Precision Modulation and the Shadow Blister Phenomenon in Optical Diffraction Using StraightEdge Apertures
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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 nonlinear 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.
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 straightedge experimental setup, denoted as the primary barrier (B_{1}) in this context. This setup encompasses essential components, including a coherent light source, an additional movable barrier closer to the observation screen (B_{2}) 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 B_{2} in obstructing a significant intensity transmitted by B_{1}.
 ‘d’: Distance between the concave lens of the Galilean beam expander to the primary barrier = 100 mm.
 ‘p’: Distance from B_{1} (fixed position barrier) to B_{2} (movable barrier) = 30 mm.
 ‘q’: Distance from B_{2} 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 lightsensitive measurements.
Pinhole
Diameter of 0.5 mm.
Slider in Zaxis
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 Xaxis
A micrometer slider was employed to finely adjust and control the position of the secondary barrier along the Xaxis. 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 longexposure 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.
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, B_{2} is located at a distance of p = 30 mm from B_{1} while xB = 3.0 mm, and then it is shifted to the geometrical shadow region of B_{1}, 0.1 mm at a time along the negative Xaxis and continues to xB = −6.0 mm.
Fig. 5 illustrates an asymmetric diffraction pattern on the observation screen ‘S’ at a distance of 880 mm from B_{2}. Fig. 5a illustrates the diffraction pattern for xB = 1.0 mm to xB = −0.3 mm, and Fig. 5b is about xB = 3.0 mm to xB = 1.0 mm. Fig. 5c shows the diffraction pattern at xB = 3.0 mm.
As illustrated in Fig. 5, the diffraction pattern displays a configuration reminiscent of the blister effect for xB ≤ 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 xB around 1 mm. This stabilization may be attributed to the prevalence of diffraction from B_{1} (a rectangular barrier) when xB is relatively large. This dominance essentially overshadows the contribution from the second barrier, as if B_{2} is absent, and gradually shifts as xB decreases. Additionally, fringes are visible on either side of the diffraction pattern for xB > 0.2 mm.
Furthermore, the diffraction pattern appears similar to the diffraction by a straight edge, denoted as B_{1}, because there is no interaction with B_{2} when xB > 1.0 mm, and notably when xB ≥ 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 ≤ xB ≤ 0.
Displacement (xB) of secondary barrier in the Xaxis (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 nonlinear decay graph for the measured intensity.
Given that the displacement domain of B_{2} corresponds to the geometrical shadow produced by B_{1}, 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)I=12I0{(12+∫0wcos(π2t2)dt)2+(12+∫0wsin(π2t2)dt)2}
Substituting the terms of the Maclaurin series, we obtain: (2)I=12I0{(12+∑k=0∞(−1)kπ2k22k(2k)!w4k+1(4k+1))2+(12+∑k=0∞(−1)kπ2k+122k+1(2k+1)!w4k+3(4k+3))2}
This formulation leads to the Euler spiral, also known as the Cornu spiral, expressed by: (3)C(w)=∫0wcos(π2t2)dt,S(s)=∫0wsin(π2t2)dt
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].
w  C(w)  S(w)  I/I_{0}  [I/I_{0}]_{(n+1)}/[I/I_{0}]_{(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 nonlinear decay trend for the intensity of Fresnel integrals similar to Fig. 6.
Fig. 9 visually highlights the positive decay similarities between the intensity ratio ηFr=[I/I0](n+1)/[I/I0](n) relevant to the Fresnel integrals and the measured intensity ratio denoted as η=I(xB+0.1)/I(xB) for the comparison of the maximum intensity at xB + 0.1 mm with the maximum intensity at xB (refer to Table I).
Furthermore, the graphical representation of the Fresnel integrals is presented in Fig. 7. As depicted in this figure, the intensity decreases nonlinearly from 0.25I/I0 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 B_{2}, as previously described for xB ≤ 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 KirchhoffFresnel 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 xB range. Notably, except for some instances near xB ≈ 1, the bands shift towards the right side (negative Xaxis) within the illuminated region of B_{2}.
