Abstract

The optical aberrations induced by imaging through skin can be predicted using formulas for Seidel aberrations of a plane-parallel plate. Knowledge of these aberrations helps to guide the choice of numerical aperture (NA) of the optics we can use in an implementation of Gabor domain optical coherence microscopy (GD-OCM), where the focus is the only aberration adjustment made through depth. On this basis, a custom-designed, liquid-lens enabled dynamic focusing optical coherence microscope operating at 0.2 NA is analyzed and validated experimentally. As part of the analysis, we show that the full width at half-maximum metric, as a characteristic descriptor for the point spread function, while commonly used, is not a useful metric for quantifying resolution in non-diffraction-limited systems. Modulation transfer function (MTF) measurements quantify that the liquid lens performance is as predicted by design, even when accounting for the effect of gravity. MTF measurements in a skinlike scattering medium also quantify the performance of the microscope in its potential applications. To guide the fusion of images across the various focus positions of the microscope, as required in GD-OCM, we present depth of focus measurements that can be used to determine the effective number of focusing zones required for a given goal resolution. Subcellular resolution in an onion sample, and high-definition in vivo imaging in human skin are demonstrated with the custom-designed and built microscope.

© 2010 Optical Society of America

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    [CrossRef]
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2010 (1)

2009 (1)

2008 (2)

2007 (2)

2006 (1)

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600nm,” Phys. Med. Biol. 51, 1479–1490 (2006).
[CrossRef] [PubMed]

2005 (5)

2004 (4)

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128 (2004).
[CrossRef]

M. Schwertner, M. Booth, and T. Wilson, “Characterizing specimen induced aberrations for high NA adaptive optical microscopy,” Opt. Express 12, 6540–6552 (2004).
[CrossRef] [PubMed]

S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85, 1128–1130 (2004).
[CrossRef]

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE , 5523, 100–109 (2004).
[CrossRef]

2003 (1)

J. Sandby-Moller, T. Poulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits,” Acta Derm. Venereol. 83, 410–413 (2003).
[CrossRef] [PubMed]

1999 (1)

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

1998 (1)

M. Gniadecka and G. B. E. Jemec, “Quantitative evaluation of chronological ageing and photo ageing in vivo: studies on skin echogenicity and thickness,” Br. J. Dermatol. 138, 815–821(1998).
[CrossRef]

1997 (1)

J. M. Schmitt, S. L. Lee, and K. M. Yung, “An optical coherence microscope with enhanced resolving power in thick tissue,” Opt. Commun. 142, 203–207 (1997).
[CrossRef]

1993 (2)

P. Corcuff, C. Bertrand, and J. L. Leveque, “Morphometry of human epidermis in vivo by real-time confocal microscopy,” Arch. Dermatol. Res. 285, 475–481 (1993).
[CrossRef] [PubMed]

A. R. Knuettel, J. M. Schmitt, R. Barnes, and J. R. Knutson, “Spatial localization using interfering photon density waves: contrast enhancement and limitations,” Proc. SPIE , 1888, 322–333 (1993).
[CrossRef]

Ahrens, G.

Akcay, A. C.

Bachman, M.

A. Divetia, T. H. Hsieh, J. Zhang, Z. Chen, M. Bachman, and G. P. Li, “Dynamically focused optical coherence tomography for endoscopic applications,” Appl. Phys. Lett. 86, 103902 (2005).
[CrossRef]

Barnes, R.

A. R. Knuettel, J. M. Schmitt, R. Barnes, and J. R. Knutson, “Spatial localization using interfering photon density waves: contrast enhancement and limitations,” Proc. SPIE , 1888, 322–333 (1993).
[CrossRef]

Bauer, S.

Bertrand, C.

P. Corcuff, C. Bertrand, and J. L. Leveque, “Morphometry of human epidermis in vivo by real-time confocal microscopy,” Arch. Dermatol. Res. 285, 475–481 (1993).
[CrossRef] [PubMed]

Booth, M.

Booth, M. J.

M. J. Booth, “Adaptive optics in microscopy,” Philos. Trans. R. Soc. London Ser. A 365, 2829–2843 (2007).
[CrossRef]

Cable, A.

Cable, A. E.

Chen, Y.

Chen, Z.

A. Divetia, T. H. Hsieh, J. Zhang, Z. Chen, M. Bachman, and G. P. Li, “Dynamically focused optical coherence tomography for endoscopic applications,” Appl. Phys. Lett. 86, 103902 (2005).
[CrossRef]

Clarkson, E.

