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Refraction holodiagrams between two birefringent materials
Optik 121 (2010) 2057–2061
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Refraction holodiagrams between two birefringent materials He ctor Rabal a,, Nelly Cap b a b
Centro de Investigaciones Opticas, CIOp, CCT la Plata CONICET and UID OPTIMO, Facultad de Ingenier´ıa, Universidad Nacional de La Plata, P.O. Box 124, La Plata 1900, Argentina Comisi´ on de Investigaciones Cient´ıficas de La Provincia de Buenos Aires, Argentina
a r t i c l e in fo
abstract
Article history: Received 5 March 2009 Accepted 21 June 2009
Holodiagrams developed by Abramson and known as tools that permit the analysis of the geometrical aspects related to different optical phenomena are here extended to include the more general case when the elements that change the direction of light propagation are refractive and are composed by two birefringent uniaxial materials. The constructed holodiagrams consider the cases for the extraordinary rays for the most general orientations of both optical axes. The curves obtained, representing the loci of equal optical path between a source point in one medium and an observation point in the other, show a complicated shape that differs from that already known for isotropic materials. Some peculiar non-trivial facts are found in this way, such as surfaces of no deviation and generalized Fresnel zone plates to conjugate object points in one medium with image points in the other or in the same in the presence of arbitrarily shaped interfaces. The total reflection phenomenon in the interface between such materials is also considered. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Holodiagram Birefringence Fresnel zone plates
1. Introduction Abramson proposed and showed the practical utility of a diagram, named the holodiagram (HD) [1,2], that depicts the loci of equal optical path between two points assumed to be a light source and an observation point in an isotropic medium for successive increments in the total optical path. Originally designed as a useful tool for the optimal use of available coherence length in holography, current applications of this holodiagram include holography, holographic interferometry, light-in flight registers, interferometry, DSPI, Doppler velocimetry, Bragg diffraction, diffractive elements, etc. In interferometry, the successive increments of the parameter are usually multiples of half the wavelength. Probably the most known use is due to the fact that it gives a straightforward idea of the sensitivity to displacements in holographic interferometry. It is also useful as a teaching element for several optical phenomena [3]. A diagram where the magnitude depicted is the difference in optical path has a similar utility when either the source or the observation point is virtual and is called Young’s fringes holodiagram [4]. Since then, several additional approaches have been proposed to include other situations, such as refractive elements, spatial degree of coherence and the generalization of the spherical wavefront. The applications of the holodiagram concept to an interface between two media lead to the proposal of the refraction
Corresponding author. Tel.: + 54 221 4842957; fax: + 54 221712771.
E-mail address: [email protected] (H. Rabal). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.06.007
holodiagram (RHD) [5]. When the two media are isotropic, they are in the shape of Descartes ovals of the egg and apple types. Snell’s law is fulfilled in all their points [6] when points corresponding to total reflection are excluded. The limiting region of the physically meaningful region of these holodiagrams, when it applies, is an arc of circumference. The concept was also useful for the design of reflecting [7] and refracting [8] profiles with the possibility of the synthesis of different wave fronts or their recognition. Also recently, the possibility of the description of light propagation of the extraordinary ray in birefringent media using these diagrams has been proposed [9]. It was shown that the resulting holodiagram shows the shape of the mirror that conjugates the source with the observation point following the particular reflection law that applies in such media. When a single birefringent medium was considered, the use of the HD provided insight for the uncovering of some features such as the existence of spherical surfaces that conjugate by reflection pairs of points and the loci of regions where total reflection occurs in interfaces between such materials with an isotropic one [10]. In this work we develop a more general case to include refraction of the extraordinary rays in the interface between two, eventually different, birefringent media or the same with different orientation of the optical axis. Some peculiarities can be predicted, as the existence of non-deviation surfaces and the shape of generalized Fresnel zone plates appropriate to conjugate points in one medium into images in the other. This approach takes into account only the geometrical aspects. The results still require the electromagnetic treatment for the verification of the existence of such waves and the calculation of their amplitudes.
