## Perfect focusing of scalar wave fields in three dimensions

Optics Express, Vol. 18, Issue 8, pp. 7650-7663 (2010)

http://dx.doi.org/10.1364/OE.18.007650

Acrobat PDF (323 KB)

### Abstract

A method to design isotropic inhomogeneous refractive index distribution is presented, in which the scalar wave field solutions propagate exactly on an eikonal function (*i.e.*, remaining constant on the Geometrical Optics wavefronts). This method is applied to the design of “dipole lenses”, which perfectly focus a scalar wave field emitted from a point source onto a point absorber, in both two and three dimensions. Also, the Maxwell fish-eye lens in two and three dimensions is analysed.

© 2010 OSA

## 1. Introduction

*k*=

*ω*/

*c*is very large. In that limit, the propagation of the scalar field can be calculated with good approximation using rays. There are, however, some trivial cases in which the ray trajectories guide the scalar field in an exact manner (

*i.e.*with no restriction to large

*k*) so that the field is constant on the Geometrical Optics wavefronts. For instance, a (monopole) point source emitting from the center of a spherical symmetric refractive index distribution

*n*(

**r**) will generate a field which will depend on the radial coordinate only. In this paper we are going to discuss about isotropic inhomogeneous refractive index distributions that propagate nontrivial scalar fields exactly on eikonals (

*i.e.*, remaining constant on the Geometrical Optics wavefronts), and we will find media that produce perfect focusing of rays and waves, perfect in the sense explained next.

**P**onto an image point

**Q**when any ray trajectory emitted from

**P**through the optical system will pass through

**Q**in an exact way. Such points

**P**and

**Q**are said to be perfect conjugates [1]. A device is called an Absolute Instrument in Geometrical Optics if it produces perfect focusing of rays not just from a single object point, but of all points in a three-dimensional domain (

*i.e.*, one with non-null volume) [1].

**P**and

**Q**coincident with the foci of the ellipsoid (

*i.e.*, it is not an Absolute Instrument). Non-trivial examples of Absolute Instruments are based on non-homogeneous media [2], the most famous being the Maxwell fish-eye lens [1]. Unlike conventional imaging optics systems, in the Maxwell fish-eye lens the refractive index in the volume containing the object and image points is inhomogeneous, (

*i.e.*, spatially varying) . There are, however, examples of Absolute Instruments in which the refractive index distribution in that volume is homogeneous [3

3. J. C. Miñano, “Perfect imaging in a homogeneous threedimensional region,” Opt. Express **14**(21), 9627–9635 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-21-9627. [CrossRef] [PubMed]

4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

*n*=

*ε*=

*μ*= –1, the capacity of which for perfect imaging has been explained by the amplification of evanescent waves in the negative index material.

5. U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. **11**(9), 093040 (2009). [CrossRef]

6. U. Leonhardt and T. G. Philbin, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A **81**(1), 011804 (2010). [CrossRef]

*n*=

*ε*=

*μ*). These results are especially relevant because it uses isotropic positive refractive index.

**P**and

**Q**coincide with the foci, it is well known that it does not focus waves perfectly (neither in two nor in three dimensions) due to the coma of the mirror.

**P**onto a point drain at

**Q**. This means that the local behavior of the field around

**Q**will coincide asymptotically with a spherical converging wave in three dimensions. We will also consider the two dimensional case, valid for cylindrical waves in cylindrical symmetric optics.

