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Refraction holodiagrams and Snell's law
Optik 114, No. 2 (2003) 89–94 ª 2003 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
N. Cap et al., Refraction holodiagrams and Snell’s law
89
International Journal for Light and Electron Optics
Refraction holodiagrams and Snell’s law N. Cap, B. Ruiz, H. Rabal Centro de Investigaciones O´pticas (CONICET-CIC), P.O. Box 124, 1900 La Plata, Argentina also with OPTIMO, Depto. de Fisicomatema´ticas, Facultad de Ingenierı´a, Universidad Nacional de La Plata, Argentina
Abstract: The refraction holodiagram RHD is analyzed here with respect to the law of refraction. Particularly, we study the surface that exactly conjugates by refraction a virtual point source with a real image or conversely. By using the total optical path as a parameter we build a diagram that consists in a family of Descartes ovals of the apple type that contains the Pascal’s limac¸on as a particular extreme case and the spherical surface with the Weierstrass points as another. These representations permit the straightforward application of Fermat’s principle in the case of arbitrary refracting surfaces and show the shape of generalized Fresnel’s zones in the intersections with any surface. Snell’s law is applied to rays incident on the apple type surfaces to find out the conditions for exact conjugation. Sensitivity to optical path variations is also discussed. The RHD curves family can be represented in a Cartesian way where the ovals appear as equally spaced straight lines. Key words: Holodiagram – Snell’s law – refraction – Descartes ovals
1. Introduction The Holodiagram (HD), proposed by Abramson [1] for the optimal making and evaluation of holograms, permits the optimization of the available coherence length to build holograms of opaque objects of relative big size. It consists in a family of revolution ellipsoids with common foci and the total optical path as a parameter. As usual, each ellipsoid is defined by a distance d equal to the sum of the distances of a generic point on it to the foci. Current applications of the HD include [2] holography (holographic interferometry, light in flight recording, etc.), interferometry (classical, white light, Doppler velocimetry, etc.), the development of diffractive elements, miscellaneous applications in astronomy, relativity, as a teaching tool [3], etc. When it is used in interferometry, it is usual to choose the increments in the d parameter to be multiples of half the used wavelength.
Received 4 November 2002; accepted 4 March 2003. Correspondence to: H. Rabal E-mail: [email protected]
Probably the best-known use of the HD is that it gives an immediate estimation of the sensitivity to displacements in holographic interferometry experiments [4]. Recently, one of us proposed the use of the Young’s fringes graph as a holodiagram when either the source or the observation point is virtual (but not both) [5]. Besides [6], some of us proposed the use of a representation of the surfaces of equal optical path sum between two points O and O0 in different optical media and called it the refraction holodiagram (RHD). It consists in a family of Descartes ovals with the total optical path as parameter and is obtained by increasing in equal steps its value. Two branches of this family are solutions. One of them resembles an egg and the other an apple and we are going to identify them using that resemblance. As optical path is stationary along points on these surfaces it represents the surface that conjugates by refraction O into O0 and conversely via the Fermat’s principle. In this work we are going to treat Snell’s refraction law as applied to the rays coming from O and refracted to O0 for different values of the parameter L representing the total optical path. We are going to briefly summarize first how the equations for the surfaces are found. Then, Snell’s law will be investigated on the apple like surfaces to find that exact conjugation corresponds to the case when either O or O0 is virtual (but not both). Sensitivity to displacements will be treated next.
2. Construction of a holodiagram including refraction and a virtual source In the next paragraphs we are going to follow a similar procedure to that already employed in a previous work [6] to obtain the refraction holodiagram RHD when both the source and the image were real. Let us suppose a spherical wave converging to the virtual point O, and let us call O0 to its image after refraction on a surface. This image may be inside the material as shown in fig. 1. Now we look for the shape of the surface such that the optical path length travelled by the light is the 0030-4026/03/114/02-089 $ 15.00/0
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N. Cap et al., Refraction holodiagrams and Snell’s law
Convergent wave
R
n1
n2
P
r2
r1
θ
O
L
O’
Refracting surface
Fig. 1. Geometry of a converging spherical wave towards a refracting surface.
same for all points P on the surface, considering that it is the interface between two media with refractive indexes n1 6¼ n2.