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2. Refraction holodiagram in the interface between two birefringent media We will consider in this work the shape of the surface that exactly conjugates by refraction a point in a uniaxial birefringent medium with its image into another one for the extraordinary ray. The source A is located in a birefringent uniaxial medium with refractive indices ne1 and no1 for the extraordinary and ordinary rays, respectively, and the observation point B is in a birefringent uniaxial medium with indices ne2 and no2 for the extraordinary and ordinary rays, respectively. Let us consider a generic point P. We are going to calculate the total optical path for a wave going from A to P traveling along the first medium and then from P to B traveling along the second birefringent material as extraordinary ray in both cases. If we call r1 to the distance from A to P and r2 to the distance from P to B, then the total optical path L is r1 n1 7r2 n2 ¼ L
ð1Þ
To calculate each RHD, cos j1 and cos j2 must be found for the chosen direction of the two axes. Revolution symmetry is not valid any more for most of the cases. To that end, we are going to look for a unit vector uPA in the direction joining P with A and a unit vector u1 in the direction of optical axis 1(see Fig. 1) u 1 ¼ ð0; cos a; sin aÞ ~ a ¼ ð0; 0; aÞ ~ a 0 ¼ ð0; 0; aÞ ~ P ¼ ðr cos y sin f1 ; r cos y cos f1 ; r sin yÞ then PA ¼ jr1 j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 cos2 y þ a2 2ar sin y
and cos j1 ¼ u PA u 1 ¼
r ða r sin yÞ sin a1 cos y cos f1 cos a1 þ PA PA
where the upper sign corresponds to both object and image, real or both virtual, and the lower sign corresponds to one of them real and the other virtual, and
for medium 1 and
ðn1 Þ2 ¼ n2e1 þ ðn2o1 n2e1 Þðu PA u 1 Þ2
for medium 2. With a similar calculation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PB ¼ jr2 j ¼ r2 cos2 y þ a2 þ2ar sin y
ð2Þ
and ðn2 Þ2 ¼ n2e2 þ ðn2o2 n2e2 Þðu PB u 2 Þ2
ð3Þ
ð4Þ
cos j2 ¼
r ða þ r sin yÞ sin a2 cos y cos f2 cos a2 PB PB
ð5Þ
ð6Þ
ð7Þ
with
This calculation can be replaced using the theorem of the cosines as we can also write
ðu PA u 1 Þ2 ¼ cos2 j1
r22 ¼ r12 þ 4a2 4ar1 cos y
and
By introducing Eqs. (8), (6) and (4) in Eq. (1) the rather involved explicit expression of the geometrical loci can be found. But the explicit calculation of the complete geometrical locus is not necessary for the construction of the holodiagram. The loci of the points with the same value for the total optical path L in space represent a surface of the holodiagram and by continuous variation of the value of the parameter L the whole holodiagram can be constructed. 2D cuts of it can be obtained by numerically plotting as images the function
ðu PB u 2 Þ2 ¼ cos2 j2 where j1 and j2 are the angles between PA and the optical axis in medium 1 and PB and the optical axis in medium B, respectively, u PA is a unit vector in the direction from P to A and u PB is a unit vector in the direction from P to B. u 1 and u 2 are unit vectors in the direction of the optical axis in medium 1 and medium 2, respectively. It can be seen that if birefringence is zero in both media the refraction HD for isotropic materials [5] is recovered. The optical axis in medium 1, the coordinate’s origin O and point A determine a plane p1 (see Fig. 1). The plane where the HD will be calculated and p1 form an angle f1. The optical axis 2, the coordinates origin O and point B determine a plane p2. The plane p where the HD will be calculated and p2 form an angle f2
Fig. 1. The definition of the geometric variables for the refraction HD between two birefringent materials.
ð8Þ
I ¼ cosðkLÞ with k being a scale constant. With the obtained results the RHD can be numerically calculated. Fig. 2 shows one example. The source and the observation point are indicated as bright points. As ordinary
Fig. 2. The RHD with fictitious materials of exaggerated birefringence. White lines indicate the boundary of the physically meaningful region a1 = a2 = f1 = f2 = 30.
