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Journal of the Optical Society of America A

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  • Editor: Franco Gori
  • Vol. 30, Iss. 8 — Aug. 1, 2013
  • pp: 1627–1631
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Formation of circular fringes by interference of two boundary diffraction waves using holography

Raj Kumar and D. P. Chhachhia  »View Author Affiliations


JOSA A, Vol. 30, Issue 8, pp. 1627-1631 (2013)
http://dx.doi.org/10.1364/JOSAA.30.001627


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Abstract

The theory of boundary diffraction waves (BDWs) is gaining importance due to its simplicity and physically appealing nature. The present work reports formation of circular fringes far away from the geometrically illuminated region by interference of two BDWs. One BDW is reconstructed from the hologram while the second is coming directly from the knife-edge. The uniqueness of the fringes is that their position can be controlled on the screen at will and fringes can be produced with bright as well as dark central fringe. These results could play an important role in understanding the nature of diffraction of light.

© 2013 Optical Society of America

1. INTRODUCTION

The problem of diffraction is one of the important problems not only in optics but in all the branches of sciences and engineering dealing with any type of wave propagation. Diffraction of light, which limits the image quality, has been one of the major issues in optical instrumentation. Over the years, researchers have extensively investigated diffraction and have developed various theories to explain it. Among these, the theory of boundary diffraction waves (BDWs) is one of the earliest and most physically appealing. The theory was proposed by Thomas Young in 1802, earlier than the celebrated work of Fresnel, which is based on the assumptions of secondary wavelets proposed by Huygens [1

1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

]. According to the theory of BDWs, a diffraction pattern arises due to superposition of two waves in which one is a directly traveling light known as a geometrical wave that propagates beyond the aperture undisturbed by the presence of diffracting apertures, while the second is known as a BDW and is generated by the interaction of incident light with the boundaries and edges of diffracting apertures. Young’s idea was put on sound mathematical foundation by Maggi and Rubinowich independently and proved that mathematically the two theories (Fresnel’s theory of diffraction and the BDW theory) are equivalent [1

1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

]. An additional advantage of the BDW theory is that in computational work it requires only one line integral extended over the illuminated boundary of the diffracting aperture instead of solving two integrals, used in Fresnel’s theory. The BDW theory is more physically appealing also, as it directly relates the diffraction phenomenon to its true cause, i.e., physical interaction of light with diffracting apertures as compared to assumptions of the existence of Huygens’ secondary wavelets. Many researchers have significantly contributed toward the theoretical and experimental studies on the BDW [1

1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

8

8. C. K. G. Piyadasa, “Detection of a cylindrical boundary diffraction wave emanating from a straight edge by light interaction,” Opt. Commun. 285, 4878–4883 (2012). [CrossRef]

], and references therein. Applications of BDWs have also been reported on various problems of physical importance [9

9. S. Kimura and C. Munakata, “Method for measuring the spot size of a laser beam using a boundary-diffraction wave,” Opt. Lett. 12, 552–554 (1987). [CrossRef]

12

12. P. Piksarv, P. Bowlan, M. Lõhmus, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort Gaussian pulses within the framework of boundary diffraction wave theory,” J. Opt. 14, 015701 (2012). [CrossRef]

].

Recently, we have reported use of BDWs arising from a knife-edge illuminated with a focused laser beam for realization of a new type of interferometer called diffraction Lloyd’s mirror interferometer [13

13. Raj Kumar, “Diffraction Lloyd mirror interferometer,” J. Opt. (India) 39, 90–101 (2010). [CrossRef]

]. In this configuration, a Lloyd’s mirror is used on a knife-edge diffracted field to form straight interference fringes due to superposition of two BDWs. In the present work we report our further investigations on BDWs. Here the knife-edge diffracted field is recorded into a hologram. During reconstruction, illumination of a hologram with both beams used for hologram recording, interference fringes are generated by superposition of two BDWs. One BDW is a knife-edge diffracted field reconstructed from the hologram, and the second BDW is directly coming from the knife-edge. It has been shown, to our knowledge for the first time, that with proper alignment of the hologram during repositioning, a system of circular interference fringes can be generated deep inside the geometrical shadow region. By giving tilt and translation to the hologram, the position of circular fringes can be changed on the observation plane/screen at will. The central fringe of this interference pattern can be bright or dark depending on the path difference between the interfering BDWs.

