Abstract

The physical optics analysis of a gradient-index (GRIN) rod and a GRIN lens array with aberrations is presented. We investigated the optical path length and aberration of a GRIN rod without definition of the stop plane. We also defined an effective aberration transmission function to include aberrations into physical optics analysis. Our theoretical results of impulse responses agree excellently with experiments. For a single GRIN rod, we obtained a theoretical value of 10.3μm and an experimental value of 10.4μm for the full width at half-maximum of the intensity point-spread function. For a GRIN array, the theoretical value of 19.2μm and the experimental measurement of 19.9μm agree to within 4%. This physical optics methodology with aberrations included can be applied to optical design software. The resolution difference in xerographic process for test material along the “parallel-to-perpendicular” directions is observed. It agrees with the theoretical result for the intensity impulse response of the GRIN array calculated with a second-order correction.

© 2008 Optical Society of America

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References

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2003 (1)

1985 (1)

1982 (1)

1980 (2)

1976 (3)

1975 (1)

1971 (5)

1970 (2)

1969 (1)

P. Baues, “Huygens' principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 37-44 (1969).
[CrossRef]

1968 (1)

Agrawal, G. P.

G. P. Agrawal, “Imaging characteristics of square law media,” Nouv. Rev. Opt. 7, 299-303 (1976).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics (North-Holland, 1973), Vol. 11.
[CrossRef]

Bao, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Baues, P.

P. Baues, “Huygens' principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 37-44 (1969).
[CrossRef]

Chen, X.

X. Chen and N. George, “Fourier optical analysis of gradient-index array imaging,” Appl. Opt. 42, 4434-4444 (2003).
[CrossRef] [PubMed]

X. Chen, “Gradient-index fiber array for imaging,” Ph.D.dissertation (University of Rochester, 2006).

Conte, S. D.

S. D. Conte and C. de Boor, Elementary Numerical Analysis, 3rd ed. (McGraw-Hill, 1980), Chap. 7.

de Boor, C.

S. D. Conte and C. de Boor, Elementary Numerical Analysis, 3rd ed. (McGraw-Hill, 1980), Chap. 7.

George, N.

Ghatak, A. K.

Gomez-Reino, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Iga, K.

K. Iga, “Theory for gradient-index imaging,” Appl. Opt. 19, 1039-1043 (1980).
[CrossRef] [PubMed]

K. Iga, Y. Kokubun, and M. Oikawa, Fundamentals of Microoptics: Distributed-Index, Microlens, and Stacked Planar Optics (Academic, 1984).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998), pp. 380-381.

Kokubun, Y.

K. Iga, Y. Kokubun, and M. Oikawa, Fundamentals of Microoptics: Distributed-Index, Microlens, and Stacked Planar Optics (Academic, 1984).

Kumar, D. Vizia

Lama, W.

Longhurst, R. H.

R. H. Longhurst, Geometrical and Physical Optics (Longman's, Green and Company, Ltd., 1957).

Marchand, E. W.

Montagnino, L.

Moore, D. T.

Oikawa, M.

K. Iga, Y. Kokubun, and M. Oikawa, Fundamentals of Microoptics: Distributed-Index, Microlens, and Stacked Planar Optics (Academic, 1984).

Paxton, K. B.

Perez, M. V.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Rees, J. D.

Rimmer, M. P.

M. P. Rimmer, “Ray tracing in inhomogeneous media,” Optical System Design, Analysis, and Production Conference, SPIE, Geneva, Switzerland, April 1983, Vol. 399.

Sands, P. J.

Seigman, A. E.

A. E. Seigman, Lasers (University Science Books, 1986).

Sharma, A.

Smith, W. J.

W. J. Smith, Modern Optical Engineering: The Design of Optical Systems (McGraw-Hill, 1966).

Streifer, W.

Yariv, A.

Appl. Opt. (8)

J. Opt. Soc. Am. (8)

Nouv. Rev. Opt. (1)

G. P. Agrawal, “Imaging characteristics of square law media,” Nouv. Rev. Opt. 7, 299-303 (1976).
[CrossRef]

Opto-electronics (1)

P. Baues, “Huygens' principle in inhomogeneous, isotropic media and a general integral equation applicable to optical resonators,” Opto-electronics 1, 37-44 (1969).
[CrossRef]

Other (12)

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics (North-Holland, 1973), Vol. 11.
[CrossRef]

A. E. Seigman, Lasers (University Science Books, 1986).

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

X. Chen, “Gradient-index fiber array for imaging,” Ph.D.dissertation (University of Rochester, 2006).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998), pp. 380-381.

