- Email: [email protected]
Lenses that provide the transformation between two given wavefronts
Optics Communications 380 (2016) 233–238
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Lenses that provide the transformation between two given wavefronts C. Criado a,n, N. Alamo b a b
Departamento de Fisica Aplicada I, Universidad de Malaga, 29071 Malaga, Spain Departamento de Algebra, Geometria y Topologia, Universidad de Malaga, 29071 Malaga, Spain
art ic l e i nf o
a b s t r a c t
Article history: Received 15 February 2016 Received in revised form 12 May 2016 Accepted 3 June 2016 Available online 15 June 2016
We give an original method to design four types of lenses solving the following problems: focusing a given wavefront in a given point, and performing the transformation between two arbitrary incoming and outgoing wavefronts. The method to design the lenses profiles is based on the optical properties of the envelopes of certain families of Cartesian ovals of revolution. & 2016 Elsevier B.V. All rights reserved.
Keywords: Geometric optical design Refracting profile Focusing profile Descartes oval Wavefront
1. Introduction Suppose that we have given a lens L with refractive index n2 shaped by two profiles R1 and R2, and a point F in the exterior region with refractive index n1. A classical optical problem is to determine the resulting wavefront from the data of the source point F and the refracting profiles R1 and R2. It can be solved by using Snell's law or, equivalently, Fermat's principle. Let us remark that the lenses we are considering here are a kind of general aspheric lenses, without any axial symmetry, so we do not have optical axis. We consider now the inverse problem, that is, given a wavefront W and a source point F, we want to design the profiles of the lens L such that focuses the wavefront into the point F. Of course, the same lens produces W from the source point F, after refracting in the lens profiles. In this paper, we provide a method to design such lenses based on the optical properties of the envelopes of certain families of Cartesian ovals of revolution (see [1]). A similar method was used in [2] to design mirrors with specific characteristics as envelopes of families of conics of revolution, considering the optical properties of the conics. Furthermore, we solve here the general problem of designing the profiles for lenses that transform a given wavefront into another with arbitrary shape. The only condition to carry out these constructions is that the lens profiles should not contain points of the caustic of the wavefront. Similar types of general problems have been studied in other n
Corresponding author. E-mail address: [email protected] (C. Criado).
http://dx.doi.org/10.1016/j.optcom.2016.06.010 0030-4018/& 2016 Elsevier B.V. All rights reserved.
articles. For example, the well-known supporting quadric method ([3] and more recently [4,5]) is also based on a similar representation of the optical surfaces that uses quadrics to design reflecting and refracting surfaces suitable to reshaping the energy radiation intensity of a given source into a prescribed output irradiance distribution. A technique for computation of freeform optical elements with two refractive surfaces generating the required intensity distribution is proposed in [6], where is studied the particular case in which the freeform elements transform a spherical wavefront in a plane one. The final two smooth refractive surfaces are obtained after using splines on piecewise smooth surfaces composed by the adequate surfaces of the freeform elements. Another geometric method for designing freeform for illuminating purpose is proposed in [7], where the freeform is represented by a discrete set of Cartesian ovals segments with appropriated parameters. Other optics systems designing optical surfaces for generating prescribed irradiance distribution can be found in [8–11]. The paper is organized as follows. In Section 2 we revise the concepts and the optical properties about envelopes of Cartesian ovals of revolution given in [1]. In Section 3 we use these results to design four types of lenses that provide the desired transformations. The section is divided into two subsections according to the type of profiles that we use to design the lenses.
2. Refracting profiles For an arbitrary wavefront W and any source point F, it is possible to construct a family {Ra} of refracting profiles with the
234
C. Criado, N. Alamo / Optics Communications 380 (2016) 233–238
Fig. 1. Planar section of the wavefront W and the profiles, containing the optical path of a ray orthogonal to W at a point x, for the case of F exterior to the ovals. In (a) the profiles R1a and R2a are the envelopes of the interior Cartesian ovals Oxa, i , whereas in (b) R3a and R4a are the envelopes of the exterior Cartesian ovals Oxa, e . These ovals have foci F and x ∈ W . Only R1a and R4a have physical sense, focusing the wavefront W into a real focus F and to a virtual focus F, respectively. In (a) the third focus x′ and the caustic * of W that has singularities of the profile R4a are also shown. The parameter a is such that 2a shows the stationary optical length between F and x.
property of producing normal rays to W after refracting the rays that emerge from F at each profile Ra. The parameter a of the family is any non-negative real number. If light propagation direction is reversed, then the same family of profiles has the property of focusing the wavefront W in the point F. From now on we suppose that the refracting profile separates two media with refractive indices n1 and n2, F being in the region of n1, and W in the region of n2. We give now an outline of the method developed in [1] to design a suitable profile Ra for the wavefront W and the point F. Consider the Cartesian oval of revolution Oxa with foci F and x, where x ∈ W and the parameter a is such that 2a is the stationary optical path length between F and x (see Fig. 1(a) and (b)). Then Ra is taken as the envelope of the family {Oxa}x ∈ W . Indeed, let us recall that a Cartesian oval is the locus of the points from which the distances r1 and r2 to two fixed points F1 and F2, called foci, verify the bipolar equation
a1r1 + a2r2 = k,
(1)
where a1, a2 and k are constants. A Cartesian oval has a third focus F3, and the oval can be defined by any two of the foci. In particular when |a1/a2| goes to 1, the third focus goes to infinity and then we get the conics (the ellipse for k > 0 and a1 = a2 > 0, and the hyperbola for a1 = − a2 > 0). A complete Cartesian oval is the set of curves associated to the bipolar equation
a1r1 ± a2r2 = ± k
(k > 0).
(2)
Only two of the four equations obtained from these double signs are not empty. They are closed curves, one interior to the other (see Fig. 1(a) and (b) with F1 = F and F2 = x ). These two curves intersect only when two foci coincide. In this case we get the socalled Limaçon of Pascal. When one of the foci is exterior to the ovals, as in Fig. 1(a) and (b), we have the following condition of compatibility:
2a/a2 < |x| < 2a/a1
(3)
that must be satisfied in order to the existence of the ovals. For the case of both foci in the interior of the ovals (see Fig. 2), 2a/a2 and 2a/a1 are both greater than |x|, with a2 > a1. See [12] for a detailed study of these ovals. In the context of the geometrical optics, the coefficients a1 and a2 correspond to the refractive indices n1 and n2, so that Eq. (1) gives the stationary optical path length between the foci F1 and F2 (see [13,14]). In what follows we will use complete Cartesian ovals of revolution, that is, the surfaces obtained when the above ovals are rotated around their focal axes. Let Oxa denote the complete
Fig. 2. Case of F interior to the ovals. R1a and R2a are envelopes of the interior ovals, whereas R3a and R4a are envelopes of the exterior ovals. Only R3a and R4a have physical sense, focusing the wavefront W to the virtual focus F. To illustrate the ray tracing, we have drawn three contiguous rays.
