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Direct visualization of Young’s boundary diffraction wave
Optics Communications 276 (2007) 54–57 www.elsevier.com/locate/optcom
Direct visualization of Young’s boundary diffraction wave Raj Kumar, Sushil K. Kaura, D.P. Chhachhia, A.K. Aggarwal
*
Coherent Optics Division, Central Scientific Instruments Organisation, Sector 30, Chandigarh 160 030, India Received 22 November 2006; received in revised form 13 March 2007; accepted 4 April 2007
Abstract Experimental investigations on Young’s boundary diffraction wave are presented where a wavefront division interferometric scheme is used on diffracted wavefront to generate two-beam interference fringes in the geometrically shadow region. These fringes have good visibility and are observable in the whole space, strongly advocating the physical existence of Young’s boundary diffraction wave as a separate entity. Analysis of these fringes may provide vital information about the structure and nature of boundary diffraction wave e.g. existence in whole space, dependence of amplitude on obliquity factor etc. 2007 Elsevier B.V. All rights reserved. OCIS codes: 050.1940; 120.3180
The study of exact nature of diffraction phenomenon is very crucial as it plays an important role in various branches of science and technology. Two main theories [1] generally used to explain the diffraction phenomena are: Fresnel–Kirchhoff theory and the boundary diffraction wave (BDW) theory. Fresnel–Kirchhoff theory is based on the Huygens assumptions of secondary wavelets while the idea of an intrinsically simple and physically appealing BDW theory, relating diffraction directly to the true cause of its origin, namely, the presence of the boundary of a diffracting body, was first introduced by Young. The BDW theory could simplify the process of calculating light distribution in the complicated images formed by an optical system (normally performed by using Fresnel–Kirchhoff diffraction formula involving the evaluation of very many double integrals) and may prove an efficient tool in the newly emerging fields of modern optics, e.g. coherent computer-driven optical systems. According to BDW theory the edge of a diffracting aperture is the origin of a diffracted wave and the disturbance at any point behind the aperture is due to interference between a direct or geometrical wave
and the BDW. The idea was later put on a sound mathematical basis by Maggi and Rubinowicz, independently, and showed that Kirchhoff’s surface integral, which is mathematical formulation of Huygens–Fresnel diffraction theory, reduces to line integral consisting of two terms. One term represents a wave originating in every point of the boundary of the aperture (called boundary diffraction wave) and the other term represents the geometrical wave. This proved the basic correctness of Young’s idea about BDW. According to BDW theory, the field behind a diffracting aperture is given by [1] U ðP Þ ¼ U g ðP Þ þ U d ðP Þ;
ð1Þ
where expðjkRÞ when P is in the direct beam R ¼ 0 when P is on the geometrical shadow
U g ðP Þ ¼
and U d ðP Þ ¼
1 4p
Z C
ð2Þ
expðjkrÞ expðjksÞ cosð~ n;~ sÞ sinð~ r; ~ dlÞdl; r s ½1 þ cosð~ s;~ rÞ ð3Þ
*
Corresponding author. Tel.: +91 1722637492; fax: +91 172 2657267/ 082. E-mail address: aka1945@rediffmail.com (A.K. Aggarwal). 0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.04.009
where R is the distance from source to the point of observation, P; s is the distance between a typical point on the
R. Kumar et al. / Optics Communications 276 (2007) 54–57
knife-edge K and P; C denotes the boundary of illuminated part of K; dl is an infinitesimal element situated on C; ~ n is unit vector perpendicular to the edge of the aperture and to the incident light, ~ s is unit vector from observation point P towards the integrated r is unit vector of incip point on C; ~ dent light and j = 1. The first factor in the integrand exp(jkr)/r represents the amplitude and phase of the wave incident on the edge, the second factor exp(jks)/s corresponds to the phase of the elementary spherical BDW starting from a point on the aperture edge C at the observation point P and the third factor cosð~n;~sÞ sinð~ r; ~ dlÞdl determines
55
OP L1 L
L2
M S
SF
K S′
M
OP
½1þcosð~ s;~ rÞ
