## Design of polarization gratings for broadband illumination

Optics Express, Vol. 13, Issue 8, pp. 3055-3067 (2005)

http://dx.doi.org/10.1364/OPEX.13.003055

Acrobat PDF (149 KB)

### Abstract

Design of broadband diffractive elements is studied. It is shown that dielectric polarization gratings can be made to perform the same optical function over a broad band of wavelengths. Any design of paraxial-domain diffractive elements can be realized as such broadband elements that may, e.g., give constant diffraction efficiencies over the wavelength band while the field propagation after the elements remains wavelength-dependent. Furthermore, elements producing symmetric signals are shown to work with arbitrarily polarized or partially polarized incident planar broadband fields. The performance of the elements is illustrated by numerical examples and some practical issues related to their fabrication are discussed.

© 2005 Optical Society of America

## 1. Introduction

2. H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. **36**, 1566–1572 (1997). [CrossRef] [PubMed]

4. D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. **227**, 49–55 (2003). [CrossRef]

5. Y. Kanamori, M. Sasaki, and K. Hane, “Broadband antireflection gratings fabricated upon silicon substrates,” Opt. Lett. **24**, 1422–1424 (1999). [CrossRef]

9. C. Sauvan, P. Lalanne, and M.-S. L. Lee, “Broadband blazing with artificial dielectrics,” Opt. Lett. **29**, 1593–1595 (2004). [CrossRef] [PubMed]

10. F. Gori, “Measuring the Stokes parameters by means of a polarization grating,” Opt. Lett. **24**, 584–586 (1999). [CrossRef]

13. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A **20**, 282–289 (2003). [CrossRef]

11. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. **27**, 1141–1143 (2002). [CrossRef]

17. C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and J. Adachi, “Polarizing diffraction-grating triplicators,” Opt. Lett. **26**, 1651–1653 (2001). [CrossRef]

12. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. **25**, 785–786 (2000). [CrossRef]

18. F. Wyrowski, “Upper bound of efficiency of diffractive phase elements,” Opt. Lett. **16**, 1915–1917 (1991). [CrossRef] [PubMed]

19. H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. **29**, 803–805 (2004). [CrossRef] [PubMed]

## 2. Broadband diffractive polarization elements

19. H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. **29**, 803–805 (2004). [CrossRef] [PubMed]

*θ*(

*x*,

*y*) that denotes the local rotation angle of the direction of the grating grooves, measured in respect to the x-axis. The polarization modulating element may hence be represented by the Jones matrix [20]

*t*

_{‖}and

*t*

_{⊥}are the complex-amplitude transmittances of the local electric field components parallel and perpendicular, respectively, to the optical axis. In general cases also

*t*

_{‖}and

*t*

_{⊥}could be spatially varying but in this paper we will concentrate on binary structures for which only

*θ*is a slowly varying function of the coordinates

*x*and

*y*.

*y*-invariant grating with period

*d*, depth

*h*, and fill factor

*f*=

*g*/

*d*are illustrated in Fig. 1. The functions describing the transmittances of the field components parallel and perpendicular to the grating grooves can be written as

*h*is the depth of the structure,

*k*=2

*π*/

*λ*is the wave number and

*n*

_{‖}and

*n*

_{⊥}are the effective refractive indices that depend on the grating parameters and the refractive indices of the used materials. If the period of the subwavelength grating is considerably smaller than the wavelength of incident light,

*d*≪λ, the effective refractive indices can be determined using approximate formulas. However, if the period is close to the wavelength, rigorous diffraction theory [21

21. R. Petit, *Electromagnetic Theory of Gratings* (Springer-Verlag, Berlin, 1980). [CrossRef]

*t*

_{‖}|=|

*t*

_{⊥}|=1. In such a case the subwavelength grating modulates only the phase of the field introducing a phase difference

*k*. However, the rigorously determined effective refractive indices are also wavelength-dependent, and it has been shown that by optimizing the grating parameters, i.e. the period, depth, and the fill factor, the dispersion of the effective indices can be employed to keep the phase difference approximately the same over a broad wavelength band [2

2. H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. **36**, 1566–1572 (1997). [CrossRef] [PubMed]

2. H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. **36**, 1566–1572 (1997). [CrossRef] [PubMed]

4. D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. **227**, 49–55 (2003). [CrossRef]

*ϕ*(

*λ*)=

*ϕ*

_{0}, over some wavelength band Δ

*λ*. In such a case the optical function of the element depends on the local phase difference between the electric field components that is controlled by the fringe orientation angle

*θ*(

*x*,

*y*) only and thus it remains the same for all the wavelengths within Δλ. To illustrate this possibility more closely we consider in the following the transmission of an elliptically polarized plane wave through the polarization modulating element characterized by Eq. (1).