Cartesian oval of revolution with foci F and x ∈ W , and let Oxa, i (respectively Oxa, e ) denote the interior (respectively exterior) oval (see Figs. 1 and 2). If we assume that F is the origin, then for any points x ∈ W and y ∈ Oxa , the vectors Fx and Fy with origin F and extreme x and y respectively, denoted again by x and y, verify:
n1|y| ± n2|y − x| = ± 2a. Specifically, the interior oval
(4) Oxa, i
verifies n1|y| + n2|y − x| = 2a , and
the exterior oval Oxa, e verifies −n1|y| + n2|y − x| = 2a . Let Ra be the envelope of the family {Oxa}x ∈ W (see [15,16]). To give an explicit parametrization of the profiles Ra, we suppose that W is oriented and we take u(x) to be the unitary normal vector to W at x in the direction of the half space containing F. Then Ra can be decomposed in four sheets Ra = R1a ∪ R2a ∪ R3a ∪ R4a , where Ria is parametrized by
y = x + λ i(x)u(x),
(5)
where, from the condition that y ∈
λi = and
Oxa ,
n12(x·u) + ( − 1)i 2an2 − ( − 1)i Δia n22 − n12
,
i = 1, 2,
(6)
C. Criado, N. Alamo / Optics Communications 380 (2016) 233–238
λi =
n12(x·u) + ( − 1)i 2an2 + ( − 1)i Δia n22 − n12
,
i = 3, 4,
235
(7)
where
(
Δia = n12(x·u) +
(
i
− 1) 2an2
2
)
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ 4a2 − n12x2⎟, ⎝ ⎠⎝ ⎠ (8)
i = 1, 2, 3, 4. Δia
Observe that > 0 because we have assumed a2 = n2 > a1 = n1. Figs. 1 and 2 illustrate these four sheets in a 2-dimensional example for F in the exterior and in the interior of the ovals, respectively. The following result was proved in [1]. Suppose that Ra does not contain any center of principal curvature of W. Then Ra is an immersed surface of 3. The singularities of the refracting profiles Ra are the points y ∈ Ra which are centers of principal curvature of W (see [17]). The locus * , of the centers of principal curvature of the wavefront W, or equivalently, the envelope of the normal lines to W, is called the caustic of W. In what follows we only consider wavefronts whose caustics are non-degenerate in the sense that each point c of the caustic corresponds to a unique point x of the wavefront. It is known that when the wavefront propagates its singularities slide along the caustic, see [18]. A similar result regarding the singularities of the reflecting profiles was proved in [2]. For the refracting profiles Ra holds the following result that was proved in [1]. For each c ∈ * of W there are two refracting profiles Ra1 and Ra2 such that c is a singularity of both of them. So, the singularities of the refracting profiles {Ra}a ≥ 0 sweep out the caustic * of W twice. In Fig. 1(a) is shown part of the caustic C of W. We can observe that in this case the singularities of the profiles Ra sweep out the caustic * . We also must observe that for the profiles to have physical sense, we have to exclude the points y of the profiles such that the corresponding incidence angle θ2 does not give a refracted ray. It occurs when light is propagated from an optically denser medium into one that is less optically dense, i.e. when n2 > n1, provided that the incidence angle θ2 exceeds the critical value θc given by sinθc = n1/n2, see Fig. 1(a) and (b).
3. Design of the lenses We will first give an explicit parametrization of the family of complete Cartesian ovals {Oxa}x ∈ W that we will use in the design of one of the profiles of focusing lenses. We can reduce the problem to the 2-dimensional case by considering for every x ∈ W the plane π(F , x ) determined by F, x, and u(x). Then if we take F as pole and Fx as polar axis, the polar equation r = r (x, φ) for a point y ∈ Oxa is determined by the condition:
n2r2 ± n1r = 2a
(9)
where r = |y|, r2 = |y − x|, and n2 > n1. Moreover, r, r2, and |x| are the sides of a triangle, see Fig. 3. Thus we have:
r22 = r 2 + |x|2 − 2r|x|cosφ.
(10)
Using (9)–(10) we get four solutions:
ri(x, φ) = and
n22|x|cosφ
i
−
− 2an1 + ( − 1) Δ (x, φ) n22 − n12
,
i = 1, 2,
(11)
Fig. 3. Planar section of the wavefront and the ovals. The points A and B, intersection between the tangent lines to the interior oval Oxa, i , passing through F, divide this oval into two parts, Oxa, i1 and Oxa, i2 , and the envelopes of these families, when x ∈ W , give rise to the profiles R1a and R2a, respectively. Analogously, the exterior oval Oxa, e decomposes into Oxa, e and Oxa, e , and the envelopes of these families, when 2 1 x ∈ W , give rise to the profiles R3a and R4a, respectively.
ri(x, φ) =
n22|x|cosφ + 2an1 + ( − 1)i Δ+ (x, φ) n22 − n12
,
i = 3, 4,
(12)
where
(
2
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ n22|x|2 − 4a2⎟, ⎝ ⎠⎝ ⎠
(13)
2
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ n22|x|2 − 4a2⎟. ⎝ ⎠⎝ ⎠
(14)
)
Δ− (x, φ) = n22|x|cosφ − 2an1
(
)
Δ+ (x, φ) = n22|x|cosφ + 2an1
If we consider a Cartesian coordinate system of the plane π (F , x ) , {F , X , Y}, with origin at F, then the Cartesian equation associated to ri(x, φ) (i = 1, 2, 3, 4), is:
yi = (ri(x, φ)cos(φ + θ ), ri(x, φ)sin(φ + θ )),
(15)
where θ is the angle determined by x and FX. The points y1 and y2 verify the equation n2|y − x| + n1|y| = 2a of the interior oval Oxa, i , whereas y3 and y4 verify the equation n2|y − x| − n1|y| = 2a of the exterior oval Oxa, e . For the case that F is exterior to the ovals, as in Fig. 1(a) and (b), we have the condition of compatibility 2a/n2 < |x| < 2a/n1. The values of φ that make Δ− > 0 and Δ+ > 0 in Eqs. (11) and (12) are respectively −φ1 < φ < φ1 and −φ2 < φ < φ2, with
φj = arccos
2an1 − ( − 1) j
(n
2 2
)(
),
− n12 n22|x|2 − 4a2
n22|x|
j = 1, 2.
(16)
The angles −φ1, φ1 are precisely the angles between Fx and the two tangent lines to the interior oval Oxa, i passing through F (see Fig. 3). The two points of tangency divide the oval into two parts, the nearest part to F, Oxa, i1 that is given by y1(x, φ) , φ ∈ [ − φ1, φ1], and
the farthest part to F, Oxa, i2 that is given by y2 (x, φ) , φ ∈ [ − φ1, φ1]. Analogously, the angles −φ2, φ2 are precisely the angles between Fx and the two tangent lines to the exterior oval Oxa, e passing through F. The two points of tangency divide the oval into two parts, the nearest part to F, Oxa, e1 that is given by
y3 (x, φ) , φ ∈ [ − φ2, φ2], and the farthest part to F, Oxa, e2 that is given by y4 (x, φ) , φ ∈ [ − φ2, φ2] (see Fig. 3). In fact, yi , i = 1 ,..., 4 of Eq. (15) give only planar sections. The surfaces whose envelopes give the surface profiles are obtained by rotating this planar section around the axis Fx. Therefore, ri, i = 1, … , 4 of Eqs. (11) and (12)
236
C. Criado, N. Alamo / Optics Communications 380 (2016) 233–238
give, in spherical polar coordinates, the equations of these four revolution surfaces Oxa, i1, Oxa, i2, Oxa, e1, and Oxa, e2.
On the other hand, the profile of the lens nearest to W is given by one element of the family {R1a}a > 0 constructed as envelopes of the family {Oxa, i1}x ∈ W . Its explicit equation is given by (see Eqs. (5), (6), and (8)):
{Oxa, e1}x ∈ W , and {Oxa, e2}x ∈ W , respectively. Taking F in the interior of the ovals, as in Fig. 2, the condition of compatibility 2a/n2, 2a/n1 > |x| is guaranteed and so Δ and Δ þ are never null. Then, the total exterior oval Oxa, e can be given by y4 (x, φ), with φ ∈ [ − π , π ]. It can also be divided into two parts, indeed for φ ∈ [ − φ1, φ1], and φ ∈ [φ1, 2π − φ1], 0 < φ1 < 2π , we get the part of the oval farthest to F, Oxa, e2, and nearest to F, Oxa, e2, respectively.
y = x + λ(x)u(x),
With this notation, the profiles R1a, R2a, R3a , and R4a defined in Section 2 are the envelopes of the families {Oxa, i1}x ∈ W , {Oxa, i2}x ∈ W ,
Analogously, we get the total interior oval Oxa, i by y2 (x, φ), with φ ∈ [ − π , π ], see Fig. 3.
