Abstract

The optimal structural parameters for an antireflective structure in high resistive float zone silicon are deduced for a rectangular and a hexagonal structure. For this the dependence of the effective index from the filling factor was calculated for both grating types. The structures were manufactured by the Bosch®-process. The required structural parameters for a continuous profile require an adaption of the fabrication process. Challenges are the depth and the slight positive slope of the structures. Starting point for the realization of the antireflective structures was the manufacturing of deep binary gratings. A rectangular structure and a hexagonal structure with period 50 μm and depth 500 μm were realized. Measurements with a THz time domain spectroscopy setup show an increase of the electric field amplitude of 15.2% for the rectangular grating and 21.76% for the hexagonal grating. The spectral analysis shows that the bandwidth of the hexagonal grating reaches from 0.1 to 2 THz.

© 2009 Optical Society of America

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References

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2008

C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tünnermann, "Design and evaluation of a THz time domain imaging system using standard optical design software," Appl. Opt. 47, 4994-5006 (2008).
[CrossRef] [PubMed]

C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tünnermann, "Design and analysis of quasioptical THz time domain imaging systems," Proc. SPIE 7100, 71000S (2008).
[CrossRef]

2007

2005

2004

J. Dai, J. Zhang, W. Zhang, and D. Grischkowsky, "Terahertz time-domain spectroscopy characterization of the far-infrared absorption and index of refraction of high-resistivity, float-zone silicon," J. Opt. Soc. Am. B 21, 1379-1386 (2004).
[CrossRef]

A. Gombert, B. Bläsi, C. Bühler, and P. Nitz, "Some application cases and related manufacturing techniques for optically functional microstructures on large areas," Opt. Eng. 43, 2525-2533 (2004).
[CrossRef]

2003

2000

A. J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yngvesson, "An anti-reflection coating for silicon optics at terahertz frequencies," IEEE Microwave Guided Wave Lett. 10, 264-266 (2000).
[CrossRef]

J. Bischoff and R. Brunner, "Numerical investigation of the resolution in solid immersion lens systems," Proc. SPIE 4099, 1-11 (2000).

1998

1997

1996

P. Lalanne and D. Lemercier-Lalanne, "On the effective medium theory of subwavelength periodic structures," J. Mod. Opt. 43, 2063-2085 (1996).
[CrossRef]

1995

1994

1990

1973

P. B. Clapham and M. C. Hutley, "Reduction of lens reflection by the 'Moth Eye' principle," Nature 244, 281-282 (1973).
[CrossRef]

1956

S. Rytov, "Electromagnetic Properties of a Finely Stratified Medium," Soviet Physics JETP 2, 466-475 (1956).

Abbott, D.

W. Withayachumnankul, B. M. Fischer, S. P. Mickan, and D. Abbott, "Retrofittable antireflection coatings for T-rays," Microwave Opt. Technol. Lett. 49, 2267-2270 (2007).
[CrossRef]

Bischoff, J.

J. Bischoff and R. Brunner, "Numerical investigation of the resolution in solid immersion lens systems," Proc. SPIE 4099, 1-11 (2000).

Bläsi, B.

A. Gombert, B. Bläsi, C. Bühler, and P. Nitz, "Some application cases and related manufacturing techniques for optically functional microstructures on large areas," Opt. Eng. 43, 2525-2533 (2004).
[CrossRef]

Brückner, C.

Brunner, R.

J. Bischoff and R. Brunner, "Numerical investigation of the resolution in solid immersion lens systems," Proc. SPIE 4099, 1-11 (2000).

Bühler, C.

A. Gombert, B. Bläsi, C. Bühler, and P. Nitz, "Some application cases and related manufacturing techniques for optically functional microstructures on large areas," Opt. Eng. 43, 2525-2533 (2004).
[CrossRef]

Cheville, R. A.

Clapham, P. B.

P. B. Clapham and M. C. Hutley, "Reduction of lens reflection by the 'Moth Eye' principle," Nature 244, 281-282 (1973).
[CrossRef]

Dai, J.

Darmo, J.

Fattinger, C.

Fischer, B. M.