*β*are defined by equations

*E*

_{ix}and

*E*

_{iy}are the

*x*and

*y*components of the incident plane wave, and they are normalized as |

*α*|

^{2}+|

*β*|

^{2}=|

*E*

_{ix}|

^{2}+|

*E*

_{iy}|

^{2}=1. Furthermore, we will now assume that

*α*and

*β*are wavelength-invariant, so that the polarization state of the incident field is the same at all the wavelengths in Δ

*λ*.

11. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. **27**, 1141–1143 (2002). [CrossRef]

15. U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, “Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation,” Opt. Express **12**, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

**E**

_{R}and

**E**

_{L}imply only to the polarization state of the incident field components, and not to the properties of the transmitted fields themselves.

*θ*(

*x*,

*y*). We note that the phase difference

*ϕ*

_{0}was assumed to be constant over the considered wavelength band Δ

*λ*. The phase term arg

*t*

_{‖}in Eq. (14) may be wavelength-dependent, but usually such spatially invariant phase terms can be ignored since they will not affect the optical effect of the element. Thus, the polarization element will perform the same optical function for all the wavelengths of the certain band Δλ. This means that the transmitted field right behind the element has the same form given by Eqs. (11)–(13) at all the different wavelengths. However, the further propagation of the field will still be wavelength-dependent as discussed in Section 1.

## 3. Designs for polarized fields

11. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. **27**, 1141–1143 (2002). [CrossRef]

15. U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, “Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation,” Opt. Express **12**, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

*λ*. While the considerations above are valid for any polarization modulating element regardless of the periodicity of the structure, in the following we will restrict our considerations into periodic polarization gratings and examine the produced far-field signals by means of diffraction efficiencies.

*d*

_{x}and

*d*

_{y}denote the grating periods in the

*x*- and

*y*-directions, respectively, and

*m*,

*n*). We assume that the grating period is large compared to the wavelengths of the incident field, and thus the diffraction orders are paraxial. The diffraction efficiencies can be simply defined as

**J**

_{i}‖

^{2}=1.

*t*

_{‖}and

*t*

_{⊥}and fringe rotation angle

*θ*(

*x*,

*y*). The Jones vectors associated with the diffraction orders generated from an arbitrarily polarized incident field can be expressed in the form

*δ*

_{m,n}is the Dirac delta function, and

*θ*(

*x*,

*y*)]. The diffraction efficiencies are obtained from Eq. (18) by using definition (17). In general cases the result may be expressed as

*T*

_{m,n}=0 if (

*m*,

*n*)≠(0,0) and a simple form

15. U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, “Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation,” Opt. Express **12**, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

*λ*.

*α*=1 and

*β*=0, Eq. (22) reduces to the form

*θ*(

*x*,

*y*) and an additional zeroth order contribution, in proportion to the coefficients determined by

*t*

_{‖}and

*t*

_{⊥}. If we choose the parameters of the subwavelength grating so that it works as an ideal half-wave plate, i.e.

*ϕ*

_{0}=

*π*and |

*t*

_{‖}|=|

*t*

_{⊥}|=1, over the wavelength band Δλ, the additional zeroth order vanishes and the polarization grating corresponds directly to the phase element. In the same way as suggested for elements designed for a single wavelength [15

**12**, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

*ϕ*

_{0}introduced by the subwavelength grating.

19. H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. **29**, 803–805 (2004). [CrossRef] [PubMed]

18. F. Wyrowski, “Upper bound of efficiency of diffractive phase elements,” Opt. Lett. **16**, 1915–1917 (1991). [CrossRef] [PubMed]

*t*

_{‖}and

*t*

_{⊥}are spatially invariant. The groove direction varies according to equation [10

10. F. Gori, “Measuring the Stokes parameters by means of a polarization grating,” Opt. Lett. **24**, 584–586 (1999). [CrossRef]

12. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. **25**, 785–786 (2000). [CrossRef]

*D*. This design produces only three nonvanishing diffraction orders the efficiencies of which are given by [12

