Abstract

The coupling efficiency of starlight into single and few-mode fibres fed with lenslet arrays to provide a continuous field of view is investigated. The single-mode field of view (FOV) and overall transmission is a highly complicated function of wavelength and fibre size leading to a continuous sample only in cases of poor throughput. Significant improvements are found in the few-mode regime with a continuous and efficient sample of the image plane shown to be possible with as few as 4 modes. This work is of direct relevance to the coupling of celestial light into photonic instrumentation and the removal of image scrambling and reduction of focal ratio degradation (FRD) using multi-mode fibre to single-mode fibre array converters.

©2009 Optical Society of America

1. Introduction

Inherently single-moded photonic devices are likely to play a significant role in the design of the next generation of astronomical instrumentation. An optical fibre feed into such an instrument is therefore, optimally, single-moded. However, Horton … Bland-Hawthorn [1] note that a few-mode fibre feed increases throughput and field of view but at the cost of spectral efficiency in an array waveguide grating spectrograph, for instance.

If very high spectral resolution must be maintained, one route forward might be to use a multi-mode fibre to single-mode fibre array to split power into single mode fibres with each fibre then sent to its own photonic spectrograph, on the assumption that they become cheap to mass produce - Fig. 1(a).

 figure: Fig. 1.

Fig. 1. (a). Single spatial element feeding multiple, mass produced, photonic spectrographs, (b) Single spatial element feeding a single catadioptric spectrograph via a single mode converter to remove image scrambling

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Alternatively, if the single-mode fibres in the array are sent directly to a catadioptric spectrograph - Fig. 1(b) - then image scrambling issues are removed since only the fundamental modes of the single-mode fibres illuminate the detector. Further, FRD is reduced because the output numerical aperture of each single mode fibre is far less than that of the input numerical aperture of the feeding multi-mode fibre. The spectra are then added in post processing yielding complete image scrambling at high throughput. The efficiency of such schemes, such as the comparison between image scrambling noise and increased detector noise in adding spectra from multiple fibres will be reported at a later date. For now, whether feeding a photonic instrument directly with a few-mode fibre or a few-mode fibre feeding a single-mode array, the minimisation of the number of modes in the feeding fibre is of significant interest.

Coupling of starlight directly into single-mode fibres has been covered by Shaklan … Roddier [2] and into a large mode area photonic crystal fibre directly by Corbett et al [3] and via a field lenslet by Corbett … Allington-Smith [4]. The coupling of celestial light directly into a few-mode fibre is covered by Horton … Bland-Hawthorn [5]. However, the filling factor, the ratio of the fibre core diameter to the jacket diameter, of a typical single or few-mode fibre is of the order of a few percent. Integral field spectroscopy, often (but not exclusively) requires a continuous sample of the sky. In this paper the coupling efficiency and subsequent sampling performance of a 2 dimensional array of fibres fed with a continuous lenslet array in the telescope image plane is investigated. The optical model and sampling are defined in Section 2. Single-mode fibre arrays are discussed in Section 3 and the highly chromatic results of this analysis lead naturally to the analysis of a few-mode array - Section 4.

Looking forward, in both the single-mode and few-mode regimes, it is found that critical and oversampling of the diffraction limited telescope point spread function (PSF) by the lenslet array are associated with poor throughput. That is, the multi-moded regime is found to be required to provide a continuous and efficient sample of the image plane even in the best performance case of no atmospheric .See Sections 3.2 and 4.4. However, when undersampling the diffraction limited PSF, as would be most likely when sampling an atmospherically (or otherwise) aberrated PSF, the total throughput is given by integrating the coupling efficiency of the diffraction limited case over the entire lenslet field of view. See Sections 3.3 and 4.5. The analysis is then greatly simplified, with attention only to the diffraction limited coupling required but with applicability of the most interesting results to the more general case.

2. The optical model

2.1 The optical geometry

The optical geometry in Fig. 2 is used throughout. As is typically the case, to avoid overfilling the numerical aperture of off-axis fibres, the telescope is assumed to feed its image plane telecentrically (not shown). In that case, the spherical image surface [6] of the telescope is flattened to coincide with the plane of the lenslet array. The part of the field imaged by the lenslet can then be taken as the telescope PSF, an Airy pattern, multiplied by the aperture function of the lenslet. dT is the telescope exit pupil diameter and FT the telescope focal ratio into its image plane. dL is the lenslet size in the image plane and FL is taken as shown.

 figure: Fig. 2.

Fig. 2. The telescope, lenslet array and fibre geometry

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dp is the diameter of the geometrical image of the telescope exit pupil placed on the fibre end-face by the field lenslet. df is the fibre core size. The fibre is assumed to be immersed by the lenslet. χ is the lenslet sample size. The numerical aperture (NA) of a single-mode fibre is usually quoted as the 1/e 2 intensity point of the fibre mode as it propagates into free space and so is irrelevant in this analysis. The NA of the few-mode case is discussed in Section 4.

