Abstract

Microstructured optical products are becoming more widespread due to advances in manufacturing. Many of these structures contain faceted surfaces with steep slopes. Adequate metrology for such surfaces is lacking. We describe an interferometric technique that combines plane wave illumination with an index matching liquid to achieve high quality, high speed measurements of such faceted microstructures. We account for refraction at the interfaces, rather than consider only optical path length changes due to the index liquid, and this significantly improves the facet angle measurement. We demonstrate the technique with the measurement of an array of micropyramids and show that our results are in good agreement with measurements taken on a contact profilometer. We also extend the technique to measure opaque microcorner cubes by implementing an intermediate replication step.

© 2010 Optical Society of America

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    [CrossRef]

2008

E. Brinksmeier, R. Gläbe, and C. Flucke, “Manufacturing of molds for replication of micro cube corner retroreflectors,” Prod. Eng. Res. Devel. 2, 33-38 (2008).
[CrossRef]

E. Peiner, M. Balke, L. Doering, and U. Brand, “Tactile probes for dimensional metrology with microcomponents at nanometer resolution,” Meas. Sci. Technol. 19, 064001 (2008).
[CrossRef]

K. Seong and J. Greivenkamp, “Surface figure measurement based on the transmitted wavefront with reverse raytracing,” Opt. Eng. 47, 043602 (2008).
[CrossRef]

2007

E. Manske, T. Hausotte, R. Mastylo, T. Machleidt, K.-H. Franke, and G. Jager, “New applications of the nanopositioning and nanomeasuring machine by using advanced tactile and non-tactile probes,” Meas. Sci. Technol. 18, 520-527 (2007).
[CrossRef]

J. Kuhn, F. Charrière, T. Colomb, E. Cuche, Y. Emery, and C. Depeursinge, “Digital holographic microscopy for nanometric quality control of micro-optical components,” Proc. SPIE 6475, 64750V (2007).
[CrossRef]

2006

K. Ryu, J. Rhee, K. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy Mater. 80, 1580-1587 (2006).
[CrossRef]

C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135-140 (2006).
[CrossRef] [PubMed]

2005

2004

2002

K. Kim, S. Park, J.-B. Lee, H. Manohara, Y. Desta, M. Murphy, and C. H. Ahn, “Rapid replication of polymeric and metallic high aspect ratio microstructures using PDMS and LIGA technology,” Microsyst. Technol. 9, 5-10 (2002).
[CrossRef]

J. Yuan, S. Chang, S. Li, and Y. Zhang, “Design and fabrication of micro-cube-corner array retro-reflectors,” Opt. Commun. 209, 75-83 (2002).
[CrossRef]

2000

L. Lin, T. Shia and C. Chiu, “Silicon-processed plastic micropyramids for brightness enhancement applications,” J. Micromech. Microeng. 10, 395-400 (2000).
[CrossRef]

1999

G. Seward and P. Cort, “Measurement and characterization of angular reflectance for cube-corners and microspheres,” Opt. Eng. 38, 164-169 (1999).
[CrossRef]

1995

1987

Ahn, C. H.

K. Kim, S. Park, J.-B. Lee, H. Manohara, Y. Desta, M. Murphy, and C. H. Ahn, “Rapid replication of polymeric and metallic high aspect ratio microstructures using PDMS and LIGA technology,” Microsyst. Technol. 9, 5-10 (2002).
[CrossRef]

Balke, M.

E. Peiner, M. Balke, L. Doering, and U. Brand, “Tactile probes for dimensional metrology with microcomponents at nanometer resolution,” Meas. Sci. Technol. 19, 064001 (2008).
[CrossRef]

Ballinger, T.

T. Ballinger and J. Huedepohl, “Advantages of using Dektak surface profilers with Vision 3D analysis software,” Application Note 542 (Veeco Dektak, 2008).

Bothe, T.

T. Bothe, W. Li, C. Kopylow, and W. Juptner, “Fringe reflection for high resolution topometry and surface description on various lateral scales,” in Fringe 2005: The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2005), pp. 362-371.