H. Rabal, N. Cap / Optik 121 (2010) 2057–2061
materials show low birefringences and then some changes are subtle and hard to perceive in the graphs for such materials, we chose (only for this figure and Fig. 5b) two fictitious materials with unnaturally increased birefringences named hyper vaterite (no = 1.55, ne = 3.65) and hyper calcite (no = 3.65, ne = 1.48). They were simulated to make different RHD appearances more evident. In Fig. 2 it can be seen that the appearance of the holodiagram is similar to that obtained in the interface between isotropic materials, that is, it looks like Descartes ovals [5], in this case of the egg type, but slightly deformed. The apple-type holodiagram can also be obtained using the minus sign in Eq. (1). That is to say that the birefringence of the materials introduces only slight differences with respect to the isotropic situation. The greater effects (departures of the isotropic case) can be expected for those regions of the space where cos j1 and cos j2 in absolute value take their higher values. That is to say, it happens when vectors PA and/or PB are in the direction of the optical axis in medium 1 and 2, respectively. The smaller deformation effect happens when the values of the cosines are close to zero. A continuous variation between these extremes is found for other values of the cosines, that is, for other regions of the holodiagram. The deformation for negative materials is in the direction of the holodiagram that would be obtained with a greater refractive index, that is fringe spacing is smaller and the converse is true for positive materials Not all regions of these diagrams are physically meaningful. Internal total reflection restricts their useful regions. Where the diagram is physically adequate, its meaning is similar to other holodiagrams. For example, its shape indicates the shape of an object that fulfils in all its points the Fermat principle. As such, that object conjugates source A with point B by refraction of the extraordinary ray. In the case of an arbitrary shape for the second medium, the locus where its interface is tangent to any curve of the holodiagram is the place where the refraction of the extraordinary ray occurs in the direction of point B. This property could be conceptually used for ray tracing in these types of interfaces although in practice it may not be precise enough .We are going to look now for the total reflection condition to establish the limits of the HD, where they are physically meaningful. From the point of view of the holodiagram concept, the point where the critical angle occurs can be found by looking for the point R, where a ray from A (or B) is tangent to a curve of the RHD. The angle between the ray from B (or A) and the ray normal to the surface is the limit angle in that condition. With this defining concept in mind, simple inspection of the holodiagram also gives indication of when a total reflection angle is to be expected. When the total optical path is allowed to change, the position of the points where total reflection occurs also changes and this establishes a boundary of the limited region where the holodiagram gives meaningful, i.e. physical, information. Simple inspection of the holodiagram also gives indication of when a total reflection angle is to be expected. In the holodiagram in Fig. 2, the corresponding boundary has been indicated with white curves. In Fig. 3, the results are shown for real crystals: calcite (ne = 1.48 and no =1.65) and vaterite (ne = 1.65 and no = 1.55). Parts a and b correspond to the egg shape and c and d to the apple shape. Fig. 3a and c show non-symmetric cuts while 3b and 3d show the symmetric ones.
3. Non-deviation surfaces There is a case that has probably not been explored before that consists of a surface separating two media but shaped in such a way that the rays do not deviate in it. That is, rays coming from a
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Fig. 3. Egg-type HD between calcite and vaterite: (a) f1 = f2 =01, a1 = a2 = 451 (nonsymmetric cut) and (b) f1 = f2 = 901, a1 = a2 = 451 (symmetric cut); apple-type HD between calcite and vaterite: (c) f1 = f2 = 01, a1 = a2 = 451 (non-symmetric case) and (d) f1 = f2 =901, a1 = a2 =451 (symmetric case).
point source in one of the media are imaged in the same place where the source is. In the case of isotropic media, spherical surfaces centered in the source are trivially obtained, but between birefringent media the shape of the surface is not at all trivial and in the most general case the resulting surface does not even show any symmetry. The RHD shows the shape that would have an object, a surface, that conjugates by refraction point A with point B separated by a distance 2a. If it is calculated using the minus sign (that is as a difference of optical path lengths) it corresponds to the experimental situation where even A or B is virtual (but not both) [4]. The singular case is when a tends to zero. Then, A and B coincide and it corresponds to rays that are emitted by A but do not deviate in the limiting surface between the two media. Fig. 4 shows some examples. In Fig. 4a, a1 = a2 = 0 means that both axes are parallel and f1 = f2 =901 means that the observed plane is perpendicular to both axes. Consequently, the non-deviation surfaces’ cut shows circular symmetry as in the isotropic case. That symmetry is broken in Fig. 4b when the same situation is observed for a different orientation of both axes (a1 = a2 =451) with respect to the observation plane. Fig. 4c shows a cut in a more general case where the axes are parallel and the cut is with an inclined plane.
4. Fresnel zone plates Let us assume that an arbitrarily shaped surface limits the separation of the two media. In the intersection of that surface with the HD, if alternate zones of the HD are imagined to be painted black in a binary approximation, light arriving from the remaining transparent zones to point B is then in phase. This means that the intersection of the HD with an arbitrarily shaped surface is a generalized Fresnel zone plate (GFZP) [1,2] or hologram of a point that produces a bright point at B.