3. Geometry to find the curve described by a generic point P such that the optical path length O–P–O0 is the same for all points P Let us consider the optical path length (OPL) L between a point in the incident spherical wavefront towards O (with radius R) and a point O0, after passing by a generic point P. We have n1 ðR r1 Þ þ n2 r2 ¼ L ¼ constant ;
ð1Þ
with R 0 and r1 and r2 the distances from point P to O and O0 respectively. To reduce the number of parameters we define the n1 and the relative optical relative refractive index n ¼ n2 path length c ¼ L=n . 2
Introducing these definitions in eq. 1 and taking the limit R ! 0, we can write r2 nr1 ¼ c ;
ðn2 1Þ r2 ðÞ þ 2ðL cos þ ncÞ rðÞ þ c2 L2 ¼ 0 : ð5Þ This equation corresponds to the so-called “Cartesian oval” [7]. As it is a second-degree equation in r, it results in two curves or branches (ovals) for each set of the involved parameters. From eq. 5 we obtain rðÞ ¼ ðL cos þ ncÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðL cos þ ncÞ2 ðn2 1Þ ðc2 L2 Þ n2 1
:
(6)
One curve shows the shape of an apple and the other of an egg. The apple solution is shown in fig. 2.
4. Refraction holodiagram ð3Þ
ðc < 0 is equivalent to exchange O by O0 and n1 by n2 ). We now are going to find the curve fulfilling the condition that the optical path OPO0 is the same for all its points P. We must include the fact that O, P and O0 form a triangle r22 ¼ r12 þ L2 2r1 L cos ;
where L is the distance between O and O0 and q is the angle between the vector OO0 and the vector OP (see fig. 1). Then, we take the pole in the focus O and obtain, in polar coordinates ðr; qÞ,
ð2Þ
where c 0;
Fig. 2. The apple type Descartes oval and the trajectory of a ray pointing towards point O before refraction and towards O0 after being refracted once at the surface.
ð4Þ
To define the refraction holodiagram (RHD) we are going to consider the apple-like solution as limit of two n1 media with the relative refractive index n ¼ . In this n2 case, we define the refraction holodiagram relative to the foci O and O0 as the set of apple like ovals obtained by increasing the relative optical path length in constant steps dc, it is, c, c þ dc, c þ 2dc, etc. It can be depicted as a gray levels image by plotting cos ðr1 nr2 Þ as fig. 3 shows.
N. Cap et al., Refraction holodiagrams and Snell’s law
91
O00 verifies the equation r1 c þ r3 L ¼ ðc2 L2 Þ n=ð1 nÞ ;
where r1 and r3 are the distances from P to O and O00, respectively. The curve described by P with the optical path length OPO00 constant coincides with that of eq. 2, that is, this curve is the same Cartesian oval described by eq. 5. We ask for the location of O, O0 and O00 for two of them to be conjugated pairs. Before answering this we point out five properties that permit to simplifly the analysis. 1. Two of the three foci are always inside the apple and the remaining one is outside [7]. 2. To exchange O by O0 is equivalent to exchange n1 by n2 and the shape of the oval remains unchanged (see eq. 2). 3. If either n > 1, or n < 1 and L < c, then O is inside the curve. It follows from pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nc ðn2 1Þ ðc2 L2 Þ ; ð11Þ cos ¼ L
Fig. 3. The refraction holodiagram.
The RHD can be obtained as a moire´ pattern, as is the case of the classical HD2. In the case c ¼ 0, the surfice is a sphere that conjugates the points O and O0 (one of them virtual), these points are called Weierstrass points.
5. The refraction holodiagram and Snell’s law We assume the points O, O0, inside the apple (see fig. 2), and start by assuming Snell’s law with a relative refractive index n, then we look for the value of n. That is sin b ¼ n sin a :
ð7Þ
If m0 is the slope of the tangent to the curve in P0 ðx0 ; y0 Þ, then tg a ¼
y0 m0 þ x0 ; m0 x0 y0
tg b ¼
ð10Þ
L x0 m0 y0 : ð8Þ y0 þ m0 ðL x0 Þ
Taking into account that P0 ðx0 ; y0 Þ is on the curve (5) and replacing (8) in (7), it results n1 : ð9Þ n¼n¼ n2 Then, Snell’s law is verified for all points in the surface of the apple (it is, rays pointing towards O are steered towards O0 ) provided that the condition that both O and O0 are inside the surface is fulfilled.
