## Coupling starlight into single-mode photonic crystal fiber using a field lens

Optics Express, Vol. 13, Issue 17, pp. 6527-6540 (2005)

http://dx.doi.org/10.1364/OPEX.13.006527

Acrobat PDF (390 KB)

### Abstract

We determine the coupling characteristics of a large mode area (LMA) photonic crystal, single-mode fiber when fed with an on-axis field lens used to place an image of the telescope exit pupil at the fiber input. The maximum field of view is found to be approximately the same as that of feeding the fiber directly with the telescope PSF in the image plane. However, the field lens feed can be used to provide a flat, maximised coupling response over the entire visible-NIR which is not possible using either the highly wavelength dependent direct feed coupling to the LMA fiber or the attenuation spectrum limited step index fiber cases.

© 2005 Optical Society of America

## 1. Introduction

### 1.1 Large mode area (LMA) photonic crystal fiber.

1. T. Birks, J. Knight, and P St. J Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. , **22** p 961-963 (1997) [CrossRef] [PubMed]

1. T. Birks, J. Knight, and P St. J Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. , **22** p 961-963 (1997) [CrossRef] [PubMed]

3. A. Bjarklev, J. Broeng, and A.S. Bjarklev, *Photonic crystal fibers*, (Kluwer, 2003) [CrossRef]

*endlessly single-moded*over all wavelengths at which silica (from which most commercially available fiber are made) is transparent.

### 1.2 Single-mode fibers in interferometry

_{cutoff}region where λ

_{cutoff}is the singlemode cutoff wavelength, below which the fiber becomes multimoded. In the VIS this equates to a useable spectral range of about 200nm or so and hence a range of fibers and feed optics is required for broad band interferometry [5

5. G. Perrin, O. Lai, P. J. Lena, and V. Coude du Foresto, “Fibered large interferometer on top of Mauna Kea: OHANA, the Optical Hawaiian Array for Nanoradian Astronomy,” in *Interferometry in Optical Astronomy*, P. J. Lena and A. Quirrenbach, eds.,Proc. SPIE **4006**, 708-714 (2000). [CrossRef]

6. M.D. Nielson, J.R. Folkenberg, N.A. Mortenson, and A. Bjarklev, “Bandwidth comparison of photonic crystal fibers and conventional single-mode fibers,” Opt. Express **12**, 430 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-430 [CrossRef]

3. A. Bjarklev, J. Broeng, and A.S. Bjarklev, *Photonic crystal fibers*, (Kluwer, 2003) [CrossRef]

7. M.D. Nielsen, N.A. Mortensen, M. Albersen, J.R. Folkenberg, A. Bjarklev, and D. Bonacinni, “Predicting macrobending loss for large-mode area photonic crystal fibers,” Opt. Express **12**, 1775 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775 [CrossRef] [PubMed]

## 2. Coupling model

*i*’ places an image of the telescope exit pupil on the endface of the fiber. The field assignments for the lenslet feed in Fig. 2(b) assume that the telescope and its foreoptics, which provides magnification,

*M*, are subsumed into one element such that

**E**

_{d}is the electric field incident on the telescope entrance pupil, and since we consider only unabberrated optical components,

**E**’

_{d}is the resulting spherical field converging toward the image plane,

*i*’ and it is this field that acts as the coherent object for the field lenslet.

*n*is the refractive index of the space immediately in front of the fiber.

**E**is the spherical wavefront diverging from some arbitrary point

*(ε,η)*in

**E**

_{d}and

**E**’ is the spherical wavefront converging from the lenslet to the point

*(p,q)*in the image field

**E**

_{i}at the fiber endface. Throughout this work, bold type indicates a complex field and is retained for fields that can be taken as completely real for brevity.

*i*’.