Brand, U.

E. Peiner, M. Balke, L. Doering, and U. Brand, “Tactile probes for dimensional metrology with microcomponents at nanometer resolution,” Meas. Sci. Technol. 19, 064001 (2008).
[CrossRef]

Brinksmeier, E.

E. Brinksmeier, R. Gläbe, and C. Flucke, “Manufacturing of molds for replication of micro cube corner retroreflectors,” Prod. Eng. Res. Devel. 2, 33-38 (2008).
[CrossRef]

Bruning, J. H.

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501-598.

Chang, S.

J. Yuan, S. Chang, S. Li, and Y. Zhang, “Design and fabrication of micro-cube-corner array retro-reflectors,” Opt. Commun. 209, 75-83 (2002).
[CrossRef]

Charrière, F.

J. Kuhn, F. Charrière, T. Colomb, E. Cuche, Y. Emery, and C. Depeursinge, “Digital holographic microscopy for nanometric quality control of micro-optical components,” Proc. SPIE 6475, 64750V (2007).
[CrossRef]

Chiu, C.

L. Lin, T. Shia and C. Chiu, “Silicon-processed plastic micropyramids for brightness enhancement applications,” J. Micromech. Microeng. 10, 395-400 (2000).
[CrossRef]

Colomb, T.

J. Kuhn, F. Charrière, T. Colomb, E. Cuche, Y. Emery, and C. Depeursinge, “Digital holographic microscopy for nanometric quality control of micro-optical components,” Proc. SPIE 6475, 64750V (2007).
[CrossRef]

Cort, P.

G. Seward and P. Cort, “Measurement and characterization of angular reflectance for cube-corners and microspheres,” Opt. Eng. 38, 164-169 (1999).
[CrossRef]

Cuche, E.

J. Kuhn, F. Charrière, T. Colomb, E. Cuche, Y. Emery, and C. Depeursinge, “Digital holographic microscopy for nanometric quality control of micro-optical components,” Proc. SPIE 6475, 64750V (2007).
[CrossRef]

Davies, A.

D. Purcell, A. Suratkar, A. Davies, and F. Farahi, “Measuring the wavefront distortion of a microlens array using an index matching liquid,” presented at the Optical Fabrication and Testing Topical Meeting, Rochester, New York, USA (9-11 October 2006).

Demma, N.

Depeursinge, C.

J. Kuhn, F. Charrière, T. Colomb, E. Cuche, Y. Emery, and C. Depeursinge, “Digital holographic microscopy for nanometric quality control of micro-optical components,” Proc. SPIE 6475, 64750V (2007).
[CrossRef]

Desmet, L.

L. Desmet, “Fabrication and design of micro-optical interconnect systems with fanout,” Ph.D. dissertation (Vrije Universiteit of Brussel, 2007).

Desta, Y.

K. Kim, S. Park, J.-B. Lee, H. Manohara, Y. Desta, M. Murphy, and C. H. Ahn, “Rapid replication of polymeric and metallic high aspect ratio microstructures using PDMS and LIGA technology,” Microsyst. Technol. 9, 5-10 (2002).
[CrossRef]

Doering, L.

E. Peiner, M. Balke, L. Doering, and U. Brand, “Tactile probes for dimensional metrology with microcomponents at nanometer resolution,” Meas. Sci. Technol. 19, 064001 (2008).
[CrossRef]

Emery, Y.

J. Kuhn, F. Charrière, T. Colomb, E. Cuche, Y. Emery, and C. Depeursinge, “Digital holographic microscopy for nanometric quality control of micro-optical components,” Proc. SPIE 6475, 64750V (2007).
[CrossRef]

Farahi, F.

D. Purcell, A. Suratkar, A. Davies, and F. Farahi, “Measuring the wavefront distortion of a microlens array using an index matching liquid,” presented at the Optical Fabrication and Testing Topical Meeting, Rochester, New York, USA (9-11 October 2006).

Flucke, C.