6. General case We are going to analyze the cases where either O or O0 may be outside the apple. To this end we take into account a third point O00, named third focus. The point
(see eq. 6) that determines the points of tangency of OP with the curve. 4. If n < 1, L c, the point O is exterior to the curve or it is in the frontier (the prove is as in property 3). 5. If n > 1, L c, the point O0 is exterior to the curve (we exchange n1 by n2 in property 4). Then, to analyze all the possibilities of location of the points O, O0 and O00 it is enough to consider n > 1, it is, the apples where the focus O is inside the curve. From the results described above, with L > 0, and using the fact that the distance between the foci O and O00 is N¼
c2 L 2 ; Lðn2 1Þ
ð12Þ
the following cases result 1. c ¼ nL, it is N ¼ L and O0 O00. 2. c > nL, it is N > L and O0 inside the curve. 3. c < nL < cn, it is 0 < N < L and O00 inside the curve. 4. c < nc < nL, it is N < 0 and O00 inside the curve. 5. c ¼ L, it is N ¼ 0 and O O00. For the two first cases exact conjugation occurs (see § 5). When the optical path length value is c ¼ nL the surface is a particular case called Pascal’s limac¸on. In this case, the two ovals (apple and egg types) obtained are tangent in one point O0 O00. Abramson [2] had already obtained this surface as a holodiagram for a different physical problem. In cases 3 and 4 exact conjugation occurs only for c ¼ nL, that is, for isolated values of the total optical path length. It is then not possible to build a RHD. In case 5 the oval collapses in the point O, and Snell’s law is trivially verified only for n ¼ 1. Extreme cases are c ¼ 0 and L ¼ 0. The case c ¼ 0 corresponds to a spherical surface that conjugates the
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points O and O0 (one of them virtual); these points are called Weierstrass points. The case L ¼ 0 corresponds to O O0 and the solutions obtained are concentric spherical surfaces that leave the converging spherical wavefront shape unchanged.
Due to the way the RHD is constructed, they inherit some nice properties of the classical HD. As these surfaces show loci of constant OPL between O and O0, any surface that is tangent to one of them in a point has there a stationary point, thus there obeying Fermat’s variational principle. It is then very easy to determine the trajectory of rays that are refracted in an arbitrarily shaped surface by constructing a RHD with the appropriate parameters and looking for the points of tangency. Besides, if the intersection of the RHD with a refracting surface is determined and alternate regions are painted black (it is, are blocked out), light from the non-blocked regions arrives to O0 approximately in phase. The result is, by its definition, a generalized Fresnel zone plate that conjugates O and O0 by both refraction and interference.
8. Sensitivity The HD provides information to phase changes produced by small displacements. The RHD can be expected to show the same property provided that the refractive index is incorporated to its definition. Phase changes are due to variations in optical path length due to changes in the position of the points P, O and O0. Each length PO, PO0, is affected by his corresponding refractive index. The analytical calculation of the locus of the points with equal value of the sensitivity vector modulus is quite involved. It is, on the other hand, very simple to numerically plot this locus. In ordinary holographic interferometry, the phase change due to the movement of a point P when it is displaced in D is given by 2p 2p 0 ðs1 þ s2 Þ D ¼ S D; l l
2p 2p ðs2 s1 Þ D ¼ S D; l l
Fig. 4. How the sensitivity vector S is obtained.
The movement of a point along the RHD does not change the OPL and consequently the phase (it is, the RHD is the direction of minimum sensitivity), while moving in the direction perpendicular to the former produces the highest phase changes. The modulus of S and S0 depends on only the angle j between the vectors s1 and s2 . Then, the locus of the points of equal sensitivity modulus is a spherical surface. To numerically obtain this sensitivity plot we follow these steps: – Determine the points O and O0 assigning their coordinates and choose the value of the relative refractive index n. – For every point P in the plot, calculate the angle j between OP and O0 P (see fig. 4). Then, by using the cosine theorem, determine the modulus S0 and S of the vector that is the sum (or difference) of a unitary vector s1 from P to O plus (or minus) another vector s2 from P to O0 with modulus n. – Finally, we plot cos S or cos S0 . Fig. 5 shows sensitivity plots obtained in this way for S and S0 .