**E**

_{i}(whether the Airy pattern of the direct feed or the electric field of the pupil image) and fiber mode field

**E**

_{f}is given by the coupling integral (1),

*(x-y)*plane transverse to the optical axis (z). Full derivations of this equation are given for smoothly varying or step index fibers in references [2] and [8]. While a complete analysis of the validity of this formula for LMA type fibers is beyond the scope of this paper, this approach can be applied to commercially available LMA fibers which have many rings of holes in their cross sectional structure surrounding the central defect and thus have well bound guided modes. In this case, A, may be taken as the real plane transverse to the fiber axis and located at

*z*=0, the fiber endface. All modes (guided and radiated) can be shown to be orthogonal over

*A*and this condition together with the requirement that the tangential components of the

**E**and

**H**fields are continuous across the fiber endface boundary can be used to isolate the power coupled into the transverse modal fields of the fiber. This paper deals only with the case where the back reflected field

**E**

_{r}and therefore its effect on the distribution of the input field

**E**

_{i}is negligible. In the case where

*n*

_{core}, where n is the refractive index in the plane of

**E**

_{i}, and

*n*

_{core}that of the material from which the fiber is made, the overall coupling must be multiplied by the Fresnel transmission coefficient,

*T*= 4

*n*.

*n*

_{core}/(

*n*+

*n*

_{core})

^{2}.

**r**=

*(p,q)*and

*F*is the feed focal ratio into

*(p,q)*the plane,

*k*is the local wavenumber at the fiber endface and

*ε*is the obscuration due to the telescope secondary such that

*ε*∈ [0,1].

**E**

_{i}and

**E**

_{d}can be derived using the coherent imaging Eq. (3) as derived by Goodman[9],

**h**is the system transfer function,

*M*is the magnification of the foreoptics and lenslet with respect to the telescope exit pupil and its (geometric) image on the fiber endface and

*A(x,y)*is the aperture stop of the lenslet.

*Z*and

_{u}*Z*are defined in Fig. 2 and the constant amplitude and phase terms have been ignored since they cancel in (1) and

_{v}*k*

_{o}is the free space wavelength.

*f*is distance between the exit pupil of the telescope/foreoptics combination and the plane

*i*’ and the exponential in Eq. (5) cancels exactly with the quadratic phase term in

*(ε,η)*in Eq. (4).

*(p,q)*describes the projection of the spherical image space, centered on the exit pupil of the lenslet, onto the

*(p,q)*plane.

**r**∣

^{2}= ∣

*p*

^{2}+

*q*

^{2}∣ > 1.65

*Λ*have magnitude less than 10

^{-4}the peak value at the centre of the mode and thus changes in

**E**

_{i}at this point have a minimal effect on the numerator in (1). By computing its value over the endface of the fiber, the change in the (p,q) quadratic phase term can be shown to be negligible at ∣

**r**∣ ≤ 1.65

*Λ*if the focal length of the lenslet,

*f*>16

*Λ*, when the lenslet is in contact with the fiber (

*n*=1.45) and

*f*> 13

*Λ*for

*n*= 1.00. The fastest lens(lets) considered in this work have focal ratios of

*f*/2 implying a minimum aperture size of ~185μm for the LMA-35, ~105μm for the LMA-20 and ~45μm for the LMA-8 for the

*n*=1.45 case, so the restriction is of little practical interest unless the very smallest of lenslets is used.

*(p,q)*quadratic phase term in the electric field will affect the maximum coupling efficiency by introducing a phase mismatch between

**E**

_{i}and the transverse component of

**E**

_{f}.

**h**

*(p,q)*of the exit aperture,

*P(x,y)*with the geometric image

**E**

_{g}

*(p,q)*(Fig.2(b)), of

**E**

_{d}

*(ε,η)*in the

*(p,q)*plane,

**E**fields were computed on a 1000

^{2}array and after numerical trials a simple midpoint integration routine was found to yield the most stable and accurate results matching analytical examples to within +/-1%. An analytical approximation to the mode field of the fiber exists[14

14. N.A. Mortensen and J.R. Folkenberg, “Near-field to far-field transition of photonic crystal fibers: symmetries and interference phenomena,” Opt. Express **10**, 475
(2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-475 [PubMed]

10. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L.C. Botton, “Multipole method for microstructured optical fiber. I Formulation,” J. Opt. Soc. Am. B **19** , No 10 (2002) [CrossRef]

11. B.T. Kuhlmey, T.P. White, G. Renversez, D. Maystre, L.C. Botton, C. Martijn de Sterke, and R.C McPhedran, “Multipole method for microstructured optical fiber. II Implementation and Results,” J. Opt. Soc. Am. B **19** , No 10 (2002) [CrossRef]

1. T. Birks, J. Knight, and P St. J Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. , **22** p 961-963 (1997) [CrossRef] [PubMed]

3. A. Bjarklev, J. Broeng, and A.S. Bjarklev, *Photonic crystal fibers*, (Kluwer, 2003) [CrossRef]

^{-6}per nm and is therefore considered negligible.