the rather complicated angular dependence of the BDW due to which the diffracted spherical wave has an asymmetric structure. Method of stationary phase states that only those points on the edge-boundary contributes substantially to the integral at which the phase is stationary with respect to translation along the boundary and the contribution of all other portions of the boundary are small of higher order because of interference with the contributions of neighboring line elements. 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 boundary diffraction wave. Since U is continuous function of position, from Eq. (2) it follows that the boundary wave is discontinuous across the edge of geometrical shadow so as to compensate for the discontinuity in the geometrical wave Ug. Here in the geometrical shadow region the diffracted wave Ud is in phase with the geometrical wave Ug while it is out of phase in the directly illuminated region. Over the years, many researchers contributed to the further development of the Young’s BDW theory [2–4] (and references therein) and showed theoretically its existence. On the other hand, though few researchers reported experimental investigations [4–7] on BDW but its physical existence still remains an important issue. This communication reports new investigations where experimental results directly demonstrate the physical existence of the BDW. The Lloyd’s mirror setup is used to generate high contrast interference fringes in the geometrically shadow region by superposition of two portions of the diffracted wavefront. The experimental arrangement is schematically shown in Fig. 1. A He–Ne laser L (Coherent Inc. 35 mW output at 632.8 nm) 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 positioned horizontally in proximity of the focus such that a single diffraction fringe covers the field of view as shown in Fig. 2. At this position knife-edge diffracts light from the Airy disk [7,8] and thereby boundary diffraction wave has maximum amplitude. The geometrical wave and the BDW are shown in Fig. 1 by solid and dotted lines, respectively. The geometrical wave equals to the undisturbed incident wave at all points lying in the illuminated region and its amplitude is zero outside this region called the geometrical shadow region. To show that BDW exists in the whole
S K
Fig. 1. Schematic experimental arrangement; solid line represents the geometrical wave while dotted line represents the boundary diffraction wave; OP is observation plane; K is knife-edge and M is mirror. Close-up around point S explains the interference between two boundary diffraction waves.
Fig. 2. A typical knife-edge diffraction pattern where single diffraction fringe covers the field of view.
space i.e. in geometrical illuminated, defined by focusing lens aperture, as well as in geometrical shadow region a Lloyd’s mirror M (20 mm · 50 mm · 1 mm, SiO2 protected front surface silver coated, reflectivity 94%) is used to superimpose two portions of the diffracted wavefront, generating high contrast interference fringes which run through both the regions. The interference pattern can be described as resulting from superposition of radiations emitted by a point source S (situated on knife-edge) and
56
R. Kumar et al. / Optics Communications 276 (2007) 54–57
its virtual image S 0 , analogous to the well-known Lloyd’s mirror interferometer. The inter-fringe separation b at the observation plane is given by b ¼ kD=d;
ð4Þ
where k is the wavelength of light used, D is the distance between the plane of the two point sources (focal point and its virtual image S and S 0 respectively) and the observation plane OP and d is separation between the point sources. Fig. 3 shows schematically various positions of the observation plane and corresponding Lloyd’s mirror arrangements, in terms of angular and azimuth angles h and / respectively, where interference patterns due to superposition of two portions of the diffracted wavefront are studied. A typical interferogram obtained due to superposition of two portions of the diffracted wavefront recorded at angle / 0 (OP1) is shown in Fig. 4, where high contrast interference fringes are easily observable in the geometrically shadow region on either side of the direct beam. These fringes also run through the geometrically illuminated region, where fringe contrast gets reduced due to presence of direct beam as background noise. The interference fringes are observable in a region of h 180 in the space. The fringes have maximum intensity near the geometrically illuminated region, which gradually goes on decreasing away from it thereby establishing dependence of fringe intensity on the obliquity factor. Similar interference fringes due to superposition of two portions of the backward diffracted wave are obtained at angle / 170 (OP2) as shown in Fig. 5. In this interferogram a circular patch seen in the bottom portion is due to a circular hole made in the screen to allow the incident light on the knife-edge. Thus, results presented in Figs. 4 and 5 conclusively show that there exists a strong BDW in a cone (defined by the focusing lens aperture) with its apex at the knife-edge, extending to a region throughout h = 360. The boundary diffraction wave also exists outside this cone but has very small amplitude resulting in weak interference
Fig. 4. A typical interferogram due to superposition of two portions of the forward traveling knife-edge diffracted wavefront.
