_c\left(\omega \right)$ is given by Eq. (17). Where $n_c\left(\omega \right)=n\left(\omega \right)-\text{i}k\left(\omega \right)$, z is the length of travel in the medium, and c is the velocity of light [12].

(17) $$\begin{array}{r}E\left(t\right)=\frac{1}{2\pi }\int E(\omega -{\omega }_0)exp\left[i\omega (t-n_c\left(\omega \right)\frac{z}{c})\right]d\omega \end{array}$$

Conclusion

Upon thorough examination of the cross sections and diffraction patterns resulting from two parallel straight edges, which may lead to the formation of a shadow blister, three distinct boundaries become evident, corresponding to four conditions based on the transverse distance between the two edges along the X-axis.

In cases where the transverse distance is sufficiently large, “Fresnel Diffraction” through a straight edge proves to be a valid approach for evaluating intensity at any arbitrary point on the observation plane and determining the position of the fringes. Moreover, the diffraction pattern maintains a fixed position due to the fixed position of the primary barrier.

However, in the second stage, as the transverse distance reduces to approximately a millimeter, the validity of the “Fresnel Integral” diminishes. In this scenario, fringe displacement experiences non-linear behavior with positive acceleration, corresponding to the constant velocity of the secondary barrier, until the transverse distance reaches a smaller value, considered a transition point.

Subsequently, in the third stage, the displacement of fringes, corresponding to the steady speed of the secondary barrier, undergoes non-linear behavior with negative acceleration until the slit width reaches zero, and the “Fresnel Integral” remains invalid.

In the final stage, as the transverse distance approaches zero, and simultaneously when the secondary barrier overlaps the primary barrier, the “Fresnel Integral” becomes valid once again. Moreover, the displacement of fringes, corresponding to the steady speed of the secondary barrier in this region, is linear with a constant speed and zero acceleration.

Notably, complexity arises when the transverse distance is very small. In this condition, the Fourier transform is valid only if we consider a complex refractive index, suggesting an inhomogeneous fractal space with a variable refractive index near the surface of the obstacles. This variable refractive index causes a time delay in the temporal domain, leading to a particular dispersion region that underlies the diffraction phenomenon.

I am grateful to Dr. Steffen Kühn for his valuable information regarding the diffraction of electromagnetic waves by a straight edge. This guidance significantly contributed to my understanding and helped me discern differences between the experimental results in optics.

Conflict of Interest

Conflict of Interest: The author declares no conflict of interest.