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H. Rabal, N. Cap / Optik 121 (2010) 2057–2061
When the surface where the GFZP is calculated coincides with the RHD surface, it results in a single transparent region (equivalent to the first classical Fresnel region in isotropic media). As the refractive surface departs the RHD shape additional, more Fresnel zones are included. If the departure is not very significant the spatial frequency of the GFZP would remain low. So, lowfrequency masks with the shape indicated by the GFZP could be used to correct small departures from the exact surface required to construct a surface that conjugates one point in one birefringent medium into another in other. Plane cuts of the RHD at planes normal to the direction AB are shown in Fig. 5. The generalized Fresnel zone plate concept could also be applied as a particular case to the non-deviation condition and a zone plate can be constructed that conjugates a point with itself in the presence of an interface with another birefringent medium. Fig. 6 shows cuts with planes of the RHD corresponding to nondeviation surfaces (a= 0, minus sign in Eq. 1) that would act as Fresnel zone plates for this case. One of them is a cut perpendicular to both axes so that the zones are circular symmetric and the other is a cut with a plane in another direction and symmetry is broken.
5. Conclusions Fig. 4. Non-deviation surfaces for a vaterite and calcite interface. (a) f1 = f2 = 901 , a1 = a2 =01 (symmetric cut), (b) f1 = f2 = 901,a1 = a2 = 451 (a non-symmetric cut) and (c) f1 =601, f2 = 901, a1 = 301,a2 = 601.
We have calculated the surfaces of equal optical path between two points in different birefringent media. It was done for the extraordinary ray for different values of the total optical path and
Fig. 5. Generalized Fresnel zone plates (2a= 150, plus sign in Eq. (1), z =7 plane) a1 = a2 =901, Df =451 The FZP is symmetric. a1 = 301, a2 = 1001, Df = 501 (Fictitious materials to enhance the asymmetry)).
Fig. 6. Fresnel Zone plate for the non-deviation condition (a = 0, minus sign in Eq. (1), z =7 plane). (a) a1 = a2 = 301, Df = 2001 (a non-symmetric case) and (b) a1 = a2 = 01, Df =451 (a symmetric case).
H. Rabal, N. Cap / Optik 121 (2010) 2057–2061
using several geometrical parameters. The results were shown as images of geometrical loci named refraction holodiagrams. The resulting RHD resembles a slightly deformed variation of the Descartes ovals that are characteristic of the isotropic case. The rather simple concept permits some insight into new features, such as obtaining non-deviation surfaces, loci of total reflection and the shape of Fresnel zone plates to conjugate a point source in one medium in an image in the other or in the same. A Fresnel zone plate that corrects the wavefront to achieve the non-deviation condition can be useful to match the propagation of a wave coming from a certain medium to the propagation in a second medium with an arbitrarily shaped interface. Setting both birefringences to zero, the concept can be adapted also to design a Fresnel zone plate that cancels deviation and spherical aberration in the interface between two isotropic materials. Estimation of the physically meaningful regions also indicates total reflection limits; a result that requires otherwise considerable calculations [11] can be qualitatively estimated by looking for the places where the refracted ray is tangent to the RHD. The shape of the RHD can be used to design a surface that conjugates by refraction a point A in a medium 1 with B in a different medium 2. The idea can be continued with another surface that conjugates B with a third point in the same medium 1 as A or in another medium 3. The compound object would behave approximately as a lens designed to conjugate points in birefringent media.
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Acknowledgements This work was supported by Faculty of Engineering, Universidad Nacional de La Plata, Argentina and by CONICET, Argentina.