12. J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. **25**, 785–786 (2000). [CrossRef]

*ϑ*=arg

*E*

_{ix}-arg

*E*

_{iy}. If we assume that all the light is transmitted completely and the phase difference introduced by the subwavelength grating is

*π*, the associated complex amplitude transmittances satisfy

*t*

_{‖}=-

*t*

_{⊥}and |

*t*

_{‖}|=|

*t*

_{⊥}|=1. In this case the zeroth diffraction order disappears and energy distribution between the ±1 orders is determined by the polarization state of the incident field. For linearly polarized light

*η*

_{1}=

*η*

_{-1}=1/2, and a diffractive 1→2 beam splitter with 100% total efficiency is obtained. On the other hand, assuming that arg

*t*

_{⊥}-arg

*t*

_{‖}=cos

^{-1}(-1/3) and the incident light is still linearly polarized,

*η*

_{0}=

*η*

_{±1}=1/3 and the grating acts thus an ideal 1→3 beam splitter [12

**25**, 785–786 (2000). [CrossRef]

**27**, 1141–1143 (2002). [CrossRef]

16. J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. **26**, 587–589 (2001). [CrossRef]

17. C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and J. Adachi, “Polarizing diffraction-grating triplicators,” Opt. Lett. **26**, 1651–1653 (2001). [CrossRef]

**29**, 803–805 (2004). [CrossRef] [PubMed]

*t*

_{‖}and

*t*

_{⊥}are spatially varying besides the rotation angle

*θ*, and the extension to the elements with broadband performance is not straightforward. However, designs for polarization-grating beam splitters can also be obtained by using iterative algorithms [13

13. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A **20**, 282–289 (2003). [CrossRef]

**12**, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

13. J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A **20**, 282–289 (2003). [CrossRef]

## 4. Polarization-insensitive designs

**12**, 5345–5355 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345. [CrossRef] [PubMed]

*m*,

*n*)≠(0,0) we see from Eq. (23) that the efficiencies equal the sum of the diffraction efficiencies associated with the complex amplitudes

*β*. It is also noted from Eq. (20) that the quantities

*m*,

*n*)≠(0,0), and the generated signal is the same for any incident polarization state. With the assumed symmetry Eq. (22) gives an expression

*t*

_{‖}|=|

*t*

_{⊥}|=1 in which case the zeroth diffraction order is also polarization-insensitive. This shows that if the desired signal is symmetric, scalar phase elements may be replaced with polarization gratings with broadband performance in the same way as discussed in the previous Section for circularly polarized light, but now the elements will work similarly with arbitrarily polarized incident fields.

23. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A **21**, 2205–2215 (2004). [CrossRef]

**T**is the Jones matrix of the element given by Eq. (1),

**J**

_{i}is the polarization matrix of the incident field, and the dagger denotes the adjoint. Some studies considering the transmission of partially polarized and partially coherent fields through polarization gratings using this formalism can be found [24

24. G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A **18**, 1399–1405 (2001). [CrossRef]

25. G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. **195**, 339–350 (2001). [CrossRef]

26. L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. **81**, 1104–1114 (1963). [CrossRef]

27. J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astr. Soc. **34**, 403–419 (1973). [CrossRef]

**J**

_{1}and

**J**

_{2}representing polarized fields can be studied alternatively using the Jones vector formalism as in the previous Sections. Assuming a periodic polarization grating, the produced far-field signal is characterized by diffraction orders with efficiencies

*η*

_{1m,n}and

*η*

_{2m,n}are the diffraction efficiencies generated by the polarized field components.

*η*

_{1m,n}=

*η*

_{2m,n}and Eq. (34) reduces to the form

## 5. Numerical examples

28. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. **157**, 13–17 (1998). [CrossRef]

*h*

_{0}is the modulation depth of the surface-relief grating. It can be shown that the diffraction efficiencies of this kind of element are given by

*J*

_{m}(

*x*) is the Bessel function of order

*m*. Now

*η*

_{0}(

*λ*

_{0})=

*η*±

_{1}(

*λ*

_{0}) at some single wavelength if we choose the grating depth

*h*

_{0}so that

*Φ*(

*λ*

_{0})≈0.91

*π*, but at other wavelengths the efficiencies change. At that certain wavelength the total diffraction efficiency of this design is

*η*≈90%, which is relatively good also when compared to the maximum obtainable efficiency

*η*≈92.6% of the optimal scalar triplicator [28

28. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. **157**, 13–17 (1998). [CrossRef]

*ϕ*=

*π*. In this case the fringe rotation angle is chosen as

*θ*(

*x*)=

*t*(

*x*,

*λ*

_{0})/2, and it naturally is wavelengthin-dependent. The maximum efficiency is the same as that of the phase element, but the polarization grating works as a uniform triplicator over a wider band of wavelengths. Since the signal is symmetric, the element will work in the same manner if illuminated by any polarized or partially polarized field as discussed in the previous Section.