2.2 The throughput model

The field on the end-face of the fibre, Ei, which resides in the (x,y) plane, is the Fourier transform of the telescope PSF and the aperture function of the lenslet [6],

Ei(x,y)=exp(ik02fL(x2+y2))g(x,y)E(x,y)

where fL = dLFL is the focal length of the lenslet and k0 the free space wave-number. Eq. (1) is the convolution of the geometrical image of the telescope exit pupil, E, with the point spread function of the hexagonal lenslet g(x,y), where

E(x,y)=P.exp(k0nL)

and

g(x,y)=12π2xy
×{3x3x+y[cos(23πnLλoFLy3x1+2)cos(23πnLλoFLy3y1+2)]
+3x3x+y[cos(23πnLλoFL2y1+2)cos(23πnLλoFLy3x1+2)]}

P is a disc of value unity inside dp and zero outside. nL is the refractive index of the lenslet and θ is the angle of tilt of the imaged wavefront at the fibre in the medium nL. The exponential term in Eq. (1) is the projection of the spherical image space onto the (x,y) plane containing the fibre end-face. Shown for completeness, in most cases of practical interest it is possible to choose dL such that fL is large enough for this term to be negligible. Corbett and Allington-Smith [4] show that the changes in coupling efficiency due to rotating the incident field Ei(x,y) with respect to the non-azimuthally symmetric mode field of a large mode area (LMA), single mode photonic crystal fibre (PCF) (one of the fibres considered below) are negligible in all cases of practical interest i.e. even if the secondary support structure is included. Polarisation in the direction of x, in Eq. (2), is then chosen arbitrarily for convenience.

The fibre coupling efficiency, ρF, of Ei into the fibre mode hf is given by,

ρF=(AdAEi×hf*.μz)Re[(AdAEi×Hf*.μz)(AdAef×hf*.μz)]

where A is the infinite (telescope image) plane (x,y) and μz is the z-directed unit vector as shown in Fig. 2.

The fraction of the total energy in the telescope PSF passed through the lenslet to Ei, ρL, is given by,

ρL=HexdAIPSFAdAIPSF

where IPSF is the energy distribution of the telescope PSF in the telescope image plane and Hex is the lenslet aperture within the same plane. The total throughput, from telescope entrance aperture to fibre output (assuming negligible attenuation within the fibre) is then the product of Eq. (4) and Eq. (5).

Ei, h and I were computed on a 10002 grid and h, for the LMA fibre computed using the multipole method [8].

2.3 The sampling model

From Fig. 2, the Smith-Hemholtz invariant tells us immediately that,

χ.dT=1FLdpnL

The sampling constant S, is defined using,

dL=S×1.22FTλo

where 1.22FTλo is the distance between the peak and first minimum in intensity of an Airy pattern - such as the diffraction limited (circular and unobscured) telescope PSF. λo is the free space wavelength. Hence, S = 1 Nyquist samples the telescope PSF out to the first minimum in intensity. Throughout, the lenslet array is considered hexagonal and dL is taken as across flats. S is defined with respect to the un-obscured case. Combining Eq. (6) and Eq. (7) then yields,

S=dpnL1.22FLλo

where nL remains constant at 1.45 throughout.

In the single mode case, only LMA PCF’s are considered, since dp remains constant at any input wavelength and so does the mode size of such fibres [4]. Therefore the coupling efficiency is only affected by the wavelength if there are significant diffractive features in the exit pupil image. i.e. when the width of g is a significant fraction of dp. We note further, that in the regime where the geometrical image dominates (i.e. the width of g << dp), maximised coupling efficiency into an LMA PCF occurs at dp = 1.33, where is the characteristic size of the LMA fibre [4]. The correct fibre size, , must be chosen for the desired waveband [8]. FL would then vary in order to retain the desired sampling constant S at the design wavelength, λo.

In the few-mode case only step index fibres are considered since few-mode LMA fibres are not commercially available.

3. Arrays of single-mode fibres

3.1 Single-mode coupling

Since Ei is the Fourier transform of the telescope PSF vignetted by the lenslet it is immediately apparent that sampling of a diffraction limited (or even near diffraction limited) telescope PSF by more than one lenslet implies some loss of spatial frequency information in the pupil image on each fibre end-face. The coupling into a single fibre mode is highly sensitive to the form of the input field and some key values of S and their effect on the exit pupil image on the fibre end-face are shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Section through the exit pupil image, Ei, at various sampling constants, S

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 figure: Fig. 4.

Fig. 4. Normalised coupling efficiency into an LMA-20 fibre fed with the telescope pupil image via a field lens.

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When nearly all of the telescope PSF passes through the lenslet it is centred on, the width of g << dp and the image of the telescope exit pupil is formed with minimal diffractive broadening but with some remaining low magnitude Gibbs phenomena, such as ringing. The coupling efficiency in this minimally diffracted regime is well described by the fit [4]

ρf=ρmax·exp[(nL∧θξλo)2]

where ρmax is the maximum, on axis (θ = 0), coupling value and ξ = 28.364. Experimental verification of Eq. (9) was performed using the geometry in Fig. 1 with nL = 1.0,∧= 13.2μm (Crystal fibre A/S, LMA-20) and the values of λo shown in Fig. 4. The telescope PSF was a tiny fraction of the size of the lens feeding the fibre, thereby yielding no effect from ρL, the vignetting of the telescope PSF by the lenslet aperture. Illumination of the telescope entrance aperture was provided by a collimated/expanded laser.

Figure 5 shows the computed coupling efficiency - Eq. (4) - as S varies. Eq. (9) is also shown for four relevant cases. As S → 2, significant fractions of the telescope PSF are passed to different fibres. The width of g starts to become a significant fraction of dp and diffractive broadening of the image starts to occur - Fig. 3. The diffractive features have a significant impact on both the maximum (on-axis) coupling efficiency (generally reducing it), and on the field of view. The panels in Fig. 5, show a cross section through Ei (Green) and fibre mode (Red) on the same axis at the various θ of the S = 2.0 curve. The cosine term in Eq. (2) and the periodically varying features in g are seen to interfere or beat with each other resulting in a variation (with θ) of the width of the diffracted image.

 figure: Fig. 5.

Fig. 5. ρf (θ)for various S - Abscissa in direction A

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 figure: Fig. 6.