E. Brinksmeier, R. Gläbe, and C. Flucke, “Manufacturing of molds for replication of micro cube corner retroreflectors,” Prod. Eng. Res. Devel. 2, 33-38 (2008).
[CrossRef]

Franke, K.-H.

E. Manske, T. Hausotte, R. Mastylo, T. Machleidt, K.-H. Franke, and G. Jager, “New applications of the nanopositioning and nanomeasuring machine by using advanced tactile and non-tactile probes,” Meas. Sci. Technol. 18, 520-527 (2007).
[CrossRef]

Gläbe, R.

E. Brinksmeier, R. Gläbe, and C. Flucke, “Manufacturing of molds for replication of micro cube corner retroreflectors,” Prod. Eng. Res. Devel. 2, 33-38 (2008).
[CrossRef]

Greivenkamp, J.

K. Seong and J. Greivenkamp, “Surface figure measurement based on the transmitted wavefront with reverse raytracing,” Opt. Eng. 47, 043602 (2008).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501-598.

Grivenkamp, J. E.

Hausotte, T.

E. Manske, T. Hausotte, R. Mastylo, T. Machleidt, K.-H. Franke, and G. Jager, “New applications of the nanopositioning and nanomeasuring machine by using advanced tactile and non-tactile probes,” Meas. Sci. Technol. 18, 520-527 (2007).
[CrossRef]

Hoo, J. Soo

A. O'Neill, J. Soo Hoo, and G. Walker, “Chips & Tips: Rapid curing of PDMS for microfluidic applications,” http://www.rsc.org/Publishing/Journals/lc/PDMS_curing.asp.

Huang, Y.

Huedepohl, J.

T. Ballinger and J. Huedepohl, “Advantages of using Dektak surface profilers with Vision 3D analysis software,” Application Note 542 (Veeco Dektak, 2008).

Jager, G.

E. Manske, T. Hausotte, R. Mastylo, T. Machleidt, K.-H. Franke, and G. Jager, “New applications of the nanopositioning and nanomeasuring machine by using advanced tactile and non-tactile probes,” Meas. Sci. Technol. 18, 520-527 (2007).
[CrossRef]

Juptner, W.

T. Bothe, W. Li, C. Kopylow, and W. Juptner, “Fringe reflection for high resolution topometry and surface description on various lateral scales,” in Fringe 2005: The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2005), pp. 362-371.

Kim, J.

K. Ryu, J. Rhee, K. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy Mater. 80, 1580-1587 (2006).
[CrossRef]

Kim, K.

K. Kim, S. Park, J.-B. Lee, H. Manohara, Y. Desta, M. Murphy, and C. H. Ahn, “Rapid replication of polymeric and metallic high aspect ratio microstructures using PDMS and LIGA technology,” Microsyst. Technol. 9, 5-10 (2002).
[CrossRef]

Klimant, I.

C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135-140 (2006).
[CrossRef] [PubMed]

Kopylow, C.

T. Bothe, W. Li, C. Kopylow, and W. Juptner, “Fringe reflection for high resolution topometry and surface description on various lateral scales,” in Fringe 2005: The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2005), pp. 362-371.

Kuhn, J.

J. Kuhn, F. Charrière, T. Colomb, E. Cuche, Y. Emery, and C. Depeursinge, “Digital holographic microscopy for nanometric quality control of micro-optical components,” Proc. SPIE 6475, 64750V (2007).
[CrossRef]

Lee, J.-B.

K. Kim, S. Park, J.-B. Lee, H. Manohara, Y. Desta, M. Murphy, and C. H. Ahn, “Rapid replication of polymeric and metallic high aspect ratio microstructures using PDMS and LIGA technology,” Microsyst. Technol. 9, 5-10 (2002).
[CrossRef]

Li, S.

J. Yuan, S. Chang, S. Li, and Y. Zhang, “Design and fabrication of micro-cube-corner array retro-reflectors,” Opt. Commun. 209, 75-83 (2002).
[CrossRef]

Li, W.