9. Cartesian representation in bipolar coordinates There is an alternative way to represent these functions in which the bipolar coordinates r1 and r2 are shown as if they were Cartesian coordinates [7]. In this representation, the condition that r1 and r2 must be the sides of a triangle, it is, where simultaneously
ð13Þ
with l the vacuum wavelength of the light. The vector S0 is called sensitivity vector and is given by the vector addition of two unitary vectors from P, one of them s1 in the direction of the illuminating source, and the other s2 , in the direction of the observation point. In the case where the source is virtual, the vector in the direction of the illumination source is s1. Then the phase change is written Dj ¼
O´
O
7. Properties of the RHD
Dj ¼
P
ð14Þ
r1 þ r2 > L ;
r1 r2 < L ;
and r2 r1 < L ;
ð15Þ
restricts the useful area to the inside of a rectangle (shown as dotted line in fig. 6) surrounded by the segments indicated as boundaries in broken lines at 45 and 135 degrees with the r2 axis. In this region, the Cartesian ovals (as given by eq. 2) appear as straight lines with ordinate at the origin c=n and abscissa c. The egg type ovals are developed in one direction (with negative slope) and the apple type in other (with positive slope). P indicates a point on
N. Cap et al., Refraction holodiagrams and Snell’s law
93
a) b) Fig. 5. a) The locus of points of equal value of the modulus S 0 ; b) The locus of points of equal value of the modulus S.
the egg type oval and Q one in the apple type oval. The intersection of each line with the shortest boundary segment, when it exists, indicates the point where the oval cuts the OO0 axis. If such is the case, both the source and the image are real. The longest branch represents the apple oval and the shortest the egg oval. When c is varied in equal steps, a family of ovals is generated and if the regions between consecutive ovals are alternatively painted black and white, the RHD is obtained. The RHD results to be, in this representation, an equally spaced ruling. Abramson’s HD (that corresponds to the particular case n1 ¼ n2 ) consisting in ellipses would appear in this representation as a ruling forming 135 degrees with the
Fig. 6. Cartesian representation of the Descartes ovals in bipolar coordinates r 1 and r 2 .
abscissas axis and the Young’s fringes HD as a grid at 45 degrees with that axis [4]. Fig. 7 shows this type of representation. While, on one side, the representation is very simple, because ovals are mapped on straight lines, on the other hand all information concerning sensitivity to optical path length variations is lost (the straight lines are here equally spaced).
10. Conclusions We have presented and developed a new concept: the refraction holodiagram with virtual source (or virtual
Fig. 7. The apple RHD in this representation. The arrow shows the case corresponding to Weierstrass points.
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image) that complements a previous one with both source and image being real. It consists of surfaces of the family of the Descartes ovals of the apple type (in the cases indicated in this work,) for which Snell’s law in every point of the surface indicates that rays converging to a point O are steered towards other O0, after refraction. That is to say, point O is exactly conjugated into O0 (one of them virtual). By combining apples and other refracting surfaces of the same family, the rays that converge to O0 after the first refraction can be made to converge to another (or the same) point in a different medium. Even if the surface consists of a single complete apple, after converging to the point O0, the rays are refracted again in the same surface as coming from O (see fig. 2). The apple behaves then as a, so-called, “do nothing” machine and the concepts before developed for such machines can be applied to it [8]. This RHD permits ray tracing in arbitrary shaped refracting surfaces, helps in the understanding of Fermat’s principle and permits the generation of generalized Fresnel zone plates combining refraction with interference. As is the case with the HD, the diagram also shows the sensitivity to optical path changes through the local orientation and spacing of its fringes. It is, the fringes of the RHD develop in the direction where the optical path length does not change. Conversely, the regions where the RHD are more densely packed show both, the sensitivity vector direction (perpendicular to the
fringes) and its local value (inversely proportional to the local spacing of the fringes). Although its application in holographic interferometry is difficult to imagine, the applications of the diagram to elementary ray tracing, determination of optical path error tolerances, computer generated holography combining refraction and diffraction, interferometry and other propagation optical phenomena can be expected. Acknowledgements. This work was partially supported by CONICET and Faculty of Engineering, Universidad Nacional de La Plata, Argentina.