Corcuff, P.

P. Corcuff, C. Bertrand, and J. L. Leveque, “Morphometry of human epidermis in vivo by real-time confocal microscopy,” Arch. Dermatol. Res. 285, 475–481 (1993).
[CrossRef] [PubMed]

Cotran, R. S.

V. Kumar and R. S. Cotran, Robbins Basic Pathology(Saunders, 2003).

Delemos, T.

Dickensheets, L. D.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128 (2004).
[CrossRef]

Ding, H.

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600nm,” Phys. Med. Biol. 51, 1479–1490 (2006).
[CrossRef] [PubMed]

Divetia, A.

A. Divetia, T. H. Hsieh, J. Zhang, Z. Chen, M. Bachman, and G. P. Li, “Dynamically focused optical coherence tomography for endoscopic applications,” Appl. Phys. Lett. 86, 103902 (2005).
[CrossRef]

Drexler, W.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Fercher, A. F.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Fujimoto, J. G.

Gniadecka, M.

M. Gniadecka and G. B. E. Jemec, “Quantitative evaluation of chronological ageing and photo ageing in vivo: studies on skin echogenicity and thickness,” Br. J. Dermatol. 138, 815–821(1998).
[CrossRef]

Gorczynska, I.

Gordon, L. M.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128 (2004).
[CrossRef]

Götzinger, E.

Grützner, G.

Hendriks, B. H.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE , 5523, 100–109 (2004).
[CrossRef]

Hendriks, B. H. W.

S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85, 1128–1130 (2004).
[CrossRef]

Himmer, A. P.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128 (2004).
[CrossRef]

Hirschberg, A. M.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE , 5523, 100–109 (2004).
[CrossRef]

Hitzenberger, C.

Hitzenberger, C. K.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Hsieh, T. H.

A. Divetia, T. H. Hsieh, J. Zhang, Z. Chen, M. Bachman, and G. P. Li, “Dynamically focused optical coherence tomography for endoscopic applications,” Appl. Phys. Lett. 86, 103902 (2005).
[CrossRef]

Hu, X. H.

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600nm,” Phys. Med. Biol. 51, 1479–1490 (2006).
[CrossRef] [PubMed]

Huber, R.

Huijbregts, L. J.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE , 5523, 100–109 (2004).
[CrossRef]

Jain, A.

J. P. Rolland, P. Meemon, S. Murali, A. Jain, N. Papp, K. Thompson, and K. S. Lee, “Gabor domain optical coherence microscopy,” Proc. SPIE , 7139, 71390F (2008).
[CrossRef]

Jemec, G. B. E.

M. Gniadecka and G. B. E. Jemec, “Quantitative evaluation of chronological ageing and photo ageing in vivo: studies on skin echogenicity and thickness,” Br. J. Dermatol. 138, 815–821(1998).
[CrossRef]

Jiang, J.

Jiang, J. Y.

Knuettel, A. R.

A. R. Knuettel, J. M. Schmitt, R. Barnes, and J. R. Knutson, “Spatial localization using interfering photon density waves: contrast enhancement and limitations,” Proc. SPIE , 1888, 322–333 (1993).
[CrossRef]

Knutson, J. R.

A. R. Knuettel, J. M. Schmitt, R. Barnes, and J. R. Knutson, “Spatial localization using interfering photon density waves: contrast enhancement and limitations,” Proc. SPIE , 1888, 322–333 (1993).
[CrossRef]

Kragel, P. J.

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600nm,” Phys. Med. Biol. 51, 1479–1490 (2006).
[CrossRef] [PubMed]

Kuiper, S.

S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85, 1128–1130 (2004).
[CrossRef]

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE , 5523, 100–109 (2004).
[CrossRef]

Kumar, V.

V. Kumar and R. S. Cotran, Robbins Basic Pathology(Saunders, 2003).

Lee, K. S.

Lee, S. L.

J. M. Schmitt, S. L. Lee, and K. M. Yung, “An optical coherence microscope with enhanced resolving power in thick tissue,” Opt. Commun. 142, 203–207 (1997).
[CrossRef]

Leveque, J. L.

P. Corcuff, C. Bertrand, and J. L. Leveque, “Morphometry of human epidermis in vivo by real-time confocal microscopy,” Arch. Dermatol. Res. 285, 475–481 (1993).
[CrossRef] [PubMed]

Lexer, F.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Li, G. P.