2. THEORY

To generate an interference pattern by superposition of two BDWs the principle of holography is used. To record the hologram one beam consists of knife-edge diffracted field, designated as object wave O and the second is a collimated beam serving as reference beam R. The object beam is generated by illuminating a knife-edge K with a converging beam of light
A(x,y)=A0r1exp(jkr1),
(1)
where k is wave propagation vector r1 is radius of curvature of the spherical wavefront and j=1. According to the theory of BDWs, the knife-edge diffracted field, which serves as object beam O(x,y) to record a hologram, is given by [1

1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

]
O(x,y)=Ug(x,y)+Ud(x,y).
(2)
Here, Ug propagates according to the laws of geometrical optics and is known as the geometrical wave, while Ud is generated from every point of the illuminated boundary of the knife-edge and is called the BDW. On observation plane (hologram recording plate in our case to record a hologram) Ug is available only in the geometrically illuminated region, while Ud propagates in all the directions after the diffracting aperture but it has strong amplitude only in the Keller cone. Thus, in the geometrically illuminated region both the geometrical wave and BDW are present while in the geometrically shadow region only BDW is present. We aligned the knife-edge at the focus of the laser beam so that BDW have maximum amplitude, making it easy to record and reconstruct the hologram of BDW. The detailed expression of BDW is given in [1

1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

] and here we are also following the same notations.

To record a hologram of the knife-edge diffracted field O(x,y), a collimated beam of light serving as a reference beam R(x,y) is incident at the recording plate at an angle η with respect to the normal of recording plate.
R(x,y)=R0exp(jkxsinη),
(3)

Interference of the object (diffracted) O(x,y) and reference R(x,y) beams results in an intensity distribution
I(x,y)=|O(x,y)+R(x,y)|2.
(4)

This intensity distribution is used to expose the holographic recording material to form an off-axis hologram. After chemical processing the hologram is repositioned and is simultaneously illuminated by both the initial recording beams, i.e., object beam O and reference beam R, resulting in an intensity distribution
U=(R+O)tR|O|2+R|R|2+O|R|2+O*R2+O|O|2+O|R|2+O2R*+R|O|2,
(5)
where t is transmittance of the processed hologram and includes the effect of recording emulsion properties and chemical processing. In expression (5), the first, third, sixth, and eighth terms are of interest for the present discussion. The first and eighth terms represent two reference beams R (one reconstructed from the hologram upon illumination with object beam O and the another a directly transmitted beam) traveling along the same direction and whose amplitude is modified by the term |O|2. These waves superimpose and generate a set of interference fringes. Similarly, the third and sixth terms represent two object beams O (one reconstructed from the hologram upon illumination with reference beam R and another directly transmitted beam) which are propagating along the same direction and whose amplitude is modified by the term |R|2. These beams also superimpose and generate another set of interference fringes. Here we are interested in studying the properties of BDWs and thus our discussion is limited to fringes formed by the waves due to the third and sixth terms. The shape of these fringes depends on the angular separation of the sources of BDWs with respect to the point of observation and the path difference between the interfering beams. Let us select the point of observation P away from the direct beam (DC term) so that no geometrical light is present there and only BDWs are reaching point P. Further, we have selected a point P in the Keller cone where the amplitude of BDWs is maximum to generate high visibility interference fringes. As the diameter of the focus spot is small, the diffracted wavefront can be considered as spherical [14

14. G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” in Extreme Ultraviolet Lithography, G. D. Kubiak and D. R. Kania, eds., Vol. 4 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1996), pp. 108–112.