12. M.J Steel, T.P White, C. Martijn de Sterke, R.C. McPhedran, and L.C. Botton, “Symmetry and degeneracy in microstructered optical fibers,” Opt. Lett. **26** No.8 (2001). [CrossRef]

^{-4}worst case with typical differences of 10

^{-5}. Hence we take the LMA fiber to be completely polarisation independent in this respect.

**E**

_{i}with respect to the polarisation doublet

**E**

_{f}and thus the polarisation of

**E**

_{i}is ignored throughout. Further, the effect of rotation of spiders of width ≤ 5% of the diameter of the telescope primary mirror has also been found to be negligible.

## 3. Characteristics of the field lens feed

### 3.1 On axis source

*f*/2 lenslet at

*λ*=400nm and an

*f*/4 lenslet at

*λ*=2000nm, respectively. The aperture of the lenslet will partially vignette the PSF of the telescope causing a loss in total throughput. However this does not effect the coupling efficiency at the fiber and hence is explained fully and quantified in Sections 3.2 and 4 and ignored for clarity here.

*O*in Fig. 2(b), as a function of telescope exit pupil image size, d

_{p}, telescope obscuration and lateral offset of the pupil image wrt to the mode field of the fiber. These figures represent maximum coupling efficiencies since off axis sources couple less efficiently due to the phase mismatch between

**E**

_{d}and

**E**

_{f}. (Section 3.2).

**h**

*(p,q)*as it scales with respect to

**E**

_{g}. The least diffracted curve peaks at 1.33

*Λ*with the most diffracted image of interest, taken as

*ρ*

_{max}>50%, showing a very weak dependence on image diameter, due to the dominance of the lenslet PSF over the image scale in these cases.

*Λ*lateral shift and for an LMA-8 fiber this equates to an alignment precision between fiber and lenslet axes of ~1.6μm rising to a few microns for the LMA-20. Alignment tolerances of this order are discussed by Liu

*et al*[13] and whilst technologically challenging, are possible.

#### 3.1.1 Gaussian apodisation

14. N.A. Mortensen and J.R. Folkenberg, “Near-field to far-field transition of photonic crystal fibers: symmetries and interference phenomena,” Opt. Express **10**, 475
(2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-475 [PubMed]

*d*(each pair of curves) and of increasing the size of

_{p}*d*wrt to the static mode field of the LMA fiber (the three sets of curves). Both the transmission and coupling curves are asymptotic to the x-axis and hence Gaussian apodisation works to reduce the overall coupling.

_{p}### 3.2 - Off axis source.

*ρ*(θ) falls below some absolute value.

_{f}**E**

_{i}for instance the amplitude of

**h**or

**E**

_{g}cancel top and bottom in Eq. (1) and the power relationship between

**E**

_{d}and

**E**

_{f}is lost leaving only

*ρ*the

_{f}*ratio*of the powers in

*E*

_{d}and

**E**

_{f}. For instance if

*ρ*= 0.5 then 50% of the energy incident at the endface of the fiber is coupled into the fiber. The power ratio between

_{f}**E**

_{d}and

**E**

_{i}is then given by

*ρ*and the total throughput from telescope entrance pupil to fiber mode by Eq. (7). The separation of the two terms on the RHS of Eq. (7) is therefore justified and further, whose individual derivations are physically relevant and instructive, see sections 3.2.1/3.2.2.

_{s}*ρ(θ)*itself is investigated in section 4.

^{2}is negligible [7

7. M.D. Nielsen, N.A. Mortensen, M. Albersen, J.R. Folkenberg, A. Bjarklev, and D. Bonacinni, “Predicting macrobending loss for large-mode area photonic crystal fibers,” Opt. Express **12**, 1775 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775 [CrossRef] [PubMed]

#### 3.2.1 - Coupling due to the fiber, *ρ*_{f}(θ)

_{f}(θ)

*ρ*(θ) characteristics for coupling into an LMA-8 and LMA-15 fiber.