The region 0 < xB ≤ 1 mm displays a nonlinear behavior in the variation of dark and bright fringe positions corresponding to the constant velocity of B_{2}, indicating sensitivity to changes in experimental parameters. In contrast, the region xB ≤ 0 consistently exhibits a linear response, unaffected by the steady movement of B_{2}, suggesting stability and predictability in the observed diffraction pattern.
As xB 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 Xaxis (geometrical shadow region of B_{2}). 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 xB, may vary depending on the experimental setup.
Moreover, when xB = 0, where the edges of both barriers meet the optical axis, a noticeable change occurs in the diffraction pattern’s behavior. Shifting B_{2} to xB < 0 causes the first dark band to move linearly in the same direction. This movement becomes nonlinear for displacements within 0 < xB ≤ 1 mm.
Table III presents the results of the displacement (xn′min,n=1) of the first dark band in the negative Xaxis on the observation screen ‘S’, corresponding to the displacement of B_{2} along the Xaxis for 0.1 ≤ xB ≤ 1 mm.
Displacement (xB) of the secondary barrier in the Xaxis (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 Xaxis, xn′ (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 Xaxis, where B_{2} is in the shadow region of B_{1}, this behavior becomes entirely linear. During this experiment, we may consider the displacement of the first dark band in the negative Xaxis 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 Xaxis corresponding to B_{2} displacement along the Xaxis for −1 ≤ xB ≤ 0.
Displacement (xB) of the secondary barrier in the Xaxis (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 Xaxis, xn (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 nonlinear behavior of the first dark band displacement (xn′) in the Xaxis for 0 < xB ≤ 1, based on the data presented in Table III, and a straight line representing the linear displacement of the first dark band (xn) for xB ≤ 0, derived from the data in Table IV.
Fig. 11 illustrates the displacement ratio of the first dark band for −1 ≤ xB ≤ 1, relevant to the comparison of the position for xB + 0.1 mm with the position for xB, denoted as (xn:(xB+0.1)min/xn:(xB)min) for every 0.1 mm displacement of B_{2}.
Considering the displacement ratio of the first dark band (xn′min,andxnmin,n=1) relative to the displacement of B_{2} within −1 ≤ xB ≤ 1 referenced to the position of band with the maximum intensity at xB = 1 (refer to Fig. 5), introduces more definitions of the behavior of the diffraction fringe displacement in the Xaxis. As shown in Fig. 11, the transition between behaviors (refer to Fig. 10) occurs at xB = 0 and xB ≈ 0.20 mm, considered transition points xZ and xT.
Analysis of the Diffraction Pattern by Unplanar Straight Edges
As previously discussed, exploring the system’s dynamics based on the constant velocity of B_{2} reveals a linear movement of the diffraction pattern for xB ≤ 0 and a nonlinear behavior for 0 < xB ≤ 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 B_{2} within −6 ≤ xB ≤ 3 mm.
Here, we are going to analyze the diffraction patterns in four regions: xB ≤ 0, 0 < xB ≤ xT, xT < xB ≤ 1 mm and xB > 1 mm.
The Structure of the Diffraction Pattern for xB ≤ 0
Table IV demonstrates that with every 0.1 mm movement of B_{2}, xn 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 B_{2}.
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 xB = 0.2 mm. Moreover, the number of these bands on the negative Xaxis seems uncountable, whereas there is only a single bright fringe on the positive Xaxis.
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 Xaxis 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 B_{2}.
The Structure of the Diffraction Pattern for xT < xB ≤ 1
Fig. 14a presents the diffraction pattern at xB = 0.3 mm. Similarly to the diffraction pattern for 0 < xB ≤ xT, 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 Xaxis 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 Xaxis, except for the band on the optical axis, as if the graph starts beyond the optical axis (see the upper graph).
Fig. 14b presents the diffraction pattern at xB = 0.8 mm. As observed, with the increasing value of xB, 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 Xaxis, especially for the smaller value of xB, except for the band with the maximum intensity, it fails to adjust with the bands on the positive Xaxis for xT < xB ≤ 1.