References [1] N. Abramson, The Making and Evaluation of Holograms, Academic Press, London, 1981. [2] N. Abramson, Light in Flight or the Holodiagram, the Columbus Egg of Optics, SPIE Optical Engineering Press, Washington, 1996. [3] N. Abramson, Holography as a teaching tool, Opt. Eng. 32 (3) (1993) 508. [4] H. Rabal, The holodiagram with virtual sources, Optik (Stuttgart) 11 (2001) 487–492. [5] G. Baldwin, F. De Zela, H. Rabal, Refraction holodiagrams, Optik (Stuttgart) 1 (12) (2001) 555–560. [6] N. Cap, B. Ruiz, H. Rabal, Refraction holodiagrams and Snell’s law, Optik (Stuttgart) 114 (2001) 89–94. [7] H. Rabal, N. Alamo, C. Criado, Reflecting profiles and generalized holodiagrams, Optik (Stuttgart) 114 (2003) 370–374. [8] N. Alamo, C. Criado, H. Rabal, Refracting profiles and generalized holodiagrams, Opt. Commun. 236 (4–6) (2004) 271–277. [9] H. Rabal, K. Gottschalk, M. Simon, Holodiagrams in birrefringent media, Appl. Opt. 42 (2003) 5825–5830. [10] K. Gottschalk, H. Rabal, M. Simon, refraction holodiagrams with birrefringent elements, fifth Iberoamerican meeting on optics and eight Latin American meeting on optics, lasers, and their applications, in: A. Marcano O., J.L. Paz (Eds.), Proc. SPIE, vol. 5622, SPIE, Bellingham, WA, 2004, pp. 1506–1511. [11] Mar´ıa C. Simon, Karina B. Bastida, Karin V. Gottschalk, Total reflection in a uniaxial crystal–uniaxial crystal interface, Optik 116 (2005) 586–594.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Refraction holodiagrams between two birefringent materials He ctor Rabal a,, Nelly Cap b a b
Centro de Investigaciones Opticas, CIOp, CCT la Plata CONICET and UID OPTIMO, Facultad de Ingenier´ıa, Universidad Nacional de La Plata, P.O. Box 124, La Plata 1900, Argentina Comisi´ on de Investigaciones Cient´ıficas de La Provincia de Buenos Aires, Argentina
a r t i c l e in fo
abstract
Article history: Received 5 March 2009 Accepted 21 June 2009
Holodiagrams developed by Abramson and known as tools that permit the analysis of the geometrical aspects related to different optical phenomena are here extended to include the more general case when the elements that change the direction of light propagation are refractive and are composed by two birefringent uniaxial materials. The constructed holodiagrams consider the cases for the extraordinary rays for the most general orientations of both optical axes. The curves obtained, representing the loci of equal optical path between a source point in one medium and an observation point in the other, show a complicated shape that differs from that already known for isotropic materials. Some peculiar non-trivial facts are found in this way, such as surfaces of no deviation and generalized Fresnel zone plates to conjugate object points in one medium with image points in the other or in the same in the presence of arbitrarily shaped interfaces. The total reflection phenomenon in the interface between such materials is also considered. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Holodiagram Birefringence Fresnel zone plates
1. Introduction Abramson proposed and showed the practical utility of a diagram, named the holodiagram (HD) [1,2], that depicts the loci of equal optical path between two points assumed to be a light source and an observation point in an isotropic medium for successive increments in the total optical path. Originally designed as a useful tool for the optimal use of available coherence length in holography, current applications of this holodiagram include holography, holographic interferometry, light-in flight registers, interferometry, DSPI, Doppler velocimetry, Bragg diffraction, diffractive elements, etc. In interferometry, the successive increments of the parameter are usually multiples of half the wavelength. Probably the most known use is due to the fact that it gives a straightforward idea of the sensitivity to displacements in holographic interferometry. It is also useful as a teaching element for several optical phenomena [3]. A diagram where the magnitude depicted is the difference in optical path has a similar utility when either the source or the observation point is virtual and is called Young’s fringes holodiagram [4]. Since then, several additional approaches have been proposed to include other situations, such as refractive elements, spatial degree of coherence and the generalization of the spherical wavefront. The applications of the holodiagram concept to an interface between two media lead to the proposal of the refraction
Corresponding author. Tel.: + 54 221 4842957; fax: + 54 221712771.
E-mail address: [email protected] (H. Rabal). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.06.007
holodiagram (RHD) [5]. When the two media are isotropic, they are in the shape of Descartes ovals of the egg and apple types. Snell’s law is fulfilled in all their points [6] when points corresponding to total reflection are excluded. The limiting region of the physically meaningful region of these holodiagrams, when it applies, is an arc of circumference. The concept was also useful for the design of reflecting [7] and refracting [8] profiles with the possibility of the synthesis of different wave fronts or their recognition. Also recently, the possibility of the description of light propagation of the extraordinary ray in birefringent media using these diagrams has been proposed [9]. It was shown that the resulting holodiagram shows the shape of the mirror that conjugates the source with the observation point following the particular reflection law that applies in such media. When a single birefringent medium was considered, the use of the HD provided insight for the uncovering of some features such as the existence of spherical surfaces that conjugate by reflection pairs of points and the loci of regions where total reflection occurs in interfaces between such materials with an isotropic one [10]. In this work we develop a more general case to include refraction of the extraordinary rays in the interface between two, eventually different, birefringent media or the same with different orientation of the optical axis. Some peculiarities can be predicted, as the existence of non-deviation surfaces and the shape of generalized Fresnel zone plates appropriate to conjugate points in one medium into images in the other. This approach takes into account only the geometrical aspects. The results still require the electromagnetic treatment for the verification of the existence of such waves and the calculation of their amplitudes.
