Abstract

We propose a numerical analysis method for evaluating GRIN lenses using the Monte Carlo method. Actual measurements of the modulation transfer function (MTF) of a GRIN lens using this method closely match those made by conventional methods. Experimentally, the MTF is measured using a square wave chart, and is then calculated based on the distribution of output strength on the chart. In contrast, the general method using computers evaluates the MTF based on a spot diagram made by an incident point light source. However the results differ greatly from those from experiments. We therefore developed an evaluation method similar to the experimental system based on the Monte Carlo method and verified that it more closely matches the experimental results than the conventional method.

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    [CrossRef]
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    [CrossRef]
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  9. H. W. Jensen, S. R. Marschner, M. Levoy, and P. Hanrahan, “A Practical Model for Subsurface Light Transport,” in Proceedings of the 28th annual conference on Computer graphics and interactive techniques, L. Pocock, ed. (Los Angeles, Calif., 2001), pp. 511–518.
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    [CrossRef]
  13. M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multi-layer coating by continuous withdrawal of a thin plastic fiber through polymer solution,” Kagaku Kogaku Ronbunshu 34(1), 187–193 (2008).
    [CrossRef]

2008

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multilayer coating by drawing a thin plastic fiber through a polymer solution,” Asia Pac. J. Chem. 3(1), 63–69 (2008).
[CrossRef]

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multi-layer coating by continuous withdrawal of a thin plastic fiber through polymer solution,” Kagaku Kogaku Ronbunshu 34(1), 187–193 (2008).
[CrossRef]

1998

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998).
[CrossRef]

1988

1980

1970

E. G. Rawson, D. R. Herriott, and J. McKenna, “Analysis of refractive index distributions in cylindrical graded-index glass rods (GRIN Rods) used as image relays,” Appl. Opt. 9(3), 753–759 (1970).
[CrossRef] [PubMed]

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6(10), 606–612 (1970).
[CrossRef]

1954

A. Fletcher, T. Murphy, and A. Young, “Solutions of Two Optical Problems,” Proc. R. Soc. Lond. A Math. Phys. Sci. 223(1153), 216–225 (1954).
[CrossRef]

J. W. Coltman, “The Specification of Imaging Properties by Response to a Sine Wave Input,” J. Opt. Soc. Am. 44(6), 468–469 (1954).
[CrossRef]

Coltman, J. W.

Fletcher, A.

A. Fletcher, T. Murphy, and A. Young, “Solutions of Two Optical Problems,” Proc. R. Soc. Lond. A Math. Phys. Sci. 223(1153), 216–225 (1954).
[CrossRef]

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6(10), 606–612 (1970).
[CrossRef]

Furuta, T.

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multi-layer coating by continuous withdrawal of a thin plastic fiber through polymer solution,” Kagaku Kogaku Ronbunshu 34(1), 187–193 (2008).
[CrossRef]

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multilayer coating by drawing a thin plastic fiber through a polymer solution,” Asia Pac. J. Chem. 3(1), 63–69 (2008).
[CrossRef]

Herriott, D. R.

Iga, K.

Kawazu, M.

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6(10), 606–612 (1970).
[CrossRef]

Koike, Y.

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6(10), 606–612 (1970).
[CrossRef]

Matsumoto, M.

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998).
[CrossRef]

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6(10), 606–612 (1970).
[CrossRef]

McKenna, J.

Murphy, T.

A. Fletcher, T. Murphy, and A. Young, “Solutions of Two Optical Problems,” Proc. R. Soc. Lond. A Math. Phys. Sci. 223(1153), 216–225 (1954).
[CrossRef]

Nishimura, T.

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998).
[CrossRef]

Nishizawa, K.

Oda, M.

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multilayer coating by drawing a thin plastic fiber through a polymer solution,” Asia Pac. J. Chem. 3(1), 63–69 (2008).
[CrossRef]

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multi-layer coating by continuous withdrawal of a thin plastic fiber through polymer solution,” Kagaku Kogaku Ronbunshu 34(1), 187–193 (2008).
[CrossRef]

Ogura, Y.

Ohtsuka, Y.

Rawson, E. G.

Suga, S.

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multi-layer coating by continuous withdrawal of a thin plastic fiber through polymer solution,” Kagaku Kogaku Ronbunshu 34(1), 187–193 (2008).
[CrossRef]

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multilayer coating by drawing a thin plastic fiber through a polymer solution,” Asia Pac. J. Chem. 3(1), 63–69 (2008).
[CrossRef]

Takezawa, Y.

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6(10), 606–612 (1970).
[CrossRef]

Yamamoto, N.

Yoshii, H.

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multi-layer coating by continuous withdrawal of a thin plastic fiber through polymer solution,” Kagaku Kogaku Ronbunshu 34(1), 187–193 (2008).
[CrossRef]

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multilayer coating by drawing a thin plastic fiber through a polymer solution,” Asia Pac. J. Chem. 3(1), 63–69 (2008).
[CrossRef]

Young, A.