(19)
with x ∈ W , u(x) the unitary normal vector to W at x in the direction of the half space containing O, and
λ(x) =
n12(x·u) − 2an2 + n22
−
n12
In this section we will show how to design lenses using profiles of type R1a. First, note that a profile could have physical sense as refracting or focusing profile only when F and W are at different sides of the tangent plane to that profile at any point of it, and a point y being in that profile must verify n1|y| + n2|y − x| = 2a . We have the following result that was proved in [1]. Let W, F, and R1a be as defined above. Then the profile R1a refracts the rays emerging from F into the normal rays to W. In other words this means that the wavefront W can be obtained from a spherical wavefront emerging from F after refraction at R1a, or equivalently the profile R1a focuses the wavefront W in F. 3.1.1. Focusing lenses of type LA1 One way to design lenses that focuses a given wavefront W in a given point F can be obtained using refracting profiles of type R1a and OFa,′i1, see Fig. 4. Of course, the same lenses can be used to obtain a given wavefront W from a point source F. To design these profiles we choose a point O between F and W, and we use it as the origin of our coordinate system. The profile of the lens nearest to F is a surface of revolution given by one element of the above constructed family {OFa,′i1}a′> 0 . Its equation in spherical polar coordinates with pole O and polar semi-axis in the direction of OF is given by (see Eqs. (11), (13) and (15)):
ri(φ) =
n22|OF |cosφ − 2an1 + ( − 1)i Δ− (φ) n22 − n12
,
i = 1, 2,
(17)
where
(
2
)
Δ− (φ) = n22|OF |cosφ − 2an1
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ n22|OF |2 − 4a2⎟. ⎝ ⎠⎝ ⎠
,
(20)
where
(
Δ1a = n12(x·u) − 2an2 3.1. Lenses of class LA
Δ1a
2
)
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ 4a2 − n12x2⎟. ⎝ ⎠⎝ ⎠
(21)
The lens designed in this way will be denoted by LA1(O, a, a′), since it depends on the choice of the point O between F and W, and on two positive numbers a, and a′ which must verify the conditions of compatibility for interior ovals. By the properties of the constructed profiles, R1a focuses the wavefront W in O, and the profile OFa,′i1 focuses the spherical wavefront emanating from O into F. Then we have the following result. Given a wavefront W and a point F, if the profile R1a does not contain any center of principal curvature of W, then the lens LA1(O, a, a′) focuses the wavefront W into F (and if F is a source point this lens produces W). Fig. 4 shows a planar section of the wavefront W, the profiles OFa,′i1 and R1a with its associated ovals. This planar section contains the optical path way xyOy′. Note that for a general wavefront W, the ray from y′ to F is not contained in the above plane, but in the plane that contains the optical path yy′F . Both planes intersect in the right line that passes through y and y′. We can ensure that the above planes coincide only when W is a surface of revolution around the axis passing through F and O. In this last case, the profile R1a is also a surface of revolution around the same axis. 3.1.2. Lenses of type LA2 To design lenses with the property of transforming a given wavefront W in other given wavefront W′ we use two profiles of type R1a, see Fig. 5. We choose, as before, a point O between the wavefronts, and use it as the origin of the coordinate system. R1a is the lens profile nearest to W, and its equations are given
(18)
Fig. 4. Planar section of a lens of type LA1 by a plane that contains the optical path way xyOy′. The figure also shows the point F that, in general, is not contained in this plane.
Fig. 5. Planar section of a lens of type LA2 by a plane that contains the optical path way xyOy′. The figure also shows the planar section of W′ and O xa′′ by a plane that contains the optical path yOy′x′. Both planes intersect in the right line that passes through y and y′.
C. Criado, N. Alamo / Optics Communications 380 (2016) 233–238
237
by (19)–(21), with x ∈ W and a > 0. On the other hand, the profile nearest to W′ is again given by the same equations, but now with x′ ∈ W′, and a′ > 0. The lens designed in this way will be denoted by LA2(O, a, a′). By the property of R1a, this lens refracts any ray orthogonal to W in a ray passing through O, and on the other hand R1a′ refracts any ray emanating from O into a ray orthogonal to W′. Then we obtain the following result. Let W and W′ be two wavefronts and suppose that the profiles R1a and R1a′ do not contain any principal center of curvature of W and W′ respectively. Then the lens LA2(O, a, a′) transforms the above wavefronts W and W′ one into the other. Note that, as in the case of the lenses LA1, the optical path way xyy′x′ (see Fig. 5) is not, in general, contained in a plane, so Fig. 5 shows a planar section of a lens of type LA2, for a plane that contains xyOy′ but not y′x′. The planar section of W′ and O xa′ that contains yOy′x′ is in another plane, and both planes intersect in the right line that passes through y and y′. The above two planes coincide when both wavefronts, W and W′, are surfaces of revolution around a common axis of symmetry that passes through O. We have drawn the planar sections of the figures of this paper using arcs of ellipses for the wavefronts. So, for example, in Fig. 5 the parametric equations of the wavefronts W and W′ are given and respectively by x = (8 + 4cost , 1.5 + 5sint ), x′ = (8 + 4cost , 1.3 + 10sint ), where the horizontal and vertical lines passing through O are the Cartesian axes. Therefore, in this figure there is not any axis of symmetry. The figures have been drawn only to illustrate the geometric design of the surfaces profiles. However, the expressions obtained for the lenses profiles are completely general and do not presume any symmetry of the wavefronts, so the refracting profiles do not have any symmetry, neither there is optical axis. 3.2. Lenses of class LB In this section we will show how to design lenses using profiles of type R1a in combination with profiles of type R3a. The profiles R3a have F in the interior of the ovals, so they have physical sense only if we consider F as a virtual source point (see Fig. 2). In fact, let us assume that the profile R3a separates two media with refractive indices n2 in the side containing F, and n1 in the other side, and that F and W are both in the same side of the tangent plane to the profile for any point of it. When a satisfies the condition of compatibility for exterior ovals, we have the following result that was proved in [1]. The wavefront W, when propagates towards R3a and after refraction at it, produces a spherical divergent wavefront with center F. The above result can equivalently be stated as follows: “A spherical wavefront convergent to F, when it propagates towards R3a, and after refraction at it, produces the wavefront W.” In this case, we say that F is a virtual source of W for the refracting profile R3a (see Fig. 2). 3.2.1. Focusing lenses of type LB1 We can design focusing lenses different from LA1 using together profiles of type R1a and OFa,′e1 with F in the interior of the ovals (see Fig. 6). The profile of the lens nearest to F is a surface of revolution given by one element of the above constructed family {OFa,′e1}a′> 0 . Its equation in spherical polar coordinates with pole O and polar axis in the direction OF is given by (see Eqs. (12) and (14)):
Fig. 6. Planar section of a lens of type LB1 by a plane containing x, y, y′, and O. The figure also shows the point F that, in general, is not contained in this plane.
r (φ) =
n22|OF |cosφ + 2an1 − n22 − n12
Δ+ (φ)
,
(22)
where
Δ+ (φ) = (n22|OF |cosφ + 2an1)2 − (n22 − n12)(n22|OF |2 − 4a2).