Notably, the width of the successive fringes, both maxima and minima, on the positive Xaxis is increasing in the farfield. Conversely, the width of the successive fringes on the negative Xaxis decreases in the farfield. Therefore, the Fraunhofer diffraction equation and RayleighSommerfeld 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 nearfield.
Furthermore, as illustrated in Fig. 11, the displacement of the diffraction pattern, aligning with the constant movement of B_{2}, exhibits a positive acceleration in this region.
The Structure of the Diffraction Pattern for xB > 1
As xB increases, the interaction with B_{2} 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 B_{1} in the absence of B_{2} when xB > 1 mm.
Finally, due to the diminishing influence of B_{2} as xB increases, the diffraction pattern in this region maintains a fixed position, primarily influenced by the fixed primary barrier (B_{1}).
Mathematical Analysis of Diffraction Patterns
Let ‘d’ represent the distance from the light source to B_{1} (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 B_{1}) on the plane of B_{2}, we have [5]: (4)χn=2p(d+p)Δdwhere χn is the distance of the nth maxima or minima for the values of n = 0, 1, 2, 3,…, and ∆ is (2n+1)λ/2 for maxima and nλ for minima. Correspondingly, the band with the maximum intensity appears at approximately χ0max=0.1440 mm, and the subsequent minima occur at approximately χ1min=0.2037 mm, both of which are considered potential candidates for the transition point, xT (refer to Fig. 16).
As discussed earlier regarding the structure of the diffraction pattern for different zones, an eminent progression occurs as xB approaches a potential transition points xZ and xT (around 0.1440 mm and 0.2037 mm relevant to the specific dimensions in the experiment).
At point xT, the bright bands on the left merge and produce a single stretched band, while at point xZ, the central band with the maximum intensity disappears, as illustrated in Fig. 5.
As xB continues to increase beyond xT (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 Xaxis to the negative Xaxis on the observation screen ‘S’. Conversely, as xB decreases from xT, these multiple bright bands are effectively blocked, leading to a gradual decrease in the number of visible fringes on the left, on the positive Xaxis.
Particularly, for the displacement ratio of the first dark band within 0 ≤ xB ≤ 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 xB, 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: xB ≤ 0, 0 < xB ≤ xT, xT < xB ≤ 1 mm and xB > 1 mm) to comprehensively explore the underlying dynamics.
Analysis of Linear Behavior within xB ≤ 0
When xB = 0, as discussed earlier, the band with the maximum intensity on the screen ‘S’ may appear on the negative Xaxis. As illustrated in Fig. 17, the path OP_{2} in the illuminated region of B_{1} is blocked by B_{2}, causing Q1 to take on the role of the light source.
Let’s consider a wavefront, WQ1WQ1′ diverging from Q1, the edge of B_{1}. Hence, ∆, the path difference Q2M2−P3M2 for minima equals nλ, and (2n+1)λ/2 for maxima. Thus, we have: Q2M2−(Q1M2−Q1P3)=Δ and Q1P3=Q1Q2=p (q2+xn2)12−[((p+q)2+xn2)12−p]=Δ (5)∑k=01/2(1/2k)(q2)1/2−k(xn2)k−∑k=01/2(1/2k)((p+q)2)12−k(xn2)k+p=Δ
By applying binomial expansion and neglecting higherorder terms in Eq. (5) for the values of n = 0, 1, 2, 3,…, similar to Eq. (4) we have: xn=2q(p+q)Δp
Hence, the minima and maxima are as follows: (6)xnmin=2q(p+q)nλp (7)xnmax=q(p+q)(2n+1)λp
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 Q1 to B_{2} (30 mm) and q as 880 mm, we have x1min=5.329 mm, as determined by Eq. (6). This numerical value closely matches the experimental results exhibited in Table IV for xB = 0.
Additionally, by moving B_{2} further, Q2 goes completely into the shadow region of B_{1}.
