## Spectral uniformity of two- and four-level diffractive optical elements for spectroscopy.

Optics Express, Vol. 17, Issue 12, pp. 10206-10222 (2009)

http://dx.doi.org/10.1364/OE.17.010206

Acrobat PDF (2729 KB)

### Abstract

We present simulations and characterization of gold coated diffractive optical elements (DOEs) that have been designed and fabricated in silicon for an industrial application of near-infrared spectroscopy. The DOE design is focusing and reflecting, and two-level and four-level binary designs were studied. Our application requires the spectral response of the DOE to be uniform over the DOE surface. Thus the variation in the spectral response over the surface was measured, and studied in simulations. Measurements as well as simulations show that the uniformity of the spectral response is much better for the four-level design than for the two-level design. Finally, simulations and measurements show that the four-level design meets the requirements of spectral uniformity from the industrial application, whereas the simulations show that the physical properties of diffraction gratings in general make the simpler two-level design unsuitable.

© 2009 Optical Society of America

## 1. Introduction

1. S. Ura, F. Okayama, K. Shiroshita, K. Nishio, T. Sasaki, H. Nishihara, T. Yotsuya, M. Okano, and K. Satoh, “Planar reflection grating lens for compact spectroscopic imaging system,” Appl. Opt. **42**, 175–180 (2003).
[CrossRef] [PubMed]

2. S. Grabarnik, A. Emadi, E. Sokolova, G. Vdovin, and R. F. Wolffenbuttel, “Optimal implementation of a microspectrometer based on a single flat diffraction grating,” Appl. Opt. **47**, 2082–2090 (2008).
[CrossRef] [PubMed]

3. R. F. Wolffenbuttel, “State-of-the-art in integrated optical microspectrometers,” IEEE Trans. Instrum. Meas. **53**, 197–202 (2004).
[CrossRef]

4. O. Løvhaugen, I.-R. Johansen, K. A. H. Bakke, B. G. Fismen, and S. Nicolas, “Dedicated spectrometers based on diffractive optics: design, modelling and evaluation,” J. Mod. Opt. **51**, 2203–2222 (2004).
[CrossRef]

*direct*simulation of the diffracted waves from the DOEs that we are interested in. However the scalar approximation does not describe polarization, and its validity is limited to small angles of incidence and diffraction, and structures where the feature sizes are typically much larger than the wavelength of the incident wave. Our design does not fulfill these conditions, so more rigorous methods must be used. To this end, we show that some important qualitative features of the optical filter response can be explained with the help of a local linear grating (LLG) [5

5. E. Noponen, J. Turunen, and A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J.Opt. Soc. Am. A **10**, 434–443 (1993).
[CrossRef]

## 2. DOE function and surface profile

^{2}. They are designed so that the incident light is split into five focal lines each containing a spectrum centered on one out of five design wavelengths, as shown in Fig. 4. The spatial positions of the focal lines are such that a simple rotation of the DOE moves the lines across the detector. The separation into several focal lines allows one to easily know which band is on the detector without knowing the exact scan angle. The separation comes at the expense of light intensity, since the incoming optical power is divided between the five focal lines.

4. O. Løvhaugen, I.-R. Johansen, K. A. H. Bakke, B. G. Fismen, and S. Nicolas, “Dedicated spectrometers based on diffractive optics: design, modelling and evaluation,” J. Mod. Opt. **51**, 2203–2222 (2004).
[CrossRef]

*λ*, so that if the desired reflection coefficient has a phase

_{avg}*ϕ*(

*x*,

*y*), the corresponding depth profile

*f*(

*x*,

*y*) for the DOE surface is

*k*=2

_{avg}*π*/

*λ*. For a DOE element reflecting and focusing a single wavelength, the phase function

_{avg}*ϕ*(

*x*,

*y*) is given by the formula

*k*is the wavenumber, and

*r*and

_{s}*r*are the distances from (

_{d}*x*,

*y*) to the source and detector, and the mod function ensures that

*ϕ*(

*x*,

*y*) is between 0 and 2

*π*. The reflection coefficient of a single wavelength DOE corresponds to that of a Fresnel lens, with a constant amplitude

*A*and a phase

_{n}*ϕ*(

_{n}*x*,

*y*). For an

*n*-wavelength DOE, the total complex reflection coefficient is found by adding the reflection coefficients of each Fresnel lens:

*A*(

_{tot}*x*,

*y*) is approximated to unity. The phase of the resulting total reflection coefficient is converted to a surface depth using the average of the