].

The path difference between two beams (one originating at knife-edge K and the second reconstructed from the hologram and appearing to originate from the virtual image of knife-edge K) at point P, which is on the observation plane kept perpendicular to the line joining the sources, is given by [15

15. J. E. Greivenkamp, “Interference,” in OSA Handbook of Optics (OSA, 1995), Vol. 1.

]

δ=d(ρ2d2r2)
and the corresponding phase difference is
Δφ=2πλ(dρ2d2r2),
(6)
where d=r2r3 is the path difference between the two sources, r2 and r3 are the distances of point P from K and K, respectively, r is an average distance, and ρ=(x2+y2). The first term shows phase difference due to the linear distance between the sources and the second is a quadratic term generating interference fringes symmetric around the line joining the sources, i.e., form circular fringes similar to the Newton’s rings. The total optical path difference between the BDWs dictates whether the central fringe will be bright (phase difference integer multiple of 2π) or dark (phase difference integer multiple of π). This we observed in our experiments that by changing d one can change the central bright to central dark fringe. When d becomes zero both the sources will coincide and there will not be any fringes in the field of view resulting in uniform illumination on the screen.

Fig. 1. Schematic representation of the generation of circular fringe pattern by superposition of two BDWs. (a) Change in position of the fringe pattern due to change in the position of hologram and (b) change in optical path length between two sources of BDWs (knife-edge K and its virtual image K or K) by change in position of the hologram during reconstruction.

Thus, by changing angle α, the path difference h between BDWs from two sources K and K changes, resulting in circular fringes of smaller fringe-width in another direction. Here d is the distance between sources along their original direction of light propagation, i.e., the direction along which light from the focused laser spot diverges. α is the angle between the line joining the original position of point sources K and K and the line joining reconstructed source K after translating the hologram and K. The radius of the mth bright fringe is given by Δϕ=2mπ
2πλ(dρ2d2r2)=2mπ,ρm=2r2(1mλ/d).
(8)

It is clear from Eq. (8) that the radius and hence the fringe width of the circular fringes changes with a change in separation between point sources which in fact changes with angle α.

3. EXPERIMENT

The experimental arrangement is schematically shown in Fig. 2. A He–Ne laser L (Coherent Inc. 35 mW output at 632.8 nm) is divided into two beams using a variable beam splitter. One beam is expanded and a telescopic system of lenses L1 and L2 is used to generate the diffraction limited focus spot. A knife-edge K (good quality razor blade) is aligned vertically in proximity of the focus spot such that a single diffraction fringe covers the whole field of view. At this position the knife-edge diffracts light from the Airy disk and thereby the BDW has maximum amplitude. The geometrical wave and the BDW are shown by solid and dotted lines, respectively. The second beam is expanded and collimated to act as a reference beam. The shear plate interferometric technique was used to ensure the optical quality of the collimated beams and the Ronchi test technique was used for the optical correction of converging beams for astigmatism and coma, which would otherwise be introduced by the off-axis arrangement. A hologram is recorded by interference of the knife-edge diffracted field and the collimated reference wave. After processing, the hologram is replaced in its original position and is illuminated by both the recording beams simultaneously. This results in generation of four beams from the hologram out of which two object beams and two reference beams travel collinearly generating two sets of interference patterns in two different directions. The hologram plate holder has the provision of controlled tilts and translations. With proper positioning of the hologram with respect to the incident beams, circular fringes can be generated in the geometrical shadow region. The knife-edge was also mounted on stage having X–Y translation. Agfa-Gevaert 8E75HD plates have been used for hologram recording and standard Kodak D-19 developer and R-9 bleach bath solutions have been used for its chemical processing. The results presented here have been captured frame-by-frame with a Canon S-50 Power Shot digital camera (1024×768 pixels) in white-balance settings.

Fig. 2. Schematic of the experimental setup for recording the hologram of BDW.

