## Transformation-designed optical elements

Optics Express, Vol. 15, Issue 22, pp. 14772-14782 (2007)

http://dx.doi.org/10.1364/OE.15.014772

Acrobat PDF (402 KB)

### Abstract

We describe transformation design of optical elements which, in addition to image transfer, perform useful operations. For one class of operations, including translation, rotation, mirroring and inversion, an image can be generated that is ideal in the sense of the perfect lens (combining both near- and far-field components in a flat, unit transfer function, up to the limits imposed by material imperfection). We also describe elements that perform magnification, free from geometric aberrations, even while providing free-space working distance on both the input and output sides. These magnifying elements also operate in the near- and far-field, allowing them to transfer near field information into the far field, as with the hyper lens and other related devices, however in contrast to those devices, insertion loss can be much lower, due to the matching properties accessible with transformation design. The devices here described inherently require dispersive materials, thus chromatic aberration will be present, and the bandwidth limited.

© 2007 Optical Society of America

## 1. Introduction

1. J. Pendry, D. Schurig, and D. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–2 (2006). [CrossRef] [PubMed]

2. D. Schurig, J. Pendry, and D. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794–9804 (2006). [CrossRef] [PubMed]

3. G. Milton, M. Briane, and J. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. **8**, 248 (2006). [CrossRef]

4. S. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. **9**, 45 (2007). [CrossRef]

5. B. Wood and J. Pendry, “Metamaterials at zero frequency,” J. Phys., Condens. Matter. **19**, 076208 (2007). [CrossRef]

6. W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, “Optical cloaking with metamaterials,” **1**, 224–227Nature Photonics (2007). [CrossRef]

7. M. Silveirinha, A. Alu, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. **75**, 36603 (2007). [CrossRef]

8. F. Teixeira, “Closed-form metamaterial blueprints for electromagnetic masking of arbitrarily shaped convex PEC objects,” IEEE Antennas Wirel. Propag. Lett. **6**, 163–4 (2007). [CrossRef]

9. F. Zolla, S. Guenneau, A. Nicolet, and J. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. **32**, 1069–71 (2007). [CrossRef] [PubMed]

10. D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science **314**, 977–80 (2006). [CrossRef] [PubMed]

11. J. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–9 (2000). [CrossRef] [PubMed]

12. J. Pendry and S. Ramakrishna, “Near-field lenses in two dimensions,” J. Phys., Condens. Matter. **14**, 8463–79 (2002). [CrossRef]

13. J. Pendry, “Perfect cylindrical lenses,” Opt. Express **11**, 755–760 (2003). [CrossRef] [PubMed]

14. Z. Jacob, L. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–56 (2006). [CrossRef] [PubMed]

15. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev., B, Condens, Matter Mater. Phys. **74**, 75103 (2006). [CrossRef]

16. D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. **90**, 077405 (2003). [CrossRef] [PubMed]

17. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science **315**, 1686 (2007). [CrossRef] [PubMed]

18. I. Smolyaninov, Y.-J. Hung, and C. Davis, “Magnifying superlens in the visible frequency range,” Science **315**, 1699–701 (2007). [CrossRef] [PubMed]

19. G. Shvets, S. Trendafilov, J. Pendry, and A. Sarychev, “Guiding, focusing, and sensing on the subwavelength scale using metallic wire arrays,” Phys. Rev. Lett. **99**, 053903 (2007). [CrossRef] [PubMed]

## 2. Optical design geometry

*z*-axis in our Cartesian coordinate system. Optical elements are considered to extend in the transverse (

*x*,

*y*) directions sufficiently far that the electromagnetic fields at the transverse edges of an element are assumed to be negligible. To verify this condition is a matter of practical importance, but an encumbrance to be shed in a first analysis. This assumption provides a particular freedom when using the transformation method to design perfectly matched, (i.e. zero reflecting for all angles of incidence), elements. With this method one must use a coordinate transformation that is everywhere continuous to achieve perfectly-matched behavior. Since a finite sized material object is by definition completely surround by free space, and free space is described by a “flat” Cartesian coordinate system, a finite-sized, perfectly-matched material object must be described by a coordinate transformation that is continuous with flat Cartesian space, everywhere along its outer boundary. We will relax this condition in the transverse direction of our optical elements, and design elements that are perfectly matched for electromagnetic fields that are near the optic axis and sufficiently far from the transverse edges, using coordinate transformations that cannot be continuous to flat space on their transverse edges. This extra freedom allows for some new functionality in manipulating fields along the optic axis.