*n*wavelengths. This procedure for incorporating the five Fresnel lenses into a single surface yields a complex pattern, including the bifurcations seen in Fig. 5 below. Note that the design algorithm in [4

4. O. Løvhaugen, I.-R. Johansen, K. A. H. Bakke, B. G. Fismen, and S. Nicolas, “Dedicated spectrometers based on diffractive optics: design, modelling and evaluation,” J. Mod. Opt. **51**, 2203–2222 (2004).
[CrossRef]

*levels, fabricated through*

^{N}*N*etch steps) to the sawtooth profile as shown in Fig. 3. The simplest binary approximation is the two-level profile (Fig. 3(a)) with one etch step in the DOE fabrication process, etching half the depth of the corresponding sawtooth profile. As given in [8] and [7], the maximum efficiency is achieved when the peak to peak phase variation introduced by the grating depth is exactly 2

*π*. For normal incidence this is achieved by a maximal depth of

*λ*/2 for the sawtooth profile. Thus, as in scalar theory the optimum depth varies slowly with incidence angle (see [7]), we chose this depth for the grating even if we do not have normal incidence. However, it is clear that the design height could be further optimized, for instance using numerical optimization algorithms based on rigorous diffraction theory. For our two-level profile (Fig. 3(a)) the depth is then

_{avg}## 3. Simulation model

5. E. Noponen, J. Turunen, and A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J.Opt. Soc. Am. A **10**, 434–443 (1993).
[CrossRef]

*P*is the period of the grating,

*θ*is the incidence angle,

_{i}*θ*

_{d,m}is the diffraction angle of diffraction order

*m*, and

*λ*is the wavelength of the incident wave. For each DOE position the simulation is run for a linear grating with parameters determined as described here. In the following sections we present results for three wavelengths,

*λ*

_{1}=1.736 µm,

*λ*

_{2}=1.685 µm, and

*λ*

_{3}=1.649 µm.

*θ*in Fig. 8(a)). At each aperture position we compute the diffraction efficiency at four positions within the aperture and average over the results to take into account the finite aperture.

_{d}9. GD-Calc, http://software.kjinnovation.com/GD-Calc.html.

10. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A **13**, 1024–1035 (1996).
[CrossRef]

11. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A **3**, 1780–1787 (1986).
[CrossRef]

*λ*=1.67 µm), the material beneath the coating does not influence the diffraction efficiency. The thickness of the metal coating on the measured DOEs was much greater than 25 nm, so we present only the results from the simulations on DOEs in bulk gold.

## 4. Results from simulations

*λ*

_{1}=1.736

*µ*m,

*λ*

_{2}=1.685 µm, and

*λ*

_{3}=1.649 µm for the two-level DOE. For the simulations of the four-level DOE the wavelengths

*λ*

_{1}=1.733 µm,

*λ*

_{2}=1.685 µm, and

*λ*

_{3}=1.649 µm were used.

*λ*

_{4}=1.694 µm and

*λ*

_{5}=1.712 µm. The height of the two-level DOE surface profile was determined taking into account all the five wavelengths, giving a maximal height of

*h*

_{2}=

*λ*/2≈424 nm. For the four-level DOE, we again present the results for

_{avg}*λ*

_{1},

*λ*

_{2}and

*λ*

_{3}. However, as mentioned the wavelength

*λ*

_{1}was changed to

*λ*

_{1}=1.733 µm, and also

*λ*

_{4}=1.58 µm and

*λ*

_{5}=1.715 µm. Thus the maximal height of the four-level DOE is

*h*

_{4}=3

*λ*/8≈314 nm. This minor change in operating wavelengths has a very small effect on the diffraction efficiency, and does not affect the discussion in this work.

_{avg}*x*,

_{i}*y*) on the DOE. The indices

_{j}*i*and

*j*increase in the direction of the x- and y-axis respectively as shown in Fig. 8(a). When discussing results for the different wavelengths, one must remember that with a change in wavelength there is also a small change in the scan angle of the DOE. The scan angles

*α*for the three wavelengths are

_{k}*α*

_{1}=1.1°,

*α*

_{2}=0° and

*α*

_{3}=-1.65°. The diffraction efficiency presented here is given for “unpolarized” light, i.e. the average of the results from TE and TM illumination. All the presented simulation results are for the order

*m*=-1. To compute the diffraction efficiency of order

*m*=-1 at a position (

*x*,

_{i}*y*) on the DOE, the incidence and diffraction angles from the experimental setup (the incidence angles for each position are shown in Fig. 7(a)) are used in the grating equation Eq. (6) to determine the period of the corresponding representative straight grating. Figure 9 shows the absolute diffraction efficiency for the order

_{j}*m*=-1, found in simulations for the DOEs with two-level profile. The corresponding diffraction efficiencies for the four-level profile is given in Fig. 10. We immediately see that the two-level profile gives much larger variations in diffraction efficiency, both for each wavelength when comparing between the profiles, and when comparing the variations with wavelength for each profile.