4. RESULTS AND DISCUSSION

A typical interference pattern formed due to the superposition of two BDWs, one coming from the knife-edge and the second reconstructed from the hologram, is shown in Fig. 3. In this figure, the left portion shows a bright circular patch which is due to geometrical light directly reaching to the screen while in the right side, in the Keller cone of the knife-edge diffracted light (horizontal bright stripe of light), there are circular interference fringes. It may be noted that the fringes are generated far away from the patch of geometrical light in the geometrically shadow region. Here the fringe pattern has a central bright fringe which is generated by waves interfering constructively with a path difference of zero or a multiple of a whole wavelength. If the path difference between the beams reaching the point corresponding to central fringe at the observation screen is a half-multiple of a wavelength of light then the central fringe will be a dark fringe due to destructive interference between the waves. We further aligned the hologram such that beams reaching the center interfere destructively and therefore generate another set of circular fringes with central dark fringe. Close-up views of interference patterns with central bright and dark fringes are shown in Figs. 4(a) and 4(b), respectively. We experimentally observed that by changing the position of the hologram by giving translation and tilts the fringe-width of the fringes can be changed and their position on the screen, both horizontally as well as vertically, can also be selected according to choice. This observation is in accordance with Eqs. (7) and (8) and is experimentally demonstrated through a series of interferograms shown in Figs. 5(a)5(d) where change in position of the circular fringe pattern with respect to the geometrical patch of light can be noticed. In Fig. 5(a), the circular fringe pattern is formed beside the geometrical patch and a portion of it is masked by the geometrical light while in Figs. 5(b)5(d) circular fringes are formed away from the geometrical patch with increasing distance from it. In these photographs a small portion of geometrical patch, less than half instead of a full geometrical patch, is included into the results to facilitate the accommodation of more circular fringe patterns away from the geometrical patch in the shadow region maintaining a fixed position of the recording camera. Here it may be noted that the fringe width increases when the fringe pattern moves toward the directly transmitted geometrical light while fringe width decreases when it is moved away from the patch of geometrical light. This is due to the fact that in the direction of geometrical light the path difference between BDWs coming from the knife-edge and the hologram decreases due to the decrease in angle α while moving away from the patch of geometrical light the path difference between these interfering waves increases resulting in a narrow fringe width. These results also support our earlier idea that knife-edge, in particular, and diffracting apertures, in general, act as real sources of BDWs and not as virtual or fictitious sources as is generally treated in theory [17

17. A. Sommerfeld, Optics (Academic, 1954).

,18

18. Y. Z. Umul, “Fictitious diffracted waves in the diffraction theory of Kirchhoff,” J. Opt. Soc. Am. A 27, 109–115 (2010). [CrossRef]

]. This concept of diffracting apertures as real sources of BDWs proposed by us in our earlier papers [13

13. Raj Kumar, “Diffraction Lloyd mirror interferometer,” J. Opt. (India) 39, 90–101 (2010). [CrossRef]

,19

19. Raj Kumar, S. K. Kaura, D. P. Chhachhia, and A. K. Aggarwal, “Direct visualization of Young’s boundary diffraction wave,” Opt. Commun. 276, 54–57 (2007). [CrossRef]

] has also been investigated by Ganci in an alternative manner [20

20. S. Ganci, “On the physical reality of edge sources,” Optik 123, 100–103 (2012). [CrossRef]

].

Fig. 3. Experimental results showing a geometrical patch of light (left) and a circular fringe pattern formed by superposition of BDWs.
Fig. 4. Photographs of circular fringes due to the interference of two BDWs with (a) bright central fringe and (b) dark central fringe.
Fig. 5. Experimental results showing control over the position of circular fringes in the observation plane at will.