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H. Rabal, N. Cap / Optik 121 (2010) 2057–2061
2. Refraction holodiagram in the interface between two birefringent media We will consider in this work the shape of the surface that exactly conjugates by refraction a point in a uniaxial birefringent medium with its image into another one for the extraordinary ray. The source A is located in a birefringent uniaxial medium with refractive indices ne1 and no1 for the extraordinary and ordinary rays, respectively, and the observation point B is in a birefringent uniaxial medium with indices ne2 and no2 for the extraordinary and ordinary rays, respectively. Let us consider a generic point P. We are going to calculate the total optical path for a wave going from A to P traveling along the first medium and then from P to B traveling along the second birefringent material as extraordinary ray in both cases. If we call r1 to the distance from A to P and r2 to the distance from P to B, then the total optical path L is r1 n1 7r2 n2 ¼ L
ð1Þ
To calculate each RHD, cos j1 and cos j2 must be found for the chosen direction of the two axes. Revolution symmetry is not valid any more for most of the cases. To that end, we are going to look for a unit vector uPA in the direction joining P with A and a unit vector u1 in the direction of optical axis 1(see Fig. 1) u 1 ¼ ð0; cos a; sin aÞ ~ a ¼ ð0; 0; aÞ ~ a 0 ¼ ð0; 0; aÞ ~ P ¼ ðr cos y sin f1 ; r cos y cos f1 ; r sin yÞ then PA ¼ jr1 j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 cos2 y þ a2 2ar sin y
and cos j1 ¼ u PA u 1 ¼
r ða r sin yÞ sin a1 cos y cos f1 cos a1 þ PA PA
where the upper sign corresponds to both object and image, real or both virtual, and the lower sign corresponds to one of them real and the other virtual, and
for medium 1 and
ðn1 Þ2 ¼ n2e1 þ ðn2o1 n2e1 Þðu PA u 1 Þ2
for medium 2. With a similar calculation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PB ¼ jr2 j ¼ r2 cos2 y þ a2 þ2ar sin y
ð2Þ
and ðn2 Þ2 ¼ n2e2 þ ðn2o2 n2e2 Þðu PB u 2 Þ2
ð3Þ
ð4Þ
cos j2 ¼
r ða þ r sin yÞ sin a2 cos y cos f2 cos a2 PB PB
ð5Þ
ð6Þ
ð7Þ
with
This calculation can be replaced using the theorem of the cosines as we can also write
ðu PA u 1 Þ2 ¼ cos2 j1
r22 ¼ r12 þ 4a2 4ar1 cos y
and
By introducing Eqs. (8), (6) and (4) in Eq. (1) the rather involved explicit expression of the geometrical loci can be found. But the explicit calculation of the complete geometrical locus is not necessary for the construction of the holodiagram. The loci of the points with the same value for the total optical path L in space represent a surface of the holodiagram and by continuous variation of the value of the parameter L the whole holodiagram can be constructed. 2D cuts of it can be obtained by numerically plotting as images the function
ðu PB u 2 Þ2 ¼ cos2 j2 where j1 and j2 are the angles between PA and the optical axis in medium 1 and PB and the optical axis in medium B, respectively, u PA is a unit vector in the direction from P to A and u PB is a unit vector in the direction from P to B. u 1 and u 2 are unit vectors in the direction of the optical axis in medium 1 and medium 2, respectively. It can be seen that if birefringence is zero in both media the refraction HD for isotropic materials [5] is recovered. The optical axis in medium 1, the coordinate’s origin O and point A determine a plane p1 (see Fig. 1). The plane where the HD will be calculated and p1 form an angle f1. The optical axis 2, the coordinates origin O and point B determine a plane p2. The plane p where the HD will be calculated and p2 form an angle f2
Fig. 1. The definition of the geometric variables for the refraction HD between two birefringent materials.