5. U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. **11**(9), 093040 (2009). [CrossRef]

**P**upon a point drain

**Q**in three dimensions. Finally, Section 5 discusses resolution limits.

3. J. C. Miñano, “Perfect imaging in a homogeneous threedimensional region,” Opt. Express **14**(21), 9627–9635 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-21-9627. [CrossRef] [PubMed]

5. U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. **11**(9), 093040 (2009). [CrossRef]

## 2. Helmholtz fields that propagate exactly on eikonals

### 2.1 Statement of the problem

*U*(

**r**) ∈ C,

**r**∈

*D*⊂ R

^{3}in a medium with refractive index distribution

*n*(

**r**):where

*k*=

*ω*/

*c*. As with [5

**11**(9), 093040 (2009). [CrossRef]

7. M. A. Alonso and G. Forbes, “Stable aggregates of flexible elements give a stronger link between rays and waves,” Opt. Express **10**(16), 728–739 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-16-728. [PubMed]

*U*(

**r**) is referred to a Helmholtz scalar wave field. Note that the name Helmholtz equation is sometimes reserved to the case

*n*= constant (i.e., the case of homogeneous media), while Eq. (1) is formally equivalent to the time-independent Schrödinger equation. This equation is relevant in other areas of physics, such as acoustics or optics. In optics, this equation in 2D is exact for describing TE polarized light in cylindrical media (in which electric field vector

**E**points orthogonal to the cross section of the cylinder). It is not exact but approximate, however, for describing electromagnetic fields in 3D.

*S*(

**r**) ∈ R that is a particular solution of the eikonal equation in the domain

*D*:

*S*(

**r**) are the geometrical optics rays associated with the eikonal function S(

**r**), which measures the advance of the optical path length along the rays. We can create the tri-orthogonal curvilinear coordinates (

*S*,

*u*,

*v*) where

*u*and

*v*are the coordinates defined on the surfaces

*S*(

**r**) = constant. This coordinate system is used to express the transport equations in the Geometrical Optics approximation, and has been specifically used in Electromagnetic Optics by Stavroudis [8]. Additionally, it is also commonly applied in the method of characteristics for solving the time-dependent wave equation. In these new coordinates

*S = S*(

**r**),

*u = u*(

**r**),

*v*=

*v*(

**r**), Eq. (1) becomes:

*U*=

*U*(

*S*,

*u*,

*v*) and

*n*=

*n*(

*S*,

*u*,

*v*). If there were solutions of

*U*depending on

*S*only, then (3) reduces to:

*U*(

*S*) will be a solution of the following ordinary differential equation:

*U*depending solely on

*S*if (and only if) S fulfills:

*U*=

*U*(

*S*) exists and the refractive index is calculated as

*S*. Therefore, in these fields

**I**is tangent to the rays (as

### 2.2 Solutions of Eq. (7)

*S*is a solution of functional Eq. (7) in the domain

*D*if and only if there is a real increasing function

*V*(

*S*) which is harmonic in

*D*,

*i.e.*, whose Laplacian is zero (∆

*V*= 0):

*V*is the argument of a logarithm, it must be positive, which implies that

_{S}*V*(

*S*) is monotonically increasing (and therefore invertible).

*U*exactly on eikonals. The design method comprises the following steps:Field

*U*(

**r**) can finally be obtained by solving Eq. (6) for

*U*(

*S*), and then calculating

*U*(

*S*(

*V*(

**r**))).

### 2.3. Equivalent statements

- a) Find a scalar wave field
*U*(**r**) particular solution of the Helmholtz equation which is a function*U = U*(*V*) of a solution*V*(**r**) of Laplace equation. The necessary and sufficient condition is that the solution takes the form*V = V*(*S*), where*S*(**r**) is a solution of the eikonal equation. - b) Find a particular solution
*S*(**r**) of the eikonal equation which is a function*S = S*(*V*) of a solution*V*(**r**) of Laplace equation. When that solution is found, then a field*U*(**r**) is a solution of the Helmholtz equation of the form*U = U*(*S*) and can be found by solving Eq. (6).

