Abstract

We study the use of individual multimode fibers for the purposes of microendoscopy. We discuss the question of image decomposition in the several modes propagating over the fiber and their scattering at the truncated fiber end. We derive analytically the scattering matrix of the “fiber-to-air” interface, we quantify the extent of intermodal coupling, and we evaluate the radiation diagram from the fiber end. Results show that intermodal coupling is weak, so that it appears possible to “capture” an external image and transmit the same through the fiber, after appropriate phase correction, without excessive distortion.

© 2007 Optical Society of America

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References

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  1. M. A. Bolshtyansky and B. Y. Zel'dovich, "Transmission of the image signal with the use of a multimode fiber," Opt. Commun. 123, 629-636 (1996).
    [CrossRef]
  2. A. Yariv, Quantum Electronics (Wiley, 1989).
  3. T. Ogasawara, M. Ohno, K. Karaki, K. Nishizawa, and A. Akiba, "Image transmission with a pair of graded-index optical fibers and a BaTiO3 phase-conjugate mirror," J. Opt. Soc. Am. B 13, 2193-2197 (1996).
    [CrossRef]
  4. M. Fukui and K. Kitayama, "Real-time restoration method for image transmission in a multimode optical fiber," Opt. Lett. 15, 977-979 (1990).
    [CrossRef] [PubMed]
  5. A. M. Tai, "Two-dimensional image transmission through a single optical fiber by wavelength-time multiplexing," Appl. Opt. 22, 3826-3832 (1983).
    [CrossRef] [PubMed]
  6. C. Y. Wu, A. R. D. Somervell, and T. H. Barnes, "Direct image transmission through a multi-mode square optical fiber," Opt. Commun. 157, 17-22 (1998).
    [CrossRef]
  7. L. Pirodda, "Transmission of one-dimensional images through a single optical fiber by time-integrated holography," Opt. Express 11, 1949-1952 (2003).
    [CrossRef] [PubMed]
  8. T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides, IEE Electromagnetic Waves Series (IEE, 1997).
    [CrossRef]
  9. B. E. A. Saleh and M. C. Teich, Fundamental of Photonics (Wiley, 1991).
    [CrossRef]
  10. R. T. Hammond, "Modal structure from far-field measurements," IEEE J. Quantum Electron. 24, 2352-2354 (1988).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).
  13. C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 1997).
  14. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

2003 (1)

1998 (1)

C. Y. Wu, A. R. D. Somervell, and T. H. Barnes, "Direct image transmission through a multi-mode square optical fiber," Opt. Commun. 157, 17-22 (1998).
[CrossRef]

1996 (2)

1990 (1)

1988 (1)

R. T. Hammond, "Modal structure from far-field measurements," IEEE J. Quantum Electron. 24, 2352-2354 (1988).
[CrossRef]

1983 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Akiba, A.

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 1997).

Barnes, T. H.

C. Y. Wu, A. R. D. Somervell, and T. H. Barnes, "Direct image transmission through a multi-mode square optical fiber," Opt. Commun. 157, 17-22 (1998).
[CrossRef]

Bolshtyansky, M. A.

M. A. Bolshtyansky and B. Y. Zel'dovich, "Transmission of the image signal with the use of a multimode fiber," Opt. Commun. 123, 629-636 (1996).
[CrossRef]

Fukui, M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Hammond, R. T.

R. T. Hammond, "Modal structure from far-field measurements," IEEE J. Quantum Electron. 24, 2352-2354 (1988).
[CrossRef]

Karaki, K.

Kitayama, K.

Mongiardo, M.

T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides, IEE Electromagnetic Waves Series (IEE, 1997).
[CrossRef]

Nishizawa, K.

Ogasawara, T.

Ohno, M.

Pirodda, L.

Rozzi, T.

T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides, IEE Electromagnetic Waves Series (IEE, 1997).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamental of Photonics (Wiley, 1991).
[CrossRef]

Somervell, A. R. D.

C. Y. Wu, A. R. D. Somervell, and T. H. Barnes, "Direct image transmission through a multi-mode square optical fiber," Opt. Commun. 157, 17-22 (1998).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Tai, A. M.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamental of Photonics (Wiley, 1991).
[CrossRef]

Wu, C. Y.

C. Y. Wu, A. R. D. Somervell, and T. H. Barnes, "Direct image transmission through a multi-mode square optical fiber," Opt. Commun. 157, 17-22 (1998).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, 1989).

Zel'dovich, B. Y.