(23)
The profile OFa,′e1 has physical sense when we consider O as a virtual point: if a spherical wavefront converges to O, after refracting in ′
OFa, e1,it focuses in F. On the other hand, as in the lenses of type LA1, the profile of the lens nearest to W is given by one element of the family {R1a}a > 0 (see Eqs. (19)–(21)). The lens designed in this way will be denoted by LB1(O, a, a′), with a and a′ being two positive numbers which must verify the conditions of compatibility for interior and exterior ovals respectively. The profile R1a focuses the wavefront W on O, and OFa,′e1 refocuses this spherical wavefront into F. Then, when a and a′ are positive numbers that verify the conditions of compatibility for interior and exterior ovals respectively, we have the following result. Given a wavefront W and a point F, and supposed that the profile R1a does not contain any center of principal curvature of W, then the lens LB1(O, a, a′) focuses the wavefront W into F (and is F is a source point this lens produces W). Fig. 6 shows a planar section of a lens of type LB1 that contains the optical path way xyy′. This plane contains the point O, but, in general, does not contain the point F. We can ensure that F is in this plane only when the wavefront W is a surface of revolution with axis that passes through F and O. 3.2.2. Lenses of type LB2 that provide the transformation between two given wavefronts To design lenses different from LA2 providing the transformation between two given wavefronts we use one profile of type R1a and other of type R3a′. As in the design of lenses of type LA2, we choose a point O between the wavefronts W and W′, and use it as the coordinates origin (see Fig. 7). The profile of type R1a is the nearest to W and its equations are given by (19)–(21), with x ∈ W and a > 0. On the other hand the profile of type R3a′ is the nearest to W′ and is constructed as the envelope of the family {O xa′, e1}x′∈ W′. Its explicit equation is given by (see Eqs. (5), (7) and (8)):
238
C. Criado, N. Alamo / Optics Communications 380 (2016) 233–238
ovals, for the case of R4a (see Figs. 1(b) and 2). We have used R1a and R3a to design the lenses LA and LB, but due to the specific configuration of the profiles with respect to the focus F and the wavefront W, the rest of possibilities for lenses, namely, (R1a, R4a ), (R3a, R3a ), (R3a, R4a ), (R4a, R4a ), have had to be discarded.
4. Conclusions We have used the refracting profiles of type R1a and R3a, among the four possibilities of profiles constructed as envelopes of Cartesian ovals of revolution, in order to design lenses of two classes, LA and LB. Each one of these classes has two kinds of lenses, one that focuses a given wavefront W in a given point F, LA1 and LB1, and other that transforms two wavefronts W and W′ one into the other, LA2 and LB2. The lenses LA2 are designed with two profiles of the same type, R1a and R1a′, and the lenses LB2 with one profile of type R1a and other of type R3a′. On the other hand, for the design of the focusing lenses LA1 we use the profiles R1a and OFa,′i1, whereas for the lenses LB1 we use the Fig. 7. Planar section of a lens of type LB2 by a plane that contains x, y, y′, and O, which, in general, does not contain x′. The figure also shows the planar section of W′ and O xa′′ by a plane that contains x′, y′, and y. Both planes intersect in the right line that passes through O and y.
y′ = x′ + λ(x′)u(x′),
(24)
with x′ ∈ W′, u(x′) the unitary normal vector to W′ at x′ in the direction of the half space containing F, and
λ(x′) =
n12(x′·u) − 2a′n2 − n22 − n12
Δ3a′
(
2
)
milies whose envelopes are R1a and R3a′, respectively. In the four cases we have freedom to choose the point O and the positive numbers a and a′ provided they meet the condition of compatibility associated to the ovals used in the construction, and that the profiles do not contain any principal center of curvature of the wavefronts.
Acknowledgments
,
(25)
where
Δ3a′ = n12(x′·u) − 2a′n2
profiles R1a and OFa,′e1, where OFa,′i1 and OFa,′e1 are profiles of the fa-
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ 4a′ 2 − n12x′ 2⎟. ⎝ ⎠⎝ ⎠
This work was partially supported by the Junta de Andalucia grants RNM-1399 (C. Criado), FQM-192 (C. Criado) and FMQ-213 (N. Alamo), and by the Ministerio de Economía y Competitividad grant MTM2013-41768-P (N. Alamo).
(26)
The lens designed in this way will be denoted by LB2(O, a, a′). By the optical properties of the profiles R1a and R3a′ (see Sections 3.1 and 3.2) we know that R1a refracts any ray orthogonal to W in a ray passing through O, whereas R3a′ refracts any ray convergent to O into a ray orthogonal to W′ (see Fig. 7). Then, when a and a′ are positive numbers that verify the conditions of compatibility for interior and exterior ovals respectively, we have the following result. Let W and W′ two wavefronts and suppose that the profiles R1a and R3a′ do not contain any principal center of curvature of W and W′ respectively. Then the lens LB2(O, a, a′) transforms the wavefront W and W′ one into the other. Fig. 7 shows a planar section of a lens of type LB2 by a plane that contains x, y, y′, and O, but in general does not contain x′. The figure also shows the planar section of W′ and O xa′′ that contains x′, y′, and y. The intersection of both planes is the right line that passes through O and y′. We can ensure that the above planes coincide only when the wavefronts W and W′ are surfaces of revolution with a common axis that passes through O. In this last case, the profiles R1a and R3a′ are also surfaces of revolution with the same axis. Remark 3.1. The profiles R1a, R2a, R3a, and R4a do not allow to design any other type of lenses apart from the above LA and LB. In fact, the only type of profiles with physical sense as refracting or focusing profiles is R1a with the focus in the exterior of the ovals. Moreover, R3a and R4a have also physical sense but now as focusing to a virtual focus F, which is in the interior of the ovals for the case of R3a (see Fig. 1(b)), and both, in the exterior and interior of the
References [1] N. Alamo, C. Criado, A method to construct refracting profiles, Inverse Probl. 20 (2004) 229–238. [2] C. Criado, N. Alamo, Optical properties of conics: a method for obtaining reflecting and focusing profiles, Opt. Commun. 167 (1999) 881–889. [3] V.I. Oliker, Mathematical aspects of design of beam shaping surfaces in geometrical optics, in: V.I. Oliker, M. Kirkilionis, S. Krömker, R. Rannacher, F. Tomi (Eds.), Trends in Nonlinear Analysis, Springer, Berlin, 2003, pp. 197–224. [4] V. Oliker, J. Rubinstein, G. Wolansky, Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light, Adv. Appl. Math. 62 (2015) 160–183. [5] L.L. Doskolovich, M.A. Moiseev, E.A. Bezus, V. Oliker, On the use of the supporting quadric method in the problem of the light field eikonal calculation, Opt. Express 23 (2015) 19605–19617. [6] M.A. Moiseev, E.V. Byzov, S.V. Kravchenko, L.L. Doskolovich, Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces, Opt. Express 23 (2015) A1140–A1148. [7] D. Michaelis, P. Schreiber, A. Bräuer, Cartesian oval representation of freeform optics in illumination systems, Opt. Lett. 36 (2011) 918–920. [8] M.A. Moiseev, L.L. Doskolovich, Design of TIR optics generating the prescribed irradiance distribution in the circle region, J. Opt. Soc. Am. A 29 (2012) 1758–1763. [9] P.W. Rhodes, D.L. Shealy, Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis, Appl. Opt. 19 (1980) 3545–3553. [10] C.F. Chen, C.H. Lin, H.T. Jan, Y.L. Yang, Design of a solar concentrator combining paraboloidal and hyperbolic mirrors using ray tracing method, Opt. Commun. 282 (2009) 360–366. [11] C.Y. Tsai, Design of free-form reflector for vehicle LED low-beam headlamp, Opt. Commun. 372 (2016) 1–13. [12] E.H. Lockwood, A Book of Curves, Cambridge University Press, Cambridge, 1961. [13] M. Born, E. Wolf, Principles of Optics, Pergamon Press, London, 1970. [14] F.A. Jenkins, H.E. White, Fundamentals of Optic, McGraw-Hill, New York, 1976. [15] R. Courant, Differential and Integral Calculus, Wiley-Interscience, New Jersey, 1968. [16] E. Goursat, A Course in Mathematical Analysis, vol. I, Dover Publications, New York, 1959. [17] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey, 1976. [18] V.I. Arnold, Singularities of Caustics and Wave Fronts, Springer-Verlag, Berlin, 1990.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Lenses that provide the transformation between two given wavefronts C. Criado a,n, N. Alamo b a b
Departamento de Fisica Aplicada I, Universidad de Malaga, 29071 Malaga, Spain Departamento de Algebra, Geometria y Topologia, Universidad de Malaga, 29071 Malaga, Spain
art ic l e i nf o
a b s t r a c t
Article history: Received 15 February 2016 Received in revised form 12 May 2016 Accepted 3 June 2016 Available online 15 June 2016
We give an original method to design four types of lenses solving the following problems: focusing a given wavefront in a given point, and performing the transformation between two arbitrary incoming and outgoing wavefronts. The method to design the lenses profiles is based on the optical properties of the envelopes of certain families of Cartesian ovals of revolution. & 2016 Elsevier B.V. All rights reserved.