W. Withayachumnankul, B. M. Fischer, S. P. Mickan, and D. Abbott, "Retrofittable antireflection coatings for T-rays," Microwave Opt. Technol. Lett. 49, 2267-2270 (2007).
[CrossRef]

Gatesman, A. J.

A. J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yngvesson, "An anti-reflection coating for silicon optics at terahertz frequencies," IEEE Microwave Guided Wave Lett. 10, 264-266 (2000).
[CrossRef]

Gombert, A.

A. Gombert, B. Bläsi, C. Bühler, and P. Nitz, "Some application cases and related manufacturing techniques for optically functional microstructures on large areas," Opt. Eng. 43, 2525-2533 (2004).
[CrossRef]

Grann, E. B.

Grischkowsky, D.

Harmon, S. A.

Hosako, I.

Hutley, M. C.

P. B. Clapham and M. C. Hutley, "Reduction of lens reflection by the 'Moth Eye' principle," Nature 244, 281-282 (1973).
[CrossRef]

Iwata, K.

Ji, M.

A. J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yngvesson, "An anti-reflection coating for silicon optics at terahertz frequencies," IEEE Microwave Guided Wave Lett. 10, 264-266 (2000).
[CrossRef]

Keiding, S.

Kikuta, H.

Kröll, J.

Kubo, H.

Lalanne, P.

P. Lalanne, and D. Lemercier-Lalanne, "Depth dependence of the effective properties of subwavelength gratings," J. Opt. Soc. Am. A 14, 450-458 (1997).
[CrossRef]

P. Lalanne and D. Lemercier-Lalanne, "On the effective medium theory of subwavelength periodic structures," J. Mod. Opt. 43, 2063-2085 (1996).
[CrossRef]

Lemercier-Lalanne, D.

P. Lalanne, and D. Lemercier-Lalanne, "Depth dependence of the effective properties of subwavelength gratings," J. Opt. Soc. Am. A 14, 450-458 (1997).
[CrossRef]

P. Lalanne and D. Lemercier-Lalanne, "On the effective medium theory of subwavelength periodic structures," J. Mod. Opt. 43, 2063-2085 (1996).
[CrossRef]

Mickan, S. P.

W. Withayachumnankul, B. M. Fischer, S. P. Mickan, and D. Abbott, "Retrofittable antireflection coatings for T-rays," Microwave Opt. Technol. Lett. 49, 2267-2270 (2007).
[CrossRef]

Moharam, M. G.

Müller, R.

C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tünnermann, "Design and analysis of quasioptical THz time domain imaging systems," Proc. SPIE 7100, 71000S (2008).
[CrossRef]

C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tünnermann, "Design and evaluation of a THz time domain imaging system using standard optical design software," Appl. Opt. 47, 4994-5006 (2008).
[CrossRef] [PubMed]

Musante, C.

A. J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yngvesson, "An anti-reflection coating for silicon optics at terahertz frequencies," IEEE Microwave Guided Wave Lett. 10, 264-266 (2000).
[CrossRef]

Nitz, P.

A. Gombert, B. Bläsi, C. Bühler, and P. Nitz, "Some application cases and related manufacturing techniques for optically functional microstructures on large areas," Opt. Eng. 43, 2525-2533 (2004).
[CrossRef]

Notni, G.

Ohira, Y.

Pommet, D. A.

Pradarutti, B.

Reiten, M. T.

Riehemann, S.

Rytov, S.

S. Rytov, "Electromagnetic Properties of a Finely Stratified Medium," Soviet Physics JETP 2, 466-475 (1956).

Steinkopf, R.

Stenzel, O.

Tünnermann, A.

Unterrainer, K.

van Exter, M.

Waldman, J.

A. J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yngvesson, "An anti-reflection coating for silicon optics at terahertz frequencies," IEEE Microwave Guided Wave Lett. 10, 264-266 (2000).
[CrossRef]

Withayachumnankul, W.

W. Withayachumnankul, B. M. Fischer, S. P. Mickan, and D. Abbott, "Retrofittable antireflection coatings for T-rays," Microwave Opt. Technol. Lett. 49, 2267-2270 (2007).
[CrossRef]

Yngvesson, S.

