## Rigorous concept for the analysis of diffractive lenses with different axial resolution and high lateral resolution

Optics Express, Vol. 11, Issue 17, pp. 1987-1994 (2003)

http://dx.doi.org/10.1364/OE.11.001987

Acrobat PDF (362 KB)

### Abstract

We present a rigorous electromagnetic design and analysis of two-dimensional diffractive lenses (DLs) with different axial resolution and high lateral resolution. Without paraxial approximation, focusing characteristics of two kinds of DL, one with a long focal depth and a high lateral resolution, the other with high axial resolution and high lateral resolution, for *f*-numbers of 0.6, 1.0, 1.5, and 2.0 have been determined including the actual focal depth, the ratio between the focal depth of the designed DL and the focal depth of the conventional quadratic lens, and the spot size of the central lobe at the actual focal plane. Numerical and graphic results show that the designed DLs indeed have a long focal depth and a high lateral resolution, or high axial resolution and high lateral resolution by use of different preset focal depths.

© 2003 Optical Society of America

## 1. Introduction

1. J. Sochacki, S. Bara, Z. Jaroszewicz, and A. Kolodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. **17**, 7–9 (1992). [CrossRef] [PubMed]

2. N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. **16**, 523–525 (1991). [CrossRef] [PubMed]

3. P. Varga, “Use of confocal microscopes in conoscopy and ellipsometry. 1. Electromagnetic theory,” Appl. Opt. **39**, 6360–6365 (2000). [CrossRef]

4. C. Yang and J. Mertz, “Transmission confocal laser scanning microscopy with a virtual pinhole based on nonlinear detection,” Opt. Lett. **28**, 224–226 (2003). [CrossRef] [PubMed]

5. P. Torok and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. **137**, 127–135 (1997). [CrossRef]

6. J. S. Ye, B. Z. Dong, B. Y. Gu, G. Z. Yang, and S.-T. Liu, “Analysis of a closed-boundary axilens with long focal depth and high transverse resolution based on rigorous electromagnetic theory,” J. Opt. Soc. Am. A **19**, 2030–2035 (2002). [CrossRef]

7. B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, and J. Wang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A **18**, 1465–1470 (2001). [CrossRef]

8. B. Lichtenberg and N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. **33**, 3518–3526 (1994). [CrossRef]

9. K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A **13**, 2219–2231 (1996). [CrossRef]

10. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in *Diffractive and Holographic Optics Technology II*, I. Cindrich and S. H. Lee, eds., Proc. SPIE2404, 28–39 (1995). [CrossRef]

11. D. W. Prather and S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A **16**, 1131–1142 (1999). [CrossRef]

13. S. D. Mellin and G. P. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express **8**, 705–722 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-13-705. [CrossRef] [PubMed]

## 2. Theoretical formulas

*E*,

_{z}*H*,

_{x}*H*are the electric field and magnetic field components, respectively, which are the functions of

_{y}*x, y*(space components), and

*t*(time component),

*σ*and

*σ** are the medium’s electric conductivity and magnetic conductivity, respectively. In general, the medium of DLs is not magnetic, so the relationship is satisfied as follows:

*µ*

_{0}=

*µ*,

*σ**=0. By use of Yee’s grids and applying the central different expressions, Eq. (1) can be calculated by a digital computer. Because the extent of the FDTD region (a computational region in the FDTD method) is limited and to be able to simulate unbounded free space, absorbing boundary conditions must be along the FDTD region in which the DL is totally embedded. The perfectly matched layer absorbing boundary conditions proposed by Berenger [14

14. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

16. D. W. Prather, S. Y. Shi, and J. Sonsrtoem, “Electromagnetic analysis of finite-thickness diffractive elements,” Opt. Eng. **41**, 1792–1796 (2002). [CrossRef]

12. J. Jiang and G. P. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express **7**, 237–242 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-6-237. [CrossRef] [PubMed]

13. S. D. Mellin and G. P. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express **8**, 705–722 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-13-705. [CrossRef] [PubMed]

*y*=

*y*

_{0}) making a phase delay caused by the transform function as in Eq. (2):

*f*and

_{x}*cos*(

*α*) are the AS spatial frequency and the direction cosine in the

*x*direction, respectively. By use of powerful tools such as the FDTD method and the AS propagation method, we can design and analyze two kinds of DL as described in the following section.