M K
y
OP1
x OP2 OP2
OP1
θ
S
K
M
φ
Fig. 5. A typical interferogram due to superposition of two portions of the backward traveling knife-edge diffracted wavefront.
K OP3
K M OP3
Fig. 3. Schematic representations of different positions OP1, OP2, and OP3 of the observation planes (on the left) and corresponding arrangements of the Lloyd’s mirror (on the right). Arrows around point O represent that a point-diffracted wave is spherical in nature.
fringes. The quality of results presented for this case in Fig. 6, / 110 (OP3), has considerably been enhanced digitally. This decrease in intensity of BDW outside the cone is in accordance with the stationary phase method These observations on diffracted wavefront through the
R. Kumar et al. / Optics Communications 276 (2007) 54–57
57
diffraction wave as a physically separate entity. Experimental investigations confirm the existence of boundary diffraction wave in whole space and dependence of its amplitude on the obliquity factor. It may be noted that the BDW theory [1] suggests that the obliquity factor should follow a cosine law, which seems to be little different from the results. Here it may be noted that the amplitude of the BDW never becomes zero as expected. The experimental results presented are in accordance with the Sommerfeld’s [9] observation – ‘‘the diffraction angle has the period 4p’’. Further analysis on this issue is under investigations and a fuller description will be published in due course. Further analysis of these interferograms, due to superposition of two boundary diffraction waves, may be useful to study the exact nature of Young’s BDW to solve the intricacies about the diffraction phenomenon. Acknowledgements
Fig. 6. A typical interferogram due to superposition of two portions of the knife-edge diffracted wavefront at an angle / 110.
described setup make it profusely clear that the diffracted wave exists in the whole space with its origin at knife-edge, and the BDW is physically a separate entity to that of the geometrical wave, as suggested by Young. Boundary diffraction wave (BDW) behind the diffracting aperture may be considered as a physically separate entity as it is well known that in the geometrical shadow region, the geometrical wave does not exist and fringes reported in this region in our experiment are because of the presence of BDW in that region. This is also in agreement to the earlier reported work on schlieren diffraction interferometry [8,9] where behind the diffracting aperture, geometrical test beam interferes with the boundary diffraction wave serving as reference beam to give interferometric test results. In summary, interference between two portions of the diffracted wavefront using Lloyd’s mirror generates high contrast fringes in the geometrically shadow region which directly confirms the existence of the Young’s boundary
The authors are grateful to Director, CSIO, Chandigarh for his constant encouragement, and support. They thank Mr. Amit K. Sharma at CSIO, Chandigarh and Prof. D. Mohan at Guru Jambheshwar University of Science and Technology, Hisar for helpful discussions. Financial support (Emeritus Scientist Scheme) by Council of Scientific and Industrial Research, New Delhi is greatly acknowledged. References [1] M. Born, E. Wolf, Principles of Optics, fourth ed., Pergamon, Oxford, 1970, p. 370. [2] A. Rubinowicz, The Miyamoto–Wolf diffraction wave, in: E. Wolf (Ed.), Progress in Optics, vol. 4, North-Holland, Amsterdam, 1965, p. 199. [3] G. Otis, J.L. Lachambre, J.W.Y. Lit, P. Lavigne, J. Opt. Soc. Am. 67 (1977) 551. [4] A.I. Khizhnyak, S.P. Anokhov, R.A. Lymarenko, M.S. Soskin, M.V. Vasnetsov, J. Opt. Soc. Am. A 17 (2002) 2199. [5] S. Ganci, Am. J. Phys. 57 (1989) 370. [6] Z.L. Horvath, J. Klebniczki, G. Kurdi, A.P. Kovacs, Opt. Commun. 239 (2004) 243. [7] R. Kumar, S.K. Kaura, A.K. Sharma, D.P. Chhachhia, A.K. Aggarwal, Opt. Laser Technol. 39 (2007) 256. [8] R. Kumar, D.P. Chhachhia, A.K. Aggarwal, Appl. Opt. 45 (2006) 6708. [9] A. Sommerfeld, Optics, Acadamic Press, 1954, p. 264.