Fig. 6. ρL for various values of S - Abscissa in direction A

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The underlying geometrical image remains constant in size, of course, but this beating causes significant changes in the coupling efficiency as θ varies. Eq. (9) no longer adequately describes the response and the coupling field of view is generally increased. However, the actual field of view is generally dominated by ρL as shown for the same five values of S in Fig. 6. The dotted vertical line in Fig. 5 shows the 1 NA point in the B direction highlighting the very small effect the extra distance to the NA in this direction has on the analysis. Unless otherwise stated, all subsequent data are shown in the A direction.

Coupling variation with telescope exit pupil obscuration by the secondary is only relevant when the image is dominated by dp and this case is covered in reference [4]. Note that the vignetted energy is not exactly 0.5 at the lenslet boundary because of the hexagonal form of the aperture. The size of both dp - Eq. (8) - and g are proportional to FLλo. Hence, any increase in scale of dp is accompanied by an identical increase in scale of g and the image of the telescope exit pupil is identically diffracted at the same S for any combination of ∧, λo and FL that satisfy Eq. (8). Therefore Eq. (8) is valid even when the image is badly diffracted and Figure 4 is completely general for any system that obeys it.

Two-dimensional datasets of ρL and ρf were computed for the hexagonal lenslet and multiplied together. The total throughput over all illuminated fibres over a two dimensional array is then plotted as a function of position of the telescope PSF centre in the telescope image plane, along direction A, in Fig. 7.

 figure: Fig. 7.

Fig. 7. Total throughput (over all illuminated fibres in the 2D array) in direction A for S = (a) 0.5 - Red, (b) 1.0 - Green, (c) 4.0 - Blue. The grey and black lines are commented on below.

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3.2 Critically and oversampled case

Since the total energy coupled into the array is plotted, where significant overlap between the throughput curves of individual fibres occurs the overall throughput is smoothed out and increased, such as in the S = 0.5 and S = 1.0 cases. For instance, the S = 0.5 case, shows that the response of the array is relatively continuous in the diffraction limit but only at the cost of a 8%–10% transmission. The S = 1.0 case couples into the array with only 10% overall transmission if the PSF is sat directly between two lenslets - as it ought to be, to be sampled correctly - but with 45% efficiency when centred on one lenslet. These conclusions are not significantly effected by either direction A or B in Fig. 5.

3.3 Undersampled case

Where the fibre traces minimally overlap, the total and per fibre throughput are the same. At the diffraction limit the undersampled case, such as S = 4.0, above, is well described by Eq. (9), yielding a field of view in each lenslet of λ/DT. This regime would be required in the case where the telescope PSF is atmospherically aberrated - and therefore somewhat larger than its diffraction limited counterpart. In this case the key quantity is the integral of Eq. (9) over the surface of the lenslet - i.e. how much of the energy incident on the entire lenslet aperture is transmitted to the fibre output. Figure 8 shows this throughput as a function of S, highlighting the in-efficiency of such a regime in the single-mode case.

 figure: Fig. 8.

Fig. 8. Total throughput (Hexagonal lenslet) as a function of S.

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However, if ultra-fine (λ/DT) discontinuous spatial samples of the image field are required, then, as can be seen from Fig. 7(c), the on axis coupling efficiency at S ≥ 4.0 is very high and so, consequently, is the integral of Eq. (9) over λ/DT, the field of view. Further, single-mode fibres do not transmit modal noise, do not suffer from image scrambling and LMA polarisation maintaining fibres are available with the same mode field properties as used throughout this analysis.

3.4 Reducing the size of dp in the diffracted regime.

If the fibre end-face is placed directly in the telescope image plane, it is known that a diffraction limited Airy pattern couples into a single-mode LMA fibre with maximum theoretical efficiency of ≈82% (minus Fresnel losses) if the fibre and Airy pattern scales are matched. Defining dp = α∧, is it possible therefore to reduce α such that the g dominated exit pupil image couples into the fibre more efficiently than in the geometrical (α= 1.33) case?

Optimal direct coupling of the telescope PSF into the fibre occurs when [3]

λopt0.7911FT

The ‘opt’ for optimal subscript reflects the fact that peak coupling can only occur at one wavelength. Setting FT = FL in Eq. (10) and λo = λopt in Eq. (8), combining and re-arranging yields,

SoptnLα
 figure: Fig. 9.

Fig. 9. (a) Peak coupling efficiency (ρfat θ= 0) as a function of S, for α= 0.334 (Sopt=0.5), 0.665 (Sopt = 1.0) and α= 1.33 (Geometrical optimum), (b) Coupling field of view of various S when α= 0.665 and α= 1.33.

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Figure 9(a) shows that the peak efficiency (θ= 0) agrees relatively well with the set value in both S = 0.5 and S = 1.0 cases (nL = 1.45). The coupling field of view is also increased as the S = 1.0 case in Fig. 9(b) shows, especially for S = 0.5 on the S = 1.0 optimal df value of 0.665∧ The grey continuous line on Fig. 7(b) is the S = 1.0, α= 0.665 total throughput curve and the dotted grey line in Fig. 7(a) the S = 0.5, α= 0.665 case. These two curves show that sampling at much higher throughput can be achieved when sampling the diffraction limited PSF but that the spatial variation in total throughput is not removed. The continuous black line in Fig. 7(a) is the S = 0.5 optimised trace showing that this PSF can be sampled with a near continuous spatial response from the lenslet array but with a peak throughput of only approximately 40%.

4. Arrays of few-mode fibres

The results of the previous section raise the question: how does the sampling alter as the number of modes supported within the fibre increases? Conversely, the minimisation of the number of modes is key for the new generation of photonic astronomical instrumentation. LMA PCF’s are not (currently) commercially available in few-mode form and we concern ourselves only with few-mode step index fibres.