## 4.1. Normalization of data

*λ*at one aperture position (

_{k}*x*,

_{i}*y*) is denoted

_{j}*γ*. The coordinates take values

_{ijk}*and Δ*

_{x}*are the distances in the x- and y-directions between two adjacent DOE positions. The wavelengths are labeled*

_{y}*λ*, where

_{k}*k*=(1, …,

*n*).

_{λ}*λ*is the mean of the efficiency for all the positions for this wavelength:

_{k}*, and not absolute diffraction efficiencies. Taking the mean of Γ*

_{ijk}*over the wavelengths, gives*

_{ijk}*η*is equal to one for the whole DOE. This is the unrealistic ideal situation, because then at each point the relationship between the components is the same. In reality

_{ijk}*η*is different from one because the diffraction efficiency varies with

_{ijk}*λ*. This variation is a result of the actual change in wavelength, and also due to the rotation of the DOE through scan angles as illustrated in Fig. 1(b). So a change in wavelength

_{k}*λ*is also accompanied with a change in DOE positioning. The scan angles

_{k}*α*for the three wavelengths are

_{k}*α*

_{1}=1.1°,

*α*

_{2}=0°, and

*α*

_{1}=-1.65°. The goal of the design is then to find a profile which minimizes |

*η*-1| over the DOE surface.

_{ijk}## 4.2. Spectral uniformity

*η*-1| (see Eq. (12)) across the DOE at a given x-position number

_{ijk}*i*. So for instance for wavelength

*λ*at x-position

_{k}*x*we find the maximum of |

_{i}*η*-1| for the positions (

_{ijk}*x*,

_{i}*y*

_{1}), (

*x*,

_{i}*y*

_{2})…(

*x*,

_{i}*y*

_{9}). This does not give information about which y-position gives the maximum deviation, but ultimately the most interesting figure for this discussion is the worst case scenario, i.e. the maximal deviation in spectral uniformity. The results from simulations are given in Fig. 11(a) for the two-level profile and Fig. 11(b) for the four-level profile. In order to take into account the finite aperture being used in the measurement setup, the simulation results are averaged over four positions at distances of 0.25mm from the (

*x*,

_{i}*y*) coordinate on the DOE.

_{j}## 4.3. Discussion of simulation results

*x*

_{3}and

*x*

_{6}. These correspond to Rayleigh and resonance type anomalies respectively. The fact that the anomalies have an impact on the spectral uniformity is due to the wavelength dependence of the excitation of the anomalies. For the four-level profile in Fig. 11(b), we see that the resulting grating groove profile yields much better spectral uniformity than the two-level profile. The anomaly around position

*x*

_{3}is also seen in the simulations for the four-level profile, but it is not as pronounced as for the two-level profile. The resonance anomaly is not excited for this type of profile. The two-level profile is clearly a less desirable choice due to the large spectral variations. However it is an important design example because it is much simpler to fabricate than the four-level DOE. Not only does the four-level DOE have groove profiles with smaller line-widths, but the fabrication also requires very precise alignment of two masks. Furthermore, our simulation results for the two-level profile happen to show clearly some vectorial optical field effects that must be taken into account in reflective DOE design in general. Below we will discuss in detail the two different anomalies.

## 4.3.1. Rayleigh’s anomaly

*x*

_{2}and

*x*

_{3}correspond to incidence angles approximately between 18° and 20°, as seen in Fig. 7 above. In Fig. 12, the absolute value of the diffraction angle for the beams of order

*m*=1 and

*m*=-2 is shown as a function of incidence angle for three different wavelengths, and a grating period of

*P*=2.5 µm. We see that the increase in diffraction efficiency coincides with the transition of the diffracted beam of order

*m*=1 (Fig. 12(a)) from a propagating to an evanescent wave. The studied diffraction efficiency of the beam of order

*m*=-1 decreases at higher incidence angles when the diffracted beam of order

*m*=-2 (Fig. 12(b)) appears at a slightly higher angle of incidence. The anomaly around x-position

*x*

_{2}and

*x*

_{3}is hence

*m*=1 into the remaining propagating orders. This sudden change in the diffraction efficiency is wavelength dependent, and this leads to poor spectral uniformity for the positions where this occurs.