Keywords: Geometric optical design Refracting profile Focusing profile Descartes oval Wavefront
1. Introduction Suppose that we have given a lens L with refractive index n2 shaped by two profiles R1 and R2, and a point F in the exterior region with refractive index n1. A classical optical problem is to determine the resulting wavefront from the data of the source point F and the refracting profiles R1 and R2. It can be solved by using Snell's law or, equivalently, Fermat's principle. Let us remark that the lenses we are considering here are a kind of general aspheric lenses, without any axial symmetry, so we do not have optical axis. We consider now the inverse problem, that is, given a wavefront W and a source point F, we want to design the profiles of the lens L such that focuses the wavefront into the point F. Of course, the same lens produces W from the source point F, after refracting in the lens profiles. In this paper, we provide a method to design such lenses based on the optical properties of the envelopes of certain families of Cartesian ovals of revolution (see [1]). A similar method was used in [2] to design mirrors with specific characteristics as envelopes of families of conics of revolution, considering the optical properties of the conics. Furthermore, we solve here the general problem of designing the profiles for lenses that transform a given wavefront into another with arbitrary shape. The only condition to carry out these constructions is that the lens profiles should not contain points of the caustic of the wavefront. Similar types of general problems have been studied in other n
Corresponding author. E-mail address: [email protected] (C. Criado).
http://dx.doi.org/10.1016/j.optcom.2016.06.010 0030-4018/& 2016 Elsevier B.V. All rights reserved.
articles. For example, the well-known supporting quadric method ([3] and more recently [4,5]) is also based on a similar representation of the optical surfaces that uses quadrics to design reflecting and refracting surfaces suitable to reshaping the energy radiation intensity of a given source into a prescribed output irradiance distribution. A technique for computation of freeform optical elements with two refractive surfaces generating the required intensity distribution is proposed in [6], where is studied the particular case in which the freeform elements transform a spherical wavefront in a plane one. The final two smooth refractive surfaces are obtained after using splines on piecewise smooth surfaces composed by the adequate surfaces of the freeform elements. Another geometric method for designing freeform for illuminating purpose is proposed in [7], where the freeform is represented by a discrete set of Cartesian ovals segments with appropriated parameters. Other optics systems designing optical surfaces for generating prescribed irradiance distribution can be found in [8–11]. The paper is organized as follows. In Section 2 we revise the concepts and the optical properties about envelopes of Cartesian ovals of revolution given in [1]. In Section 3 we use these results to design four types of lenses that provide the desired transformations. The section is divided into two subsections according to the type of profiles that we use to design the lenses.
2. Refracting profiles For an arbitrary wavefront W and any source point F, it is possible to construct a family {Ra} of refracting profiles with the
234
C. Criado, N. Alamo / Optics Communications 380 (2016) 233–238
Fig. 1. Planar section of the wavefront W and the profiles, containing the optical path of a ray orthogonal to W at a point x, for the case of F exterior to the ovals. In (a) the profiles R1a and R2a are the envelopes of the interior Cartesian ovals Oxa, i , whereas in (b) R3a and R4a are the envelopes of the exterior Cartesian ovals Oxa, e . These ovals have foci F and x ∈ W . Only R1a and R4a have physical sense, focusing the wavefront W into a real focus F and to a virtual focus F, respectively. In (a) the third focus x′ and the caustic * of W that has singularities of the profile R4a are also shown. The parameter a is such that 2a shows the stationary optical length between F and x.
property of producing normal rays to W after refracting the rays that emerge from F at each profile Ra. The parameter a of the family is any non-negative real number. If light propagation direction is reversed, then the same family of profiles has the property of focusing the wavefront W in the point F. From now on we suppose that the refracting profile separates two media with refractive indices n1 and n2, F being in the region of n1, and W in the region of n2. We give now an outline of the method developed in [1] to design a suitable profile Ra for the wavefront W and the point F. Consider the Cartesian oval of revolution Oxa with foci F and x, where x ∈ W and the parameter a is such that 2a is the stationary optical path length between F and x (see Fig. 1(a) and (b)). Then Ra is taken as the envelope of the family {Oxa}x ∈ W . Indeed, let us recall that a Cartesian oval is the locus of the points from which the distances r1 and r2 to two fixed points F1 and F2, called foci, verify the bipolar equation
a1r1 + a2r2 = k,
(1)
where a1, a2 and k are constants. A Cartesian oval has a third focus F3, and the oval can be defined by any two of the foci. In particular when |a1/a2| goes to 1, the third focus goes to infinity and then we get the conics (the ellipse for k > 0 and a1 = a2 > 0, and the hyperbola for a1 = − a2 > 0). A complete Cartesian oval is the set of curves associated to the bipolar equation
a1r1 ± a2r2 = ± k
(k > 0).
(2)
Only two of the four equations obtained from these double signs are not empty. They are closed curves, one interior to the other (see Fig. 1(a) and (b) with F1 = F and F2 = x ). These two curves intersect only when two foci coincide. In this case we get the socalled Limaçon of Pascal. When one of the foci is exterior to the ovals, as in Fig. 1(a) and (b), we have the following condition of compatibility:
2a/a2 < |x| < 2a/a1
(3)
that must be satisfied in order to the existence of the ovals. For the case of both foci in the interior of the ovals (see Fig. 2), 2a/a2 and 2a/a1 are both greater than |x|, with a2 > a1. See [12] for a detailed study of these ovals. In the context of the geometrical optics, the coefficients a1 and a2 correspond to the refractive indices n1 and n2, so that Eq. (1) gives the stationary optical path length between the foci F1 and F2 (see [13,14]). In what follows we will use complete Cartesian ovals of revolution, that is, the surfaces obtained when the above ovals are rotated around their focal axes. Let Oxa denote the complete
Fig. 2. Case of F interior to the ovals. R1a and R2a are envelopes of the interior ovals, whereas R3a and R4a are envelopes of the exterior ovals. Only R3a and R4a have physical sense, focusing the wavefront W to the virtual focus F. To illustrate the ray tracing, we have drawn three contiguous rays.