Thus, similarly to what was discussed earlier for xB = 0, a wavefront, WQ1WQ1′diverging from Q1, as shown in Fig. 18. Hence, ∆, the path difference Q2M3−P3M3 for minima equals nλ, and (2n+1)λ/2 for maxima. Thus, we have:
Q2M3−(Q1M3−Q1Q2)=Δ (q2+(xn−xB)2)12−((p+q)2+xn2)12+(p2+xB2)12=Δ (8)∑k=01/2(1/2k)(q2)12−k((xn−xB)2)k−∑k=01/2(1/2k)((p+q)2)12−k(xn2)k+∑k=01/2(1/2k)(p2)12−k(xB2)k=Δ
Finally, on application of binomial expansion and neglecting the higherorder terms in Eq. (8), we achieve a quadric equation: (9)u1xn2+u2xn+u3=0where u1=p2 u2=−2p(p+q)xB u3=(p+q)[(p+q)xB2−2Δpq]
Correspondingly, in addition for xB = 0, the position of the first dark band (x1min) for xB < 0 appears as shown in Table V:
Displacement (xB) of the secondary barrier in the Xaxis (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 Xaxis, xn (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 xB = 0, 0.034 mm for xB = −0.1 mm, and 0.334 mm for xB = −1 mm. Additionally, the mathematical rate of displacement for the first dark band in the negative Xaxis, with every 0.1 mm movement of B_{2}, 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 xB = 0. This emphasizes xZ as a transition point once again.
Notably, the wavefront WQ1WQ1′ diverging from Q1, the edge of B_{1} (refer to Figs. 17 and 18), singularly governs the diffraction process. Consequently, the phase difference (refer to Eq. (10)) remains constant.
(10)Δ∅=2πλΔr
This constancy in phase difference ensures a consistent ratio of fringe displacement, and correspondingly, the movement of fringes relative to the displacement of B_{2} maintains a steady velocity with zero acceleration (a = 0). Similarly, the results of Tables IV and V as well as Fig. 10 within xB ≤ 0 underpin this linear behavior, providing a clear understanding of the relationship between fringe dynamics and the movement of the secondary barrier.
Analysis of NonLinear 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 B_{2} within 0 < xB ≤ xT.
It is notable that here we are going to consider the displacement behavior of the diffraction pattern within a small movement of B_{2}, 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.
Thus, we need to consider the wavefront WOWO′ at Q1, at the edge of B_{1}, and the diffracted light at this point causing wavefront WQ1WQ1′, which arrives at the edge of B_{2} in the points Q2,Q2′,andQ2" through the paths Q1Q2,Q1Q2′,andQ1Q2". Additionally, we should consider the wavefront WO1WO1′ at the points Q2,Q2′,andQ2" through the paths OQ2,OQ2′,andOQ2".
Eq. (2) reveals a nonlinear intensity decay pattern for Fresnel integrals when the edge of B_{2} aligns with the points Q2,Q2′,andQ2", within 0 < xB ≤ xT as Q2Q2′=Q2′Q2"=xQ (refer to Fig. 19). Consequently, Table I anticipates a similar nonlinear decay trend for the intensity of relevant bright fringes.
On the other hand, if we connect points Q1 and Q2 and extend this line to intersect with the screen (refer to Fig. 16), the distance from M_{0}, the intersection point, to the optical axis will be 6.066 mm when xB = 0.2 mm. This value closely matches the experimental result for the solitary bright band on the positive Xaxis (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 B_{2} within the zone between two transition points, xZ and xT. Refer to Fig. 19, at point Q2 we have: Δr=OQ2−Q1Q2
Thus: Δr=[(d+p)2+xB2]12−(p2+xB2)12 Δr=∑k=01/2(1/2k)((d+p)2)12−k(xB2)k−∑k=01/2(1/2k)(p2)12−k(xB2)k
Applying binomial expansion and neglecting higherorder terms, we get: (11)Δr=d−βxB2andβ=d2p(d+p)
As depicted in Fig. 19, the wavefront from light source ‘O’ (WO1WO1′) is obstructed at the optical axis due to the shadow region produced by B_{1}. Consequently, the fringes at points M_{4}, M_{5}, and M_{6} result from the paths originating from Q1 with phase ∅, and those from Q2,Q2′,andQ2" with phases ∅1, ∅2, and ∅3. Therefore, for the maxima x_{M4}, x_{M5}, and x_{M6} at points M_{4}, M_{5}, and M_{6}, respectively:
 If Δ∅2,3=Δ∅1,2thenΔxM5,M6=ΔxM4,M5, resulting in constant fringe displacement (a = 0).
