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.

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References

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  1. M. Born, and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975, 5th ed)
  2. S. Cornbleet, Microwave and Geometrical Optics, (Academic, London, 1994)
  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]
  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]
  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]
  8. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, (Willey-VCH, Weinheim, 2006)
  9. Yu. A. Kratsov, and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media, p.23, Springer Verlag, Berlin Heidelberg, 1990.
  10. R. K. Luneburg, Mathematical Theory of Optics, (University of California Press, Los Angeles 1964)
  11. P. M. Morse, and H. Feshbach, Methods of Theoretical Physics (New York, McGraw-Hill, 1953)
  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 .
  13. http://www.youtube.com/watch?v=bG9XSY8i_q8
  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. 52(6), 638–647 (2006).
    [CrossRef]

2010 (1)

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

2009 (1)

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

2006 (2)

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

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]

2002 (1)

2000 (1)

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

Alonso, M. A.

Bobrovnitskii, Yu. I.

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

Forbes, G.

Leonhardt, U.

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

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

Miñano, J. C.

Pendry, J. B.

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

Philbin, T. G.

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

Acoust. Phys. (1)

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

N. J. Phys. (1)

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

Opt. Express (2)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (1)

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

Other (9)

M. Born, and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975, 5th ed)

S. Cornbleet, Microwave and Geometrical Optics, (Academic, London, 1994)

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

Supplementary Material (1)

» Media 1: MOV (1094 KB)     

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Figures (4)

Fig. 1
Fig. 1

Cross section at z = constant of the σ (in blue) and τ (in red) isosurfaces of the decentered cylindrical bipolar coordinate system.

Fig. 2
Fig. 2

Section at y = 0 showing the wavefronts (in red) and rays (in blue) associated with the 3D dipole lens. The system has rotational symmetry with respect to the z axis. Unlike in the 2D dipole lens, neither is it true that the ray trajectories are arc of circumferences nor that the wavefront surfaces are spheres.

Fig. 3
Fig. 3

Normalized refractive index distribution (a/L)n(ρ,z) of the selected example.

Fig. 4
Fig. 4

(Media 1) Real part of U(S) of selected 3D dipole lens with ωt = 1.033π, a = 1, L = 1, k = 9.5 on the plane y = 0. Due to the rotational symmetry, this graph also applies for any other plane containing the z axis.

Equations (54)