_{f}*ρ*> 0.3, and lenslet

*F*< 4 [17] is overplotted in Fig. 7, given by

_{L}*ρ*

_{max}=

*ρ*(θ=0) is the maximum coupling value, n is the refractive index at the fiber endface, n

_{f}_{core}is the refractive index of the fiber core,

*Λ*is the characteristic length of the photonic structure in the fiber,

*Λ*is the free space wavelength and

_{o}*β*is a numerical constant of value 13.832 when θ is in degrees. As noted in section 3.1,

*ρ*

_{max}changes due to the wavelength dependent scaling of

**h**with respect to

**E**

_{g}in Eq. (6).

*ρ*(θ) for various pupil image sizes,

*d*, on the fiber endface. From the plot,

_{p}*ρ*(θ) can be increased/decreased slightly with a corresponding decrease/increase in image size. However, the FOV on the sky is a function of

*ρ*(θ) and

*d*(See Eq. (12)) and the optimum

_{p}*ρ*

_{max}×

*ρ;*(θ) is found at a

*d*of 133

_{p}*Λ*.

*γ*∈[0.0,1.0]:

*γ*<

*ρ*

_{max}}, (8) can be rearranged to yield,

*on the sky*for the given telescope configuration. In Fig. 2(a), the Smith-Hemholtz invariant can be used to derive the angular scale between the telescope exit pupil and its geometrical image such that

_{fiber}, the critical focal ratio associated with a maximum coupling loss of

*ρ*

_{max}-

*γ*yields and

*d*= 1.33

_{p}*Λ*,

*sample size on the sky is independent of the size of the fiber core*, associated with the characteristic size of the fiber,

*Λ*. This result can be understood on the basis that a larger core implies a larger ∣

*y*

_{max}∣ in exp(

*ikny*θ) and therefore requires a smaller θ for the same value of this term. Therefore as the core grows, the maximum input angle (for the same

*λ*) gets proportionately smaller and hence

*χ*remains constant.

*i*to

*i*′) magnification of the foreoptics,

*M*can also be computed using

_{T}is the focal length of the telescope and substituting in (13) yields

_{L}is the total aperture size in metres of the lenslet of focal ratio

*F*, where the approximation that the minimum focal ratio into the pupil image on the endface of the fiber is that of the lenslet itself, since

_{L}*Z*<<

_{v}*Z*in Fig. 2.

_{u}*M*to a reasonable figure, say ≤10 and assuming that

*F*≤2 then for an LMA-8 fiber,

_{L}*d*≤ 260μm for

_{L}*F*= 7 and

_{T}*d*≤ 600μm for

_{L}*F*= 16, assuming that

_{T}*n*= 1.00 and

*d*≤ 600μm for

_{L}*F*= 7 and

_{T}*d*≤ 1400μm for

_{L}*F*= 16 for the LMA-20. So significant foreoptics magnifications are required to implement the pupil imaging scheme with a macroscopic field lens with lower

_{T}*M*required for smaller lenslets of course.

#### 3.2.2 - Coupling due to the lenslet, *p*_{s}(*θ*)

_{s}

*i*′ it is stopped by the finite aperture of the field lens and this creates an extra term

*ρ*in the off axis coupling efficiency.

_{s}*ρ*can be computed for any aperture using,

_{s}*U*is the Airy pattern as specified by (2), and δ

**r**is the offset in

*i*′ of the centre of

*U*from the telescope axis, dropped from the denominator since integrated over the infinite plane.

## 4. Comparison of direct and field lens coupling

*ρ*and

_{f}*ρ*on the coupling into the fiber using a field lenslet and compare these results with the case where the telescope PSF is fed directly into the fiber.

_{s}*i*′ Rayleigh lengths is given by Eq. (13) and so the sample size of the lenslet in

*i*′can be usefully expressed as

*S*is the number of Rayleigh lengths of the telescope PSF free space wavelength,

*λ*, sampled

_{o}*by the lenslet*. The numerical constant in Eq. (17) subsumes the values of 1.33

*d*,

_{p}*π*and the conversion from degrees to radians. Note that since Eq. (16) is the sample size of the lenslet then it natural that it should have no dependence on

*γ*or

*ρ*

_{max}.