## 3. Numerical experiments

### 3.1 Profile distribution of diffractive lenses

*n*

_{1}and

*n*

_{2}are the refractive indices of the medium and the air of lenses, respectively;

*n*

_{1}>

*n*

_{2};

*λ*=

*λ*

_{0}/

*n*

_{2};

*λ*

_{0}is the incident wavelength in vacuum;

*m*is the number of zones;

*D*and

*f*are the diameter and the focal length of DLs, respectively. The focal length is usually a constant for conventional DLs, but, to design DLs with different axial resolutions, we set the focal length as a continuous function that was suggested by Davidson

*et al.*[2

2. N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. **16**, 523–525 (1991). [CrossRef] [PubMed]

*f*

_{0}and

*d*are the beginning focal length and the preset focal depth. So substituting Eq. (4) into Eq. (3), we obtained the profile distribution of DLs in the nonparaxial approximation as

_{f}*df*equals zero. By using different preset focal depths, we obtained a DL with a different actual focal depth, which can be verified as described below.

### 3.2 Numerical simulations

7. B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, and J. Wang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A **18**, 1465–1470 (2001). [CrossRef]

*n*

_{1}=1.5) and the outside medium is air (refractive index

*n*

_{2}=1.0). The FDTD method requires sampling the calculated region at λ/20 intervals, and wavelength λ in air is 1

*µm*. The

*f*-number of a DL is defined as the ratio of its beginning focal length

*f*

_{0}to its diameter

*D*(

*f*/#=

*f*

_{0}/

*D*). We designed four families of DLs with

*f*-numbers of 0.6, 1.0, 1.5, and 2.0, all having a 12-

*µm*diameter, resulting in DL focal lengths of 7.2, 12, 18, and 24

*µm*. To obtain lenses with a different axial resolution, we set the preset focal length of

*df*as ±0.3

*f*

_{0}and ±0.6

*f*

_{0}. With a positive value we obtained a DL with low axial resolution and with a negative value we obtained a DL with high axial resolution. So a higher axial resolution means a smaller (<1)

*Ra*value, and a lower axial resolution means a larger (>1)

*Ra*value.

*f*/1.0 by using the FDTD method and the AS method as shown in Fig. 1 for different values of preset focal depths

*df*. For comparison, the axial intensity distribution of a conventional DL (

*df*=0

*µm*) was also calculated and is represented by curve a in Fig.1. Curve b corresponds to

*df*=-3.6

*µm*and curve c corresponds to

*df*=+3.6

*µm*(0.3

*f*

_{0}). It is obvious that the shape of curve b is narrower than that of curve a, and the shape of curve c is wider than that of curve a. This means that, when setting a positive

*df*, we can design a DL with a longer focal depth, i.e., a lower axial resolution, and this conclusion is in good agreement with that in Ref. 7

7. B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, and J. Wang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A **18**, 1465–1470 (2001). [CrossRef]

*df*is negative, the designed DL will have a shorter focal depth (i.e., a higher axial resolution). The focal depths of curves a, b, and c are 4.2, 3.4, and 6.8

*µm*, respectively. Values of

*Ra*for two lenses are 0.80 and 1.62. The actual focal planes, meaning the position that corresponds to the maximal intensity peak for curves a, b, and c in Fig. 1, are located at

*y*=10.2, 7.9, and 11.3

*µm*, respectively. The lateral intensity distributions at the focal planes for these three DLs are shown in Fig. 2. The sizes of the central lobes are 1.14, 0.97, and 1.24

*µm*, which are almost the same as the diffraction-limited spot size

*λf*

_{0}/

*D*=1

*µm*. The lateral intensity profiles of the designed DLs at different observation planes within the focal depth region are plotted in Fig. 3: (a)

*df*=-3.6

*µm*and (b)

*df*=+3.6

*µm*(dashed curves correspond to

*y*=6.6 and 8.6

*µm*, solid curves represent the actual focal plane, and the dotted curves correspond to

*y*=9.8 and 15.0

*µm*). The plots clearly demonstrate that the designed DLs represent good focusing characteristics, indicating that the total intensity profiles of the electric fields are always focused on the central lobe, which illustrates that the designed DLs have high lateral resolutions. To obtain a regional view of both the axial resolution and the lateral resolution, we display the propagation plots of electric field intensity over a region surrounded by focal points in Fig. 4: a,

*df*=-3.6

*µm*; b,

*df*=0

*µm*(a conventional lens); c,

*df*=+3.6

*µm*. The bright regions correspond to high field values, and the dark regions correspond to low values. As meant by figures, we believe that our designed DLs have special functions of different axial resolutions and high lateral resolutions.

*f*-number at

*f*/2.0 for this design, and the important features of the DL remained unchanged. Figure 5 shows the axial direction intensity of the DL for

*f*/2.0 with a different value for

*df*: curve a, a conventional DL (

*df*=0

*µm*); curve b,

*df*=-14.4

*µm;*curve c, +14.4

*µm*(0.6

*f*

_{0}). The actual focal depths for three DLs are 11.1, 4.1, and 17.2 µm, and the

*Ra*values for curves b and c are 0.37 and 1.55, respectively.