T. Bothe, W. Li, C. Kopylow, and W. Juptner, “Fringe reflection for high resolution topometry and surface description on various lateral scales,” in Fringe 2005: The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2005), pp. 362-371.

Lin, L.

L. Lin, T. Shia and C. Chiu, “Silicon-processed plastic micropyramids for brightness enhancement applications,” J. Micromech. Microeng. 10, 395-400 (2000).
[CrossRef]

Machleidt, T.

E. Manske, T. Hausotte, R. Mastylo, T. Machleidt, K.-H. Franke, and G. Jager, “New applications of the nanopositioning and nanomeasuring machine by using advanced tactile and non-tactile probes,” Meas. Sci. Technol. 18, 520-527 (2007).
[CrossRef]

Manohara, H.

K. Kim, S. Park, J.-B. Lee, H. Manohara, Y. Desta, M. Murphy, and C. H. Ahn, “Rapid replication of polymeric and metallic high aspect ratio microstructures using PDMS and LIGA technology,” Microsyst. Technol. 9, 5-10 (2002).
[CrossRef]

Manske, E.

E. Manske, T. Hausotte, R. Mastylo, T. Machleidt, K.-H. Franke, and G. Jager, “New applications of the nanopositioning and nanomeasuring machine by using advanced tactile and non-tactile probes,” Meas. Sci. Technol. 18, 520-527 (2007).
[CrossRef]

Mastylo, R.

E. Manske, T. Hausotte, R. Mastylo, T. Machleidt, K.-H. Franke, and G. Jager, “New applications of the nanopositioning and nanomeasuring machine by using advanced tactile and non-tactile probes,” Meas. Sci. Technol. 18, 520-527 (2007).
[CrossRef]

Mayr, T.

C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135-140 (2006).
[CrossRef] [PubMed]

Mocker, H.

Moser, C.

C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135-140 (2006).
[CrossRef] [PubMed]

Murphy, M.

K. Kim, S. Park, J.-B. Lee, H. Manohara, Y. Desta, M. Murphy, and C. H. Ahn, “Rapid replication of polymeric and metallic high aspect ratio microstructures using PDMS and LIGA technology,” Microsyst. Technol. 9, 5-10 (2002).
[CrossRef]

Olson, G.

O'Neill, A.

A. O'Neill, J. Soo Hoo, and G. Walker, “Chips & Tips: Rapid curing of PDMS for microfluidic applications,” http://www.rsc.org/Publishing/Journals/lc/PDMS_curing.asp.

Ottevaere, H.

H. Ottevaere and H. Thienpont, Encyclopedia of Modern Optics, D. Steel and B. Guenther, eds. (Academic, 2004), pp. 21-43.

Park, K.

K. Ryu, J. Rhee, K. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy Mater. 80, 1580-1587 (2006).
[CrossRef]

Park, S.

K. Kim, S. Park, J.-B. Lee, H. Manohara, Y. Desta, M. Murphy, and C. H. Ahn, “Rapid replication of polymeric and metallic high aspect ratio microstructures using PDMS and LIGA technology,” Microsyst. Technol. 9, 5-10 (2002).
[CrossRef]

Peiner, E.

E. Peiner, M. Balke, L. Doering, and U. Brand, “Tactile probes for dimensional metrology with microcomponents at nanometer resolution,” Meas. Sci. Technol. 19, 064001 (2008).
[CrossRef]

Purcell, D.

D. Purcell, A. Suratkar, A. Davies, and F. Farahi, “Measuring the wavefront distortion of a microlens array using an index matching liquid,” presented at the Optical Fabrication and Testing Topical Meeting, Rochester, New York, USA (9-11 October 2006).

Reichelt, S.

Rhee, J.

K. Ryu, J. Rhee, K. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy Mater. 80, 1580-1587 (2006).
[CrossRef]

Ross, J.

Ryu, K.

K. Ryu, J. Rhee, K. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy Mater. 80, 1580-1587 (2006).
[CrossRef]

Seong, K.