References [1] Abramson N: The holo-diagram, a practical device for the making and evaluation of holograms. Appl. Opt. 8 (1969) 1235–1240 [2] Abramson N: Light in flight or the holodiagram, the Columbus egg of optics. SPIE Optical Engineering Press, Washington 1996 [3] Abramson N: The holodiagram as a teaching tool. Opt. Eng. 32 (1993) 508–513 [4] Abramson N: The making and evaluation of holograms. Academic Press, London 1981 [5] Rabal H: The holodiagram with virtual sources. Optik 112 (2001) 487–492 [6] Baldwin G, De Zela F, Rabal H: Refraction holodiagrams. Optik 112 (2002) 555–560 [7] Lockwood EH: A book of curves. pp. 187ff. Cambridge University Press, Cambridge 1961 [8] Caulfield HJ: The Alvarez-Lohmann lens as a do-nothing machine. Opt. Laser Technol. 34 (2002) 1–5
N. Cap et al., Refraction holodiagrams and Snell’s law
89
International Journal for Light and Electron Optics
Refraction holodiagrams and Snell’s law N. Cap, B. Ruiz, H. Rabal Centro de Investigaciones O´pticas (CONICET-CIC), P.O. Box 124, 1900 La Plata, Argentina also with OPTIMO, Depto. de Fisicomatema´ticas, Facultad de Ingenierı´a, Universidad Nacional de La Plata, Argentina
Abstract: The refraction holodiagram RHD is analyzed here with respect to the law of refraction. Particularly, we study the surface that exactly conjugates by refraction a virtual point source with a real image or conversely. By using the total optical path as a parameter we build a diagram that consists in a family of Descartes ovals of the apple type that contains the Pascal’s limac¸on as a particular extreme case and the spherical surface with the Weierstrass points as another. These representations permit the straightforward application of Fermat’s principle in the case of arbitrary refracting surfaces and show the shape of generalized Fresnel’s zones in the intersections with any surface. Snell’s law is applied to rays incident on the apple type surfaces to find out the conditions for exact conjugation. Sensitivity to optical path variations is also discussed. The RHD curves family can be represented in a Cartesian way where the ovals appear as equally spaced straight lines. Key words: Holodiagram – Snell’s law – refraction – Descartes ovals
1. Introduction The Holodiagram (HD), proposed by Abramson [1] for the optimal making and evaluation of holograms, permits the optimization of the available coherence length to build holograms of opaque objects of relative big size. It consists in a family of revolution ellipsoids with common foci and the total optical path as a parameter. As usual, each ellipsoid is defined by a distance d equal to the sum of the distances of a generic point on it to the foci. Current applications of the HD include [2] holography (holographic interferometry, light in flight recording, etc.), interferometry (classical, white light, Doppler velocimetry, etc.), the development of diffractive elements, miscellaneous applications in astronomy, relativity, as a teaching tool [3], etc. When it is used in interferometry, it is usual to choose the increments in the d parameter to be multiples of half the used wavelength.
Received 4 November 2002; accepted 4 March 2003. Correspondence to: H. Rabal E-mail: [email protected]
Probably the best-known use of the HD is that it gives an immediate estimation of the sensitivity to displacements in holographic interferometry experiments [4]. Recently, one of us proposed the use of the Young’s fringes graph as a holodiagram when either the source or the observation point is virtual (but not both) [5]. Besides [6], some of us proposed the use of a representation of the surfaces of equal optical path sum between two points O and O0 in different optical media and called it the refraction holodiagram (RHD). It consists in a family of Descartes ovals with the total optical path as parameter and is obtained by increasing in equal steps its value. Two branches of this family are solutions. One of them resembles an egg and the other an apple and we are going to identify them using that resemblance. As optical path is stationary along points on these surfaces it represents the surface that conjugates by refraction O into O0 and conversely via the Fermat’s principle. In this work we are going to treat Snell’s refraction law as applied to the rays coming from O and refracted to O0 for different values of the parameter L representing the total optical path. We are going to briefly summarize first how the equations for the surfaces are found. Then, Snell’s law will be investigated on the apple like surfaces to find that exact conjugation corresponds to the case when either O or O0 is virtual (but not both). Sensitivity to displacements will be treated next.
2. Construction of a holodiagram including refraction and a virtual source In the next paragraphs we are going to follow a similar procedure to that already employed in a previous work [6] to obtain the refraction holodiagram RHD when both the source and the image were real. Let us suppose a spherical wave converging to the virtual point O, and let us call O0 to its image after refraction on a surface. This image may be inside the material as shown in fig. 1. Now we look for the shape of the surface such that the optical path length travelled by the light is the 0030-4026/03/114/02-089 $ 15.00/0
90
N. Cap et al., Refraction holodiagrams and Snell’s law
Convergent wave
R
n1
n2
P
r2
r1
θ
O
L
O’
Refracting surface
Fig. 1. Geometry of a converging spherical wave towards a refracting surface.
same for all points P on the surface, considering that it is the interface between two media with refractive indexes n1 6¼ n2.