 If Δ∅2,3>Δ∅1,2thenΔxM5,M6>ΔxM4,M5, resulting in positive acceleration fringe displacement (a > 0).
 If Δ∅2,3<Δ∅1,2thenΔxM5,M6<ΔxM4,M5, resulting in negative acceleration fringe displacement (a < 0).
Thus, based on the Eqs. (10) and (11) we have: Δ∅=2πλ[((d−βxB(Q2")2)−((d−βxB(Q2′)2)))−((d−βxB(Q2′)2)−(d−βxB(Q2)2))]where Q2′Q2"=Q2Q2′=xQ
Consequently: xB(Q2′)=xB−xQandxB(Q2")=xB−2xQ
Hence: Δ∅=2πβλ[−(xB−2xQ)2+2(xB−xQ)2−xB2]
And finally we have: (12)Δ∅=−4πβxQ2λ
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 < xB ≤ xT. 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 B_{2} results in an additional phase difference from B_{1}. Therefore, the positions of the maxima and minima displace corresponding to the normal incidence, and the interference pattern shifts by qsinθ2. 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 Xaxis [6].
On the other hand, the path difference Q2M1−P2M1 for the maxima at M_{1} and the path difference Q2M4−P3M4 for the maxima at M_{4} result in an asymmetrically widened central band with maximum intensity (refer to Figs. 13 and 19).
Analysis of NonLinear Behavior within xT < xB ≤ 1
As discussed earlier, complexities arise within the range of 0 < xB ≤ 1, particularly concerning the distance between the two barriers. Additionally, an examination of the diffraction behavior in the narrow zone 0 < xB ≤ xT 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 xT < xB ≤ 1, especially as several bright fringes produced by B_{1} come into play at point Q2 (refer to Fig. 21) when xB > xT.
One significant observation in Fig. 5 is an asymmetric central band in the range xT < xB ≤ 1, which becomes less widened as xB increases. On the other hand, if we connect points Q1 and Q2 and extend this line to intersect with the screen (refer to Fig. 20), the distance from M_{0}, the intersection point to the optical axis will be 30.33 mm when xB = 1 mm. This allows several fringes to appear on the positive Xaxis, produced by the primary barrier.
Furthermore, according to Eq. (4), the distance from the band with the maximum intensity (xn′) for n=0 produced by B_{1} on the positive Xaxis to the optical axis is 2.21 mm. Similarly, the distance from the band with the maximum intensity (xn) for n=0 produced by B_{2} on the negative Xaxis to the optical axis is 1.9 − 1 = 0.9 mm when xB = 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 Xaxes. The distance from B_{1} 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 Xaxis, as presented in Table III.
As depicted in Fig. 21, for point M_{3} on the positive Xaxis, different sources contribute. Therefore, the path OM_{3} may arrive at M_{3} without contributing to B_{1} and B_{2}, especially when xB is closer to 1 mm. In addition, Q1 takes the role of the light source with the wavefront WOWO′, and the consequence arrives at Q2 as a diffraction pattern following Eq. (1) in the form of the wavefront WQ1WQ1′. Finally, the path OQ2 and, following that, the path Q2M3 is the third contribution.
For the point M_{5} on the negative Xaxis, the wavefront from ‘O’ (WO1WO1′) is obstructed at the optical axis. Therefore, Q1 and Q2 take the role of the light source.
As B_{2} moves from Q2 to Q2′ and then to Q2" at a constant velocity (since Q2Q2′=Q2′Q2"=xQ, refer to Fig. 21), the number of fringes on the positive Xaxis reduces rapidly, and the fringes on the negative Xaxis move further away from the optical axis, as discussed earlier, considering the first dark band, with positive acceleration. When the edge of B_{2} reaches the transition point xT, the acceleration changes to negative.