## 3. Transformation design

1. J. Pendry, D. Schurig, and D. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–2 (2006). [CrossRef] [PubMed]

2. D. Schurig, J. Pendry, and D. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express **14**, 9794–9804 (2006). [CrossRef] [PubMed]

*x*are the reference coordinates, (usually representing flat space), and

^{i}*x*are the coordinates of the manipulated space. In these new coordinates Maxwell’s equations are identical in form, but the linear constitutive equations describing the medium are changed. An alternative viewpoint of these changes is that one is manipulating the material properties in an unchanged, flat space. The two viewpoints can account for the same electromagnetic behavior. The manipulation-of-space viewpoint is a useful design picture and the manipulation-of-material viewpoint leads to an implementable medium specification. These material properties are given by

^{i'}## 4. Perfect lens

24. U. Leonhardt and T. Philbin, “General relativity in Electrical Engineering,” New J. Phys. **8**, 247 (2006). [CrossRef]

*d*is the thickness of the lens. The

*z*-component of this transform is shown as the solid blue line in Fig. 2. Using Eq. 3, we calculate the transformation matrix inside the lens, -

*d*/2 <

*z*' <

*d*/2, where the material properties differ from free space

*∂z'*/

*∂z*= - 1, inside the lens, but this need not be the case. Slopes differing from ± 1 lead to anisotropic material properties, but conveniently allow the combined focal distance to differ from being equal the lens thickness. In fact, the mapping need not be linear; any continuous relation not multi-valued in

*z*will lead to finite and unique material properties. The key design parameters of this relation are the values of

*z'*that map to the same

*z*value. For example, in Eq. 4c,

*z'*= -

*d*,0,

*d*all map to

*z*= 0. These points indicate the locations of one of an infinite set of object, and image planes. Below it will be necessary to align an internal image plane with a discontinuity in the transformation of the

*x*-and

*y*-coordinates.

## 5. Invariant operations

*z*coordinate as above, and

*θ*(

*z'*) is a continuous function with

*θ*(-

*d*/2) = 0 and

*θ*(

*d*/2) is the desired rotation, (Fig. 2). Note that the coordinate transformation is continuous on the left edge of the element,

*z'*= -

*d*/2, but not on the right,

*z'*=

*d*/2. However, since the properties of the space are invariant for this rotation, this discontinuity is a property of the choice of coordinates and not the space itself, thus no physical manifestation, such as reflection or refraction, can result. The mapping for a translation is shown in Fig. 1C. The operations of mirroring and inversion may present some practical problems since the most obvious transformation results in all the field energy incident on the element being mapped through a line or point respectively.

## 6. Magnification

*M*, one can have perfect lens image fidelity, when the image plane lies in a medium of refractive index

*n*/

*M*, where

*n*is the refractive index of the medium containing the object. This constraint is not convenient, as one often wishes to have the object and image planes both in free space. There is an intrinsic media mismatch when projecting a magnified image into the same medium as the object. One can however, still perform this function with no aberrations and minimal reflection by judicious positioning of the mismatch discontinuity. The best position, for reasons described below, is at a focal plane. However, choosing the object or

*external*image plane precludes having any free space working distance on the input or output side respectively. Fortunately, internal image planes can be created when negative material properties are employed. The transformation for a magnification centered on

*x*=

*y*= 0 is given by

*m*(

*z'*) varies continuously from 1 to the desired magnification,

*M*, as shown in Fig. 2, (green line). This configuration results in an image at

*z*=

*d*/2 that is magnified by a factor

*M*relative to the object at

*z*= -

*d*/2 (Fig. 1D).

25. V. Mahajan, *Optical imaging and aberrations* (SPIE Optical Engineering Press, 1998). [CrossRef]

26. D. Schurig and D. Smith, “Negative index lens aberrations,” Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. **70**, 65601 (2004). [CrossRef]

*at*the interface, it appears to be a point source from both sides - no geometric aberrations are introduced of any order (Fig. 3).

*z*-oriented plane within the lens is an internal image plane, corresponding to an external image and object plane. By positioning the discontinuity of the

*x*-

*y*coordinate transformation at a particular location (in our case

*z'*= 0), one chooses which one of these focal plane sets is going to be aberration free.