A. Fletcher, T. Murphy, and A. Young, “Solutions of Two Optical Problems,” Proc. R. Soc. Lond. A Math. Phys. Sci. 223(1153), 216–225 (1954).
[CrossRef]

ACM Trans. Model. Comput. Simul.

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998).
[CrossRef]

Appl. Opt.

Asia Pac. J. Chem.

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multilayer coating by drawing a thin plastic fiber through a polymer solution,” Asia Pac. J. Chem. 3(1), 63–69 (2008).
[CrossRef]

IEEE J. Quantum Electron.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6(10), 606–612 (1970).
[CrossRef]

J. Opt. Soc. Am.

Kagaku Kogaku Ronbunshu

M. Oda, S. Suga, H. Yoshii, and T. Furuta, “Multi-layer coating by continuous withdrawal of a thin plastic fiber through polymer solution,” Kagaku Kogaku Ronbunshu 34(1), 187–193 (2008).
[CrossRef]

Proc. R. Soc. Lond. A Math. Phys. Sci.

A. Fletcher, T. Murphy, and A. Young, “Solutions of Two Optical Problems,” Proc. R. Soc. Lond. A Math. Phys. Sci. 223(1153), 216–225 (1954).
[CrossRef]

Other

H. W. Jensen, “Global Illumination using Photon Maps,” in Proceedings of the 7th Eurographics Workshop on Rendering, P. Schröder, ed. (Porto, Portugal, 1996), pp. 21–30.

H. W. Jensen, S. R. Marschner, M. Levoy, and P. Hanrahan, “A Practical Model for Subsurface Light Transport,” in Proceedings of the 28th annual conference on Computer graphics and interactive techniques, L. Pocock, ed. (Los Angeles, Calif., 2001), pp. 511–518.

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Figures (17)

Fig. 1
Fig. 1

Characteristics of a GRIN lens.

Fig. 2
Fig. 2

Schematic diagram of a rod lens array.

Fig. 3
Fig. 3

MTF measurement system.

Fig. 4
Fig. 4

MTF measurement method.

Fig. 5
Fig. 5

How to set the working distance.

Fig. 6
Fig. 6

Outline of the optical system in the simulation.

Fig. 7
Fig. 7

Coordinate system for calculating the geometrical-optical intensity distribution from the spot diagram.

Fig. 8
Fig. 8

Light ray generated from square wave chart.

Fig. 9
Fig. 9

Flow chart for Monte Carlo light ray tracing method.

Fig. 10
Fig. 10

Image reconstruction by the integration of light rays reaching specific pixels.

Fig. 11
Fig. 11

Distribution of the light of single GRIN lens.

Fig. 12
Fig. 12

Relation between the RMSE and the number of rays.

Fig. 13
Fig. 13

Image reconstructed by the Monte Carlo method.

Fig. 14
Fig. 14

Refractive index distribution used in MTF evaluation.

Fig. 15
Fig. 15

Spot diagrams of rod lens arrays with changing the incident position. (a) Incident position is x = 0 mm (ideal condition). (b) Incident position is x = 0.04 mm (ideal condition). (c) Incident position is x = 0 mm (data no. 1). (d) Incident position is x = 0.4 mm (data no. 1).

Fig. 16
Fig. 16

Reconstructed images and intensity distribution obtained by the Monte Carlo method. (a) Data no. 1 (Z = 5.01 mm, l = 1.40 mm). (b) Data no. 2 (Z = 4.71 mm, l = 1.20 mm). (c) Data no. 4 (Z = 4.50 mm, l = 2.20 mm).

Fig. 17
Fig. 17

Reconstructed images and intensity distribution with changing spatial frequencies (Z = 4.71 mm, l = 1.20 mm). (a) 4 LP/mm. (b) 8 LP/mm.

Tables (1)

Tables Icon

Table 1 MTF evaluation results by experiment and simulation

Equations (8)

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n ( r ) = n 0 ( 1 1 2 g 2 r 2 ) .
P = 2 π / g .
OTF ( s , t ) = + + I ( x , y ) exp { 2 π i ( s x + t y ) } d x d y + + I ( x , y ) d x d y .
OTF G ( s , t ) = 1 A + + I p ( x , y ) exp { 2 π i ( s x + t y ) } d x d y .
OTF G ( s , t ) = 1 N j = 1 N exp { 2 π i ( s x j + t y j ) } .
H ( ω ) = π 4 [ R ( ω ) + 1 3 R ( 3 ω ) 1 5 R ( 5 ω ) + 1 7 R ( 7 ω ) ]
R ( ω ) = 4 π [ H ( ω ) 1 3 H ( 3 ω ) + 1 5 H ( 5 ω ) 1 7 H ( 7 ω ) + ]
e ( x ) = k π 4 ( D l ) 2 1 ( x l θ ) 2 ,

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