3. Geometry to find the curve described by a generic point P such that the optical path length O–P–O0 is the same for all points P Let us consider the optical path length (OPL) L between a point in the incident spherical wavefront towards O (with radius R) and a point O0, after passing by a generic point P. We have n1 ðR r1 Þ þ n2 r2 ¼ L ¼ constant ;
ð1Þ
with R 0 and r1 and r2 the distances from point P to O and O0 respectively. To reduce the number of parameters we define the n1 and the relative optical relative refractive index n ¼ n2 path length c ¼ L=n . 2
Introducing these definitions in eq. 1 and taking the limit R ! 0, we can write r2 nr1 ¼ c ;
ðn2 1Þ r2 ðÞ þ 2ðL cos þ ncÞ rðÞ þ c2 L2 ¼ 0 : ð5Þ This equation corresponds to the so-called “Cartesian oval” [7]. As it is a second-degree equation in r, it results in two curves or branches (ovals) for each set of the involved parameters. From eq. 5 we obtain rðÞ ¼ ðL cos þ ncÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðL cos þ ncÞ2 ðn2 1Þ ðc2 L2 Þ n2 1
:
(6)
One curve shows the shape of an apple and the other of an egg. The apple solution is shown in fig. 2.
4. Refraction holodiagram ð3Þ
ðc < 0 is equivalent to exchange O by O0 and n1 by n2 ). We now are going to find the curve fulfilling the condition that the optical path OPO0 is the same for all its points P. We must include the fact that O, P and O0 form a triangle r22 ¼ r12 þ L2 2r1 L cos ;
where L is the distance between O and O0 and q is the angle between the vector OO0 and the vector OP (see fig. 1). Then, we take the pole in the focus O and obtain, in polar coordinates ðr; qÞ,
ð2Þ
where c 0;
Fig. 2. The apple type Descartes oval and the trajectory of a ray pointing towards point O before refraction and towards O0 after being refracted once at the surface.
ð4Þ
To define the refraction holodiagram (RHD) we are going to consider the apple-like solution as limit of two n1 media with the relative refractive index n ¼ . In this n2 case, we define the refraction holodiagram relative to the foci O and O0 as the set of apple like ovals obtained by increasing the relative optical path length in constant steps dc, it is, c, c þ dc, c þ 2dc, etc. It can be depicted as a gray levels image by plotting cos ðr1 nr2 Þ as fig. 3 shows.
N. Cap et al., Refraction holodiagrams and Snell’s law
91
O00 verifies the equation r1 c þ r3 L ¼ ðc2 L2 Þ n=ð1 nÞ ;
where r1 and r3 are the distances from P to O and O00, respectively. The curve described by P with the optical path length OPO00 constant coincides with that of eq. 2, that is, this curve is the same Cartesian oval described by eq. 5. We ask for the location of O, O0 and O00 for two of them to be conjugated pairs. Before answering this we point out five properties that permit to simplifly the analysis. 1. Two of the three foci are always inside the apple and the remaining one is outside [7]. 2. To exchange O by O0 is equivalent to exchange n1 by n2 and the shape of the oval remains unchanged (see eq. 2). 3. If either n > 1, or n < 1 and L < c, then O is inside the curve. It follows from pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nc ðn2 1Þ ðc2 L2 Þ ; ð11Þ cos ¼ L
Fig. 3. The refraction holodiagram.
The RHD can be obtained as a moire´ pattern, as is the case of the classical HD2. In the case c ¼ 0, the surfice is a sphere that conjugates the points O and O0 (one of them virtual), these points are called Weierstrass points.
5. The refraction holodiagram and Snell’s law We assume the points O, O0, inside the apple (see fig. 2), and start by assuming Snell’s law with a relative refractive index n, then we look for the value of n. That is sin b ¼ n sin a :
ð7Þ
If m0 is the slope of the tangent to the curve in P0 ðx0 ; y0 Þ, then tg a ¼
y0 m0 þ x0 ; m0 x0 y0
tg b ¼
ð10Þ
L x0 m0 y0 : ð8Þ y0 þ m0 ðL x0 Þ
Taking into account that P0 ðx0 ; y0 Þ is on the curve (5) and replacing (8) in (7), it results n1 : ð9Þ n¼n¼ n2 Then, Snell’s law is verified for all points in the surface of the apple (it is, rays pointing towards O are steered towards O0 ) provided that the condition that both O and O0 are inside the surface is fulfilled.