*n*

_{0}=1.8,

*n*

_{1}=2 and

*n*

_{2}=

*n*

_{3}=1 as illustrated in Fig. 1. The period of the subwavelength grating is assumed to be

*d*=0.4

*λ*

_{0}, where

*λ*

_{0}is any fixed wavelength, the fill factor is

*f*=0.8, and the desired phase retardation

*ϕ*=

*π*is obtained with modulation depth

*h*=3

*λ*

_{0}. These parameters for the achromatic subwavelength structure are obtained from Ref. 2, but, instead of the sandwiched structure that would yield somewhat higher theoretical efficiencies [19

**29**, 803–805 (2004). [CrossRef] [PubMed]

*t*

_{‖}and

*t*

_{⊥}of the subwavelength grating are computed rigorously using the Fourier modal method [29], and the efficiency of the diffraction orders of the polarization grating is then determined by Eq. (27).

*n*

_{1}=

*n*

_{0}=1.8 and

*n*

_{2}=

*n*

_{3}=1. Thereby its diffraction efficiencies are multiplied by the Fresnel transmission coefficient that for the interface between the materials

*n*

_{0}and

*n*

_{3}at normal incidence angle is

*T*≈0.918. The results clearly show how the diffraction efficiencies of the phase element change strongly as a function of the wavelength but the polarization grating produces uniformly the same efficiency for all three orders over the whole wavelength band considered.

*ϕ*=

*π*, relatively deep subwavelength-period structures should be used, which makes the fabrication process more difficult. The required depth is decreased if high-index materials are used in the modulated region (see e.g. Ref. 30). Therefore we have chosen to study a polarization grating fabricated in a thin TiO

_{2}film on glass (SiO

_{2}) substrate. The refractive index data including material dispersion for these materials

*n*

_{1}and

*n*

_{0}, respectively, is taken from Refs. [31

31. S.-C. Chiao, B. G. Bovard, and H. A. Macleod, “Optical-constant calculation over an extended spectral region: application to titanium dioxide film,” Appl. Opt. **34**, 7355–7360 (1995). [CrossRef] [PubMed]

*n*

_{2}=

*n*

_{3}=1).

*d*=220 nm,

*f*=0.6, and

*h*=800 nm. Since the fabrication processes often introduce some errors causing deviations from the desired structure, we will also examine the sensitivity of the broadband beam splitter to such fabrication errors. Figure 3 illustrates the total diffraction efficiency of the beam splitter made of TiO

_{2}with the basic parameters and additionally with different grating depths

*h*=800±50 nm. It can be seen that 100% total diffraction efficiency is not fully reached owing to the losses at the interfaces, and the efficiency fluctuates slightly as a function of the wavelength. However, the diffraction efficiencies with all parameter combinations are still higher than the upper bound for scalar duplicators that is

*η*

_{u}≈81% over nearly the whole a 350-nm-wide wavelength band. We also emphasize that the parameters used here represent just an example and not the most optimal solution for an achromatic subwavelength structure.

*h*is changed. This feature of the broadband gratings is useful if one wants to design elements to be used with some fixed wavelengths. As another example of the parameter sensitivity of the element, the dependence of the results on the fill factor of the same grating is illustrated in Fig. 4 using deviated parameters

*f*=0.6±0.05. The performance of the grating is seen to be slightly more sensitive to errors in the fill factor than in the depth, but still no drastic deterioration is noticed within these error bounds.