Direct visualization of Young’s boundary diffraction wave Raj Kumar, Sushil K. Kaura, D.P. Chhachhia, A.K. Aggarwal
*
Coherent Optics Division, Central Scientific Instruments Organisation, Sector 30, Chandigarh 160 030, India Received 22 November 2006; received in revised form 13 March 2007; accepted 4 April 2007
Abstract Experimental investigations on Young’s boundary diffraction wave are presented where a wavefront division interferometric scheme is used on diffracted wavefront to generate two-beam interference fringes in the geometrically shadow region. These fringes have good visibility and are observable in the whole space, strongly advocating the physical existence of Young’s boundary diffraction wave as a separate entity. Analysis of these fringes may provide vital information about the structure and nature of boundary diffraction wave e.g. existence in whole space, dependence of amplitude on obliquity factor etc. 2007 Elsevier B.V. All rights reserved. OCIS codes: 050.1940; 120.3180
The study of exact nature of diffraction phenomenon is very crucial as it plays an important role in various branches of science and technology. Two main theories [1] generally used to explain the diffraction phenomena are: Fresnel–Kirchhoff theory and the boundary diffraction wave (BDW) theory. Fresnel–Kirchhoff theory is based on the Huygens assumptions of secondary wavelets while the idea of an intrinsically simple and physically appealing BDW theory, relating diffraction directly to the true cause of its origin, namely, the presence of the boundary of a diffracting body, was first introduced by Young. The BDW theory could simplify the process of calculating light distribution in the complicated images formed by an optical system (normally performed by using Fresnel–Kirchhoff diffraction formula involving the evaluation of very many double integrals) and may prove an efficient tool in the newly emerging fields of modern optics, e.g. coherent computer-driven optical systems. According to BDW theory the edge of a diffracting aperture is the origin of a diffracted wave and the disturbance at any point behind the aperture is due to interference between a direct or geometrical wave
and the BDW. The idea was later put on a sound mathematical basis by Maggi and Rubinowicz, independently, and showed that Kirchhoff’s surface integral, which is mathematical formulation of Huygens–Fresnel diffraction theory, reduces to line integral consisting of two terms. One term represents a wave originating in every point of the boundary of the aperture (called boundary diffraction wave) and the other term represents the geometrical wave. This proved the basic correctness of Young’s idea about BDW. According to BDW theory, the field behind a diffracting aperture is given by [1] U ðP Þ ¼ U g ðP Þ þ U d ðP Þ;
ð1Þ
where expðjkRÞ when P is in the direct beam R ¼ 0 when P is on the geometrical shadow
U g ðP Þ ¼
and U d ðP Þ ¼
1 4p
Z C
ð2Þ
expðjkrÞ expðjksÞ cosð~ n;~ sÞ sinð~ r; ~ dlÞdl; r s ½1 þ cosð~ s;~ rÞ ð3Þ
*
Corresponding author. Tel.: +91 1722637492; fax: +91 172 2657267/ 082. E-mail address: aka1945@rediffmail.com (A.K. Aggarwal). 0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.04.009
where R is the distance from source to the point of observation, P; s is the distance between a typical point on the
R. Kumar et al. / Optics Communications 276 (2007) 54–57
knife-edge K and P; C denotes the boundary of illuminated part of K; dl is an infinitesimal element situated on C; ~ n is unit vector perpendicular to the edge of the aperture and to the incident light, ~ s is unit vector from observation point P towards the integrated r is unit vector of incip point on C; ~ dent light and j = 1. The first factor in the integrand exp(jkr)/r represents the amplitude and phase of the wave incident on the edge, the second factor exp(jks)/s corresponds to the phase of the elementary spherical BDW starting from a point on the aperture edge C at the observation point P and the third factor cosð~n;~sÞ sinð~ r; ~ dlÞdl determines
55
OP L1 L
L2
M S
SF
K S′
M
OP
½1þcosð~ s;~ rÞ