K. Seong and J. Greivenkamp, “Surface figure measurement based on the transmitted wavefront with reverse raytracing,” Opt. Eng. 47, 043602 (2008).
[CrossRef]

Seward, G.

G. Seward and P. Cort, “Measurement and characterization of angular reflectance for cube-corners and microspheres,” Opt. Eng. 38, 164-169 (1999).
[CrossRef]

Shia, T.

L. Lin, T. Shia and C. Chiu, “Silicon-processed plastic micropyramids for brightness enhancement applications,” J. Micromech. Microeng. 10, 395-400 (2000).
[CrossRef]

Shieh, H.

Suratkar, A.

D. Purcell, A. Suratkar, A. Davies, and F. Farahi, “Measuring the wavefront distortion of a microlens array using an index matching liquid,” presented at the Optical Fabrication and Testing Topical Meeting, Rochester, New York, USA (9-11 October 2006).

Thienpont, H.

H. Ottevaere and H. Thienpont, Encyclopedia of Modern Optics, D. Steel and B. Guenther, eds. (Academic, 2004), pp. 21-43.

Walker, G.

A. O'Neill, J. Soo Hoo, and G. Walker, “Chips & Tips: Rapid curing of PDMS for microfluidic applications,” http://www.rsc.org/Publishing/Journals/lc/PDMS_curing.asp.

Wu, S.

Yuan, J.

J. Yuan, S. Chang, S. Li, and Y. Zhang, “Design and fabrication of micro-cube-corner array retro-reflectors,” Opt. Commun. 209, 75-83 (2002).
[CrossRef]

Zappe, H.

Zhang, Y.

J. Yuan, S. Chang, S. Li, and Y. Zhang, “Design and fabrication of micro-cube-corner array retro-reflectors,” Opt. Commun. 209, 75-83 (2002).
[CrossRef]

Appl. Opt.

J. Micromech. Microeng.

L. Lin, T. Shia and C. Chiu, “Silicon-processed plastic micropyramids for brightness enhancement applications,” J. Micromech. Microeng. 10, 395-400 (2000).
[CrossRef]

J. Microsc.

C. Moser, T. Mayr, and I. Klimant, “Filter cubes with built-in ultrabright light-emitting diodes as exchangeable excitation light sources in fluorescence microscopy,” J. Microsc. 222, 135-140 (2006).
[CrossRef] [PubMed]

Meas. Sci. Technol.

E. Peiner, M. Balke, L. Doering, and U. Brand, “Tactile probes for dimensional metrology with microcomponents at nanometer resolution,” Meas. Sci. Technol. 19, 064001 (2008).
[CrossRef]

E. Manske, T. Hausotte, R. Mastylo, T. Machleidt, K.-H. Franke, and G. Jager, “New applications of the nanopositioning and nanomeasuring machine by using advanced tactile and non-tactile probes,” Meas. Sci. Technol. 18, 520-527 (2007).
[CrossRef]

Microsyst. Technol.

K. Kim, S. Park, J.-B. Lee, H. Manohara, Y. Desta, M. Murphy, and C. H. Ahn, “Rapid replication of polymeric and metallic high aspect ratio microstructures using PDMS and LIGA technology,” Microsyst. Technol. 9, 5-10 (2002).
[CrossRef]

Opt. Commun.

J. Yuan, S. Chang, S. Li, and Y. Zhang, “Design and fabrication of micro-cube-corner array retro-reflectors,” Opt. Commun. 209, 75-83 (2002).
[CrossRef]

Opt. Eng.

G. Seward and P. Cort, “Measurement and characterization of angular reflectance for cube-corners and microspheres,” Opt. Eng. 38, 164-169 (1999).
[CrossRef]

K. Seong and J. Greivenkamp, “Surface figure measurement based on the transmitted wavefront with reverse raytracing,” Opt. Eng. 47, 043602 (2008).
[CrossRef]

Proc. SPIE

J. Kuhn, F. Charrière, T. Colomb, E. Cuche, Y. Emery, and C. Depeursinge, “Digital holographic microscopy for nanometric quality control of micro-optical components,” Proc. SPIE 6475, 64750V (2007).
[CrossRef]

Prod. Eng. Res. Devel.