4.1 Fibre numerical aperture

Despite the definition of single-mode numerical aperture given in Section 2, a step index single mode fibre must still, in some sense, be manufactured from a high index core surrounded by a low index cladding in order to trap and guide light. The geometrical numerical aperture may therefore be defined using

NA=ncore2nclad2

Strictly, this is a geometrical optics results which bears little relation to the 1/e 2 point of the (near) Gaussian fundamental mode as it propagates into free space. However, as the number of modes supported within the fibre increases, Eq. (12) becomes a more accurate estimate of how rays of light enter or leave the fibre. The few-mode regime sits between the Gaussian and geometrical results.

4.2 Fibre mode characteristics

Each modal field, i, supported within the step index fibre can be described by,

hi=(hx,yi+hzi)exp(iβiz)

where, z is directed along the fibre axis and the (x-y) plane, the transverse (cross-sectional) plane orthogonal to z. In weakly guiding fibres (ncorenclad) hz is negligible with respect to the transverse components hx and hy and the scalar fibre model applies. Thence, each valid mode hx,yi must satisfy,

Ψhx,yi=βi2hx,yi

where Ψ is an operator [9]. Parameters U and V are defined

U=df2(ko2ncore2β2)
V=ko(df2)NA

W 2 = V 2-U 2 and continuity of the mode field across the core-cladding boundary means that all valid modes must satisfy

UJl+1(U)Jl(U)=WKl+1(W)Jl(W)

where J is the Bessel function of the first kind and K the modified Bessel function of the second kind. Once V is set, at each l, m valid solutions UmV of Eq. (17) are found and at each l,m four mode fields are found to exist such that,

HEl+1,m(Even)hx,y=fl(Um)[sin()x̂+cos()ŷ]
HEl+1,m(Odd)hx,y=fl(Um)[cos()x̂sin()ŷ]
HEl-1,m(Even)hx,y=fl(Um)[sin()x̂cos()ŷ]
HEl-1,m(Odd)hx,y=fl(Um)[cos()x̂+sin()ŷ]

where fl is a constant and ϕ = Arg(x,y). When l = 1, even the fully vectorial model EH even and odd modes have a zero hz and are exactly transverse. For this reason they are known as TM and TE respectively. Where the few mode fibre is intended to feed a mode splitting waveguide for instance, the modes within the feeding few-mode fibre are transferred into the single-mode fibre array in order of ascending U [10,11] - Fig. 10(a) - and corrections to the scalar model to yield the fully vectorial value of U are required to correctly determine this order. Two examples of corrected groups TE01, TM01 … HE21(Odd + Even) and HE31(Odd+Even), EH11(Odd + Even) … HE21(Odd+Even) are shown in Fig. 10(b) for a strongly guiding fibre in order to amplify the differences.

 figure: Fig. 10.

Fig. 10. Plot of U against V for (a) a weakly guiding fibre (ncorenclad) - scalar model, (b) a strongly guiding fibre (ncore >> nclad) - corrected model

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Notice that near to cutoff (the line U=V) some modes cross over. For instance below V ≈ 3.75 in Fig. 10(b), TM01 has a lower U value than HE21. Therefore, the TE01, TM01 and HE11 (Odd + Even) would be transferred into a 3-mode single-mode array, whereas for V > 3.75, the TE01, HE21(Odd + Even) and HE11 (Odd + Even) would be transferred. However, it is found that the coupling into the TM0,m and HE2,m (Even) modes are identical and cross over’s between them do not affect the overall coupling efficiency into the fibre. The small δV where, for instance, the HE12 and HE31 modes cross over are commented upon below.

4.3 Few-mode sampling performance

The sampling model for a step index fibre can be derived as per Eq. (8) in the LMA single-mode fibre case. Since we are dealing with so few modes we still require optimal coupling of the undiffracted image, E, into the fundamental mode in order to concentrate the power carried within the fibre into the lowest few modes. This occurs when [9],

dp=1.6df2lnV

and

S=V2lnV1.6nL1.22πNAFL

Geometrical optics is not valid in the single- and few-mode regimes and the primary role of FL is to provide the correct image size, dp, with respect to the fundamental mode of the fibre as prescribed by Eq. (19) to maximise coupling efficiency. FL can then exceed the geometrical NA of the fibre in order to provide the correct image with respect to the fundamental mode of the fibre. An example being a 0.22 NA fibre operated at V = 3.6 (4 modes), fed with an S = 4.0 image. Eq. (20) predicts 1/2FL = 0.323. However, the coupling efficiency curves are found in such cases to fall to zero long before any input angle can actually couple into the fibre outside the geometrical fibre NA - e.g. see the 4 mode curve on Fig. 11(e). A full analysis of the numbers of modes required to sample without the requirement of Eq. (19) in the multi-moded case is given in reference 11.

Figure 11 shows the coupling efficiency, ρf(θ) as the number of modes increases for various values of S It might be expected that each curve should be associated with a specific V value because higher V implies more modes. However, it is possible to restrict attention to illuminated modes even though higher order (un-illuminated modes) might be present, or, more pertinently, to only those modes present in the few-mode fibre that are transferred into the single-mode array. Looking at Figs 11(b) and 11(f) we see that the total coupling efficiency for 1 mode, 3 modes and 4 modes for the V = 3.6 and V = 10 cases are excellent approximations of each other. The V = 2.41 curves in Fig. 11(f) are at the cutoff (U=V) of the upper 3 modes of the 4 mode case and have a significantly reduced field of view within the feeding lenslet. Inspection of Fig. 10(a) reveals that away from cutoff each U value asymptotes to some value as V increases. The coupling efficiency curves, Fig.11 (a)(e) are therefore presented only for the V = 10 case but with applicability to approximately any V assuming that the modes in question are not too close to cutoff. This is accurate enough for us to draw some general conclusions, however, where appropriate below, the resulting performance at cutoff is also shown for information.