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Δ U ( r ) + k 2 n 2 ( r ) U ( r ) = 0
( S ) 2 = n 2 ( r )
( S ) 2 2 U S 2 + ( u ) 2 2 U u 2 + ( v ) 2 2 U v 2 + Δ S U S + Δ u U u + Δ v U v + k 2 n 2 U = 0
n 2 d 2 U d S 2 + Δ S d U d S + k 2 n 2 U = 0
Δ S n 2 = F ( S )   (independent of  u  and  v )
d 2 U d S 2 + F ( S ) d U d S + k 2 U = 0
Δ S ( S ) 2 = F ( S )     (independent of  u  and  v )
( Δ S ( S ) 2 ) × S = 0
I ( r ) = 1 2 i k ( U * U U U * ) = 1 2 i k ( U * U S U U S * ) S
Δ V = V S S ( S ) 2 + V S Δ S = 0 Δ S ( S ) 2 = ( ln V S ) S
F ( S ) = ( ln V S ) S
n ( ρ , z ) = 2 a L a 2 + ρ 2
x = a ( cosh α sinh τ cosh τ cos σ sinh α ) y = a cosh α sin σ cosh τ cos σ z = z
where  σ { π 2 , π 2 } , τ , z R
= ( cosh τ cos σ a cosh α σ , cosh τ cos σ a cosh α τ , z )
Δ = ( cosh τ cos σ ) 2 a 2 cosh 2 α [ 2 σ 2 + 2 τ 2 ] + 2 z 2
n ( σ , τ , z ) = L ( cosh τ cos σ ) a cosh α cosh ( τ α )
d S d τ = ± L cosh ( τ α ) S ( τ ) = ± 2 L arctan ( exp ( τ α ) ) + C
τ ( S ) = α + ln ( tan ( S 2 L ) )
Δ S = ( cosh τ cos σ ) 2 a 2 cosh 2 α d 2 S d τ 2 = L a 2 ( cosh τ cos σ ) 2 cosh 2 α sinh ( τ α ) cosh 2 ( τ α )
Δ S ( S ) 2 = 1 L sinh ( τ α ) = 1 L sinh ( ln ( tan ( S 2 L ) ) ) = 1 L tan ( S L ) = F ( S )
d 2 U d S 2 + 1 L tan ( S L ) d U d S + k 2 U = 0
d 2 U d p 2 + 1 p d U d p + ( k L ) 2 ( 2 1 + p 2 ) 2 U = 0
V ( S ) = A d S sin ( S L ) = A L ( ln ( tan ( S 2 L ) ) + B ) = τ ( S )
V ( x , y , z ) = 1 2 ln ( ( x + a ) 2 + y 2 ( x a ) 2 + y 2 )
n = | S | = S V | V | ~ 1 S V ~ 1 | V | ~ r ~ exp ( | V | )
x = a cosh α sin σ cosh τ cos σ cos ϕ y = a cosh α sin σ cosh τ cos σ sin ϕ z = a sinh α + a cosh α sinh τ cosh τ cos σ
( S ) 2 = ( cosh τ cos σ ) 2 a 2 cosh 2 α ( ( S σ ) 2 + ( S τ ) 2 + 1 sin 2 σ ( S ϕ ) 2 ) = n 2 ( σ , τ , ϕ )
n ( r ) = 2 a L a 2 + r 2 = L ( cosh τ cos σ ) a cosh α cosh ( τ α )
Δ S = ( cosh τ cos σ ) 3 a 2 cosh 2 α τ ( 1 cosh τ cos σ d S d τ ) =     = L ( cosh τ cos σ ) a 2 cosh 2 α cosh ( τ α ) ( ( cosh τ cos σ ) sinh ( τ α ) cosh ( τ α ) + sinh τ )
V ( ρ , z ) = a ρ 2 + ( z a ) 2 a ρ 2 + ( z + a ) 2
S V ~ 1 | V | ~ r 2 ~ | V | 2
S ( V ) = L ( π 2 + arctan ( V ) )
n ( ρ , z ) = | S | = 2 L a a 4 + a 2 ( 2 ρ 2 + 6 z 2 + s ) + ( ρ 2 + z 2 ) ( ρ 2 + z 2 s ) ( s a ) 2 + 2 ( a 2 + ρ 2 + z 2 s )
F ( S ) = ( ln V S ) S = 2 L tan ( S L ) = 2 L V ( S )
d 2 U d S 2 + 2 L tan ( S L ) d U d S + k 2 U = 0
U ( S ) = C 1 sin ( κ S ) + C 2 cos ( κ S ) sin ( S L )
κ = k 1 + ( 1 k L ) 2
U ( S ) = e i κ S 4 π a sin ( S L )
I ( r ) = κ k S ( 4 π a ) 2 sin 2 ( S L )
U ( ρ , z ) = 1 4 π a 1 + V 2 ( ρ , z ) e i κ L ( π 2 + arctan ( V ( ρ , z ) ) )
U ( S ) e i κ S 4 π a ( S / L )    (around  P ) U ( S ) e i κ S 4 π a ( π ( S / L ) )    (around  Q )
S ( ρ , z ) L a ρ 2 + ( z + a ) 2    (around  P ) S ( ρ , z ) π L L a ρ 2 + ( z a ) 2    (around  Q )  
r = ρ 2 + ( z + a ) 2 r ' = ρ 2 + ( z a ) 2
U ( ρ , z ) e i κ 1 r 4 π r   (around  P ) U ( ρ , z ) e i κ π L e i κ 1 r ' 4 π r '   (around  Q )
κ 1 = κ L a
( Δ + k 2 ) e i κ 1 r 4 π r d V Δ ( e i κ 1 r 4 π r ) d V = ( e i κ 1 r 4 π r ) d S 1 4 π r 2 r d S = 1 ( Δ + k 2 ) e i κ π L i κ 1 r ' 4 π r ' d V Δ ( e i κ π L i κ 1 r ' 4 π r ' ) d V = ( e i κ π L i κ r ' 4 π r ' ) d S e i κ π L 4 π r ' 2 r ' d S = e i κ π L
Δ U + k 2 n 2 ( r ) U = δ ( r P ) e i κ π L δ ( r Q )
W ( ρ , z ) = e i κ S ( ρ , z ) 4 π a sin ( S ( ρ , z ) L ) + e i κ ( 2 π L S ( ρ , z ) ) 4 π a sin ( 2 π S ( ρ , z ) L )             = e i κ π L 2 π a sin ( κ ( π L S ( ρ , z ) ) ) sin ( S ( ρ , z ) L )
W ( ρ , z ) e i κ π L 2 π sin ( κ 1 r ' ( ρ , z ) ) r ' ( ρ , z )
| W ( Q ) | = κ 2 π
W ( ρ , z ) e i κ π L 2 π sin ( κ 1 ( π a r ( ρ , z ) ) ) r ( ρ , z )
Δ W + k 2 n 2 ( r ) W = 2 e i κ π L sin ( κ π L ) δ ( r P )
r 0 π κ 1 = λ n max 2 + ( λ π a ) 2 λ n max

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