Cartesian oval of revolution with foci F and x ∈ W , and let Oxa, i (respectively Oxa, e ) denote the interior (respectively exterior) oval (see Figs. 1 and 2). If we assume that F is the origin, then for any points x ∈ W and y ∈ Oxa , the vectors Fx and Fy with origin F and extreme x and y respectively, denoted again by x and y, verify:
n1|y| ± n2|y − x| = ± 2a. Specifically, the interior oval
(4) Oxa, i
verifies n1|y| + n2|y − x| = 2a , and
the exterior oval Oxa, e verifies −n1|y| + n2|y − x| = 2a . Let Ra be the envelope of the family {Oxa}x ∈ W (see [15,16]). To give an explicit parametrization of the profiles Ra, we suppose that W is oriented and we take u(x) to be the unitary normal vector to W at x in the direction of the half space containing F. Then Ra can be decomposed in four sheets Ra = R1a ∪ R2a ∪ R3a ∪ R4a , where Ria is parametrized by
y = x + λ i(x)u(x),
(5)
where, from the condition that y ∈
λi = and
Oxa ,
n12(x·u) + ( − 1)i 2an2 − ( − 1)i Δia n22 − n12
,
i = 1, 2,
(6)
C. Criado, N. Alamo / Optics Communications 380 (2016) 233–238
λi =
n12(x·u) + ( − 1)i 2an2 + ( − 1)i Δia n22 − n12
,
i = 3, 4,
235
(7)
where
(
Δia = n12(x·u) +
(
i
− 1) 2an2
2
)
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ 4a2 − n12x2⎟, ⎝ ⎠⎝ ⎠ (8)
i = 1, 2, 3, 4. Δia
Observe that > 0 because we have assumed a2 = n2 > a1 = n1. Figs. 1 and 2 illustrate these four sheets in a 2-dimensional example for F in the exterior and in the interior of the ovals, respectively. The following result was proved in [1]. Suppose that Ra does not contain any center of principal curvature of W. Then Ra is an immersed surface of 3. The singularities of the refracting profiles Ra are the points y ∈ Ra which are centers of principal curvature of W (see [17]). The locus * , of the centers of principal curvature of the wavefront W, or equivalently, the envelope of the normal lines to W, is called the caustic of W. In what follows we only consider wavefronts whose caustics are non-degenerate in the sense that each point c of the caustic corresponds to a unique point x of the wavefront. It is known that when the wavefront propagates its singularities slide along the caustic, see [18]. A similar result regarding the singularities of the reflecting profiles was proved in [2]. For the refracting profiles Ra holds the following result that was proved in [1]. For each c ∈ * of W there are two refracting profiles Ra1 and Ra2 such that c is a singularity of both of them. So, the singularities of the refracting profiles {Ra}a ≥ 0 sweep out the caustic * of W twice. In Fig. 1(a) is shown part of the caustic C of W. We can observe that in this case the singularities of the profiles Ra sweep out the caustic * . We also must observe that for the profiles to have physical sense, we have to exclude the points y of the profiles such that the corresponding incidence angle θ2 does not give a refracted ray. It occurs when light is propagated from an optically denser medium into one that is less optically dense, i.e. when n2 > n1, provided that the incidence angle θ2 exceeds the critical value θc given by sinθc = n1/n2, see Fig. 1(a) and (b).
3. Design of the lenses We will first give an explicit parametrization of the family of complete Cartesian ovals {Oxa}x ∈ W that we will use in the design of one of the profiles of focusing lenses. We can reduce the problem to the 2-dimensional case by considering for every x ∈ W the plane π(F , x ) determined by F, x, and u(x). Then if we take F as pole and Fx as polar axis, the polar equation r = r (x, φ) for a point y ∈ Oxa is determined by the condition:
n2r2 ± n1r = 2a
(9)
where r = |y|, r2 = |y − x|, and n2 > n1. Moreover, r, r2, and |x| are the sides of a triangle, see Fig. 3. Thus we have:
r22 = r 2 + |x|2 − 2r|x|cosφ.
(10)
Using (9)–(10) we get four solutions:
ri(x, φ) = and
n22|x|cosφ
i
−
− 2an1 + ( − 1) Δ (x, φ) n22 − n12
,
i = 1, 2,
(11)
Fig. 3. Planar section of the wavefront and the ovals. The points A and B, intersection between the tangent lines to the interior oval Oxa, i , passing through F, divide this oval into two parts, Oxa, i1 and Oxa, i2 , and the envelopes of these families, when x ∈ W , give rise to the profiles R1a and R2a, respectively. Analogously, the exterior oval Oxa, e decomposes into Oxa, e and Oxa, e , and the envelopes of these families, when 2 1 x ∈ W , give rise to the profiles R3a and R4a, respectively.
ri(x, φ) =
n22|x|cosφ + 2an1 + ( − 1)i Δ+ (x, φ) n22 − n12
,
i = 3, 4,
(12)
where
(
2
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ n22|x|2 − 4a2⎟, ⎝ ⎠⎝ ⎠
(13)
2
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ n22|x|2 − 4a2⎟. ⎝ ⎠⎝ ⎠
(14)
)
Δ− (x, φ) = n22|x|cosφ − 2an1
(
)
Δ+ (x, φ) = n22|x|cosφ + 2an1
If we consider a Cartesian coordinate system of the plane π (F , x ) , {F , X , Y}, with origin at F, then the Cartesian equation associated to ri(x, φ) (i = 1, 2, 3, 4), is:
yi = (ri(x, φ)cos(φ + θ ), ri(x, φ)sin(φ + θ )),
(15)
where θ is the angle determined by x and FX. The points y1 and y2 verify the equation n2|y − x| + n1|y| = 2a of the interior oval Oxa, i , whereas y3 and y4 verify the equation n2|y − x| − n1|y| = 2a of the exterior oval Oxa, e . For the case that F is exterior to the ovals, as in Fig. 1(a) and (b), we have the condition of compatibility 2a/n2 < |x| < 2a/n1. The values of φ that make Δ− > 0 and Δ+ > 0 in Eqs. (11) and (12) are respectively −φ1 < φ < φ1 and −φ2 < φ < φ2, with
φj = arccos
2an1 − ( − 1) j
(n
2 2
)(
),
− n12 n22|x|2 − 4a2
n22|x|
j = 1, 2.
(16)
The angles −φ1, φ1 are precisely the angles between Fx and the two tangent lines to the interior oval Oxa, i passing through F (see Fig. 3). The two points of tangency divide the oval into two parts, the nearest part to F, Oxa, i1 that is given by y1(x, φ) , φ ∈ [ − φ1, φ1], and
the farthest part to F, Oxa, i2 that is given by y2 (x, φ) , φ ∈ [ − φ1, φ1]. Analogously, the angles −φ2, φ2 are precisely the angles between Fx and the two tangent lines to the exterior oval Oxa, e passing through F. The two points of tangency divide the oval into two parts, the nearest part to F, Oxa, e1 that is given by
y3 (x, φ) , φ ∈ [ − φ2, φ2], and the farthest part to F, Oxa, e2 that is given by y4 (x, φ) , φ ∈ [ − φ2, φ2] (see Fig. 3). In fact, yi , i = 1 ,..., 4 of Eq. (15) give only planar sections. The surfaces whose envelopes give the surface profiles are obtained by rotating this planar section around the axis Fx. Therefore, ri, i = 1, … , 4 of Eqs. (11) and (12)
236
C. Criado, N. Alamo / Optics Communications 380 (2016) 233–238
give, in spherical polar coordinates, the equations of these four revolution surfaces Oxa, i1, Oxa, i2, Oxa, e1, and Oxa, e2.
On the other hand, the profile of the lens nearest to W is given by one element of the family {R1a}a > 0 constructed as envelopes of the family {Oxa, i1}x ∈ W . Its explicit equation is given by (see Eqs. (5), (6), and (8)):
{Oxa, e1}x ∈ W , and {Oxa, e2}x ∈ W , respectively. Taking F in the interior of the ovals, as in Fig. 2, the condition of compatibility 2a/n2, 2a/n1 > |x| is guaranteed and so Δ and Δ þ are never null. Then, the total exterior oval Oxa, e can be given by y4 (x, φ), with φ ∈ [ − π , π ]. It can also be divided into two parts, indeed for φ ∈ [ − φ1, φ1], and φ ∈ [φ1, 2π − φ1], 0 < φ1 < 2π , we get the part of the oval farthest to F, Oxa, e2, and nearest to F, Oxa, e2, respectively.
y = x + λ(x)u(x),
With this notation, the profiles R1a, R2a, R3a , and R4a defined in Section 2 are the envelopes of the families {Oxa, i1}x ∈ W , {Oxa, i2}x ∈ W ,
Analogously, we get the total interior oval Oxa, i by y2 (x, φ), with φ ∈ [ − π , π ], see Fig. 3.