A. J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yngvesson, "An anti-reflection coating for silicon optics at terahertz frequencies," IEEE Microwave Guided Wave Lett. 10, 264-266 (2000).
[CrossRef]

Zhang, J.

Zhang, W.

Appl. Opt.

IEEE Microwave Guided Wave Lett.

A. J. Gatesman, J. Waldman, M. Ji, C. Musante, and S. Yngvesson, "An anti-reflection coating for silicon optics at terahertz frequencies," IEEE Microwave Guided Wave Lett. 10, 264-266 (2000).
[CrossRef]

J. Mod. Opt.

P. Lalanne and D. Lemercier-Lalanne, "On the effective medium theory of subwavelength periodic structures," J. Mod. Opt. 43, 2063-2085 (1996).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Microwave Opt. Technol. Lett.

W. Withayachumnankul, B. M. Fischer, S. P. Mickan, and D. Abbott, "Retrofittable antireflection coatings for T-rays," Microwave Opt. Technol. Lett. 49, 2267-2270 (2007).
[CrossRef]

Nature

P. B. Clapham and M. C. Hutley, "Reduction of lens reflection by the 'Moth Eye' principle," Nature 244, 281-282 (1973).
[CrossRef]

Opt. Eng.

A. Gombert, B. Bläsi, C. Bühler, and P. Nitz, "Some application cases and related manufacturing techniques for optically functional microstructures on large areas," Opt. Eng. 43, 2525-2533 (2004).
[CrossRef]

Opt. Express

Proc. SPIE

C. Brückner, B. Pradarutti, R. Müller, S. Riehemann, G. Notni, and A. Tünnermann, "Design and analysis of quasioptical THz time domain imaging systems," Proc. SPIE 7100, 71000S (2008).
[CrossRef]

J. Bischoff and R. Brunner, "Numerical investigation of the resolution in solid immersion lens systems," Proc. SPIE 4099, 1-11 (2000).

Soviet Physics JETP

S. Rytov, "Electromagnetic Properties of a Finely Stratified Medium," Soviet Physics JETP 2, 466-475 (1956).

Other

H. A. Macleod, Thin-film optical filters (Institute of Physics Publ, 2002).

H. Ibach and H. Lüth, Festkörperphysik (Springer, 2002).

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Figures (11)

Fig. 1.
Fig. 1.

(a) Binary rectangular grating and top view (c) with incident wave vector k i , and grating vectors a 1 and a 2. The vectors of the reciprocal grating are g 1 and g 2. (b) Binary hexagonal grating and top view (d) with grating vectors a 1 and a 2 and the vectors of the reciprocal grating g 1 and g 2. The red dots indicate the grating points of the reciprocal grating. The hexagonal grating can also be described as a rectangular grating with basis vectors a 1 and a 2 * with different grating periods in x- and y-direction. For both grating types the TE polarization is perpendicular to the plane and the TM polarization lies in the plane spanned by the grating vector a1 and the normal to the surface (z-direction). The grating depth is d, and the widths of the pillars of the rectangular grating in x- and y-direction are wx and wy , respectively. In case of the hexagonal grating the diameter of the pillar is D. The effective indices in x-, y- and z-direction are neff,x , neff,y and neff,z , respectively.

Fig. 2.
Fig. 2.

(a) Reflectance in dependence of the refractive index for wavelength 375 μm, layer thickness 50.694 μm, refractive index of the substrate material 3.42 and refractive index of the layer 1.85, determined by thin film matrix formalism. (b) Reflectance in dependence on the filling factor for the rectangular grating in the quasi-static limit (wavelength 375 μm, grating period 0.375 μm), grating depth 50.694 μm and refractive index of the substrate material 3.42, determined by RCWA.

Fig. 3.
Fig. 3.

(a) Dependence of the effective index on the filling factor for a rectangular grating at different normalized grating periods. For the ratio Λ/λ = 0.001 the grating is in the quasi-static limit and for Λ/λ = 0.2 the zeroth-order condition is just obeyed. (b) Dependence of the effective index on the filling factor for different grating types. The curves for the 2D gratings and the zeroth-order EMT-curves for the 1D grating are valid in the quasi-static limit (Λ/λ = 0.001). The second-order EMT-curves for the 1D grating are calculated at Λ/λ = 0.2.