M. A. Bolshtyansky and B. Y. Zel'dovich, "Transmission of the image signal with the use of a multimode fiber," Opt. Commun. 123, 629-636 (1996).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

R. T. Hammond, "Modal structure from far-field measurements," IEEE J. Quantum Electron. 24, 2352-2354 (1988).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

C. Y. Wu, A. R. D. Somervell, and T. H. Barnes, "Direct image transmission through a multi-mode square optical fiber," Opt. Commun. 157, 17-22 (1998).
[CrossRef]

M. A. Bolshtyansky and B. Y. Zel'dovich, "Transmission of the image signal with the use of a multimode fiber," Opt. Commun. 123, 629-636 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (7)

A. Yariv, Quantum Electronics (Wiley, 1989).

T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides, IEE Electromagnetic Waves Series (IEE, 1997).
[CrossRef]

B. E. A. Saleh and M. C. Teich, Fundamental of Photonics (Wiley, 1991).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 1997).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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Figures (4)

Fig. 1
Fig. 1

Cylindrical coordinates system adopted. n 1 , n 2 , and n 3 are the refractive indices of the core, cladding, and air, respectively.

Fig. 2
Fig. 2

Image of a clover (a) at z = 0 and (b) at z = 1 m m , (c) 10   mm , and (d) 100   mm .

Fig. 3
Fig. 3

Different propagation steps of an image of the letter T.

Fig. 4
Fig. 4

Different propagation steps of an image of a clover.

Tables (2)

Tables Icon

Table 1 Scattering Parameters of the Modelized Fiber (Absolute Values) a

Tables Icon

Table 2 Absolute Values of S Parameters a (Azimuthal Variation n = 0) for n 1 = 1.5, 1.75, and 2

Equations (36)