S. D. Conte and C. de Boor, Elementary Numerical Analysis, 3rd ed. (McGraw-Hill, 1980), Chap. 7.

R. H. Longhurst, Geometrical and Physical Optics (Longman's, Green and Company, Ltd., 1957).

W. J. Smith, Modern Optical Engineering: The Design of Optical Systems (McGraw-Hill, 1966).

E. W. Marchand, Gradient Index Optics (Academic, 1978).

M. P. Rimmer, “Ray tracing in inhomogeneous media,” Optical System Design, Analysis, and Production Conference, SPIE, Geneva, Switzerland, April 1983, Vol. 399.

Lambda Research Corporation, OSLO Optics Reference Release 6.1 (Lambda Research Corporation, 2001).

K. Iga, Y. Kokubun, and M. Oikawa, Fundamentals of Microoptics: Distributed-Index, Microlens, and Stacked Planar Optics (Academic, 1984).

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Figures (11)

Fig. 1
Fig. 1

Imaging setup for the GRIN rod with two end surfaces 2 and 3, an object plane 1, external and internal Fourier transform planes 4 and 6, external and internal image planes 5 and 7. Plane 8 is the stop plane (S). The definition of the Fourier transform planes 4 and 6 are discussed in [22].

Fig. 2
Fig. 2

Rays form images in the GRIN rod: rays 0 to 5 from an on-axis object form an on-axis image. Other rays from the object at ( 2 R , 0 ) form an off-axis image.

Fig. 3
Fig. 3

Ray-intercept curves, transverse ray error ε x versus x 2 , for (a) an on-axis object and (b) an off-axis object at ( 2 R , 0 ) .

Fig. 4
Fig. 4

Wave aberration on plane 3 of a GRIN rod. We use e i ω t notation here.

Fig. 5
Fig. 5

Cross sections of OPD, δ ( x 3 , 0 ; x 1 , y 1 ) , along the x axis on plane 3 for (a) an axial object and (b) an off-axis object at ( 2 R , 0 ) .

Fig. 6
Fig. 6

Cross section along the x axis of the intensity PSFs for different object points: (a) an on-axis object; (b) an off-axis object at ( R , 0 ) ; (c) an off-axis object at ( 2 R , 0 ) ; and (d) an off-axis object at ( 4 R , 0 ) .

Fig. 7
Fig. 7

Single rod’s PSF measurement setup.

Fig. 8
Fig. 8

Intensity PSF of the GRIN rod, I rod ( x 5 , y 5 ) .

Fig. 9
Fig. 9

GRIN rods contributing to the imaging of the object at the center of rod 1 in a two-row GRIN array.

Fig. 10
Fig. 10

Cross sections of the intensity PSF of the GRIN array along the x axis (dotted curve) and the y axis (solid curve) by coherently summing the amplitude PSFs of the five central GRIN rods [see Eq. (A2)]. This theory includes effects of aberration and OPD but not misalignment.

Fig. 11
Fig. 11

Two-row SELFOC array imaging 18   line pairs / mm resolution target in white light.

Tables (2)

Tables Icon

Table 1 Gradient-Index Array Intensity Point-Spread Functions’ Full-Width at Half-Maximum for Different Object Points

Tables Icon

Table 2 Comparison between the Full-Width at Half-Maximum of Intensity Point-Spread Functions

Equations (38)