The 2 mode case is not considered because this would require the HE11 fundamental + either the TE01 or the TM01 mode. Since TE01 and TM01 are orthogonally polarised with respect to each other, this is tantamount to partially selecting out one polarisation state from the image.

As the number of modes increases so does the field of view of the fibre: obvious in the partially diffracted S= 4.0 case, this is far less conspicuous in the badly diffracted examples in Fig. 11(a) and Fig. 11 (b). Here, the increased numbers of modes have a minimal effect on the overall coupling efficiency except to increase it across all input in angles at each HE1,m. Co-incidentally, the scale of the abscissa of Fig. 11(e) is also a good approximation to the NA of the fibre. At V = 10, 22 modes are supported and the NA of the fibre is reached with a throughput of ≈0.55. However, even at V = 20 the coupling efficiency will only reach ≈0.60 [9] at the fibre NA. The intermediate cases S = 1.5 and S= 2.0 are more promising with a ≈0.75 coupling efficiency at the lenslet boundary with 7 modes in the S = 2.0 case and the same efficiency with 5 modes in the S = 1.5 case. However, some diffraction of the image is required for this increase in coupling at higher angles to work. Indeed, from Fig. 11(d), the deliberate diffraction model of Section 3 applies here too, with the optimal coupling at S= 2.0 - Fig. 9(a).

In Fig. 11(e), modes 6 and 7 can be swapped over in the range V ≈ 4.00 to 4.35 where the HE12 mode cuts below the HE31 curve in Fig. 9(b), but this appears to have little effect on the overall result. It is noted, therefore, that the coupling efficiency, ρf, is generally rather insensitive to these effects.

What happens to the field of view of the fibre if we choose a large V and then only select out a few lower order modes for some fixed S? By example, a V = 10 fibre feeding a single-mode array that only supported the lowest 5 modes would have the coupling response of the 5 mode curve in Fig. 11(e). i.e. Even though 22 modes are supported within the feeding multi-mode fibre, selecting out higher order modes in the adiabatic transition into the single-mode array would decrease the field of view as expected of the geometrical ray optical model [9,11].

 figure: Fig. 11.

Fig. 11. Fibre coupling efficiency, ρf, with input angle at the fibre for S = (a) 0.5, (b) 1.0, (c) 1.5, (d) 2.0 and (e) 4.0 (f) 4.0 comparison at various V - See text.

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4.4 Critically sampled case

In the S = 1.0 case, already more efficient than the SM LMA case at the lenslet boundary due to the increased coupling efficiency here, even 17 modes is only 10% or so more efficient at the lenslet boundary than the single-mode regime - Fig. 12. Undoubtedly, the 17 mode case would be more robust with regards to the coupling of an exit pupil image poorly aligned with the fibre core, for instance, but the best that can be achieved, as shown in this Fig. is a 30% total throughput when the telescope PSF is sampled at the Nyquist limit in the diffraction limited regime.

 figure: Fig. 12.

Fig. 12. Total throughput (over all illuminated fibres) for S = 1.0

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4.5 Undersampled case

The coupling efficiency curves - Fig. 11 - integrated over the surface of the hexagonal lenslet are the key values in the undersampled case. The S = 1.5 and S = 2.0 cases in Fig. 13, show that efficient and continuous sampling of the image plane is possible with a few mode fibre - As few as 4 modes will yield a total throughput of 80% or so. The requirement that the telescope exit pupil image is deliberately diffracted on the fibre end-face in order to achieve such a result is noted and the partially diffracted S = 4.0 case shows that around 15–20 modes or so are required to achieve the same throughput.

 figure: Fig. 13.

Fig. 13. Total throughput as a function of the # of modes in the undersampled case

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The curves at the modal cutoff’s highlight, as already concluded, that operating the fibre with the lower order modes too near to their cutoff should be avoided where possible in order to maximise the total throughput. These effects reduce as the number of modes increases however.

5. Discussion

The results of Sections 3.2, 3.3, 4.4 and 4.5 indicate that the few-mode/undersampled regime is the only case of interest with the result that at least 5 modes (operating away from cutoff coupling issues) are required for >80% coupling between S = 1.5 and S = 2.0 - Fig. 13. This is the central result of this paper and we explore its practical implications in this section.

5.1 Fibre core size

With reference to Fig. 10 and avoiding the cutoff region, at least 5 modes supported within the fibre limits V ≥ 5. Via Eq. (16), the limiting case of V = 5 yields fibre core diameters of 4μm, 4.5μm and 6μm at λ = 550nm, λ =1.24μm (J-band centre) and λ =1.64μm (H-band centre), respectively. This is challenging for lenslet alignment. Importantly however, there is no restriction that an n-mode fibre break out into an n-mode single-mode fibre array, so it is possible to have a highly multi-moded fibre feeding only a 5 mode single mode array, for instance. The lowest five modes of the feeding fibre are selected out by the array. Thence, the size of the feeding fibre core is restricted only to providing the minimum number of modes for some given coupling efficiency. The single-mode array example in reference [10] supported only the lowest 3% of the modes in the feeding few-mode fibre indicating that the core was over-sized by 97% providing huge scope for larger core sizes.