(19)
with x ∈ W , u(x) the unitary normal vector to W at x in the direction of the half space containing O, and
λ(x) =
n12(x·u) − 2an2 + n22
−
n12
In this section we will show how to design lenses using profiles of type R1a. First, note that a profile could have physical sense as refracting or focusing profile only when F and W are at different sides of the tangent plane to that profile at any point of it, and a point y being in that profile must verify n1|y| + n2|y − x| = 2a . We have the following result that was proved in [1]. Let W, F, and R1a be as defined above. Then the profile R1a refracts the rays emerging from F into the normal rays to W. In other words this means that the wavefront W can be obtained from a spherical wavefront emerging from F after refraction at R1a, or equivalently the profile R1a focuses the wavefront W in F. 3.1.1. Focusing lenses of type LA1 One way to design lenses that focuses a given wavefront W in a given point F can be obtained using refracting profiles of type R1a and OFa,′i1, see Fig. 4. Of course, the same lenses can be used to obtain a given wavefront W from a point source F. To design these profiles we choose a point O between F and W, and we use it as the origin of our coordinate system. The profile of the lens nearest to F is a surface of revolution given by one element of the above constructed family {OFa,′i1}a′> 0 . Its equation in spherical polar coordinates with pole O and polar semi-axis in the direction of OF is given by (see Eqs. (11), (13) and (15)):
ri(φ) =
n22|OF |cosφ − 2an1 + ( − 1)i Δ− (φ) n22 − n12
,
i = 1, 2,
(17)
where
(
2
)
Δ− (φ) = n22|OF |cosφ − 2an1
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ n22|OF |2 − 4a2⎟. ⎝ ⎠⎝ ⎠
,
(20)
where
(
Δ1a = n12(x·u) − 2an2 3.1. Lenses of class LA
Δ1a
2
)
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ 4a2 − n12x2⎟. ⎝ ⎠⎝ ⎠
(21)
The lens designed in this way will be denoted by LA1(O, a, a′), since it depends on the choice of the point O between F and W, and on two positive numbers a, and a′ which must verify the conditions of compatibility for interior ovals. By the properties of the constructed profiles, R1a focuses the wavefront W in O, and the profile OFa,′i1 focuses the spherical wavefront emanating from O into F. Then we have the following result. Given a wavefront W and a point F, if the profile R1a does not contain any center of principal curvature of W, then the lens LA1(O, a, a′) focuses the wavefront W into F (and if F is a source point this lens produces W). Fig. 4 shows a planar section of the wavefront W, the profiles OFa,′i1 and R1a with its associated ovals. This planar section contains the optical path way xyOy′. Note that for a general wavefront W, the ray from y′ to F is not contained in the above plane, but in the plane that contains the optical path yy′F . Both planes intersect in the right line that passes through y and y′. We can ensure that the above planes coincide only when W is a surface of revolution around the axis passing through F and O. In this last case, the profile R1a is also a surface of revolution around the same axis. 3.1.2. Lenses of type LA2 To design lenses with the property of transforming a given wavefront W in other given wavefront W′ we use two profiles of type R1a, see Fig. 5. We choose, as before, a point O between the wavefronts, and use it as the origin of the coordinate system. R1a is the lens profile nearest to W, and its equations are given
(18)
Fig. 4. Planar section of a lens of type LA1 by a plane that contains the optical path way xyOy′. The figure also shows the point F that, in general, is not contained in this plane.
Fig. 5. Planar section of a lens of type LA2 by a plane that contains the optical path way xyOy′. The figure also shows the planar section of W′ and O xa′′ by a plane that contains the optical path yOy′x′. Both planes intersect in the right line that passes through y and y′.
C. Criado, N. Alamo / Optics Communications 380 (2016) 233–238
237
by (19)–(21), with x ∈ W and a > 0. On the other hand, the profile nearest to W′ is again given by the same equations, but now with x′ ∈ W′, and a′ > 0. The lens designed in this way will be denoted by LA2(O, a, a′). By the property of R1a, this lens refracts any ray orthogonal to W in a ray passing through O, and on the other hand R1a′ refracts any ray emanating from O into a ray orthogonal to W′. Then we obtain the following result. Let W and W′ be two wavefronts and suppose that the profiles R1a and R1a′ do not contain any principal center of curvature of W and W′ respectively. Then the lens LA2(O, a, a′) transforms the above wavefronts W and W′ one into the other. Note that, as in the case of the lenses LA1, the optical path way xyy′x′ (see Fig. 5) is not, in general, contained in a plane, so Fig. 5 shows a planar section of a lens of type LA2, for a plane that contains xyOy′ but not y′x′. The planar section of W′ and O xa′ that contains yOy′x′ is in another plane, and both planes intersect in the right line that passes through y and y′. The above two planes coincide when both wavefronts, W and W′, are surfaces of revolution around a common axis of symmetry that passes through O. We have drawn the planar sections of the figures of this paper using arcs of ellipses for the wavefronts. So, for example, in Fig. 5 the parametric equations of the wavefronts W and W′ are given and respectively by x = (8 + 4cost , 1.5 + 5sint ), x′ = (8 + 4cost , 1.3 + 10sint ), where the horizontal and vertical lines passing through O are the Cartesian axes. Therefore, in this figure there is not any axis of symmetry. The figures have been drawn only to illustrate the geometric design of the surfaces profiles. However, the expressions obtained for the lenses profiles are completely general and do not presume any symmetry of the wavefronts, so the refracting profiles do not have any symmetry, neither there is optical axis. 3.2. Lenses of class LB In this section we will show how to design lenses using profiles of type R1a in combination with profiles of type R3a. The profiles R3a have F in the interior of the ovals, so they have physical sense only if we consider F as a virtual source point (see Fig. 2). In fact, let us assume that the profile R3a separates two media with refractive indices n2 in the side containing F, and n1 in the other side, and that F and W are both in the same side of the tangent plane to the profile for any point of it. When a satisfies the condition of compatibility for exterior ovals, we have the following result that was proved in [1]. The wavefront W, when propagates towards R3a and after refraction at it, produces a spherical divergent wavefront with center F. The above result can equivalently be stated as follows: “A spherical wavefront convergent to F, when it propagates towards R3a, and after refraction at it, produces the wavefront W.” In this case, we say that F is a virtual source of W for the refracting profile R3a (see Fig. 2). 3.2.1. Focusing lenses of type LB1 We can design focusing lenses different from LA1 using together profiles of type R1a and OFa,′e1 with F in the interior of the ovals (see Fig. 6). The profile of the lens nearest to F is a surface of revolution given by one element of the above constructed family {OFa,′e1}a′> 0 . Its equation in spherical polar coordinates with pole O and polar axis in the direction OF is given by (see Eqs. (12) and (14)):
Fig. 6. Planar section of a lens of type LB1 by a plane containing x, y, y′, and O. The figure also shows the point F that, in general, is not contained in this plane.
r (φ) =
n22|OF |cosφ + 2an1 − n22 − n12
Δ+ (φ)
,
(22)
where
Δ+ (φ) = (n22|OF |cosφ + 2an1)2 − (n22 − n12)(n22|OF |2 − 4a2).