## 7. Transfer function

*z*= 0), thus reflection can only occur at this location. One can use the following theorem to easily calculate the reflection and transmission at this interface. If material

*A*is perfectly matched to material

*B*, and material

*C*is perfectly matched to material

*D*, then the reflection and transmission properties of the interface

*A*-

*C*are identical to those of the interface

*B*-

*D*. The material on the left side of the interface (at

*z*= 0

^{-}) is perfectly matched to the isotropic, homogenous medium with

*ϵ*=

*μ*= 1 /

*M*, and the material on the right side of the interface (at

*z*= 0

^{+}) is perfectly matched to free space,

*ϵ*=

*μ*= 1. From standard boundary matching the Fresnel formula for the transmission is

*z*-component of the wave vector on the right side,

*k*

_{z-}, and left side,

*k*

_{z+}, are given by

*k*, refers to the transverse wave vector at the object plane, (not the transverse wave vector at the interface and image plane, which is a factor of

_{x}*M*smaller), and

*k*

_{0}=

*ω*/

*c*is the free space wave vector magnitude. Note that standard Fresnel reflection and transmission formulas are based on uniform plane wave solutions that occur only in homogenous media. If we allow the magnification function,

*m*(

*z'*) to vary right up to the interface as shown in Fig. 2, these formulas do not strictly apply. However, if this leads to a degradation in the transfer function one can always make this function constant near the interface, thus specifying a homogenous layer there.

*k*=

_{x}*M*

*k*

_{0}. The total theoretical transfer function is a combination of Eq. 9 and the factor 1/

*M*. This later factor results from the reflectionless magnification that occurs prior to the interface, and is derivable from conservation of energy.

## 8. Material properties

*f*, to differ from the half-thickness,

*d*/2[20

20. D. Schurig and D. Smith, “Sub-diffraction imaging with compensating bilayers,” New J. Phys. **7**, 162 (2005). [CrossRef]

*z*-

*z'*coordinate map, (Fig. 2, dashed line.) The second means is a restriction of the input aperture,

*D*, of the element, Fig. 5; the material properties are most extreme at the front surface of the element, at the aperture boundary. One can minimize the required material properties at this boundary for a given magnification,

*M*, and f-number,

*F*=

*f*/

*D*. Performed analytically, this minimization is not particularly tractable, but can be handled numerically quite easily. One finds that for small values of

*F*/

*M*, the minimizing f-number,

*f*

_{min}, is given by

^{i'}

*. Then inverting Eq. 13, to re-express it in terms of the primed coordinates (the coordinates of the material specification), one finds*

_{j}*z*-coordinate with the transverse coordinates. Using Eq. 13 and 14, one can calculate the material property tensor which is complicated and the display of which is not particularly illuminating. For further analysis, (or implementation), one must diagonalize the material property tensor. Since Eq. 3 yields real, symmetric matrices (when real valued coordinate transformations are used), this diagonalization can always be accomplished and an ortho-normal basis found. In this case, only one of the eigenvalues has a simple expression,

*n*

_{1}represents the common value of the permittivity and permeability,

*n*

_{1}=

*ϵ*

_{1}=

*μ*

_{1}. For the parameters explored, all three eigenvalues are negative, and

*n*

_{1}and

*n*

_{3}lie between -1 and 0. These values are plotted in Fig. 5A, using the three color channels (red, green, blue) to indicate their magnitude. The right hand half of the element is a uniform gray with the material properties given by Eq. 6.

## 9. Conclusion

## References and links

1. | J. Pendry, D. Schurig, and D. Smith, “Controlling electromagnetic fields,” Science |

2. | D. Schurig, J. Pendry, and D. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express |

3. | G. Milton, M. Briane, and J. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. |

4. | S. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. |

5. | B. Wood and J. Pendry, “Metamaterials at zero frequency,” J. Phys., Condens. Matter. |

6. | W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, “Optical cloaking with metamaterials,” |

7. | M. Silveirinha, A. Alu, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. |

8. | F. Teixeira, “Closed-form metamaterial blueprints for electromagnetic masking of arbitrarily shaped convex PEC objects,” IEEE Antennas Wirel. Propag. Lett. |

9. | F. Zolla, S. Guenneau, A. Nicolet, and J. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. |

10. | D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science |

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

12. | J. Pendry and S. Ramakrishna, “Near-field lenses in two dimensions,” J. Phys., Condens. Matter. |

13. | J. Pendry, “Perfect cylindrical lenses,” Opt. Express |

14. | Z. Jacob, L. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express |

15. | A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev., B, Condens, Matter Mater. Phys. |

16. | D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. |

17. | Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

18. | I. Smolyaninov, Y.-J. Hung, and C. Davis, “Magnifying superlens in the visible frequency range,” Science |

19. | G. Shvets, S. Trendafilov, J. Pendry, and A. Sarychev, “Guiding, focusing, and sensing on the subwavelength scale using metallic wire arrays,” Phys. Rev. Lett. |

20. | D. Schurig and D. Smith, “Sub-diffraction imaging with compensating bilayers,” New J. Phys. |