*df*. The actual focal planes for curves a, b, and c appear at

*y*=17.3, 10.0, and 22.5

*µm*, respectively. The lateral intensity distributions at the actual focal planes of three DLs are shown in Fig. 6, and the spot sizes of the central lobes are 1.00, 1.59, and 2.10

*µm*, which are similar to a diffraction-limited spot size of 2

*µm*. To observe the lateral intensity distribution on the plane for different distances from the DL surface, we calculated the lateral intensity of DLs for different depths

*df*at three different distances from the DL, respectively. The results are shown in Fig. 7: (a)

*df*=-14.4

*µm*(dashed curve,

*y*=9.3

*µm;*solid curves, the actual focal plane; dotted curve,

*y*=12.4

*µm*); (b)

*df*=+14.4

*µm*(dashed curve,

*y*=17.9

*µm;*solid curve, the actual focal plane; dotted curve,

*y*=32.7

*µm*). As an initial example, the plots evidently illustrate that the designed DLs can achieve long focal depths with high lateral resolution if we set a positive preset focal depth, as well as high axial resolution with high lateral resolution if we set a negative preset focal depth. Also, to present a global view of the DL focusing performance, propagation plots of the electric-field intensity are displayed in Fig. 8.

*f*-numbers (

*f*/# of 0.6, 1.0,1.5, and 2) and with different preset focal depths (

**±**0.3

*f*

_{0}and

**±**0.6

*f*

_{0}), we summarized the numerical results in Table 1, including the actual focal depth, the value of relative focal depth

*Ra*, the position of the actual focal plane, and the spot size of the central lobe at the actual focal plane. It is obvious that the actual focal depth of the DL increases when the preset focal depth increases, and the axial resolution of the DL increases when the preset focal depth is reduced for a larger

*f*-number (i.e.,

*f*/#>1). A smaller preset focal depth cannot always produce a DL with a higher axial resolution for a smaller

*f*-number (i.e.,

*f*/#<=1), and we are currently investigating this. We believe that this inexplicable difference is not caused by computational precision, because we set the space interval at λ/40, recalculated the DL for a small

*f*-number, and obtained almost the same results:

*f*/#=0.6 and

*µm*for the space interval of λ/20;

*µm*for λ/40. For

*f*/#=1.0, the

*f*-number DL will deteriorate because of some factors such as the diffractive structure shadow and the large obliquity factor for off-axis angles [17

17. D. W. Prather, D. Pustai, and S. Y. Shi, “Performance of multilevel diffractive lenses as a function of f-number,” Appl. Opt. **40**, 207–210 (2001). [CrossRef]

## 4. Conclusion

*f*-numbers of 0.6, 1.0, 1.5, and 2.0 by using different preset focal depths. In comparison with a conventional lens, the results of rigorous numerical simulations illustrate that these two kinds of DL can achieve the required focusing characteristics. The combination of numerical and graphic results have shown that the designed DLs have long focal depths if we use a positive preset focal depth and have high axial resolution if we use a negative preset focal depth. In general, the focal depth of DLs of all four

*f*-numbers increases when the preset focal depth increases, and the focal depth of DLs whose

*f*-numbers are larger than 1 decreases when the preset focal depth decreases. However, a smaller preset focal depth cannot always produce the DL with a shorter focal depth when the

*f*-number of the DL is less than 1. On the basis of the design and analysis results, our method could prove to be a useful technique for the rigorous design of DLs with different axial resolution and high lateral resolution. These DLs might also be useful for practical applications such as in optical disk readout systems and in confocal scanning microscopy systems.

## Acknowledgments

## References and links

1. | J. Sochacki, S. Bara, Z. Jaroszewicz, and A. Kolodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. |

2. | N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. |

3. | P. Varga, “Use of confocal microscopes in conoscopy and ellipsometry. 1. Electromagnetic theory,” Appl. Opt. |

4. | C. Yang and J. Mertz, “Transmission confocal laser scanning microscopy with a virtual pinhole based on nonlinear detection,” Opt. Lett. |

5. | P. Torok and T. Wilson, “Rigorous theory for axial resolution in confocal microscopes,” Opt. Commun. |

6. | J. S. Ye, B. Z. Dong, B. Y. Gu, G. Z. Yang, and S.-T. Liu, “Analysis of a closed-boundary axilens with long focal depth and high transverse resolution based on rigorous electromagnetic theory,” J. Opt. Soc. Am. A |

7. | B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, and J. Wang, “Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A |