(23)
The profile OFa,′e1 has physical sense when we consider O as a virtual point: if a spherical wavefront converges to O, after refracting in ′
OFa, e1,it focuses in F. On the other hand, as in the lenses of type LA1, the profile of the lens nearest to W is given by one element of the family {R1a}a > 0 (see Eqs. (19)–(21)). The lens designed in this way will be denoted by LB1(O, a, a′), with a and a′ being two positive numbers which must verify the conditions of compatibility for interior and exterior ovals respectively. The profile R1a focuses the wavefront W on O, and OFa,′e1 refocuses this spherical wavefront into F. Then, when a and a′ are positive numbers that verify the conditions of compatibility for interior and exterior ovals respectively, we have the following result. Given a wavefront W and a point F, and supposed that the profile R1a does not contain any center of principal curvature of W, then the lens LB1(O, a, a′) focuses the wavefront W into F (and is F is a source point this lens produces W). Fig. 6 shows a planar section of a lens of type LB1 that contains the optical path way xyy′. This plane contains the point O, but, in general, does not contain the point F. We can ensure that F is in this plane only when the wavefront W is a surface of revolution with axis that passes through F and O. 3.2.2. Lenses of type LB2 that provide the transformation between two given wavefronts To design lenses different from LA2 providing the transformation between two given wavefronts we use one profile of type R1a and other of type R3a′. As in the design of lenses of type LA2, we choose a point O between the wavefronts W and W′, and use it as the coordinates origin (see Fig. 7). The profile of type R1a is the nearest to W and its equations are given by (19)–(21), with x ∈ W and a > 0. On the other hand the profile of type R3a′ is the nearest to W′ and is constructed as the envelope of the family {O xa′, e1}x′∈ W′. Its explicit equation is given by (see Eqs. (5), (7) and (8)):
238
C. Criado, N. Alamo / Optics Communications 380 (2016) 233–238
ovals, for the case of R4a (see Figs. 1(b) and 2). We have used R1a and R3a to design the lenses LA and LB, but due to the specific configuration of the profiles with respect to the focus F and the wavefront W, the rest of possibilities for lenses, namely, (R1a, R4a ), (R3a, R3a ), (R3a, R4a ), (R4a, R4a ), have had to be discarded.
4. Conclusions We have used the refracting profiles of type R1a and R3a, among the four possibilities of profiles constructed as envelopes of Cartesian ovals of revolution, in order to design lenses of two classes, LA and LB. Each one of these classes has two kinds of lenses, one that focuses a given wavefront W in a given point F, LA1 and LB1, and other that transforms two wavefronts W and W′ one into the other, LA2 and LB2. The lenses LA2 are designed with two profiles of the same type, R1a and R1a′, and the lenses LB2 with one profile of type R1a and other of type R3a′. On the other hand, for the design of the focusing lenses LA1 we use the profiles R1a and OFa,′i1, whereas for the lenses LB1 we use the Fig. 7. Planar section of a lens of type LB2 by a plane that contains x, y, y′, and O, which, in general, does not contain x′. The figure also shows the planar section of W′ and O xa′′ by a plane that contains x′, y′, and y. Both planes intersect in the right line that passes through O and y.
y′ = x′ + λ(x′)u(x′),
(24)
with x′ ∈ W′, u(x′) the unitary normal vector to W′ at x′ in the direction of the half space containing F, and
λ(x′) =
n12(x′·u) − 2a′n2 − n22 − n12
Δ3a′
(
2
)
milies whose envelopes are R1a and R3a′, respectively. In the four cases we have freedom to choose the point O and the positive numbers a and a′ provided they meet the condition of compatibility associated to the ovals used in the construction, and that the profiles do not contain any principal center of curvature of the wavefronts.
Acknowledgments
,
(25)
where
Δ3a′ = n12(x′·u) − 2a′n2
profiles R1a and OFa,′e1, where OFa,′i1 and OFa,′e1 are profiles of the fa-
⎛ ⎞⎛ ⎞ − ⎜ n22 − n12⎟⎜ 4a′ 2 − n12x′ 2⎟. ⎝ ⎠⎝ ⎠
This work was partially supported by the Junta de Andalucia grants RNM-1399 (C. Criado), FQM-192 (C. Criado) and FMQ-213 (N. Alamo), and by the Ministerio de Economía y Competitividad grant MTM2013-41768-P (N. Alamo).
(26)
The lens designed in this way will be denoted by LB2(O, a, a′). By the optical properties of the profiles R1a and R3a′ (see Sections 3.1 and 3.2) we know that R1a refracts any ray orthogonal to W in a ray passing through O, whereas R3a′ refracts any ray convergent to O into a ray orthogonal to W′ (see Fig. 7). Then, when a and a′ are positive numbers that verify the conditions of compatibility for interior and exterior ovals respectively, we have the following result. Let W and W′ two wavefronts and suppose that the profiles R1a and R3a′ do not contain any principal center of curvature of W and W′ respectively. Then the lens LB2(O, a, a′) transforms the wavefront W and W′ one into the other. Fig. 7 shows a planar section of a lens of type LB2 by a plane that contains x, y, y′, and O, but in general does not contain x′. The figure also shows the planar section of W′ and O xa′′ that contains x′, y′, and y. The intersection of both planes is the right line that passes through O and y′. We can ensure that the above planes coincide only when the wavefronts W and W′ are surfaces of revolution with a common axis that passes through O. In this last case, the profiles R1a and R3a′ are also surfaces of revolution with the same axis. Remark 3.1. The profiles R1a, R2a, R3a, and R4a do not allow to design any other type of lenses apart from the above LA and LB. In fact, the only type of profiles with physical sense as refracting or focusing profiles is R1a with the focus in the exterior of the ovals. Moreover, R3a and R4a have also physical sense but now as focusing to a virtual focus F, which is in the interior of the ovals for the case of R3a (see Fig. 1(b)), and both, in the exterior and interior of the
References [1] N. Alamo, C. Criado, A method to construct refracting profiles, Inverse Probl. 20 (2004) 229–238. [2] C. Criado, N. Alamo, Optical properties of conics: a method for obtaining reflecting and focusing profiles, Opt. Commun. 167 (1999) 881–889. [3] V.I. Oliker, Mathematical aspects of design of beam shaping surfaces in geometrical optics, in: V.I. Oliker, M. Kirkilionis, S. Krömker, R. Rannacher, F. Tomi (Eds.), Trends in Nonlinear Analysis, Springer, Berlin, 2003, pp. 197–224. [4] V. Oliker, J. Rubinstein, G. Wolansky, Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light, Adv. Appl. Math. 62 (2015) 160–183. [5] L.L. Doskolovich, M.A. Moiseev, E.A. Bezus, V. Oliker, On the use of the supporting quadric method in the problem of the light field eikonal calculation, Opt. Express 23 (2015) 19605–19617. [6] M.A. Moiseev, E.V. Byzov, S.V. Kravchenko, L.L. Doskolovich, Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces, Opt. Express 23 (2015) A1140–A1148. [7] D. Michaelis, P. Schreiber, A. Bräuer, Cartesian oval representation of freeform optics in illumination systems, Opt. Lett. 36 (2011) 918–920. [8] M.A. Moiseev, L.L. Doskolovich, Design of TIR optics generating the prescribed irradiance distribution in the circle region, J. Opt. Soc. Am. A 29 (2012) 1758–1763. [9] P.W. Rhodes, D.L. Shealy, Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis, Appl. Opt. 19 (1980) 3545–3553. [10] C.F. Chen, C.H. Lin, H.T. Jan, Y.L. Yang, Design of a solar concentrator combining paraboloidal and hyperbolic mirrors using ray tracing method, Opt. Commun. 282 (2009) 360–366. [11] C.Y. Tsai, Design of free-form reflector for vehicle LED low-beam headlamp, Opt. Commun. 372 (2016) 1–13. [12] E.H. Lockwood, A Book of Curves, Cambridge University Press, Cambridge, 1961. [13] M. Born, E. Wolf, Principles of Optics, Pergamon Press, London, 1970. [14] F.A. Jenkins, H.E. White, Fundamentals of Optic, McGraw-Hill, New York, 1976. [15] R. Courant, Differential and Integral Calculus, Wiley-Interscience, New Jersey, 1968. [16] E. Goursat, A Course in Mathematical Analysis, vol. I, Dover Publications, New York, 1959. [17] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey, 1976. [18] V.I. Arnold, Singularities of Caustics and Wave Fronts, Springer-Verlag, Berlin, 1990.