## 3. Perfect focusing in two dimensions

### 3.1 The cylindrical Maxwell fish-eye lens

*L*and

*a*are positive real constants and

*ρ*,

*z*are cylindrical coordinates (

*ρ*

^{2}=

*x*

^{2}+

*y*

^{2}). As mentioned in the introduction, the Maxwell fish-eye lens is an Absolute Instrument in geometrical optics, and thus the rays contained in planes

*z*=

*z*

_{0}passing through any point

**P**will focus again at its conjugate point

**Q**. The position of

**Q**in the x-y plane can be obtained as the transformation from

**P**by an inversion with respect to the circle of radius

*a*centered at

**O,**followed by central symmetry with respect to

**O**. Therefore,

**P**·

**Q**= −

*a*

^{2}.

**11**(9), 093040 (2009). [CrossRef]

*S*of the rays emitted from

**P**in the cylindrical Maxwell fish-eye lens fulfills Eq. (5).

**P**=

*a*(−cosh α − sinh α,0,0) and

**Q**=

*a*(cosh α − sinh α,0,0) for a given real constant

*α*. Note that

**P**·

**Q**= −

*a*

^{2}cosh

^{2}α +

*a*

^{2}sinh

^{2}α = −

*a*

^{2}. In the case α = 0,

**P**= (−

*a*,0,0) and

**Q**= (

*a*,0,0).

*z*) [11]:

*σ*and constant

*τ*are circles that intersect at right angles. As shown in Fig. 1 , curves of constant

*σ*(in blue) correspond to circles that intersect at the two points

**P**and

**Q**, while the curves of constant

*τ*(in red) are non-intersecting circles of different radii that surround the points

**P**and

**Q**.

*z*) as:

*τ*isosurfaces coincide with Geometrical wavefronts by the fact that the eikonal equation (∇

*S*)

^{2}=

*n*

^{2}has particular solutions depending only on τ,

*i.e.*,

*S*=

*S*(

*τ*), which are calculated with Eq. (14) and Eq. (16):

*S*increases with

*τ*) and

*C*= 0. The minimum value of

*S*(

*τ*) is reached at the

**P**where

*τ→ −∞*and

*S*(

*−∞*) = 0, while the maximum value is achieved at the

**Q**where

*τ→∞*and

*S*(

*∞*) =

*πL*. Therefore 0 ≤

*S*≤

*π L*.

*S*(

*τ*) to obtain:

*U*which is function of S only, and it fulfils the ordinary differential Eq. (6):

*i.e.*,

**11**(9), 093040 (2009). [CrossRef]

*L*(which was normalized to 1 in that reference). The resolution was described in detail there but is not pertinent and so will not be continued here.

*A*= 1/

*L*and

*B*=

*α*(see Eq. (18)). This indicates that the function

*τ*(

*x*,

*y*) is harmonic, which is in fact well known, since the mapping of the planes x-y to τ-σ produced by the bipolar coordinate transformation (13) is conformal [11], and therefore both

*τ*(

*x*,

*y*) and

*σ*(

*x*,

*y*) are harmonic functions.

### 3.2 The general cylindrical case: The “2D dipole lens”

**P**at a point drain at

**Q**. For that task, consider again the following harmonic function in R

^{3}, valid except at the lines x = -

*a*; y = 0 and x = -

*a*; y = 0

*i.e.*, two line charges of equal magnitude but opposite sign crossing the z = 0 plane on the points

**P**= (-

*a*,0,0) and

**Q**= (

*a*,0,0). The function (24) coincides with function

*τ*(

*x*,

*y*) of the previous section for the case

*α*= 0 (as can be checked by inverting the mapping (13)), which is well known in electrostatic theory [11].

*S*(

*V*). We will call this family of solutions the “2D dipole lens”. Different functions

*S*(

*V*) lead to different refractive index distributions, but all of them have the same wavefronts and rays defined by the coordinate lines of the bipolar reference system (13). One distinguished solution of this family is the cylindrical Maxwell fish-eye lens discussed in the previous section. In general, the refractive index distribution will not be a function of

*ρ*only, as was the case for the cylindrical Maxwell fish-eye lens. Also unlike the Maxwell fish-eye lens, a general 2D dipole lens will not produce perfect focusing of the cylindrical scalar waves emitted by a point source different than

**P**or

**Q**(which are focused one into the other). Therefore, no perfect imaging for a volumetric region is expected in the general case.