5.2 Lenslet requirements

Taking S = 1.5 and dL = 50μm, Eq. (7) yields FT = 42, 22 and 17 at λ = 550nm, λ =1.24μm (J-band centre) and λ =1.64μm (H-band centre), respectively. Telescope final focal ratios vary, approximately, between f/2 (ELT) and >f/50 (commonly f/8 to f/16 for 4m class and very large telescopes (VLT’s)). Relatively high magnifications would, therefore, be required for a fast ELT in the visible, but more modest values (≈5 or less) are required for smaller telescopes and/or longer wavelengths. Larger lenslets, dL, require higher magnifications, of course. Again, assuming this limiting case of V = 5 in the few-mode fibre, but taking S = 2.0 and the largest geometrical NA of the few-mode fibre as 0.22 then by Eq. (20) FL ≥ 3.8. Higher V would require a slower lenslet. Commercially available lenslets of dL ≥ 50μm and speeds of ≥f/3.5 are common.

5.3 Bandwidth

The curves in Fig. 11 are a good approximation to all V for all modes away from cutoff and therefore so are the integrated curves (away from cutoff) in Fig. 13. That is, there is no need to recompute coupling efficiency curves for say S = 2.0 once FL is optimised for S =1.5. The bandwidth associated with the 80% coupling efficiency between S = 1.5 and S = 2.0 is then computed directly from the ratio 2.0/1.5. For example, if S = 1.5 at 1.15μm then S = 2.0 yields 1.53μm and the entire J band is coupled with 80% efficiency.

5.4 Throughput

Whether or not the 80% intrinsic throughput is an issue depends on the application. It is not uncommon for significantly higher losses to be associated with the many surfaces of a cata-dioptric spectrograph and/or the losses associated with feeding a slow spectrograph with a fast fibre output beam. The losses associated with a photonic spectrograph are yet to be determined and, of course, the loss in throughput might be a small price to pay for otherwise unavailable functionality such as the complete removal of image scrambling. Current image scramblers are very lossy [12].

5.5 Sampling

The utility of the optimal sample sizes of S = 1.5 to S = 2.0 depends greatly on the application, presence of an atmosphere and degree of adaptive optical correction. However, they are well matched to Strehl ratios of > 0.7 typically found observing <12 magnitude objects in the J,H,K bands using a well corrected telescope in good seeing (>10cm at 500nm), for instance. [13]

6. Conclusions

The sampling of the telescope image plane by a continuous lenslet array feeding single- and few-mode fibres has been investigated. A continuous and efficient sample of the image plane is not possible in the single-mode regime even when the exit pupil image is deliberately diffracted. In the few-mode case, the critically (and oversampled) cases do not benefit from a modest increase in the number of modes. In the undersampled case, applicable to any atmosphere, it is found that by deliberately diffracting the telescope exit pupil image, an efficient (80%) and continuous sample of the image plane is possible with as few as 4 modes but with 5 modes over the range S=1.5 to S=2.0 avoiding modal cutoff issues. The practicality of these limits is explored and is noted as physically realisable with current technology and of use to astronomy, possibly providing new functionality associated with few-mode to single-mode array converters.

A practical investigation into the coupling of a deliberately diffracted telescope exit pupil image is underway.

Acknowledgments

I gratefully acknowledge the advice of Dr. Jeremy Allington-Smith and the funding of the STFC (formerly PPARC) who supported this research.

References and links

1. J. Bland-Hawthorn and A. Horton, “Instruments without optics,” SPIE 6269, 62690N (2006). [CrossRef]  

2. S. Shaklan and F. Roddier, “Coupling starlight into single-mode fibres,” Appl. Opt. , 27, 2334 (1987). [CrossRef]  

3. J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, “The coupling performance of photonic crystal fibres in fibre stellar interferometry,” Mon. Not. R. Astron. Soc. 368, 203–210 (2006).

4. J. Corbett and J. Allington-Smith, “Coupling starlight into single-mode photonic crystal fiber using a field lens,” Opt. Express 13, 6527–6540 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6527. [CrossRef]   [PubMed]  

5. A. J. Horton and J. Bland-Hawthorn, “Coupling light into few-mode optical fibres I: The diffraction limit,” Opt. Express 15, 1443–1453 (2007). http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-4-1443 [CrossRef]   [PubMed]  

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

7. 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]  

8. 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 fibres,” Opt. Express 12, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775. [CrossRef]   [PubMed]  

9. A. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer, 1983).

10. S. G. Leon-Saval, T. Birks, J. Bland-Hawthorn, and M. Englund, “Multimoded fiber devices with singlemode performance,” Opt. Lett. 30, 2545 (2005). [CrossRef]   [PubMed]  

11. J. Corbett, T. Butterley, and J. R. Allington-Smith, “Fibre modal power distributions and their application to OH-suppression fibres,” Mon. Not. R. Astron. Soc. 378, 482–492 (2007). [CrossRef]  

12. T. Hunter and L. Ramsey “Scrambling properties of optical fibres and the performance of a double scrambler,” Proc. Astro. Soc. Pacific 104, 1244–1251 (1992). [CrossRef]  