Fig. 4.
Fig. 4.

Reflectance in dependence on the normalized depth d/λ for different grating types calculated by effective medium theory in the quasi-static limit.

Fig. 5.
Fig. 5.

Comparison of the transmittance at a surface with a binary subwavelength structure and a homogeneous layer. The respective refractive index of the homogeneous layer is given in the plots. The black dashed vertical line is the zeroth-diffraction-order limit. Above, some energy is diffracted into higher diffraction orders. The period is 50 μm and the depth is 500 μm for all calculations. (a) Rectangular grating with filling factor 0.79. (b) Hexagonal grating with filling factor 0.84. (c) Hexagonal grating with filling factor 0.7361.

Fig. 6.
Fig. 6.

SEM-images of the rectangular structure. (a) Cross section. The depth of the structure is 462.5 μm, the grating period is 52.7 μm, and the width of the pillars in the upper region of the structure is 36.8 μm (filling factor 0.7). (b) Structure under 15° tilting angle.

Fig. 7.
Fig. 7.

SEM-images of the hexagonal structure. (a) Cross section, the structural depth is 500 μm, the period is 50 μm, and the diameter of the pillars in the upper region of the structure is 37.5 μm (filling factor 0.75). The etching mask was removed completely. (b) Structure under 15° tilting angle. Nearly the whole 4-inch Si-wafer consists of the shown pillars.

Fig. 8.
Fig. 8.

THz time domain spectroscopy system (principle). The sample was placed in the collimated beam between the second set of off-axis parabolic mirrors.

Fig. 9.
Fig. 9.

(a) Measured waveform of the unstructured sample and the sample with the rectangular structure. The inset shows theoretical delay times of the secondary pulses with respect to the main pulse of the layer system with a layer with an effective index of 1.85 and a layer with a refractive index of 3.42. Both layers have a depth of 500 μm. (b) Spectral analysis: Ratio of the transmitted electric field amplitudes of structured and unstructured sample.

Fig. 10.
Fig. 10.

(a) Measured waveform of the unstructured sample and the sample with the hexagonal structure. (b) Spectral analysis: Ratio of the transmitted electric field amplitudes of structured and unstructured sample.

Fig. 11.
Fig. 11.

(a) Simulation for the hexagonal structure with effective medium theory. (b) Simulation for conic sections in a hexagonal grating with filling factor 0.3 at the top of the structure. The remaining intensity reflectance is 1%. The dashed line shows the zeroth-diffraction-order limit for a grating period of 50 μm. The inset shows the corresponding hexagonal structure with filling factor 0.3 at the top of the structure.

Tables (1)

Tables Icon

Table 1: Normalized grating period Γ/λ dependent on angle of incidence and grating type

Equations (9)

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k i , xy k uv , xy = g uv ,
Λ λ f g n 1 sin θ cos ( φ δ i ) + n 2 2 n 1 2 sin 2 θ sin ( φ δ i ) ,
Λ λ f g n 2 = f g 3.42
Λ λ f g n 1 + n 2 = f g 4.42
Λ λ f g n 1 sin θ + n 2
λ c , hexagonal = 0.866 λ c , rectangular
n eff , ( 0 ) = ( 1 f ) n 1 2 + f n 2 2 and n eff , ( 0 ) = n 1 n 2 ( 1 f ) n 1 2 + f n 2 2 ,
n eff , ( 2 ) = n eff , ( 0 ) 2 + π 2 3 ( Λ λ ) 2 f 2 ( 1 f ) 2 ( n 2 2 n 1 2 ) 2 and
n eff , ( 2 ) = n eff , ( 0 ) 2 + π 2 3 ( Λ λ ) 2 f 2 ( 1 f ) 2 ( 1 n 2 2 1 n 1 2 ) 2 ( n eff , ( 0 ) ) 6 ( n eff , ( 0 ) ) 2

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