A. Divetia, T. H. Hsieh, J. Zhang, Z. Chen, M. Bachman, and G. P. Li, “Dynamically focused optical coherence tomography for endoscopic applications,” Appl. Phys. Lett. 86, 103902 (2005).
[CrossRef]

Lu, J. Q.

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600nm,” Phys. Med. Biol. 51, 1479–1490 (2006).
[CrossRef] [PubMed]

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations, 2nd ed. (SPIE Press, 2004), Vol. 2.

Meemon, P.

J. P. Rolland, P. Meemon, S. Murali, K. P. Thompson, and K. S. Lee, “Gabor-based fusion technique for optical coherence microscopy,” Opt. Express 18, 3632–3642 (2010).
[CrossRef] [PubMed]

J. P. Rolland, P. Meemon, S. Murali, A. Jain, N. Papp, K. Thompson, and K. S. Lee, “Gabor domain optical coherence microscopy,” Proc. SPIE , 7139, 71390F (2008).
[CrossRef]

Molebny, S.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Murali, S.

Papp, N.

J. P. Rolland, P. Meemon, S. Murali, A. Jain, N. Papp, K. Thompson, and K. S. Lee, “Gabor domain optical coherence microscopy,” Proc. SPIE , 7139, 71390F (2008).
[CrossRef]

Pircher, M.

Potsaid, B.

Poulsen, T.

J. Sandby-Moller, T. Poulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits,” Acta Derm. Venereol. 83, 410–413 (2003).
[CrossRef] [PubMed]

Qi, B.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128 (2004).
[CrossRef]

Renders, C. A.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE , 5523, 100–109 (2004).
[CrossRef]

Rolland, J. P.

Sandby-Moller, J.

J. Sandby-Moller, T. Poulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits,” Acta Derm. Venereol. 83, 410–413 (2003).
[CrossRef] [PubMed]

Sattmann, H.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Schmitt, J. M.

J. M. Schmitt, S. L. Lee, and K. M. Yung, “An optical coherence microscope with enhanced resolving power in thick tissue,” Opt. Commun. 142, 203–207 (1997).
[CrossRef]

A. R. Knuettel, J. M. Schmitt, R. Barnes, and J. R. Knutson, “Spatial localization using interfering photon density waves: contrast enhancement and limitations,” Proc. SPIE , 1888, 322–333 (1993).
[CrossRef]

Schwertner, M.

Srinivasan, V. J.

Sticker, M.

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

Stifter, D.

Thompson, K.

J. P. Rolland, P. Meemon, S. Murali, A. Jain, N. Papp, K. Thompson, and K. S. Lee, “Gabor domain optical coherence microscopy,” Proc. SPIE , 7139, 71390F (2008).
[CrossRef]

Thompson, K. P.

Tuchin, V. V.

V. V. Tuchin, “Optical clearing of tissues and blood,” J. Phys. D 38, 2497–2518 (2005).
[CrossRef]

van As, M. A.

S. Kuiper, B. H. Hendriks, L. J. Huijbregts, A. M. Hirschberg, C. A. Renders, and M. A. van As, “Variable-focus liquid lens for portable applications,” Proc. SPIE , 5523, 100–109 (2004).
[CrossRef]

Vitkin, I. A.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128 (2004).
[CrossRef]

Welford, W. T.

W. T. Welford, “Thin lens aberrations,” in Aberrations of the Symmetrical Optical System (Academic, 1974), pp. 198–199.

Wiesauer, K.

Wilson, T.

Wojtkowski, M.

Wooden, W. A.

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600nm,” Phys. Med. Biol. 51, 1479–1490 (2006).
[CrossRef] [PubMed]

Wulf, H. C.

J. Sandby-Moller, T. Poulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits,” Acta Derm. Venereol. 83, 410–413 (2003).
[CrossRef] [PubMed]

Yang, X. D. V.

B. Qi, A. P. Himmer, L. M. Gordon, X. D. V. Yang, L. D. Dickensheets, and I. A. Vitkin, “Dynamic focus control in high-speed optical coherence tomography based on a microelectromechanical mirror,” Opt. Commun. 232, 123–128 (2004).
[CrossRef]

Yung, K. M.

J. M. Schmitt, S. L. Lee, and K. M. Yung, “An optical coherence microscope with enhanced resolving power in thick tissue,” Opt. Commun. 142, 203–207 (1997).
[CrossRef]

Zhang, J.