13. F. Roddier, Adaptive optics in astronomy (Cambridge, 1999). [CrossRef]  

References

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  • |

  1. J. Bland-Hawthorn and A. Horton, “Instruments without optics,” SPIE 6269, 62690N (2006).
    [Crossref]
  2. S. Shaklan and F. Roddier, “Coupling starlight into single-mode fibres,” Appl. Opt.,  27, 2334 (1987).
    [Crossref]
  3. J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, “The coupling performance of photonic crystal fibres in fibre stellar interferometry,” Mon. Not. R. Astron. Soc. 368, 203–210 (2006).
  4. J. Corbett and J. Allington-Smith, “Coupling starlight into single-mode photonic crystal fiber using a field lens,” Opt. Express 13, 6527–6540 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6527.
    [Crossref] [PubMed]
  5. A. J. Horton and J. Bland-Hawthorn, “Coupling light into few-mode optical fibres I: The diffraction limit,” Opt. Express 15, 1443–1453 (2007). http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-4-1443
    [Crossref] [PubMed]
  6. J.W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  7. 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]
  8. 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 fibres,” Opt. Express 12, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775.
    [Crossref] [PubMed]
  9. A. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer, 1983).
  10. S. G. Leon-Saval, T. Birks, J. Bland-Hawthorn, and M. Englund, “Multimoded fiber devices with singlemode performance,” Opt. Lett. 30, 2545 (2005).
    [Crossref] [PubMed]
  11. J. Corbett, T. Butterley, and J. R. Allington-Smith, “Fibre modal power distributions and their application to OH-suppression fibres,” Mon. Not. R. Astron. Soc. 378, 482–492 (2007).
    [Crossref]
  12. T. Hunter and L. Ramsey “Scrambling properties of optical fibres and the performance of a double scrambler,” Proc. Astro. Soc. Pacific 104, 1244–1251 (1992).
    [Crossref]
  13. F. Roddier, Adaptive optics in astronomy (Cambridge, 1999).
    [Crossref]

2007 (2)

A. J. Horton and J. Bland-Hawthorn, “Coupling light into few-mode optical fibres I: The diffraction limit,” Opt. Express 15, 1443–1453 (2007). http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-4-1443
[Crossref] [PubMed]

J. Corbett, T. Butterley, and J. R. Allington-Smith, “Fibre modal power distributions and their application to OH-suppression fibres,” Mon. Not. R. Astron. Soc. 378, 482–492 (2007).
[Crossref]

2006 (2)

J. Bland-Hawthorn and A. Horton, “Instruments without optics,” SPIE 6269, 62690N (2006).
[Crossref]

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, “The coupling performance of photonic crystal fibres in fibre stellar interferometry,” Mon. Not. R. Astron. Soc. 368, 203–210 (2006).

2005 (2)

2004 (1)

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 fibres,” Opt. Express 12, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775.
[Crossref] [PubMed]

2002 (1)

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]

1992 (1)

T. Hunter and L. Ramsey “Scrambling properties of optical fibres and the performance of a double scrambler,” Proc. Astro. Soc. Pacific 104, 1244–1251 (1992).
[Crossref]

1987 (1)

Albersen, M.

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 fibres,” Opt. Express 12, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775.
[Crossref] [PubMed]

Allington-Smith, J.

Allington-Smith, J. R.

J. Corbett, T. Butterley, and J. R. Allington-Smith, “Fibre modal power distributions and their application to OH-suppression fibres,” Mon. Not. R. Astron. Soc. 378, 482–492 (2007).
[Crossref]

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, “The coupling performance of photonic crystal fibres in fibre stellar interferometry,” Mon. Not. R. Astron. Soc. 368, 203–210 (2006).

Birks, T.

Bjarklev, A.

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 fibres,” Opt. Express 12, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775.
[Crossref] [PubMed]

Bland-Hawthorn, J.

Bonacinni, D.

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 fibres,” Opt. Express 12, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775.
[Crossref] [PubMed]

Botton, L. C.

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]

Butterley, T.

J. Corbett, T. Butterley, and J. R. Allington-Smith, “Fibre modal power distributions and their application to OH-suppression fibres,” Mon. Not. R. Astron. Soc. 378, 482–492 (2007).
[Crossref]

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, “The coupling performance of photonic crystal fibres in fibre stellar interferometry,” Mon. Not. R. Astron. Soc. 368, 203–210 (2006).

Corbett, J.

J. Corbett, T. Butterley, and J. R. Allington-Smith, “Fibre modal power distributions and their application to OH-suppression fibres,” Mon. Not. R. Astron. Soc. 378, 482–492 (2007).
[Crossref]

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, “The coupling performance of photonic crystal fibres in fibre stellar interferometry,” Mon. Not. R. Astron. Soc. 368, 203–210 (2006).

J. Corbett and J. Allington-Smith, “Coupling starlight into single-mode photonic crystal fiber using a field lens,” Opt. Express 13, 6527–6540 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-17-6527.
[Crossref] [PubMed]

Dabirian, A.

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, “The coupling performance of photonic crystal fibres in fibre stellar interferometry,” Mon. Not. R. Astron. Soc. 368, 203–210 (2006).

Englund, M.

Folkenberg, J. R.

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 fibres,” Opt. Express 12, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775.
[Crossref] [PubMed]

Goodman, J.W.

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

Horton, A.

J. Bland-Hawthorn and A. Horton, “Instruments without optics,” SPIE 6269, 62690N (2006).
[Crossref]

Horton, A. J.

Hunter, T.

T. Hunter and L. Ramsey “Scrambling properties of optical fibres and the performance of a double scrambler,” Proc. Astro. Soc. Pacific 104, 1244–1251 (1992).
[Crossref]

Kuhlmey, B. T.

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]

Leon-Saval, S. G.

Love, J. D.

A. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer, 1983).

Maystre, D.

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]

McPhedran, R. C.

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]

Mortensen, N. A.

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, “The coupling performance of photonic crystal fibres in fibre stellar interferometry,” Mon. Not. R. Astron. Soc. 368, 203–210 (2006).

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 fibres,” Opt. Express 12, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775.
[Crossref] [PubMed]

Nielsen, M. D.