ð8Þ
I ¼ cosðkLÞ with k being a scale constant. With the obtained results the RHD can be numerically calculated. Fig. 2 shows one example. The source and the observation point are indicated as bright points. As ordinary
Fig. 2. The RHD with fictitious materials of exaggerated birefringence. White lines indicate the boundary of the physically meaningful region a1 = a2 = f1 = f2 = 30.
H. Rabal, N. Cap / Optik 121 (2010) 2057–2061
materials show low birefringences and then some changes are subtle and hard to perceive in the graphs for such materials, we chose (only for this figure and Fig. 5b) two fictitious materials with unnaturally increased birefringences named hyper vaterite (no = 1.55, ne = 3.65) and hyper calcite (no = 3.65, ne = 1.48). They were simulated to make different RHD appearances more evident. In Fig. 2 it can be seen that the appearance of the holodiagram is similar to that obtained in the interface between isotropic materials, that is, it looks like Descartes ovals [5], in this case of the egg type, but slightly deformed. The apple-type holodiagram can also be obtained using the minus sign in Eq. (1). That is to say that the birefringence of the materials introduces only slight differences with respect to the isotropic situation. The greater effects (departures of the isotropic case) can be expected for those regions of the space where cos j1 and cos j2 in absolute value take their higher values. That is to say, it happens when vectors PA and/or PB are in the direction of the optical axis in medium 1 and 2, respectively. The smaller deformation effect happens when the values of the cosines are close to zero. A continuous variation between these extremes is found for other values of the cosines, that is, for other regions of the holodiagram. The deformation for negative materials is in the direction of the holodiagram that would be obtained with a greater refractive index, that is fringe spacing is smaller and the converse is true for positive materials Not all regions of these diagrams are physically meaningful. Internal total reflection restricts their useful regions. Where the diagram is physically adequate, its meaning is similar to other holodiagrams. For example, its shape indicates the shape of an object that fulfils in all its points the Fermat principle. As such, that object conjugates source A with point B by refraction of the extraordinary ray. In the case of an arbitrary shape for the second medium, the locus where its interface is tangent to any curve of the holodiagram is the place where the refraction of the extraordinary ray occurs in the direction of point B. This property could be conceptually used for ray tracing in these types of interfaces although in practice it may not be precise enough .We are going to look now for the total reflection condition to establish the limits of the HD, where they are physically meaningful. From the point of view of the holodiagram concept, the point where the critical angle occurs can be found by looking for the point R, where a ray from A (or B) is tangent to a curve of the RHD. The angle between the ray from B (or A) and the ray normal to the surface is the limit angle in that condition. With this defining concept in mind, simple inspection of the holodiagram also gives indication of when a total reflection angle is to be expected. When the total optical path is allowed to change, the position of the points where total reflection occurs also changes and this establishes a boundary of the limited region where the holodiagram gives meaningful, i.e. physical, information. Simple inspection of the holodiagram also gives indication of when a total reflection angle is to be expected. In the holodiagram in Fig. 2, the corresponding boundary has been indicated with white curves. In Fig. 3, the results are shown for real crystals: calcite (ne = 1.48 and no =1.65) and vaterite (ne = 1.65 and no = 1.55). Parts a and b correspond to the egg shape and c and d to the apple shape. Fig. 3a and c show non-symmetric cuts while 3b and 3d show the symmetric ones.
3. Non-deviation surfaces There is a case that has probably not been explored before that consists of a surface separating two media but shaped in such a way that the rays do not deviate in it. That is, rays coming from a
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Fig. 3. Egg-type HD between calcite and vaterite: (a) f1 = f2 =01, a1 = a2 = 451 (nonsymmetric cut) and (b) f1 = f2 = 901, a1 = a2 = 451 (symmetric cut); apple-type HD between calcite and vaterite: (c) f1 = f2 = 01, a1 = a2 = 451 (non-symmetric case) and (d) f1 = f2 =901, a1 = a2 =451 (symmetric case).
point source in one of the media are imaged in the same place where the source is. In the case of isotropic media, spherical surfaces centered in the source are trivially obtained, but between birefringent media the shape of the surface is not at all trivial and in the most general case the resulting surface does not even show any symmetry. The RHD shows the shape that would have an object, a surface, that conjugates by refraction point A with point B separated by a distance 2a. If it is calculated using the minus sign (that is as a difference of optical path lengths) it corresponds to the experimental situation where even A or B is virtual (but not both) [4]. The singular case is when a tends to zero. Then, A and B coincide and it corresponds to rays that are emitted by A but do not deviate in the limiting surface between the two media. Fig. 4 shows some examples. In Fig. 4a, a1 = a2 = 0 means that both axes are parallel and f1 = f2 =901 means that the observed plane is perpendicular to both axes. Consequently, the non-deviation surfaces’ cut shows circular symmetry as in the isotropic case. That symmetry is broken in Fig. 4b when the same situation is observed for a different orientation of both axes (a1 = a2 =451) with respect to the observation plane. Fig. 4c shows a cut in a more general case where the axes are parallel and the cut is with an inclined plane.