*S*and hence the largest effect of Eq. (16) occurs for the smallest

*Λ*. Figure 8(a) is an overplot of

*ρ*and

_{s}*ρ*for the LMA-8 fiber with

_{f}*F*= 2, with a telescope obscuration of 25%. Looking first at the on-axis, θ = 0 values, the fiber coupling

_{L}*ρ*~96% at 2000nm is due to the excellent match of the

_{F}*f*/2 PSF at this wavelength and the fiber mode however, the value of

*ρ*

_{s}at this wavelength is 76% yielding the total coupling of ~73%. However, at 400nm

*ρ*

_{F}has a value of ~75% but a

*ρ*

_{s}of 96% and so as the fiber coupling,

*ρ*

_{F}increases with increasing wavelength, the maximum (on-axis) lenslet coupling, ps decreases at approximately the same rate leaving

*ρ*(0) in Fig. 7(b) essentially flat across the whole of the visible-NIR.

5. G. Perrin, O. Lai, P. J. Lena, and V. Coude du Foresto, “Fibered large interferometer on top of Mauna Kea: OHANA, the Optical Hawaiian Array for Nanoradian Astronomy,” in *Interferometry in Optical Astronomy*, P. J. Lena and A. Quirrenbach, eds.,Proc. SPIE **4006**, 708-714 (2000). [CrossRef]

*f*/3 lenslet causes

*ρ*to fall off in the NIR for the LMA-8 but with the slower lenslet having little affect on the LMA-15.

_{s}*λ*/

*d*as the direct feed case as shown by the good overplot of the lenslet feed and 600nm direct fed curves. This is also the same FOV as the direct feed into a step index fiber [18].

_{T}*ρ*becomes more restrictive in angle at higher

_{s}*λ*, whereas

*ρ*becomes more restrictive at lower wavelength. These two terms trade off one another in (7) to yield the curves in the Fig. 8(c).

_{f}*ρ*

_{max}but at the significant cost of nearly ¼ the peak throughput of that associated with the optimum wavelength (600nm in this case) and so the lenslet feed into an LMA fiber makes the entire visible-NIR simultaneously accessible with a both constant FOV and high throughput.

## 5. Summary

12. M.J Steel, T.P White, C. Martijn de Sterke, R.C. McPhedran, and L.C. Botton, “Symmetry and degeneracy in microstructered optical fibers,” Opt. Lett. **26** No.8 (2001). [CrossRef]

## References

1. | T. Birks, J. Knight, and P St. J Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. , |

2. | E.G. Neumann, |

3. | A. Bjarklev, J. Broeng, and A.S. Bjarklev, |

4. | V. Coude de Foresto, S. Ridgway, and J.M Mariotti, “Deriving object visibilities from interferograms obtained with a fiber stellar interferometer,”A&AS |

5. | G. Perrin, O. Lai, P. J. Lena, and V. Coude du Foresto, “Fibered large interferometer on top of Mauna Kea: OHANA, the Optical Hawaiian Array for Nanoradian Astronomy,” in |

6. | M.D. Nielson, J.R. Folkenberg, N.A. Mortenson, and A. Bjarklev, “Bandwidth comparison of photonic crystal fibers and conventional single-mode fibers,” Opt. Express |

7. | M.D. Nielsen, N.A. Mortensen, M. Albersen, J.R. Folkenberg, A. Bjarklev, and D. Bonacinni, “Predicting macrobending loss for large-mode area photonic crystal fibers,” Opt. Express |

08. | A.W. Snyder and J.D. Love, |

9. | Joseph W Goodman, |

10. | T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L.C. Botton, “Multipole method for microstructured optical fiber. I Formulation,” J. Opt. Soc. Am. B |

11. | B.T. Kuhlmey, T.P. White, G. Renversez, D. Maystre, L.C. Botton, C. Martijn de Sterke, and R.C McPhedran, “Multipole method for microstructured optical fiber. II Implementation and Results,” J. Opt. Soc. Am. B |

12. | M.J Steel, T.P White, C. Martijn de Sterke, R.C. McPhedran, and L.C. Botton, “Symmetry and degeneracy in microstructered optical fibers,” Opt. Lett. |

13. | D.T. Liu, B.M. Levine, and M. Shao, “Design and fabrication of a coherent array of single-mode optical fibers for the nulling coronograph,”Proc. SPIE 5170 Techniques and Instrumentation for detection of exoplanets. (2003) |

14. | N.A. Mortensen and J.R. Folkenberg, “Near-field to far-field transition of photonic crystal fibers: symmetries and interference phenomena,” Opt. Express |

15. | Born and Wolf, |

16. | A. Snyder, “Excititation and scattering of modes on a dielectric optical fiber,” IEEE Trans. MTT V NIT- |