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 fibres,” Opt. Express 12, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1775.
[Crossref] [PubMed]

Ramsey, L.

T. Hunter and L. Ramsey “Scrambling properties of optical fibres and the performance of a double scrambler,” Proc. Astro. Soc. Pacific 104, 1244–1251 (1992).
[Crossref]

Renversez, G.

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]

Roddier, F.

Shaklan, S.

Snyder, A.

A. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer, 1983).

Sterke, C. Martijn de

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]

White, T. P.

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]

Appl. Opt. (1)

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

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]

Mon. Not. R. Astron. Soc. (2)

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, “The coupling performance of photonic crystal fibres in fibre stellar interferometry,” Mon. Not. R. Astron. Soc. 368, 203–210 (2006).

J. Corbett, T. Butterley, and J. R. Allington-Smith, “Fibre modal power distributions and their application to OH-suppression fibres,” Mon. Not. R. Astron. Soc. 378, 482–492 (2007).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Proc. Astro. Soc. Pacific (1)

T. Hunter and L. Ramsey “Scrambling properties of optical fibres and the performance of a double scrambler,” Proc. Astro. Soc. Pacific 104, 1244–1251 (1992).
[Crossref]

SPIE (1)

J. Bland-Hawthorn and A. Horton, “Instruments without optics,” SPIE 6269, 62690N (2006).
[Crossref]

Other (3)

A. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer, 1983).

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

F. Roddier, Adaptive optics in astronomy (Cambridge, 1999).
[Crossref]

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Figures (13)

Fig. 1.
Fig. 1. (a). Single spatial element feeding multiple, mass produced, photonic spectrographs, (b) Single spatial element feeding a single catadioptric spectrograph via a single mode converter to remove image scrambling
Fig. 2.
Fig. 2. The telescope, lenslet array and fibre geometry
Fig. 3.
Fig. 3. Section through the exit pupil image, E i , at various sampling constants, S
Fig. 4.
Fig. 4. Normalised coupling efficiency into an LMA-20 fibre fed with the telescope pupil image via a field lens.
Fig. 5.
Fig. 5. ρf (θ)for various S - Abscissa in direction A
Fig. 6.
Fig. 6. ρL for various values of S - Abscissa in direction A
Fig. 7.
Fig. 7. Total throughput (over all illuminated fibres in the 2D array) in direction A for S = (a) 0.5 - Red, (b) 1.0 - Green, (c) 4.0 - Blue. The grey and black lines are commented on below.
Fig. 8.
Fig. 8. Total throughput (Hexagonal lenslet) as a function of S.
Fig. 9.
Fig. 9. (a) Peak coupling efficiency (ρf at θ= 0) as a function of S, for α= 0.334 (Sopt =0.5), 0.665 (Sopt = 1.0) and α= 1.33 (Geometrical optimum), (b) Coupling field of view of various S when α= 0.665 and α= 1.33.
Fig. 10.
Fig. 10. Plot of U against V for (a) a weakly guiding fibre (ncore nclad ) - scalar model, (b) a strongly guiding fibre (ncore >> nclad ) - corrected model
Fig. 11.
Fig. 11. Fibre coupling efficiency, ρf , with input angle at the fibre for S = (a) 0.5, (b) 1.0, (c) 1.5, (d) 2.0 and (e) 4.0 (f) 4.0 comparison at various V - See text.
Fig. 12.
Fig. 12. Total throughput (over all illuminated fibres) for S = 1.0
Fig. 13.
Fig. 13. Total throughput as a function of the # of modes in the undersampled case

Equations (25)

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E i ( x , y ) = exp ( i k 0 2 f L ( x 2 + y 2 ) ) g ( x , y ) E ( x , y )
E ( x , y ) = P . exp ( k 0 n L )
g ( x , y ) = 1 2 π 2 xy
× { 3 x 3 x + y [ cos ( 2 3 π n L λ o F L y 3 x 1 + 2 ) cos ( 2 3 π n L λ o F L y 3 y 1 + 2 ) ]
+ 3 x 3 x + y [ cos ( 2 3 π n L λ o F L 2 y 1 + 2 ) cos ( 2 3 π n L λ o F L y 3 x 1 + 2 ) ] }
ρ F = ( A dA E i × h f * . μ z ) Re [ ( A dA E i × H f * . μ z ) ( A dA e f × h f * . μ z ) ]
ρ L = Hex dA I PSF A dA I PSF
χ . d T = 1 F L d p n L
d L = S × 1.22 F T λ o
S = d p n L 1.22 F L λ o
ρ f = ρ max · exp [ ( n L ∧θ ξ λ o ) 2 ]
λ opt 0.7911 F T
S opt n L α
NA = n core 2 n clad 2
h i = ( h x , y i + h z i ) exp ( i β i z )
Ψ h x , y i = β i 2 h x , y i
U = d f 2 ( k o 2 n core 2 β 2 )
V = k o ( d f 2 ) NA
U J l + 1 ( U ) J l ( U ) = W K l + 1 ( W ) J l ( W )
H E l + 1 , m ( Even ) h x , y = f l ( U m ) [ sin ( ) x ̂ + cos ( ) y ̂ ]
H E l + 1 , m ( Odd ) h x , y = f l ( U m ) [ cos ( ) x ̂ sin ( ) y ̂ ]
H E l - 1 , m ( Even ) h x , y = f l ( U m ) [ sin ( ) x ̂ cos ( ) y ̂ ]
H E l - 1 , m ( Odd ) h x , y = f l ( U m ) [ cos ( ) x ̂ + sin ( ) y ̂ ]
d p = 1.6 d f 2 ln V
S = V 2 lnV 1.6 n L 1.22 πNA F L

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