## 6. Conclusions

## Acknowledgments

## References and links

1. | J. Turunen and F. Wyrowski, eds., |

2. | H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-wave plates using the dispersion of form birefringence,” Appl. Opt. |

3. | G. P. Nordin and P. C. Deguzman, “Broadband form birefringent quarter-wave plate for the mid-infrared wavelength region,” Opt. Express |

4. | D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si -Lu, and G.-F. Jin, “Broadband achromatic phase retarder by sub-wavelength grating,” Opt. Commun. |

5. | Y. Kanamori, M. Sasaki, and K. Hane, “Broadband antireflection gratings fabricated upon silicon substrates,” Opt. Lett. |

6. | I. R. Hooper and J. R. Sambles, “Broadband polarization-converting mirror for the visible region of the spectrum,” Opt. Lett. |

7. | D. Kim and K. Burke, “Design of a grating-based thin-film filter for broadband spectropolarimetry,” Appl. Opt. |

8. | D. Yi, Y. Yan, H. Liu, S. Lu, and G. Jin, “Broadband polarizing beam splitter based on the form birefringence of a subwavelength grating in the quasi-static domain,” Opt. Lett. |

9. | C. Sauvan, P. Lalanne, and M.-S. L. Lee, “Broadband blazing with artificial dielectrics,” Opt. Lett. |

10. | F. Gori, “Measuring the Stokes parameters by means of a polarization grating,” Opt. Lett. |

11. | Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. |

12. | J. Tervo and J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. |

13. | J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A |

14. | M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,” J. Mod. Opt. |

15. | U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, “Design, fabrication and characterization of subwavelength computer-generated holograms for spot array generation,” Opt. Express |

16. | J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, “Polarization beam splitters using polarization diffraction gratings,” Opt. Lett. |

17. | C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and J. Adachi, “Polarizing diffraction-grating triplicators,” Opt. Lett. |

18. | F. Wyrowski, “Upper bound of efficiency of diffractive phase elements,” Opt. Lett. |

19. | H. Lajunen, J. Tervo, and J. Turunen, “High-efficiency broadband diffractive elements based on polarization gratings,” Opt. Lett. |

20. | D. S. Kliger, J. W. Lewis, and C. E. Randall, |

21. | R. Petit, |

22. | L. Mandel and E. Wolf, |

23. | J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A |

24. | G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A |

25. | G. Piquero, R. Borghi, A. Mondello, and M. Santarsiero, “Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,” Opt. Commun. |

26. | L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. |

27. | J. C. Samson, “Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,” Geophys. J. R. Astr. Soc. |

28. | F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili“Analytical derivation of optimum triplicator,” Opt. Commun. |

29. | J. Turunen“Diffraction theory of microrelief gratings,” in |

30. | S. Astilean, Ph. Lalanne, P. Chavel, E. Cambril, and H. Launois, High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nmOpt. Lett. |

31. | S.-C. Chiao, B. G. Bovard, and H. A. Macleod, “Optical-constant calculation over an extended spectral region: application to titanium dioxide film,” Appl. Opt. |

32. | H. R. Philipp, “Silicon Dioxide (SiO |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.1970) Diffraction and gratings : Diffractive optics

(230.1360) Optical devices : Beam splitters

(260.1440) Physical optics : Birefringence

(260.5430) Physical optics : Polarization

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 8, 2005

Revised Manuscript: April 5, 2005

Published: April 18, 2005

**Citation**

Hanna Lajunen, Jari Turunen, and Jani Tervo, "Design of polarization gratings for broadband illumination," Opt. Express **13**, 3055-3067 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-3055