*m*=-1 more closely, and thus less of the diffracted energy is in the order

*m*=1 in this case. When the energy is redistributed from the

*m*=1 order to the propagating orders it has less impact on the

*m*=-1 order which is already carrying most of the energy. One could avoid this type of anomaly by designing the setup such that the involved incidence angles and grating periods are such that they lie outside the parameter space where the number of propagating diffraction orders vary. However, there are other requirements to take into account, such as separating the focal regions of the wavelength components, and other instrument specifications. Rayleigh’s anomaly represents an undesirable additional design consideration.

## 4.3.2. Surface shape resonance

13. Lumerical FDTD Solutions, http://www.lumerical.com.

14. T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface Shape Resonances in Lamellar Metallic Gratings,” Phys. Rev. Lett. **81**, 665–668 (1998).
[CrossRef]

15. E. K. Popov, N. Bonod, and S. Enoch, “Comparison of plasmon surface waves on shallow and deep metallic 1D and 2D gratings,” Opt. Express **15**, 4224–4237 (2007).
[CrossRef] [PubMed]

## 5. Experimental results

### 5.1. Experimental setup

*n*predefined angles

_{λ}*α*at which light of

_{k}*n*wavelengths is successively focused on the detector. When collecting data from the experiments, we assume that only light of the correct wavelength is focused on the detector at the predefined scan positions. The DOE is covered by a movable screen, with a circular aperture with a diameter of 1 mm that was centered on 9×9 positions on the DOE as described above. In the analysis of measurement data we disregarded the fact that the projected area of the aperture (seen from the source or the detector) depend on the incidence and diffraction angles. Given an aperture position, our main concern is variations in diffraction efficiency between the five wavelength bands, and the difference in the illumination would only come from the small scan angles

_{λ}*α*that separate the wavelength bands, when the aperture position is given.

_{k}## 5.2. Results from measurements

*η*-1| across the DOE at a given x-position the same way as for the simulation results above.

_{ijk}## 5.3. Discussion

*x*

_{3}and

*x*

_{4}, but no large defined peak around x-position

*x*

_{6}, as was seen in the simulations in Fig. 11(a). This broad peak resembles the result from the simulations where the diffraction efficiency from gratings of three different groove widths were averaged over, Fig. 16(a). Thus we believe the reason we measure a broad peak and not the more sharply defined high-amplitude peak as predicted by the simulations in Fig. 11(b), is because of variations in grating period and groove profile in the real DOE. Furthermore, the roughness of the grating lines of the fabricated two-level DOE would also lead to scattering and may also to some extent prevent the excitation of the resonance anomaly predicted in simulations. We emphasize that for our application the spectral uniformity of the four-level DOE (see Fig. 5(a)) is found to be very good both in the simulations and measurements. The uniformity is naturally even better if larger areas of the DOE are compared, with the help of a larger aperture in the measurements. To further improve the spectral uniformity, one could perform an optimization with regards to the height and width of the steps in the profile, as reported in [16

16. J. Pietarinen, T. Vallius, and J. Turunen, “Wideband four-level transmission gratings with flattened spectral efficiency,” Opt. Express **14**, 2583–2588 (2006).
[CrossRef] [PubMed]

## 5.3.1. Variations from ideal linear gratings

*x*

_{3}is almost unchanged by the averaging over different groove widths. This illustrates the fact that the conditions for exciting this anomaly are not dependent on the detailed groove profile. The variations for the four-level profile are generally larger in the measurements, Fig. 15(b), than the simulations in Fig. 11(b). However, for the four-level profile, the measurements and simulations agree very well, to the extent that they both predict very small variations compared to the two-level results. On a relative scale, however, the agreement is quite poor, since the measured maximal variations are many times greater than the simulated variations, except in x-position 3. For the two-level profile, measurements and simulations agree much better on a relative scale, within a factor of 2 for most x-positions.

## 6. Conclusion

^{2}with a complex pattern of grooves with sawtooth-like profiles. We studied two binary approximations to this profile, with two and four levels, respectively.