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n 2 ( r ) = n 1 2 ( 1 α 2 r 2 ) , r R ,
n = d d s ( n d r d s ) ,
x ( z ) = cos ( n 1 α z n 1 2 n 1 2 α 2 ( x 2 ( 0 ) + y 2 ( 0 ) ) ( 1 γ 0 2 ) ) x ( 0 ) + 1 n 1 α sin ( n 1 α z n 1 2 n 1 2 α 2 ( x 2 ( 0 ) + y 2 ( 0 ) ) ( 1 γ 0 2 ) ) α 0 , y ( z ) = cos ( n 1 α z n 1 2 n 1 2 α 2 ( x 2 ( 0 ) + y 2 ( 0 ) ) ( 1 γ 0 2 ) ) y ( 0 ) + 1 n 1 α sin ( n 1 α z n 1 2 n 1 2 α 2 ( x 2 ( 0 ) + y 2 ( 0 ) ) ( 1 γ 0 2 ) ) β 0 ,
n d x ( z ) d s = n 1 α sin ( n 1 α z n 1 2 n 1 2 α 2 ( x 2 ( 0 ) + y 2 ( 0 ) ) ( 1 γ 0 2 ) ) x ( 0 ) + α 0 cos ( n 1 α z n 1 2 n 1 2 α 2 ( x 2 ( 0 ) + y 2 ( 0 ) ) ( 1 γ 0 2 ) ) , n d y ( z ) d s = n 1 α sin ( n 1 α z n 1 2 n 1 2 α 2 ( x 2 ( 0 ) + y 2 ( 0 ) ) ( 1 γ 0 2 ) ) y ( 0 ) + β 0 cos ( n 1 α z n 1 2 n 1 2 α 2 ( x 2 ( 0 ) + y 2 ( 0 ) ) ( 1 γ 0 2 ) ) ,
x ( z ) = cos ( α z ) x ( 0 ) + sin ( α z ) α 0 n 1 α , n d x ( z ) d s = n 1 α sin ( α z ) x ( 0 ) + cos ( α z ) α 0 .
[ A B C D ] = [ cos ( α L ) sin ( α L ) n 1 α n 1 α sin ( α L ) cos ( α L ) ] ,
[ x ( z ) n d x ( z ) d z ] = [ A B C D ] [ x ( 0 ) n ( 0 ) d x ( 0 ) d z ] ,
d s = n d z n 1 2 n 1 2 α 2 ( x 0 2 + y 0 2 ) ( 1 γ 0 2 ) ,
OPL = z 0 z n 2 ( r ( z ) ) d z n 1 2 n 1 2 α 2 ( x 0 2 + y 0 2 ) ( 1 γ 0 2 ) .
Ω = Q ¯ Q = OPL ( P Q ) OPL ( P Q ¯ ) = OPL ( P Q ) OPL ( P A ) OPL ( A Q ¯ ) ,
OPL ( A Q ¯ ) OPL ( A P * ) OPL ( Q ¯ P * ) = d 2 2 + ( x 3 x 5 * ) 2 + ( y 3 y 5 * ) 2 d 2 2 + x 5 * 2 + y 5 * 2 ,
Ω = Q ¯ Q = OPL ( P Q ) OPL ( P A ) d 2 2 + ( x 3 x 5 * ) 2 + ( y 3 y 5 * ) 2 + d 2 2 + x 5 * 2 + y 5 * 2 = OPL ( P B P * ) OPL ( P A ) d 2 2 + ( x 3 x 5 * ) 2 + ( y 3 y 5 * ) 2 ,
Ω = OPL ideal OPL ( P A ) .
δ 0 = OPL ( P A ) OPL ideal = Ω .
δ = OPL ( P A ) [ ( d 1 + n 1 * L + d 2 ) d 2 2 + ( x 3 x 5 * ) 2 + ( y 3 y 5 * ) 2 ] ,
δ = Ω + Δ l = δ 0 + Δ l ,
δ ( x 3 , y 3 ; x 1 , y 1 ) = OPL ( x 3 , y 3 ; x 1 , y 1 ) + d 2 2 + ( x 3 x 5 * ) 2 + ( y 3 y 5 * ) 2 ( d 1 + n * L + d 2 ) ,
δ ( x 3 , 0 ; 0 , 0 ) = 0.0048 ρ 4 0.0002 ρ 6 0.0002 ρ 8 0.0002 ρ 10 , | ρ | 0.4964 ( mm ) ,
T 3 ( x 3 , y 3 ; x 1 , y 1 ) = P 3 aperture ( x 3 , y 3 ; x 1 , y 1 ) × e i k δ ( x 3 , y 3 ; x 1 , y 1 ) ,