17. |
Equation (8) actually underestimates slightly ρ < 0.40 for >f/4 and λ |

18. | Olivier Guyon, “Wide-field interferometric imaging with single-mode fibers,” A&A |

**OCIS Codes**

(060.2430) Fiber optics and optical communications : Fibers, single-mode

(100.2960) Image processing : Image analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(350.1260) Other areas of optics : Astronomical optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 27, 2005

Revised Manuscript: August 11, 2005

Published: August 22, 2005

**Citation**

Jason Corbett and Jeremy Allington-Smith, "Coupling starlight into single-mode photonic crystal fiber using a field lens," Opt. Express **13**, 6527-6540 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-17-6527

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### References

- T. Birks, J. Knight and P St. J Russell, �??Endlessly single-mode photonic crystal fiber,�?? Opt. Lett., 22 p 961-963 (1997) [CrossRef] [PubMed]
- E.G. Neumann, Single-Mode Fibers �?? Fundamentals, (Springer-Verlag, 1988)
- A. Bjarklev, J. Broeng, A.S. Bjarklev, Photonic crystal fibers, (Kluwer, 2003) [CrossRef]
- Coude de Foresto, V. Ridgway S., Mariotti J.M, �??Deriving object visibilities from interferograms obtained with a fiber stellar interferometer,�??A&AS 121 379 (1997)
- G. Perrin, O. Lai, P. J. Lena, and V. Coude du Foresto, �??Fibered large interferometer on top of Mauna Kea: OHANA, the Optical Hawaiian Array for Nanoradian Astronomy,�?? in Interferometry in Optical Astronomy, P. J. Lena and A. Quirrenbach, eds., Proc. SPIE 4006, 708�??714 (2000). [CrossRef]
- M.D. Nielson, J.R. Folkenberg, N.A. Mortenson and A. Bjarklev, �??Bandwidth comparison of photonic crystal fibers and conventional single-mode fibers,�?? Opt. Express 12, 430 (2004) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-430">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-430</a> [CrossRef]
- M.D. Nielsen, N.A. Mortensen, M. Albersen, J.R. Folkenberg, A. Bjarklev, D. Bonacinni, �??Predicting macrobending loss for large-mode area photonic crystal fibers,�?? Opt. Express 12, 1775 (2004) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775</a> [CrossRef] [PubMed]
- A.W. Snyder and J.D. Love, Optical waveguide Theory, (Chapman/Kluwer, 1983/ 2000)
- Joseph W Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1996)
- T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, L.C. Botton, �??Multipole method for microstructured optical fiber. I Formulation,�?? J. Opt. Soc. Am. B 19 , No 10 (2002) [CrossRef]
- B.T. Kuhlmey, T.P. White, G. Renversez, D. Maystre, L.C. Botton, C. Martijn de Sterke, R.C. McPhedran, �??Multipole method for microstructured optical fiber. II Implementation and Results,�?? J. Opt. Soc. Am. B 19, No 10 (2002) [CrossRef]
- M.J Steel, T.P White, C. Martijn de Sterke, R.C. McPhedran, L.C. Botton, �??Symmetry and degeneracy in microstructered optical fibers,�?? Opt. Lett. 26 No.8 (2001). [CrossRef]
- D.T. Liu, B.M. Levine & M. Shao, �??Design and fabrication of a coherent array of single-mode optical fibers for the nulling coronograph,�?? Proc. SPIE 5170 Techniques and Instrumentation for detection of exoplanets. (2003)
- N.A. Mortensen and J.R. Folkenberg, �??Near-field to far-field transition of photonic crystal fibers: symmetries and interference phenomena,�?? Opt. Express 10, 475 (2002). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-475">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-475</a> [PubMed]
- Born and Wolf, Principles of optics 7th edition, (Cambridge University Press).
- A. Snyder, �??Excititation and scattering of modes on a dielectric optical fiber,�?? IEEE Trans. MTT V NIT-17 No.12 (1969)
- Equation (8) actually underestimates slightly �? < 0.40 for >f/4 and λ0 > 1000nm when coupling into the LMA-8 fiber however no simple model can account for the actual variation and direct recourse to Eq. (1) and (6) is required.
- Olivier Guyon, �??Wide-field interferometric imaging with single-mode fibers,�?? A&A 387, 366-378 (2002)

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