*S*(

*V*) care must be taken with its asymptotic values when |

*V|*→∞. This care is needed for the resulting refractive index (given by |∇

*S*|) to be bounded, considering that

*V*is unbounded around

**P**and

**Q**. In that case we must choose that when |

*V|*→∞,

*i.e.*for

**r**close to

**P**(or

**Q)**:

**r**to

**P**(or

**Q**).

## 4. Perfect focusing in three dimensions

**P**and

**Q**are placed on the

*z*axis, whereas in the previous section they were on the

*x*axis.

### 4.1 The spherical Maxwell fish-eye lens

**P**=

*a*(0,0,cosh α − sinh α) and

**Q**=

*a*(0,0,−cosh α − sinh α) define the bispherical coordinate system (σ,τ,φ) given by [11]:

*S*of Eq. (27) of the function

*S*=

*S*(

*τ*) coincides with the cylindrical Maxwell fish-eye lens [Eq. (17)], as was expected. Using the expression for the Laplacian in the (σ,τ,φ) coordinates [11] gives:

*α*→ ± ∞, in which

**P**or

**Q**are at the origin)

*S*does not fulfill Eq. (7) for any value of

*α*, which means that there is no solution of the field

*U*that is a function of

*S*only, as occurred in 2D. This means that the spherical Maxwell fish eye lens does not belong to the class of media discussed here, but this does not mean that it cannot perfectly focus Helmholtz scalar wave fields of a point source

**P**onto its image point

**Q**in three dimensions (that ability or inability is has still to be proved). Leonhardt and Philbin, however, have proven very recently [6

6. U. Leonhardt and T. G. Philbin, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A **81**(1), 011804 (2010). [CrossRef]

*n*=

*ε*=

*μ*).

**P**to

**Q**in three dimensions.

### 4.2 The general rotational case: The “3D dipole lens”

^{3}except at the points (

*x*,

*y*,

*z*) given by

**P**= (0,0,-

*a*) and

**Q**= (0,0,

*a*) (so that

*D*= R

^{3}\{

*P*,

*Q*})

*ρ*,

*z*are again cylindrical coordinates (

*ρ*

^{2}=

*x*

^{2}+

*y*

^{2}). This harmonic function is (up to a multiplicative constant) equal to the electrostatic potential created by an electric dipole (two point charges of equal magnitude but opposite sign) located on the points

**P**and

**Q**. That potential distribution has rotational symmetry with respect to the z axis, and the cross sections of the

*V*= constant surfaces and electrostatic field lines are shown in Fig. 2 (see also an animation in [13]). In our case, the

*V*= constant surfaces are the wavefronts and the electrostatic field lines correspond to the ray trajectories and energy flux lines. Note that both families of curves are not circumferences, as occurred in the 2D case. Particularly, the

*V*= constant surfaces are octics (8-degree algebraic curves) that belong to the family of Generalised Cayley’s ovals [14].

*n*= |∇

*S*| to be bounded, we must select

*S*(

*V*) with the appropriate asymptotic behavior when |

*V|*→∞, which in this 3D case is:

*S*(

*ρ*,

*z*) is only reached at

**P**,

*S*(0,–

*a*)

*=*0, while the maximum value is achieved at

**Q**,

*S*(0,

*a*)

*= πL*. Therefore 0 ≤

*S*≤

*π L*.

*z*(as expected by the symmetries of

*V*(

**r**) and of

*S*(

*V*)) and is shown in Fig. 3 . At

**P**and

**Q**,

*ρ*,

*z*) = (0,0),

**r**|,

*n*(

**r**) is smaller than one (as also occurs in the Maxwell fish eye lens).

12. A. D. Polyanin, and V. F. Zaitsev, *Handbook of Exact Solutions for Ordinary Differential Equations*, 2nd Edition, Chapman & Hall/CRC, Boca Raton, 2003. It also coincides with the change of variables *S*’ = *S/L* + *π*/2 in the equation described in http://eqworld.ipmnet.ru/en/solutions/ode/ode0235.pdf.