On the other hand, as B_{2} moves from Q2 to Q2′ and then to Q2" at a constant velocity, the rate at which the edge of B_{2} encounters the maxima and minima of the diffraction caused by B_{1} on the plane of B_{2}, as depicted in Fig. 21 based on Eq. (1), decreases nonlinearly. This suggests an increase in the time of contribution for successive fringes compared to the dark band until the edge of B_{2} reaches the transition point.
In other words, according to Eq. (4), the width of each successive fringe at the edge of B_{2} equals the difference in the position of two consecutive minima. Therefore, we have: (13)χn+1min−χnmin=2p(d+p)λd(n+1−n)
Since d, p, and λ are constants, Eq. (13) converges to zero as n→∞. Consequently, in the opposite direction, when B_{2} moves toward xT 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 B_{2} (refer to Table III and Fig. 11).
Analysis of Diffraction Pattern Behavior for xB > 1
In the last stage, as discussed earlier, xB is significant enough not to contribute to the path OM_{1}. Therefore, the diffraction pattern remains unchanged due to the fixed position of B_{1}, and it appears as the diffraction caused by a straight edge (B_{1}) through the path difference Q1M1−P1M1. Thus, the intensity at M_{1} (refer to Fig. 22) is measured by Eq. (1), and the nth maxima and minima of the diffraction pattern are calculated using Eq. (4).
Discussion
As discussed earlier, although Figs. 13 and 14 illustrate how the fringes on the negative Xaxis somehow conform to the principles of the Fresnel integrals, these integrals fail to account for the fringes on the positive Xaxis (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 Xaxis 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 Q1 to Q2 (approximately 30 mm) and xB (approximately 1 mm), which cannot be treated as a small aperture.
Similarly, employing the Fraunhofer approximation and RayleighSommerfeld 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 ‘S_{0}’ at the coordinates (ξ,η,0)—derived directly from the Fourier transform of the aperture distribution itself in the farfield—is given by Eq. (14) [7]. However, Eq. (14) fails to reproduce the fringes with the observed growth trend on the positive Xaxis. (14)U(x,y,z)=ejkzejk(x2+y2)/2zjλz∬−∞∞U(ξ,η,0)e−j2πλz(xξ+yη)dξdη
Conversely, analyzing the plot graph of the diffraction pattern within 0 < xB ≤ 1 (relevant to the intensity) leads to novel insights into diffraction phenomena. Fig. 23 illustrates the plot graph of the diffraction pattern when xB = 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.
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 Xaxis. This asymmetry is a result of the contribution of the sources O, Q1, and Q2 causing x2′−x1′<x1′−x0′.
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 upchirp (positive chirp). The analogous expansion for k(ω) for the angular frequency ω0 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)k(ω)=k(ω0)+dkdω(ω−ω0)+12d2kdω2(ω−ω0)2+…
Furthermore, Fig. 23 depicts an upchirped 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].
(16)E(t)=E0exp(−(1+iα)t22τ02)exp(−iω0t)where E0 is the peak amplitude, ω0 is the angular frequency, α is the chirp rate, and τ0 represents the FWHM. Additionally, for an upchirp α>0 and for an downchirp α<0.
Comparing the upchirped pulse illustrated in Fig. 24 with Fig. 5 reveals similarities between Fig. 24a for a larger degree of chirping with xB = 0.2 mm and Fig. 24b for a smaller degree of chirping with xB = 0.4 mm. This difference in chirping behavior can be attributed to the tighter transverse distance between B_{1} and B_{2} for xB = 0.2 mm. With this smaller xB 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, Q1, and Q2 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 < xB ≤ 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(t) in the medium with the complex refractive index nc(ω) is given by Eq. (17). Where nc(ω)=n(ω)−ik(ω), z is the length of travel in the medium, and c is the velocity of light [12]. (17)E(t)=12π∫E(ω−ω0)exp[iω(t−nc(ω)zc)]dω
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 Xaxis.
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 nonlinear 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 nonlinear 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.
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