the rather complicated angular dependence of the BDW due to which the diffracted spherical wave has an asymmetric structure. Method of stationary phase states that only those points on the edge-boundary contributes substantially to the integral at which the phase is stationary with respect to translation along the boundary and the contribution of all other portions of the boundary are small of higher order because of interference with the contributions of neighboring line elements. 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 boundary diffraction wave. Since U is continuous function of position, from Eq. (2) it follows that the boundary wave is discontinuous across the edge of geometrical shadow so as to compensate for the discontinuity in the geometrical wave Ug. Here in the geometrical shadow region the diffracted wave Ud is in phase with the geometrical wave Ug while it is out of phase in the directly illuminated region. Over the years, many researchers contributed to the further development of the Young’s BDW theory [2–4] (and references therein) and showed theoretically its existence. On the other hand, though few researchers reported experimental investigations [4–7] on BDW but its physical existence still remains an important issue. This communication reports new investigations where experimental results directly demonstrate the physical existence of the BDW. The Lloyd’s mirror setup is used to generate high contrast interference fringes in the geometrically shadow region by superposition of two portions of the diffracted wavefront. The experimental arrangement is schematically shown in Fig. 1. A He–Ne laser L (Coherent Inc. 35 mW output at 632.8 nm) 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 positioned horizontally in proximity of the focus such that a single diffraction fringe covers the field of view as shown in Fig. 2. At this position knife-edge diffracts light from the Airy disk [7,8] and thereby boundary diffraction wave has maximum amplitude. The geometrical wave and the BDW are shown in Fig. 1 by solid and dotted lines, respectively. The geometrical wave equals to the undisturbed incident wave at all points lying in the illuminated region and its amplitude is zero outside this region called the geometrical shadow region. To show that BDW exists in the whole
S K
Fig. 1. Schematic experimental arrangement; solid line represents the geometrical wave while dotted line represents the boundary diffraction wave; OP is observation plane; K is knife-edge and M is mirror. Close-up around point S explains the interference between two boundary diffraction waves.
Fig. 2. A typical knife-edge diffraction pattern where single diffraction fringe covers the field of view.
space i.e. in geometrical illuminated, defined by focusing lens aperture, as well as in geometrical shadow region a Lloyd’s mirror M (20 mm · 50 mm · 1 mm, SiO2 protected front surface silver coated, reflectivity 94%) is used to superimpose two portions of the diffracted wavefront, generating high contrast interference fringes which run through both the regions. The interference pattern can be described as resulting from superposition of radiations emitted by a point source S (situated on knife-edge) and
56
R. Kumar et al. / Optics Communications 276 (2007) 54–57
its virtual image S 0 , analogous to the well-known Lloyd’s mirror interferometer. The inter-fringe separation b at the observation plane is given by b ¼ kD=d;
ð4Þ
where k is the wavelength of light used, D is the distance between the plane of the two point sources (focal point and its virtual image S and S 0 respectively) and the observation plane OP and d is separation between the point sources. Fig. 3 shows schematically various positions of the observation plane and corresponding Lloyd’s mirror arrangements, in terms of angular and azimuth angles h and / respectively, where interference patterns due to superposition of two portions of the diffracted wavefront are studied. A typical interferogram obtained due to superposition of two portions of the diffracted wavefront recorded at angle / 0 (OP1) is shown in Fig. 4, where high contrast interference fringes are easily observable in the geometrically shadow region on either side of the direct beam. These fringes also run through the geometrically illuminated region, where fringe contrast gets reduced due to presence of direct beam as background noise. The interference fringes are observable in a region of h 180 in the space. The fringes have maximum intensity near the geometrically illuminated region, which gradually goes on decreasing away from it thereby establishing dependence of fringe intensity on the obliquity factor. Similar interference fringes due to superposition of two portions of the backward diffracted wave are obtained at angle / 170 (OP2) as shown in Fig. 5. In this interferogram a circular patch seen in the bottom portion is due to a circular hole made in the screen to allow the incident light on the knife-edge. Thus, results presented in Figs. 4 and 5 conclusively show that there exists a strong BDW in a cone (defined by the focusing lens aperture) with its apex at the knife-edge, extending to a region throughout h = 360. The boundary diffraction wave also exists outside this cone but has very small amplitude resulting in weak interference
Fig. 4. A typical interferogram due to superposition of two portions of the forward traveling knife-edge diffracted wavefront.