*C*

_{1}and

*C*

_{2}are arbitrary constants and:

*κ*→

*k*when

*k*→∞. Select the particular solution

*C*

_{2}= 1/(4

*πa*);

*C*

_{1}= i/(4

*πa*), which leads to

*S*, that is, from point

**P**to point

**Q**, with phase advancing in proportion to

*κ*. The energy flux vector of this field can be computed from Eq. (9) as:

*U*(

*S*(

*V*(

*ρ*,

*z*)), so that from Eqs. (38), (32) and (30):

*U*(

*ρ*,

*z*)exp(ω

*t*) for ω

*t*= 1.033

*π*,

*a*= 1,

*L*= 1,

*κ*= 9.5. Time evolution of the field is shown in Media 1.

**P**and

**Q,**since at

*S*(

**P**)

*=*0 and

*S*(

**Q**)

*= πL*the sine function vanishes in (38) and asymptotically behaves as:

*S*given by Eq. (32) approximates to:

*r*the distance from a point (

*ρ*,

*z*) to

**P**and

*r’*the distance to

**Q**,

*i.e.*the asymptotic behavior of the field around

**P**and

**Q**can be written as:where:

**P**emitting radiation and point drain at

**Q**receiving said radiation (because the energy flux (39) follows the ray trajectories), so no flux is radiated towards infinity.

## 5. Resolution limit

*r*and

*r’*around

**P**and

**Q**, respectively, we can confirm from (44) that:

**S**points towards the spheres exterior.

^{3}as:

**Q**is necessary for mathematical consistency. In the absence of the drain at

**Q**, the field amplitude there will not diverge, because the wave will pass through the focus

**Q**, expand from it and converge back upon

**P**. We can calculate the field

*W*(

**r**) in absence of the drain at

**Q**as:

*S(ρ,z)*is given by Eq. (30) and (32), (note that 0≤

*S*≤

*πL*). Around

**Q**(

*i.e.*,

*ρ*= 0,

*z*=

*a*) we can calculate from Eq. (48) the asymptote:

**Q**and has amplitude:

**P**(

*i.e.*,

*ρ*= 0,

*z*= −

*a*) the asymptote is:

*W*(

**r**) of Eq. (48) is then the solution of the following Helmholtz equation:

*κL = κ*

_{1}

*a*≠

*l*, with

*l*a positive integer. At the frequencies for which

*κL = κ*

_{1}

*a*=

*l*, the system is an oscillator, so that Eq. (52) becomes homogeneous and the field amplitude at

**P**equals that at

**Q**of Eq. (50).

**Q**. It resembles an Airy pattern in three dimensions, with peak amplitude given by Eq. (50), and the radius of the sphere of the first null is:

*λ*is the wavelength of the field in vacuum, and

*n*

_{max}= 2

*L*/

*a*is the maximum value of

*n*[see Eq. (33)] obtained for (

*ρ*,

*z*) = (0,0). This seems to indicate that both the amplitude at focus and the resolution capacity of the 3D dipole lens (and the Maxwell fish eye in two dimensions [5

**11**(9), 093040 (2009). [CrossRef]

15. Yu. I. Bobrovnitskiĭ, “Impedance theory of sound absorption: The best absorber and the black body,” Acoust. Phys. **52**(6), 638–647 (2006). [CrossRef]

**Q**, it will present a high field amplitude on its surface. Using these subwavelength absorbers as detectors, it should be possible to resolve two point sources separated by a distance much smaller than the wavelength.