8. | B. Lichtenberg and N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. |

9. | K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A |

10. | D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary element method for vector modeling diffractive optical elements,” in |

11. | D. W. Prather and S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A |

12. | J. Jiang and G. P. Nordin, “A rigorous unidirectional method for designing finite aperture diffractive optical elements,” Opt. Express |

13. | S. D. Mellin and G. P. Nordin, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express |

14. | J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

15. | A. Taflove, |

16. | D. W. Prather, S. Y. Shi, and J. Sonsrtoem, “Electromagnetic analysis of finite-thickness diffractive elements,” Opt. Eng. |

17. | D. W. Prather, D. Pustai, and S. Y. Shi, “Performance of multilevel diffractive lenses as a function of f-number,” Appl. Opt. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(220.3620) Optical design and fabrication : Lens system design

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 9, 2003

Revised Manuscript: August 11, 2003

Published: August 25, 2003

**Citation**

Feng Di, Yan Yingbai, Jin Guofan, and Wu Minxian, "Rigorous concept for the analysis of diffractive lenses with different axial resolution and high lateral resolution," Opt. Express **11**, 1987-1994 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-17-1987

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### References

- J. Sochacki, S. Bara, Z. Jaroszewicz, and A. Kolodziejczyk, �??Phase retardation of the uniform-intensity axilens,�?? Opt. Lett. 17, 7�??9 (1992). [CrossRef] [PubMed]
- N. Davidson, A. A. Friesem, and E. Hasman, �??Holographic axilens: high resolution and long focal depth,�?? Opt. Lett. 16, 523�??525 (1991). [CrossRef] [PubMed]
- P. Varga, �??Use of confocal microscopes in conoscopy and ellipsometry. 1. Electromagnetic theory,�?? Appl. Opt. 39, 6360�??6365 (2000). [CrossRef]
- C. Yang and J. Mertz, �??Transmission confocal laser scanning microscopy with a virtual pinhole based on nonlinear detection,�?? Opt. Lett. 28, 224�??226 (2003). [CrossRef] [PubMed]
- P. Torok and T. Wilson, �??Rigorous theory for axial resolution in confocal microscopes,�?? Opt. Commun. 137, 127�??135 (1997). [CrossRef]
- J. S. Ye, B. Z. Dong, B. Y. Gu, G. Z. Yang, and S.-T. Liu, �??Analysis of a closed-boundary axilens with long focal depth and high transverse resolution based on rigorous electromagnetic theory,�?? J. Opt. Soc. Am. A 19, 2030�??2035 (2002). [CrossRef]
- B. Z. Dong, J. Liu, B. Y. Gu, G. Z. Yang, and J. Wang, �??Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution,�?? J. Opt. Soc. Am. A 18, 1465�??1470 (2001). [CrossRef]
- B. Lichtenberg and N. C. Gallagher, �??Numerical modeling of diffractive devices using the finite element method, �?? Opt. Eng. 33, 3518�??3526 (1994). [CrossRef]
- K. Hirayama, E. N. Glytsis, T. K. Gaylord, and D. W. Wilson, �??Rigorous electromagnetic analysis of diffractive cylindrical lenses,�?? J. Opt. Soc. Am. A 13, 2219�??2231 (1996). [CrossRef]
- D. W. Prather, M. S. Mirotznik, and J. N. Mait, �??Boundary element method for vector modeling diffractive optical elements, �?? in Diffractive and Holographic Optics Technology II, I. Cindrich and S. H. Lee, eds., Proc. SPIE 2404, 28�??39 (1995). [CrossRef]
- D. W. Prather and S. Shi, �??Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements, �?? J. Opt. Soc. Am. A 16, 1131�??1142 (1999). [CrossRef]
- J. Jiang and G. P. Nordin, �??A rigorous unidirectional method for designing finite aperture diffractive optical elements,�?? Opt. Express 7, 237�??242 (2000), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-6-237.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-6-237</a> [CrossRef] [PubMed]
- S. D. Mellin and G. P. Nordin, �??Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,�?? Opt. Express 8, 705�??722 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-13-705.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-13-705</a> [CrossRef] [PubMed]
- J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185-200 (1994). [CrossRef]
- A. Taflove, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).
- D. W. Prather, S. Y. Shi, and J. Sonsrtoem, �??Electromagnetic analysis of finite-thickness diffractive elements,�?? Opt. Eng. 41, 1792�??1796 (2002). [CrossRef]
- D. W. Prather, D. Pustai, and S. Y. Shi, �?? Performance of multilevel diffractive lenses as a function of f-number,�?? Appl. Opt. 40, 207�??210 (2001). [CrossRef]

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