21. | V. Shalaev, “Optical negative-index metamaterials,” |

22. | C. Soukoulis, S. Linden, and M. Wegener, “Negative Refractive Index at Optical Wavelengths,” Science |

23. | H. Lezec, J. Dionne, and H. Atwater, “Negative refraction at visible frequencies,” Science |

24. | U. Leonhardt and T. Philbin, “General relativity in Electrical Engineering,” New J. Phys. |

25. | V. Mahajan, |

26. | D. Schurig and D. Smith, “Negative index lens aberrations,” Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. |

**OCIS Codes**

(070.4690) Fourier optics and signal processing : Morphological transformations

(100.6640) Image processing : Superresolution

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: September 14, 2007

Revised Manuscript: October 15, 2007

Manuscript Accepted: October 17, 2007

Published: October 24, 2007

**Citation**

D. Schurig, J. B. Pendry, and D. R. Smith, "Transformation-designed optical elements," Opt. Express **15**, 14772-14782 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14772

Sort: Year | Journal | Reset

### References

- J. Pendry, D. Schurig, and D. Smith, "Controlling electromagnetic fields," Science 312, 1780-2 (2006). [CrossRef] [PubMed]
- D. Schurig, J. Pendry, and D. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (2006). [CrossRef] [PubMed]
- G. Milton, M. Briane, and J. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New J. Phys. 8, 248 (2006). [CrossRef]
- S. Cummer and D. Schurig, "One path to acoustic cloaking," New J. Phys. 9, 45 (2007). [CrossRef]
- B. Wood and J. Pendry, "Metamaterials at zero frequency," J. Phys., Condens. Matter. 19, 076208 (2007). [CrossRef]
- W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, "Optical cloaking with metamaterials," 1, 224-227 Nature Photonics (2007). [CrossRef]
- M. Silveirinha, A. Alu, and N. Engheta, "Parallel-plate metamaterials for cloaking structures," Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 75, 36603 (2007). [CrossRef]
- F. Teixeira, "Closed-form metamaterial blueprints for electromagnetic masking of arbitrarily shaped convex PEC objects," IEEE Antennas Wirel. Propag. Lett. 6, 163-4 (2007). [CrossRef]
- F. Zolla, S. Guenneau, A. Nicolet, and J. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-71 (2007). [CrossRef] [PubMed]
- D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-80 (2006). [CrossRef] [PubMed]
- J. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-9 (2000). [CrossRef] [PubMed]
- J. Pendry and S. Ramakrishna, "Near-field lenses in two dimensions," J. Phys., Condens. Matter. 14, 8463-79 (2002). [CrossRef]
- J. Pendry, "Perfect cylindrical lenses," Opt. Express 11, 755-760 (2003). [CrossRef] [PubMed]
- Z. Jacob, L. Alekseyev, and E. Narimanov, "Optical hyperlens: far-field imaging beyond the diffraction limit," Opt. Express 14,8247-56 (2006). [CrossRef] [PubMed]
- A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations," Phys. Rev., B, Condens, Matter Mater. Phys. 74, 75103 (2006). [CrossRef]
- D. Smith and D. Schurig, "Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors," Phys. Rev. Lett. 90, 077405 (2003). [CrossRef] [PubMed]
- Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007). [CrossRef] [PubMed]
- I. Smolyaninov, Y.-J. Hung, and C. Davis, "Magnifying superlens in the visible frequency range," Science 315, 1699-701 (2007). [CrossRef] [PubMed]
- G. Shvets, S. Trendafilov, J. Pendry, and A. Sarychev, "Guiding, focusing, and sensing on the subwavelength scale using metallic wire arrays," Phys. Rev. Lett. 99, 053903 (2007). [CrossRef] [PubMed]
- D. Schurig and D. Smith, "Sub-diffraction imaging with compensating bilayers," New J. Phys. 7, 162 (2005). [CrossRef]
- V. Shalaev, "Optical negative-index metamaterials," 1, 41-48 Nature Photonics (2007). [CrossRef]
- C. Soukoulis, S. Linden, and M. Wegener, "Negative Refractive Index at Optical Wavelengths," Science 315, 47-9 (2007). [CrossRef] [PubMed]
- H. Lezec, J. Dionne, and H. Atwater, "Negative refraction at visible frequencies," Science 316, 430-2 (200). [PubMed]
- U. Leonhardt and T. Philbin, "General relativity in Electrical Engineering," New J. Phys. 8, 247 (2006). [CrossRef]
- V. Mahajan, Optical imaging and aberrations (SPIE Optical Engineering Press, 1998). [CrossRef]
- D. Schurig and D. Smith, "Negative index lens aberrations," Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 70, 65601 (2004). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.