6. General case We are going to analyze the cases where either O or O0 may be outside the apple. To this end we take into account a third point O00, named third focus. The point
(see eq. 6) that determines the points of tangency of OP with the curve. 4. If n < 1, L c, the point O is exterior to the curve or it is in the frontier (the prove is as in property 3). 5. If n > 1, L c, the point O0 is exterior to the curve (we exchange n1 by n2 in property 4). Then, to analyze all the possibilities of location of the points O, O0 and O00 it is enough to consider n > 1, it is, the apples where the focus O is inside the curve. From the results described above, with L > 0, and using the fact that the distance between the foci O and O00 is N¼
c2 L 2 ; Lðn2 1Þ
ð12Þ
the following cases result 1. c ¼ nL, it is N ¼ L and O0 O00. 2. c > nL, it is N > L and O0 inside the curve. 3. c < nL < cn, it is 0 < N < L and O00 inside the curve. 4. c < nc < nL, it is N < 0 and O00 inside the curve. 5. c ¼ L, it is N ¼ 0 and O O00. For the two first cases exact conjugation occurs (see § 5). When the optical path length value is c ¼ nL the surface is a particular case called Pascal’s limac¸on. In this case, the two ovals (apple and egg types) obtained are tangent in one point O0 O00. Abramson [2] had already obtained this surface as a holodiagram for a different physical problem. In cases 3 and 4 exact conjugation occurs only for c ¼ nL, that is, for isolated values of the total optical path length. It is then not possible to build a RHD. In case 5 the oval collapses in the point O, and Snell’s law is trivially verified only for n ¼ 1. Extreme cases are c ¼ 0 and L ¼ 0. The case c ¼ 0 corresponds to a spherical surface that conjugates the
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points O and O0 (one of them virtual); these points are called Weierstrass points. The case L ¼ 0 corresponds to O O0 and the solutions obtained are concentric spherical surfaces that leave the converging spherical wavefront shape unchanged.
Due to the way the RHD is constructed, they inherit some nice properties of the classical HD. As these surfaces show loci of constant OPL between O and O0, any surface that is tangent to one of them in a point has there a stationary point, thus there obeying Fermat’s variational principle. It is then very easy to determine the trajectory of rays that are refracted in an arbitrarily shaped surface by constructing a RHD with the appropriate parameters and looking for the points of tangency. Besides, if the intersection of the RHD with a refracting surface is determined and alternate regions are painted black (it is, are blocked out), light from the non-blocked regions arrives to O0 approximately in phase. The result is, by its definition, a generalized Fresnel zone plate that conjugates O and O0 by both refraction and interference.
8. Sensitivity The HD provides information to phase changes produced by small displacements. The RHD can be expected to show the same property provided that the refractive index is incorporated to its definition. Phase changes are due to variations in optical path length due to changes in the position of the points P, O and O0. Each length PO, PO0, is affected by his corresponding refractive index. The analytical calculation of the locus of the points with equal value of the sensitivity vector modulus is quite involved. It is, on the other hand, very simple to numerically plot this locus. In ordinary holographic interferometry, the phase change due to the movement of a point P when it is displaced in D is given by 2p 2p 0 ðs1 þ s2 Þ D ¼ S D; l l
2p 2p ðs2 s1 Þ D ¼ S D; l l
Fig. 4. How the sensitivity vector S is obtained.
The movement of a point along the RHD does not change the OPL and consequently the phase (it is, the RHD is the direction of minimum sensitivity), while moving in the direction perpendicular to the former produces the highest phase changes. The modulus of S and S0 depends on only the angle j between the vectors s1 and s2 . Then, the locus of the points of equal sensitivity modulus is a spherical surface. To numerically obtain this sensitivity plot we follow these steps: – Determine the points O and O0 assigning their coordinates and choose the value of the relative refractive index n. – For every point P in the plot, calculate the angle j between OP and O0 P (see fig. 4). Then, by using the cosine theorem, determine the modulus S0 and S of the vector that is the sum (or difference) of a unitary vector s1 from P to O plus (or minus) another vector s2 from P to O0 with modulus n. – Finally, we plot cos S or cos S0 . Fig. 5 shows sensitivity plots obtained in this way for S and S0 .