Sort: Journal | Reset

### References

- J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie�??Verlag, Berlin, 1997).
- H. Kikuta, Y. Ohira, and K. Iwata, �??Achromatic quarter-wave plates using the dispersion of form birefringence,�?? Appl. Opt. 36, 1566�??1572 (1997). [CrossRef] [PubMed]
- G. P. Nordin and P. C. Deguzman, �??Broadband form birefringent quarter-wave plate for the mid-infrared wave-length region,�?? Opt. Express 5, 163�??168 (1999). [CrossRef] [PubMed]
- D.-E. Yi, Y.-B. Yan, H.-T. Liu, Si-Lu, and G.-F. Jin, �??Broadband achromatic phase retarder by subwavelength grating,�?? Opt. Commun. 227, 49�??55 (2003). [CrossRef]
- Y. Kanamori, M. Sasaki, and K. Hane, �??Broadband antireflection gratings fabricated upon silicon substrates,�?? Opt. Lett. 24, 1422�??1424 (1999). [CrossRef]
- I. R. Hooper and J. R. Sambles, �??Broadband polarization-converting mirror for the visible region of the spectrum,�?? Opt. Lett. 27, 2152�??2154 (2002). [CrossRef]
- D. Kim and K. Burke, �??Design of a grating-based thin-film filter for broadband spectropolarimetry,�?? Appl. Opt. 31, 6321�??6326 (2003). [CrossRef]
- D. Yi, Y. Yan, H. Liu, S. Lu, and G. Jin, �??Broadband polarizing beam splitter based on the form birefringence of a subwavelength grating in the quasi-static domain,�?? Opt. Lett. 29, 754�??756 (2004). [CrossRef] [PubMed]
- C. Sauvan, P. Lalanne, and M.-S. L. Lee, �??Broadband blazing with artificial dielectrics,�?? Opt. Lett. 29, 1593�??1595 (2004). [CrossRef] [PubMed]
- F. Gori, �??Measuring the Stokes parameters by means of a polarization grating,�?? Opt. Lett. 24, 584�??586 (1999). [CrossRef]
- Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, �??Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,�?? Opt. Lett. 27, 1141�??1143 (2002). [CrossRef]
- J. Tervo and J. Turunen, �??Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,�?? Opt. Lett. 25, 785�??786 (2000). [CrossRef]
- J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, �??Design of space-variant diffractive polarization elements,�?? J. Opt. Soc. Am. A 20, 282�??289 (2003). [CrossRef]
- M. Honkanen, V. Kettunen, J. Tervo, and J. Turunen, �??Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction,�?? J. Mod. Opt. 47, 2351�??2359 (2000).
- U. Levy, C.-H. Tsai, H.-C. Kim, and Y. Fainman, �??Design, fabrication and characterization of subwave-length computer-generated holograms for spot array generation,�?? Opt. Express 12, 5345�??5355 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5345.</a> [CrossRef] [PubMed]
- J. A. Davis, J. Adachi, C. R. Fernández-Pousa, and I. Moreno, �??Polarization beam splitters using polarization diffraction gratings,�?? Opt. Lett. 26, 587�??589 (2001). [CrossRef]
- C. R. Fernández-Pousa, I. Moreno, J. A. Davis, and J. Adachi, �??Polarizing diffraction-grating triplicators,�?? Opt. Lett. 26, 1651�??1653 (2001). [CrossRef]
- F. Wyrowski, �??Upper bound of efficiency of diffractive phase elements,�?? Opt. Lett. 16, 1915�??1917 (1991). [CrossRef] [PubMed]
- H. Lajunen, J. Tervo, and J. Turunen, �??High-efficiency broadband diffractive elements based on polarization gratings,�?? Opt. Lett. 29, 803�??805 (2004). [CrossRef] [PubMed]
- D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990), Section 3.4.
- R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
- J. Tervo, T. Setälä, and A. T. Friberg, �??Theory of partially coherent electromagnetic fields in the space�??frequency domain,�?? J. Opt. Soc. Am. A 21, 2205�??2215 (2004). [CrossRef]
- G. Piquero, R. Borghi, and M. Santarsiero, �??Gaussian Schell-model beams propagating through polarization gratings,�?? J. Opt. Soc. Am. A 18, 1399�??1405 (2001). [CrossRef]
- G. Piquero, R. Borghi, A. Mondello and M. Santarsiero, �??Far field of beams generated by quasi-homogeneous sources passing through polarization gratings,�?? Opt. Commun. 195, 339�??350 (2001). [CrossRef]
- L. Mandel, �??Intensity fluctuations of partially polarized light,�?? Proc. Phys. Soc. 81, 1104�??1114 (1963). [CrossRef]
- J. C. Samson, �??Descriptions of the polarization states of vector processes: applications to ULF magnetic fields,�?? Geophys. J. R. Astr. Soc. 34, 403�??419 (1973). [CrossRef]
- F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, �??Analytical derivation of optimum triplicator,�?? Opt. Commun. 157, 13�??17 (1998). [CrossRef]
- J. Turunen, �??Diffraction theory of microrelief gratings,�?? in Micro-Optics: Elements, Systems, and Applications, H.-P. Herzig, ed. (Taylor & Francis, London, 1997), Chapter 2.
- S. Astilean, Ph. Lalanne, P. Chavel, E. Cambril, and H. Launois, "High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nm", Opt. Lett. 23, 552�??554 (1998). [CrossRef]
- S.-C. Chiao, B. G. Bovard, and H. A. Macleod, �??Optical-constant calculation over an extended spectral region: application to titanium dioxide film,�?? Appl. Opt. 34, 7355�??7360 (1995). [CrossRef] [PubMed]
- H. R. Philipp, �??Silicon Dioxide (SiO2) (Glass),�?? in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Orlando, 1985), pp. 749�??763.

## 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.