A. Divetia, T. H. Hsieh, J. Zhang, Z. Chen, M. Bachman, and G. P. Li, “Dynamically focused optical coherence tomography for endoscopic applications,” Appl. Phys. Lett. 86, 103902 (2005).
[CrossRef]

Acta Derm. Venereol. (1)

J. Sandby-Moller, T. Poulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits,” Acta Derm. Venereol. 83, 410–413 (2003).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

A. Divetia, T. H. Hsieh, J. Zhang, Z. Chen, M. Bachman, and G. P. Li, “Dynamically focused optical coherence tomography for endoscopic applications,” Appl. Phys. Lett. 86, 103902 (2005).
[CrossRef]

S. Kuiper and B. H. W. Hendriks, “Variable-focus liquid lens for miniature cameras,” Appl. Phys. Lett. 85, 1128–1130 (2004).
[CrossRef]

Arch. Dermatol. Res. (1)

P. Corcuff, C. Bertrand, and J. L. Leveque, “Morphometry of human epidermis in vivo by real-time confocal microscopy,” Arch. Dermatol. Res. 285, 475–481 (1993).
[CrossRef] [PubMed]

Br. J. Dermatol. (1)

M. Gniadecka and G. B. E. Jemec, “Quantitative evaluation of chronological ageing and photo ageing in vivo: studies on skin echogenicity and thickness,” Br. J. Dermatol. 138, 815–821(1998).
[CrossRef]

J. Mod. Opt. (1)

F. Lexer, C. K. Hitzenberger, W. Drexler, S. Molebny, H. Sattmann, M. Sticker, and A. F. Fercher, “Dynamic coherent focus OCT with depth-independent transversal resolution,” J. Mod. Opt. 46, 541–553 (1999).

J. Phys. D (1)

V. V. Tuchin, “Optical clearing of tissues and blood,” J. Phys. D 38, 2497–2518 (2005).
[CrossRef]

Opt. Commun. (2)

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Figures (9)

Fig. 1
Fig. 1

Converging beam through a plane-parallel plate and notations.

Fig. 2
Fig. 2

Dependence on NA at the maximum off-axis FOV for the following: (a) wave aberrations for converging beams of varying power (0.05–0.5 NA) and a plate thickness of 0.5 mm, (b) RMS wavefront error at best focus for converging beams of varying power (0.05–0.5 NA) and plates of varying thickness (0.5–4 mm), and (c) associated SRs. Units are in waves at 1 μm . Imaging with a SR of 0.8 through 2 mm of skin yields a NA of 0.16 , whereas 0.23 NA is tolerable when considering 1 mm of skin instead.

Fig. 3
Fig. 3

PSFs and associated SRs obtained on axis in CODE V for a penetration depth of 0.5, 1, 2, and 4 mm (left to right) are shown for selected NA values of 0.1, 0.3, and 0.5 (top to bottom).

Fig. 4
Fig. 4

(a) Experimental setup: PC, polarization controller; Col, collimator; DC, dispersion compensator; DF, dynamic focusing of optical coherence microscope; M, mirror. (b), (c) MTF estimation from edge-response measurements.

Fig. 5
Fig. 5

On-axis MTF curves measured in air at 1.8, 1, and 0.2 mm in the (a) horizontal configuration and (b) vertical configuration.

Fig. 6
Fig. 6

MTF at the center of the focus range, i.e., 1 mm depth in air (a) contrast at 177 and 250 line-pairs/mm cutoff frequencies for different defocus locations (b) ± 30 μm defocus, still maintaining over 20% contrast up to 250 line-pairs/mm corresponding to 2 μm resolution.

Fig. 7
Fig. 7

MTF curves measured on-axis in a skin-equivalent scattering medium (a) in focus at 0.1 mm , (b) in focus at 1 mm , and (c) at a defocus of ± 30 μm from the 1 mm focus depth. In all cases, 20% or higher contrast is obtained.

Fig. 8
Fig. 8

1 mm × 1 mm en face images of the onion cell structure starting at 50 μm under the onion skin and separated in depth by 50 μm .

Fig. 9
Fig. 9

2 mm ( lateral ) × 0.8 mm ( depth ) cross section of the human finger tip acquired at a 2 μm resolution with eight focusing zones fused into a single image.