M K
y
OP1
x OP2 OP2
OP1
θ
S
K
M
φ
Fig. 5. A typical interferogram due to superposition of two portions of the backward traveling knife-edge diffracted wavefront.
K OP3
K M OP3
Fig. 3. Schematic representations of different positions OP1, OP2, and OP3 of the observation planes (on the left) and corresponding arrangements of the Lloyd’s mirror (on the right). Arrows around point O represent that a point-diffracted wave is spherical in nature.
fringes. The quality of results presented for this case in Fig. 6, / 110 (OP3), has considerably been enhanced digitally. This decrease in intensity of BDW outside the cone is in accordance with the stationary phase method These observations on diffracted wavefront through the
R. Kumar et al. / Optics Communications 276 (2007) 54–57
57
diffraction wave as a physically separate entity. Experimental investigations confirm the existence of boundary diffraction wave in whole space and dependence of its amplitude on the obliquity factor. It may be noted that the BDW theory [1] suggests that the obliquity factor should follow a cosine law, which seems to be little different from the results. Here it may be noted that the amplitude of the BDW never becomes zero as expected. The experimental results presented are in accordance with the Sommerfeld’s [9] observation – ‘‘the diffraction angle has the period 4p’’. Further analysis on this issue is under investigations and a fuller description will be published in due course. Further analysis of these interferograms, due to superposition of two boundary diffraction waves, may be useful to study the exact nature of Young’s BDW to solve the intricacies about the diffraction phenomenon. Acknowledgements
Fig. 6. A typical interferogram due to superposition of two portions of the knife-edge diffracted wavefront at an angle / 110.
described setup make it profusely clear that the diffracted wave exists in the whole space with its origin at knife-edge, and the BDW is physically a separate entity to that of the geometrical wave, as suggested by Young. Boundary diffraction wave (BDW) behind the diffracting aperture may be considered as a physically separate entity as it is well known that in the geometrical shadow region, the geometrical wave does not exist and fringes reported in this region in our experiment are because of the presence of BDW in that region. This is also in agreement to the earlier reported work on schlieren diffraction interferometry [8,9] where behind the diffracting aperture, geometrical test beam interferes with the boundary diffraction wave serving as reference beam to give interferometric test results. In summary, interference between two portions of the diffracted wavefront using Lloyd’s mirror generates high contrast fringes in the geometrically shadow region which directly confirms the existence of the Young’s boundary
The authors are grateful to Director, CSIO, Chandigarh for his constant encouragement, and support. They thank Mr. Amit K. Sharma at CSIO, Chandigarh and Prof. D. Mohan at Guru Jambheshwar University of Science and Technology, Hisar for helpful discussions. Financial support (Emeritus Scientist Scheme) by Council of Scientific and Industrial Research, New Delhi is greatly acknowledged. References [1] M. Born, E. Wolf, Principles of Optics, fourth ed., Pergamon, Oxford, 1970, p. 370. [2] A. Rubinowicz, The Miyamoto–Wolf diffraction wave, in: E. Wolf (Ed.), Progress in Optics, vol. 4, North-Holland, Amsterdam, 1965, p. 199. [3] G. Otis, J.L. Lachambre, J.W.Y. Lit, P. Lavigne, J. Opt. Soc. Am. 67 (1977) 551. [4] A.I. Khizhnyak, S.P. Anokhov, R.A. Lymarenko, M.S. Soskin, M.V. Vasnetsov, J. Opt. Soc. Am. A 17 (2002) 2199. [5] S. Ganci, Am. J. Phys. 57 (1989) 370. [6] Z.L. Horvath, J. Klebniczki, G. Kurdi, A.P. Kovacs, Opt. Commun. 239 (2004) 243. [7] R. Kumar, S.K. Kaura, A.K. Sharma, D.P. Chhachhia, A.K. Aggarwal, Opt. Laser Technol. 39 (2007) 256. [8] R. Kumar, D.P. Chhachhia, A.K. Aggarwal, Appl. Opt. 45 (2006) 6708. [9] A. Sommerfeld, Optics, Acadamic Press, 1954, p. 264.