E. Brinksmeier, R. Gläbe, and C. Flucke, “Manufacturing of molds for replication of micro cube corner retroreflectors,” Prod. Eng. Res. Devel. 2, 33-38 (2008).
[CrossRef]

Sol. Energy Mater.

K. Ryu, J. Rhee, K. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy Mater. 80, 1580-1587 (2006).
[CrossRef]

Other

T. Ballinger and J. Huedepohl, “Advantages of using Dektak surface profilers with Vision 3D analysis software,” Application Note 542 (Veeco Dektak, 2008).

A. O'Neill, J. Soo Hoo, and G. Walker, “Chips & Tips: Rapid curing of PDMS for microfluidic applications,” http://www.rsc.org/Publishing/Journals/lc/PDMS_curing.asp.

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501-598.

T. Bothe, W. Li, C. Kopylow, and W. Juptner, “Fringe reflection for high resolution topometry and surface description on various lateral scales,” in Fringe 2005: The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2005), pp. 362-371.

L. Desmet, “Fabrication and design of micro-optical interconnect systems with fanout,” Ph.D. dissertation (Vrije Universiteit of Brussel, 2007).

D. Purcell, A. Suratkar, A. Davies, and F. Farahi, “Measuring the wavefront distortion of a microlens array using an index matching liquid,” presented at the Optical Fabrication and Testing Topical Meeting, Rochester, New York, USA (9-11 October 2006).

H. Ottevaere and H. Thienpont, Encyclopedia of Modern Optics, D. Steel and B. Guenther, eds. (Academic, 2004), pp. 21-43.

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Figures (7)

Fig. 1
Fig. 1

Interferogram of plane wave illumination on a micropyramid array (a) using no index liquid and (b) using an index liquid.

Fig. 2
Fig. 2

Schematic of the measurement principle.

Fig. 3
Fig. 3

Experimental setup of the Mach–Zehnder interferometer.

Fig. 4
Fig. 4

(a) Surface profile obtained with the Dektak surface profilometer. (b) Interferogram of a small section of the micropyramidal array using the index liquid. Interferences fringes are resolvable on all facets, although difficult to see in this image. (c) Optical path difference map obtained using the MZ interferometer in conjunction with an index matching liquid.

Fig. 5
Fig. 5

Facet angle map of the micropyramid array obtained with (a) the Dektak surface profilometer and (b) the MZ interferometer.

Fig. 6
Fig. 6

(a) SEM image of microcorner cube array fabricated at the University of Bremen. (b) Surface profile of replicated microcorner cube array obtained with the MZ interferometer in conjunction with an index matching liquid. (c) Interferogram of microcorner cube array measurement.

Fig. 7
Fig. 7

(a) Facet angle error as a function of facet angle. (b) Facet angle error map for Reflexite sample illustrating the error in the facet angle that occurs by neglecting refraction effects.

Tables (1)

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Table 1 Comparison between MZ Interferometer and Profilometer

Equations (11)

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n 1 sin θ 1 = n 2 sin θ 2 ,
n 2 sin θ 3 = n 3 sin θ 4 ,
θ 2 = θ 1 + θ 3 .
n 1 sin θ 1 = n 2 sin ( θ 1 + θ 3 ) .
n 1 sin θ 1 = n 2 sin θ 1 cos θ 3 + n 2 cos θ 1 sin θ 3 .
n 1 tan θ 1 = n 2 tan θ 1 cos θ 3 + n 2 sin θ 3 .
θ 1 = tan 1 ( n 2 sin θ 3 n 1 n 2 cos θ 3 ) ,
θ 3 = sin 1 ( sin θ 4 n 2 ) .
OPD ( x , y ) = λ ϕ ( x , y ) 2 π ,
θ 4 ( x , y ) = a tan ( OPD ( x , y ) ) .
θ 4 = tan 1 ( d x 2 + d y 2 ) .

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