V 2 ( x , y ) = i k exp ( i k L ) 2 π B d x 0 d y 0 V 1 ( x 0 , y 0 ) × exp { i k 2 B [ D ( x 2 + y 2 ) 2 ( x 0 x + y 0 y ) + A ( x 0 2 + y 0 2 ) ] } ,
( A B C D )
V 5 ( x 5 , y 5 ) = d x 0 d y 0 V 1 ( x 0 , y 0 ) h 15 ( x 5 , y 5 ; x 0 , y 0 ) .
V 1 ( x 0 , y 0 ) = δ ( x 0 x 1 , y 0 y 1 ) ,
h 15 ( x 5 , y 5 ; x 1 , y 1 ) = l exp [ i k ( d 1 + n 1 L + d 2 ) ] λ 2 d 1 B d 2 d x 3 d y 3 exp [ i k 2 d 1 ( 1 l d 1 ) ( x 1 2 + y 1 2 ) ] × exp [ i k 2 d 2 ( x 5 2 + y 5 2 ) ] exp [ i k ( D B + 1 d 2 l B 2 ) 2 ( x 3 2 + y 3 2 ) ] exp [ i k ( x 5 d 2 l x 1 d 1 B ) x 3 i k ( y 5 d 2 l y 1 d 1 B ) y 3 ] T 3 ( x 3 , y 3 ; x 1 , y 1 ) ,
l = B d 1 A d 1 + B .
D B + 1 d 2 l B 2 = 0 .
d 2 = A d 1 + B C d 1 + D .
M = x 5 x 1 = y 5 y 1 = 1 C d 1 + D .
d 2 = M [ cos ( α L ) d 1 + sin ( α L ) n 1 α ] ,
M = 1 cos ( α L ) n 1 α d 1 sin ( α L ) .
h 15 ( x 5 , y 5 ; x 1 , y 1 ) = M exp [ i k ( d 1 + n 1 L + d 2 ) ] λ 2 d 2 2 exp [ i k cos ( α L ) M ( x 1 2 + y 1 2 ) 2 d 2 ] × exp [ i k ( x 2 + y 2 ) 2 d 2 ] F 1 { T 3 ( x 3 , y 3 ; x 1 , y 1 ) } .
F 1 { T 3 ( x 3 , y 3 ; x 1 , y 1 ) } = ( λ d 2 ) 2 d f x d f y T 3 ( λ d 2 f x , λ d 2 f y ; x 1 , y 1 ) × exp [ i 2 π f x ( x M x 1 ) + i 2 π f y ( y M y 1 ) ] ,
I total ( x 6 M , y 6 M ) = I objective ( x 6 M x 5 , y 6 M y 5 ) I rod ( x 5 , y 5 ) d x 5 d y 5 ,
h 15 A ( x 5 , y 5 ; x 1 , y 1 ) = m = 1 9 | h 15 ( m ) | 2 ,
| h 15 ( m ) | 2 = M 2 | F 1 { T 3 ( m ) ( x 3 , y 3 ; x 1 , y 1 ) } | 2 ( λ d 2 ) 4 .
h 15 A = | m = 1 9 h 15 ( m ) | 2 = m = 1 9 | h 15 ( m ) | 2 + m = 1 9 n = 1 n m 9 h 15 ( m ) h 15 ( n ) * .
| ρ ( m ) ρ ( n ) | = λ 1 λ 2 λ 2 λ 1 ,
| h 15 ( 1 ) | 2 + | h 15 ( 2 ) | 2 + | h 15 ( 3 ) | 2 + | h 15 ( 6 ) | 2 + | h 15 ( 7 ) | 2 + h 15 ( 2 ) h 15 ( 3 ) * + h 15 ( 2 ) * h 15 ( 3 ) h 15 A = + h 15 ( 2 ) h 15 ( 6 ) * + h 15 ( 2 ) * h 15 ( 6 ) + h 15 ( 2 ) h 15 ( 7 ) * + h 15 ( 2 ) * h 15 ( 7 ) + h 15 ( 3 ) h 15 ( 6 ) * + h 15 ( 3 ) * h 15 ( 6 ) + h 15 ( 3 ) h 15 ( 7 ) * + h 15 ( 3 ) * h 15 ( 7 ) + h 15 ( 6 ) h 15 ( 7 ) * + h 15 ( 6 ) * h 15 ( 7 ) ,

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