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α = 2 π cos 1 [ ε 2 ε 1 2 ( ε 2 + ε 1 ) ] 1.
V = 2 π a λ ( n 1 2 n 2 2 ) 1 / 2 ,
E ( r , ϕ , z ) = { n , m α n m ψ n m ( r , ϕ ) exp ( j β n m z ) r a 0 r > a ,
ψ n m ( r , ϕ ) = J n ( k n m r ) exp ( j n ϕ ) N n m 1 / 2 .
E ( r , ϕ ) = n , m ( a n m + b n m ) J n ( k n m r ) exp ( j n ϕ ) N n m 1 / 2 = 0 + 0 2 π f ( k r , ϑ ) exp [ j k r r   cos ( ϑ ϕ ) ] 2 π × k r d k r d ϑ ,
H ( r , ϕ ) = n , m 1 z 0 n m ( a n m b n m ) J n ( k n m r ) exp ( j n ϕ ) N n m 1 / 2 = 0 + 0 2 π f ( k r , ϑ ) z 0 ( k r , ϑ ) exp [ j k r r   cos ( ϑ ϕ ) ] 2 π × k r d k r d ϑ ,
b j = a j z 0 j S H x ( r , ϕ ) ψ j ( r , ϕ ) d S ,
j a j ψ j ( r , ϕ ) = S Z ( r , ϕ ; r , ϕ ) H x ( r , ϕ ) d S ,
Z ( r , ϕ ; r , ϕ ) = 1 2 j z 0 j ψ j ( r , ϕ ) ψ j ( r , ϕ ) + 1 2 0 + 0 2 π z 0 ( k r , ϑ ) × exp [ j k r r  cos ( ϑ ϕ ) ] 2 π × exp [ j k r r cos ( ϑ ϕ ) ] 2 π k r d k r d ϑ
z 0 ( k r , ϑ ) = ω μ 0 ( n 3 k 0 ) 2 k r 2 ,
0 + 0 2 π z 0 ( k r , ϑ ) exp [ j k r r  cos ( ϑ ϕ ) ] 2 π × exp [ j k r r cos ( ϑ ϕ ) ] 2 π k r d k r d ϑ = ω μ 0 4 π 2 exp ( j n 3 k 0 R ) R ,
R = [ r 2 + r 2 + 2 r r cos ( ϕ ϕ ) ] 1 / 2 .
h l = i λ l i ψ i ( r , ϕ ) .
S j l = δ j l z 0 j λ l j .
ψ l ( r , ϕ ) = p λ l p [ 1 2 z 0 p ψ p ( r , ϕ ) j ω μ 0 2 π × 0 a 0 2 π exp ( j n 3 k 0 R ) R ψ p ( r , ϕ ) r d r d ϕ ] ,
δ l q = 1 2 z 0 q λ l q j ω μ 0 4 π p λ l p ξ p q .
ξ p q = 0 a 0 2 π 0 a 0 2 π ψ q ( r , ϕ ) exp ( j n 3 k 0 R ) R × ψ p ( r , ϕ ) r d r d ϕ r d r d ϕ .
ξ p q = j π 2 { t = 0 + 4 π 2 ( 4 t + 1 ) ( N q N p ) 1 / 2 [ ( 2 t t ) 2 2 t ] 2 I q t ( 1 ) I p t ( 2 ) q 1 = p 1 = 0 t < q 1 t + q 1 o d d 8 ( 2 t + 1 ) q 1 ( N q N p ) 1 / 2 ( q 1 t + 1 ) ( q 1 + t 1 ) ( q 1 t ) ( q 1 + t ) I q t ( 1 ) I p t ( 2 ) q 1 = p 1 0 0 q 1 p 1 ,
f ( k r , ϑ ) = l f l ( k r , ϑ ) .
f l ( k r , ϑ ) = 1 4 π 2 0 a 0 2 π ( a l + b l ) J l 1 ( k l r ) exp [ j r k r   cos ( ϕ ϑ ) ] × exp ( j l 1 ϕ ) r d r d ϕ = ( a l + b l ) 4 π 2 0 a J l 1 ( k l r ) r d r 0 2 π exp [ j r k r   cos ( ϕ ϑ ) ] × exp ( j l 1 ϕ ) d ϕ = ( a l + b l ) 2 π j l 1  exp ( j l 1 ϑ ) 0 a J l 1 ( k l r ) J l 1 ( k r r ) r d r = ( a l + b l ) 2 π j l 1 a k l J l 1 ( k l a ) J l 1 ( k r a ) k r 2 k l 2   exp ( j l 1 ϑ ) .
E ( r , ϕ , z ) = 0 + 0 2 π f ( k r , ϑ ) exp [ j r k r   cos ( ϑ ϕ ) ] × k r d k r d ϑ = 2 π ( j ) l 1 k 0 exp ( j n 3 k 0 r 2 + z 2 ) r 2 + z 2 z l f l ( k r , ϕ ) .
N tot 4 V 2 π 2 ,
N 2 V π ,
M n V π n 2 ,
exp ( j n 3 k 0 R ) R = π j 2 r r t = 0 + ( 2 t + 1 ) J t + 1 2 ( n 3 k 0 r ) × H t + 1 2 ( 2 ) ( n 3 k 0 r ) P t ( cos   θ ) ,
ξ p q = j π 2 t = 0 + ( 2 t + 1 ) ( N q N p ) 1 / 2 I q t ( 1 ) I p t ( 2 ) I q p t ( 3 ) ,
I q t ( 1 ) = 0 a J q 1 ( k q r ) J t + 1 2 ( n 3 k 0 r ) r d r ,
I p t ( 2 ) = 0 a J p 1 ( k p r ) J t + 1 2 ( n 3 k 0 r ) r d r ,
I q p t ( 3 ) = 0 2 π 0 2 π exp ( j q 1 ϕ ) exp ( j p 1 ϕ ) × P t [ cos ( ϕ ϕ + π ) ] d ϕ d ϕ .
I q p t ( 3 ) = 0 2 π d ϕ ϕ 2 π ϕ P t ( cos   α ) d α
= { 4 π [ ( t t / 2 ) 2 t ] 2 t   even 0 t   odd .
{ α = ϕ ϕ β = ϕ + ϕ ,
I q p t ( 3 ) = 2 π 0 P t ( cos   α ) d α α 4 π + α exp ( j q 1 β ) d β + 0 2 π P t ( cos   α ) d α α 4 π α exp ( j q 1 β ) d β = 4 q 1 0 π P t ( cos   α ) sin ( q 1 α ) d α = { 8 q 1 ( q 1 t + 1 ) ( q 1 t + 3 ) ( q 1 + t 1 ) ( q 1 t ) ( q 1 t + 2 ) ( q 1 + t ) if   q 1 > t   and   q 1 + t   odd 0 if   q 1 t  or  q 1 + t   even .
I q t ( 1 ) = ( k q 2 ) q 1 Γ ( q 1 + 1 ) l = 0 + ( 1 ) l F [ l ,   t 1 2 l ; q 1 + 1 ; ( k q n 3 k 0 ) 2 ] ( n 3 k 0 2 ) t + 1 2 + 2 l l ! Γ ( t + 3 2 + l ) 0 a r t + q 1 + 2 l + 1 d r = ( k q 2 ) q 1 Γ ( q 1 + 1 ) l = 0 + ( 1 ) l F [ l ,   t 1 2 l ; q 1 + 1 ; ( k q n 3 k 0 ) 2 ] ( n 3 k 0 2 ) t + 1 2 + 2 l a t + q 1 + 2 l + 2 l ! Γ ( t + 3 2 + l ) ( t + q 1 + 2 l + 2 ) ,
H t + 1 2 ( 2 ) ( x ) = J t + 1 2 ( x ) j N t + 1 2 ( x ) = J t + 1 2 ( x ) + j   sin ( π t + π 2 ) 1 J t 1 2 ( x ) ,
I p t ( 2 ) = ( k p 2 ) p 1 Γ ( p 1 + 1 ) l = 0 + { ( 1 ) l F [ l , t 1 2 l ; p 1 + 1 ; ( k p n 3 k 0 ) 2 ] ( n 3 k 0 2 ) t + 1 2 + 2 l a t + p 1 + 2 l + 2 l ! Γ ( t + 3 2 + l ) ( t + p 1 + 2 l + 2 ) + ( 1 ) t + l F [ l , t + 1 2 l ; p 1 + 1 ; ( k p n 3 k 0 ) 2 ] ( n 3 k 0 2 ) t 1 2 + 2 l a t + p 1 + 2 l + 1 l ! Γ ( t + 1 2 + l ) ( t + p 1 + 2 l + 1 ) } .

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