## 6. Conclusions

*U*(

**r**) exactly on eikonals in that medium, in the sense that that

*U*(

**r**) =

*U*(

*S*(

**r**)), where

*S*(

**r**) is a particular solution of the eikonal equation in that medium.

*n*(

**r**), the “3D dipole lens”, which in three dimensions perfectly focuses a point source at

**P,**emitting a Helmholtz scalar wave field, onto a point drain at

**Q**. To our knowledge this is the first example of such a system with a positive refractive index. Apart from the difficulty of making the 3D dipole lens (because eventually

*n*(

**r**) goes below 1), they have a clear theoretical interest. Perfect focusing of Helmholtz fields in 3D is particularly relevant to acoustics (for example, to ultrasound imaging), turning the sound waves of an explosion into an implosion.

**P**located in any arbitrary position, which makes it a perfect imaging device in two dimensions [5

**11**(9), 093040 (2009). [CrossRef]

*V*(

**r**) is selected as that produced by several electric charges.

## Acknowledgments

## References and links

1. | M. Born, and E. Wolf, |

2. | S. Cornbleet, |

3. | J. C. Miñano, “Perfect imaging in a homogeneous threedimensional region,” Opt. Express |

4. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

5. | U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. |

6. | U. Leonhardt and T. G. Philbin, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A |

7. | M. A. Alonso and G. Forbes, “Stable aggregates of flexible elements give a stronger link between rays and waves,” Opt. Express |

8. | O. N. Stavroudis, |

9. | Yu. A. Kratsov, and Yu. I. Orlov, |

10. | R. K. Luneburg, |

11. | P. M. Morse, and H. Feshbach, |

12. | A. D. Polyanin, and V. F. Zaitsev, |

13. | |

14. | http://www.mathcurve.com/courbes2d/cayleyovale/cayleyovale.shtml |

15. | Yu. I. Bobrovnitskiĭ, “Impedance theory of sound absorption: The best absorber and the black body,” Acoust. Phys. |

**OCIS Codes**

(110.2760) Imaging systems : Gradient-index lenses

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 11, 2010

Revised Manuscript: March 16, 2010

Manuscript Accepted: March 19, 2010

Published: March 29, 2010

**Citation**

Pablo Benítez, Juan C. Miñano, and Juan C. González, "Perfect focusing of scalar wave fields in three dimensions," Opt. Express **18**, 7650-7663 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-7650

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### References

- M. Born, and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975, 5th ed)
- S. Cornbleet, Microwave and Geometrical Optics, (Academic, London, 1994)
- J. C. Miñano, “Perfect imaging in a homogeneous threedimensional region,” Opt. Express 14(21), 9627–9635 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-21-9627 . [CrossRef] [PubMed]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- U. Leonhardt, “Perfect imaging without negative refraction,” N. J. Phys. 11(9), 093040 (2009). [CrossRef]
- U. Leonhardt and T. G. Philbin, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A 81(1), 011804 (2010). [CrossRef]
- M. A. Alonso and G. Forbes, “Stable aggregates of flexible elements give a stronger link between rays and waves,” Opt. Express 10(16), 728–739 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-16-728 . [PubMed]
- O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, (Willey-VCH, Weinheim, 2006)
- Yu. A. Kratsov, and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, p.23, Springer Verlag, Berlin Heidelberg, 1990.
- R. K. Luneburg, Mathematical Theory of Optics, (University of California Press, Los Angeles 1964)
- P. M. Morse, and H. Feshbach, Methods of Theoretical Physics (New York, McGraw-Hill, 1953)
- A. D. Polyanin, and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC, Boca Raton, 2003. It also coincides with the change of variables S’ = S/L + π/2 in the equation described in http://eqworld.ipmnet.ru/en/solutions/ode/ode0235.pdf .
- http://www.youtube.com/watch?v=bG9XSY8i_q8
- http://www.mathcurve.com/courbes2d/cayleyovale/cayleyovale.shtml
- Yu. I. Bobrovnitskiĭ, “Impedance theory of sound absorption: The best absorber and the black body,” Acoust. Phys. 52(6), 638–647 (2006). [CrossRef]

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