9. Cartesian representation in bipolar coordinates There is an alternative way to represent these functions in which the bipolar coordinates r1 and r2 are shown as if they were Cartesian coordinates [7]. In this representation, the condition that r1 and r2 must be the sides of a triangle, it is, where simultaneously
ð13Þ
with l the vacuum wavelength of the light. The vector S0 is called sensitivity vector and is given by the vector addition of two unitary vectors from P, one of them s1 in the direction of the illuminating source, and the other s2 , in the direction of the observation point. In the case where the source is virtual, the vector in the direction of the illumination source is s1. Then the phase change is written Dj ¼
O´
O
7. Properties of the RHD
Dj ¼
P
ð14Þ
r1 þ r2 > L ;
r1 r2 < L ;
and r2 r1 < L ;
ð15Þ
restricts the useful area to the inside of a rectangle (shown as dotted line in fig. 6) surrounded by the segments indicated as boundaries in broken lines at 45 and 135 degrees with the r2 axis. In this region, the Cartesian ovals (as given by eq. 2) appear as straight lines with ordinate at the origin c=n and abscissa c. The egg type ovals are developed in one direction (with negative slope) and the apple type in other (with positive slope). P indicates a point on
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a) b) Fig. 5. a) The locus of points of equal value of the modulus S 0 ; b) The locus of points of equal value of the modulus S.
the egg type oval and Q one in the apple type oval. The intersection of each line with the shortest boundary segment, when it exists, indicates the point where the oval cuts the OO0 axis. If such is the case, both the source and the image are real. The longest branch represents the apple oval and the shortest the egg oval. When c is varied in equal steps, a family of ovals is generated and if the regions between consecutive ovals are alternatively painted black and white, the RHD is obtained. The RHD results to be, in this representation, an equally spaced ruling. Abramson’s HD (that corresponds to the particular case n1 ¼ n2 ) consisting in ellipses would appear in this representation as a ruling forming 135 degrees with the
Fig. 6. Cartesian representation of the Descartes ovals in bipolar coordinates r 1 and r 2 .
abscissas axis and the Young’s fringes HD as a grid at 45 degrees with that axis [4]. Fig. 7 shows this type of representation. While, on one side, the representation is very simple, because ovals are mapped on straight lines, on the other hand all information concerning sensitivity to optical path length variations is lost (the straight lines are here equally spaced).
10. Conclusions We have presented and developed a new concept: the refraction holodiagram with virtual source (or virtual
Fig. 7. The apple RHD in this representation. The arrow shows the case corresponding to Weierstrass points.
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image) that complements a previous one with both source and image being real. It consists of surfaces of the family of the Descartes ovals of the apple type (in the cases indicated in this work,) for which Snell’s law in every point of the surface indicates that rays converging to a point O are steered towards other O0, after refraction. That is to say, point O is exactly conjugated into O0 (one of them virtual). By combining apples and other refracting surfaces of the same family, the rays that converge to O0 after the first refraction can be made to converge to another (or the same) point in a different medium. Even if the surface consists of a single complete apple, after converging to the point O0, the rays are refracted again in the same surface as coming from O (see fig. 2). The apple behaves then as a, so-called, “do nothing” machine and the concepts before developed for such machines can be applied to it [8]. This RHD permits ray tracing in arbitrary shaped refracting surfaces, helps in the understanding of Fermat’s principle and permits the generation of generalized Fresnel zone plates combining refraction with interference. As is the case with the HD, the diagram also shows the sensitivity to optical path changes through the local orientation and spacing of its fringes. It is, the fringes of the RHD develop in the direction where the optical path length does not change. Conversely, the regions where the RHD are more densely packed show both, the sensitivity vector direction (perpendicular to the
fringes) and its local value (inversely proportional to the local spacing of the fringes). Although its application in holographic interferometry is difficult to imagine, the applications of the diagram to elementary ray tracing, determination of optical path error tolerances, computer generated holography combining refraction and diffraction, interferometry and other propagation optical phenomena can be expected. Acknowledgements. This work was partially supported by CONICET and Faculty of Engineering, Universidad Nacional de La Plata, Argentina.
References [1] Abramson N: The holo-diagram, a practical device for the making and evaluation of holograms. Appl. Opt. 8 (1969) 1235–1240 [2] Abramson N: Light in flight or the holodiagram, the Columbus egg of optics. SPIE Optical Engineering Press, Washington 1996 [3] Abramson N: The holodiagram as a teaching tool. Opt. Eng. 32 (1993) 508–513 [4] Abramson N: The making and evaluation of holograms. Academic Press, London 1981 [5] Rabal H: The holodiagram with virtual sources. Optik 112 (2001) 487–492 [6] Baldwin G, De Zela F, Rabal H: Refraction holodiagrams. Optik 112 (2002) 555–560 [7] Lockwood EH: A book of curves. pp. 187ff. Cambridge University Press, Cambridge 1961 [8] Caulfield HJ: The Alvarez-Lohmann lens as a do-nothing machine. Opt. Laser Technol. 34 (2002) 1–5