4. Fresnel zone plates Let us assume that an arbitrarily shaped surface limits the separation of the two media. In the intersection of that surface with the HD, if alternate zones of the HD are imagined to be painted black in a binary approximation, light arriving from the remaining transparent zones to point B is then in phase. This means that the intersection of the HD with an arbitrarily shaped surface is a generalized Fresnel zone plate (GFZP) [1,2] or hologram of a point that produces a bright point at B.
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When the surface where the GFZP is calculated coincides with the RHD surface, it results in a single transparent region (equivalent to the first classical Fresnel region in isotropic media). As the refractive surface departs the RHD shape additional, more Fresnel zones are included. If the departure is not very significant the spatial frequency of the GFZP would remain low. So, lowfrequency masks with the shape indicated by the GFZP could be used to correct small departures from the exact surface required to construct a surface that conjugates one point in one birefringent medium into another in other. Plane cuts of the RHD at planes normal to the direction AB are shown in Fig. 5. The generalized Fresnel zone plate concept could also be applied as a particular case to the non-deviation condition and a zone plate can be constructed that conjugates a point with itself in the presence of an interface with another birefringent medium. Fig. 6 shows cuts with planes of the RHD corresponding to nondeviation surfaces (a= 0, minus sign in Eq. 1) that would act as Fresnel zone plates for this case. One of them is a cut perpendicular to both axes so that the zones are circular symmetric and the other is a cut with a plane in another direction and symmetry is broken.
5. Conclusions Fig. 4. Non-deviation surfaces for a vaterite and calcite interface. (a) f1 = f2 = 901 , a1 = a2 =01 (symmetric cut), (b) f1 = f2 = 901,a1 = a2 = 451 (a non-symmetric cut) and (c) f1 =601, f2 = 901, a1 = 301,a2 = 601.
We have calculated the surfaces of equal optical path between two points in different birefringent media. It was done for the extraordinary ray for different values of the total optical path and
Fig. 5. Generalized Fresnel zone plates (2a= 150, plus sign in Eq. (1), z =7 plane) a1 = a2 =901, Df =451 The FZP is symmetric. a1 = 301, a2 = 1001, Df = 501 (Fictitious materials to enhance the asymmetry)).
Fig. 6. Fresnel Zone plate for the non-deviation condition (a = 0, minus sign in Eq. (1), z =7 plane). (a) a1 = a2 = 301, Df = 2001 (a non-symmetric case) and (b) a1 = a2 = 01, Df =451 (a symmetric case).
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using several geometrical parameters. The results were shown as images of geometrical loci named refraction holodiagrams. The resulting RHD resembles a slightly deformed variation of the Descartes ovals that are characteristic of the isotropic case. The rather simple concept permits some insight into new features, such as obtaining non-deviation surfaces, loci of total reflection and the shape of Fresnel zone plates to conjugate a point source in one medium in an image in the other or in the same. A Fresnel zone plate that corrects the wavefront to achieve the non-deviation condition can be useful to match the propagation of a wave coming from a certain medium to the propagation in a second medium with an arbitrarily shaped interface. Setting both birefringences to zero, the concept can be adapted also to design a Fresnel zone plate that cancels deviation and spherical aberration in the interface between two isotropic materials. Estimation of the physically meaningful regions also indicates total reflection limits; a result that requires otherwise considerable calculations [11] can be qualitatively estimated by looking for the places where the refracted ray is tangent to the RHD. The shape of the RHD can be used to design a surface that conjugates by refraction a point A in a medium 1 with B in a different medium 2. The idea can be continued with another surface that conjugates B with a third point in the same medium 1 as A or in another medium 3. The compound object would behave approximately as a lens designed to conjugate points in birefringent media.
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Acknowledgements This work was supported by Faculty of Engineering, Universidad Nacional de La Plata, Argentina and by CONICET, Argentina.
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