5. CONCLUSION

In this work we have reported the experimental investigation that Young’s BDWs can interfere to generate a circular fringe pattern well inside the geometrical shadow region where there is no geometrical light. By changing the position of the hologram by giving tilt and translation, one can easily change the direction of the reconstructed wavefront and hence the radius of fringes as well as their position on the screen at will. The control over the position of the circular interference fringe pattern is possible in our case because the BDW, which has a spherical wavefront, spreads out in all space after the diffracting aperture. This is generally not possible in the case of interference of two geometrical wavefronts whose divergence in space is limited. The central fringe of this interference pattern can be bright or dark depending on the path difference between the interfering waves. These results also support the idea that diffracting apertures serve as real sources of BDWs and not as fictitious sources. The results could be helpful in developing further understanding about the diffraction of light.

ACKNOWLEDGMENTS

The authors are grateful to the Director, CSIO, Chandigarh, for giving permission to publish the work. The authors also thank Mr. Omendra Singh for his kind help in performing the experiments. This work has been financially supported by the Council of Scientific and Industrial Research, New Delhi under the OMEGA project.

REFERENCES

1.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

2.

A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160–162 (1957). [CrossRef]

3.

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—part I,” J. Opt. Soc. Am. 52, 615–622 (1962). [CrossRef]

4.

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—part II,” J. Opt. Soc. Am. 52, 626–636 (1962). [CrossRef]

5.

A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, “Structure of an edge-dislocation wave originating in plane-wave diffraction by a half-plane,” J. Opt. Soc. Am. A 17, 2199–2207 (2000). [CrossRef]

6.

R. Kumar, “Structure of boundary diffraction wave revisited,” Appl. Phys. B 90, 379–382 (2008). [CrossRef]

7.

Y. Z. Umul, “The theory of the boundary diffraction wave,” in Advances in Imaging and Electron Physics (Academic, 2011), Vol. 165, Chap. 6.

8.

C. K. G. Piyadasa, “Detection of a cylindrical boundary diffraction wave emanating from a straight edge by light interaction,” Opt. Commun. 285, 4878–4883 (2012). [CrossRef]

9.

S. Kimura and C. Munakata, “Method for measuring the spot size of a laser beam using a boundary-diffraction wave,” Opt. Lett. 12, 552–554 (1987). [CrossRef]

10.

Raj Kumar, D. P. Chhachhia, and A. K. Aggarwal, “Folding mirror schlieren diffraction interferometer,” Appl. Opt. 45, 6708–6711 (2006). [CrossRef]

11.

P. Janpugdee, P. H. Pathak, and R. J. Burkholder, “On the boundary diffraction wave method for the analytical prediction of the radiation from large planar phased array antennas,” Radio Sci. 42, RS6S26 (2007). [CrossRef]

12.

P. Piksarv, P. Bowlan, M. Lõhmus, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort Gaussian pulses within the framework of boundary diffraction wave theory,” J. Opt. 14, 015701 (2012). [CrossRef]

13.

Raj Kumar, “Diffraction Lloyd mirror interferometer,” J. Opt. (India) 39, 90–101 (2010). [CrossRef]

14.

G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” in Extreme Ultraviolet Lithography, G. D. Kubiak and D. R. Kania, eds., Vol. 4 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1996), pp. 108–112.

15.

J. E. Greivenkamp, “Interference,” in OSA Handbook of Optics (OSA, 1995), Vol. 1.

16.

N. L. Hecht, J. E. Minardi, D. Lewis, and R. L. Fusek, “Quantitative theory for predicting fringe pattern formation in holographic interferometry,” Appl. Opt. 12, 2665–2676 (1973). [CrossRef]

17.

A. Sommerfeld, Optics (Academic, 1954).

18.

Y. Z. Umul, “Fictitious diffracted waves in the diffraction theory of Kirchhoff,” J. Opt. Soc. Am. A 27, 109–115 (2010). [CrossRef]

19.

Raj Kumar, S. K. Kaura, D. P. Chhachhia, and A. K. Aggarwal, “Direct visualization of Young’s boundary diffraction wave,” Opt. Commun. 276, 54–57 (2007). [CrossRef]

20.