Tables (3)

Tables Icon

Table 1 Expressions of Monochromatic Seidel Aberrations and Axial and Lateral Color with Notations Defined in Fig. 1 (Adapted from [16], pp. 120–185)

Tables Icon

Table 2 Strehl ratio (SR), Peak-to-Valley (PV), RMS Wavefront Error in Waves at 1 μm , and the PSF FWHM and FW at 1 / e 2 of Maximum Intensity Determined at Different NA and Plate Thickness a

Tables Icon

Table 3 Predicted and Measured DOF

Equations (19)

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W RMS 2 = 1 12 [ ( Δ W 20 + W 040 ) + ( W 220 S + 1 2 W 222 ) H 2 ] 2 + 1 180 [ W 040 ] 2 + 1 24 [ W 222 H 2 ] 2 + 1 4 [ ( Δ W 11 + 2 3 W 131 ) H + W 311 H 3 ] 2 + 1 72 [ W 131 H ] 2 ,
W RMS 2 = 1 180 [ W 040 ] 2 + 1 24 [ W 222 H 2 ] 2 + 1 72 [ W 131 H ] 2 .
SR = e ( 2 π W RMS ) 2 ,
W = j = 0 n = 0 m = 0 W klm H k ρ l cos m ϕ ,
W lm = j = 0 W klm H k .
W = n = 0 m = 0 W lm ρ l cos m ϕ ,
W = W 00 + W 11 ρ cos ϕ + W 20 ρ 2 + W 40 ρ 4 + W 31 ρ 3 cos ϕ + W 22 ρ 2 cos 2 ϕ .
ω 2 = 1 π 0 2 π 0 1 W 2 ρ d ρ d ϕ [ 1 π 0 2 π 0 1 W ρ d ρ d ϕ ] 2 ,
W 2 = n i = 0 m i = 0 n j = 0 m j = 0 W l i m i W l j m j ρ l i + l j cos m i + m j ϕ = ( 4 ) W l i m i W l j m j ρ l i + l j cos m i + m j ϕ ,
1 π 0 2 π 0 1 W 2 ρ d ρ d ϕ = ( 4 ) W l i m i W l j m j [ 1 l i + l j + 2 ] V ( m i + m j ) ,
V ( M ) = 1 π 0 2 π cos M ϕ d ϕ .
1 π 0 2 π 0 1 W ρ d ρ d ϕ = n = 0 m = 0 W lm 0 1 ρ l + 1 d ρ 1 π 0 2 π cos m ϕ d ϕ = ( 2 ) W lm 1 l + 2 V ( m ) [ 1 π 0 2 π 0 1 W ρ d ρ d ϕ ] 2 = ( 4 ) W l i m i W l j m j [ 1 l i + 2 ] [ 1 l j + 2 ] V ( m i ) V ( m j ) .
ω 2 = ( 4 ) { V ( m i + m j ) l i + l j + 2 V ( m i ) V ( m j ) ( l i + 2 ) ( l j + 2 ) } W l i m i W l j m j .
ω 2 W 20 2 + W 40 2 + W 22 2 + 2 W 20 W 40 + 2 W 20 W 22 + 2 W 40 W 22 + W 11 2 + 2 W 11 W 31 + W 31 2 .
M V 0 2 2 1 4 3 / 4 6 5 / 8 8 35 / 64
ω 2 = 1 12 W 20 2 + 4 45 W 40 2 + 3 48 W 22 2 + 1 6 W 20 W 40 + 1 12 W 20 W 22 + 1 12 W 40 W 22 + 1 4 W 11 2 + 1 3 W 11 W 31 + 1 8 W 31 2 ,
ω 2 = 1 12 [ W 20 + W 40 + 1 2 W 22 ] 2 + 1 180 [ W 40 ] 2 + 1 24 [ W 22 ] 2 + 1 4 [ W 11 + 2 3 W 31 ] 2 + 1 72 W 31 2
ω 2 = 1 12 [ ( W 020 + W 040 ) + ( W 220 S + 1 2 W 222 ) H 2 ] 2 + 1 180 [ W 040 ] 2 + 1 24 [ W 222 H 2 ] 2 + 1 4 [ ( W 111 + 2 3 W 131 ) H + W 311 H 3 ] 2 + 1 72 [ W 131 H ] 2 .
ω 2 = 1 180 [ W 040 ] 2 + 1 24 [ W 222 H 2 ] 2 + 1 72 [ W 131 H ] 2 .

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