## Acknowledgments

## References and links

1. | S. Ura, F. Okayama, K. Shiroshita, K. Nishio, T. Sasaki, H. Nishihara, T. Yotsuya, M. Okano, and K. Satoh, “Planar reflection grating lens for compact spectroscopic imaging system,” Appl. Opt. |

2. | S. Grabarnik, A. Emadi, E. Sokolova, G. Vdovin, and R. F. Wolffenbuttel, “Optimal implementation of a microspectrometer based on a single flat diffraction grating,” Appl. Opt. |

3. | R. F. Wolffenbuttel, “State-of-the-art in integrated optical microspectrometers,” IEEE Trans. Instrum. Meas. |

4. | O. Løvhaugen, I.-R. Johansen, K. A. H. Bakke, B. G. Fismen, and S. Nicolas, “Dedicated spectrometers based on diffractive optics: design, modelling and evaluation,” J. Mod. Opt. |

5. | E. Noponen, J. Turunen, and A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J.Opt. Soc. Am. A |

6. | M. C. Hutley, |

7. | G. J. Swanson, “Binary optics technology: Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” Tech. Rep. 914 (Massachusetts Institute of Technology, Cambridge, Mass., 1991). |

8. | J. W. Goodman, |

9. | |

10. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A |

11. | M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A |

12. | E.D. Palik, |

13. | Lumerical FDTD Solutions, http://www.lumerical.com. |

14. | T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface Shape Resonances in Lamellar Metallic Gratings,” Phys. Rev. Lett. |

15. | E. K. Popov, N. Bonod, and S. Enoch, “Comparison of plasmon surface waves on shallow and deep metallic 1D and 2D gratings,” Opt. Express |

16. | J. Pietarinen, T. Vallius, and J. Turunen, “Wideband four-level transmission gratings with flattened spectral efficiency,” Opt. Express |

**OCIS Codes**

(050.1380) Diffraction and gratings : Binary optics

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1960) Diffraction and gratings : Diffraction theory

(230.3990) Optical devices : Micro-optical devices

(300.6190) Spectroscopy : Spectrometers

(300.6340) Spectroscopy : Spectroscopy, infrared

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: April 22, 2009

Revised Manuscript: May 29, 2009

Manuscript Accepted: May 29, 2009

Published: June 3, 2009

**Citation**

Hallvard Angelskår, Ib-Rune Johansen, Matthieu Lacolle, Håkon Sagberg, and Aasmund S. Sudbø, "Spectral uniformity of two- and four-level diffractive optical elements for spectroscopy," Opt. Express **17**, 10206-10222 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-12-10206

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### References

- S. Ura, F. Okayama, K. Shiroshita, K. Nishio, T. Sasaki, H. Nishihara, T. Yotsuya, M. Okano, and K. Satoh, "Planar reflection grating lens for compact spectroscopic imaging system," Appl. Opt. 42, 175-180 (2003). [CrossRef] [PubMed]
- S. Grabarnik, A. Emadi, E. Sokolova, G. Vdovin, and R. F. Wolffenbuttel, "Optimal implementation of a microspectrometer based on a single flat diffraction grating," Appl. Opt. 47, 2082-2090 (2008). [CrossRef] [PubMed]
- R. F. Wolffenbuttel, "State-of-the-art in integrated optical microspectrometers," IEEE Trans. Instrum. Meas. 53, 197-202 (2004). [CrossRef]
- O. Løvhaugen, I.-R. Johansen, K. A. H. Bakke, B. G. Fismen, and S. Nicolas, "Dedicated spectrometers based on diffractive optics: design, modelling and evaluation," J. Mod. Opt. 51, 2203-2222 (2004). [CrossRef]
- E. Noponen, J. Turunen, and A. Vasara, "Electromagnetic theory and design of diffractive-lens arrays," J.Opt. Soc. Am. A 10, 434-443 (1993). [CrossRef]
- M. C. Hutley, Diffraction gratings (Academic Press, London, 1982).
- G. J. Swanson, "Binary optics technology: Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements," Tech. Rep. 914 (Massachusetts Institute of Technology, Cambridge, Mass., 1991).
- J. W. Goodman, Introduction to Fourier optics, 2nd ed. (McGraw-Hill, 1996).
- GD-Calc, http://software.kjinnovation.com/GD-Calc.html.
- L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996). [CrossRef]
- M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of metallic surface-relief gratings," J. Opt. Soc. Am. A 3, 1780-1787 (1986). [CrossRef]
- E.D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
- Lumerical FDTD Solutions, http://www.lumerical.com.
- T. López-Rios, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, "Surface Shape Resonances in Lamellar Metallic Gratings," Phys. Rev. Lett. 81, 665-668 (1998). [CrossRef]
- E. K. Popov, N. Bonod, and S. Enoch, "Comparison of plasmon surface waves on shallow and deep metallic 1D and 2D gratings," Opt. Express 15, 4224-4237 (2007). [CrossRef] [PubMed]
- J. Pietarinen, T. Vallius, and J. Turunen, "Wideband four-level transmission gratings with flattened spectral efficiency," Opt. Express 14, 2583-2588 (2006). [CrossRef] [PubMed]

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