S. Ganci, “On the physical reality of edge sources,” Optik 123, 100–103 (2012). [CrossRef]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(090.0090) Holography : Holography
(120.3180) Instrumentation, measurement, and metrology : Interferometry

ToC Category:
Holography

History
Original Manuscript: May 28, 2013
Manuscript Accepted: June 26, 2013
Published: July 22, 2013

Citation
Raj Kumar and D. P. Chhachhia, "Formation of circular fringes by interference of two boundary diffraction waves using holography," J. Opt. Soc. Am. A 30, 1627-1631 (2013)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-8-1627


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References

  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  2. A. Rubinowicz, “Thomas Young and the theory of diffraction,” Nature 180, 160–162 (1957). [CrossRef]
  3. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—part I,” J. Opt. Soc. Am. 52, 615–622 (1962). [CrossRef]
  4. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—part II,” J. Opt. Soc. Am. 52, 626–636 (1962). [CrossRef]
  5. A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, “Structure of an edge-dislocation wave originating in plane-wave diffraction by a half-plane,” J. Opt. Soc. Am. A 17, 2199–2207 (2000). [CrossRef]
  6. R. Kumar, “Structure of boundary diffraction wave revisited,” Appl. Phys. B 90, 379–382 (2008). [CrossRef]
  7. Y. Z. Umul, “The theory of the boundary diffraction wave,” in Advances in Imaging and Electron Physics (Academic, 2011), Vol. 165, Chap. 6.
  8. C. K. G. Piyadasa, “Detection of a cylindrical boundary diffraction wave emanating from a straight edge by light interaction,” Opt. Commun. 285, 4878–4883 (2012). [CrossRef]
  9. S. Kimura and C. Munakata, “Method for measuring the spot size of a laser beam using a boundary-diffraction wave,” Opt. Lett. 12, 552–554 (1987). [CrossRef]
  10. Raj Kumar, D. P. Chhachhia, and A. K. Aggarwal, “Folding mirror schlieren diffraction interferometer,” Appl. Opt. 45, 6708–6711 (2006). [CrossRef]
  11. P. Janpugdee, P. H. Pathak, and R. J. Burkholder, “On the boundary diffraction wave method for the analytical prediction of the radiation from large planar phased array antennas,” Radio Sci. 42, RS6S26 (2007). [CrossRef]
  12. P. Piksarv, P. Bowlan, M. Lõhmus, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort Gaussian pulses within the framework of boundary diffraction wave theory,” J. Opt. 14, 015701 (2012). [CrossRef]
  13. Raj Kumar, “Diffraction Lloyd mirror interferometer,” J. Opt. (India) 39, 90–101 (2010). [CrossRef]
  14. G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” in Extreme Ultraviolet Lithography, G. D. Kubiak and D. R. Kania, eds., Vol. 4 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1996), pp. 108–112.
  15. J. E. Greivenkamp, “Interference,” in OSA Handbook of Optics (OSA, 1995), Vol. 1.
  16. N. L. Hecht, J. E. Minardi, D. Lewis, and R. L. Fusek, “Quantitative theory for predicting fringe pattern formation in holographic interferometry,” Appl. Opt. 12, 2665–2676 (1973). [CrossRef]
  17. A. Sommerfeld, Optics (Academic, 1954).
  18. Y. Z. Umul, “Fictitious diffracted waves in the diffraction theory of Kirchhoff,” J. Opt. Soc. Am. A 27, 109–115 (2010). [CrossRef]
  19. Raj Kumar, S. K. Kaura, D. P. Chhachhia, and A. K. Aggarwal, “Direct visualization of Young’s boundary diffraction wave,” Opt. Commun. 276, 54–57 (2007). [CrossRef]
  20. S. Ganci, “On the physical reality of edge sources,” Optik 123, 100–103 (2012). [CrossRef]

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