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Measurement of Focal Ratio Degradation in Optical Fibres used in Astronomy

퐴.푁.푅푎푚푎푝푟푎푘푎푠ℎ, 푀푢푑푖푡 푆ℎ푟푖푣푎푠푡푎푣푎, 푆푖푑푑푎푟푡ℎ푎 푆 푉푒푟푚푎

퐼푈퐶퐴퐴,푃푢푛푒,퐺푎푛푒푠ℎ푘ℎ푖푛푑,푃표푠푡 퐵푎푔 4,푃푢푛푒 푈푛푖푣푒푟푠푖푡푦 퐶푎푚푝푢푠, 411007, 퐼푛푑푖푎

퐷푒푝푡. 표푓 퐼푛푠푡푟푢푚푒푛푡푎푡푖표푛,퐶.푈. 푆.퐴.푇. , 682022, 퐼푛푑푖푎

ABSTRACT

A simple approach was adapted to measure the focal ratio degradation (FRD), and throughput of Silica-Silica FBP broad spectrum optical fiber. FRD being one of the predominant sources of losses in the fibers used in Astronomical Instrumentation has to quantified and minimized. The fibers tested will be used in the development of Integral Field Unit in the IUCAA Girawali Observatory. Several beam speeds were used for obtaining the results, and throughput plotted, using a white LED source. The effect of several sources of FRD, like edge and bends were taken into account and observed. The Numerical aperture (NA) of the fiber was 0.220.02, and the result was obtained with very precise conformation of actual parameters of fiber. The throughput of 85-90% was obtained at input f-numbers within a range of f-numbers from 3-5.

Keywords: optical fibers, astronomical instrumentation, focal ratio degradation

1. INTRODUCTION

Focal ratio is defined as the ratio of focal length and effective aperture diameter. It is a dimensionless quantity, and quantitative measure of lens speed. It can be denoted as f/#. Also, higher the f-number, lower will be the lens speed and vice versa.

N = f/D. (1)

A simple picture to illustrate it well is drawn below,

Figure 1 – Sketch to illustrate f-number. The one on the upper left hand side would help to understand the relation between f-number and lens speed.

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Numerical Aperture (NA) = nsin (), (2)

Also, tan () = D/2f

N = 1/ (2tan ()). (3)

From the figure on the lower left hand side, we can say that,

(Input f-number) f1 > f2 (Output f-number),

It clearly leads us to a conclusion that the emergent beam is faster. Considering the image on the lower left hand side, we can express NA of the fiber as,

NA = 푛 − 푛 . (4)

Now we can realize that, FRD manifests itself in the form of gradual decline in the value of input f/ #, as information travels from input end of the fiber to the output end.

In optical fibers, which are used for astronomical purposes, we generally come across loss of signal information, in the form of Focal Ratio Degradation (FRD). FRD is the phenomenon in which the f-number of the input beam, which is fed to the optical fiber, decreases. A given f-number gives rise to a particular modal distribution. FRD is a loss, and we generally come across two types of losses in optical fibers, i.e. mode dependent and mode independent. FRD is a mode dependent loss, which constitutes waveguide scattering by variation of core diameter and mechanical deformation. We can perceive the mechanical deformations as the change in geometry of any fiber, away from a regular cylinder. It can manifest itself in fibers in form of micro-bends (small deformations when compared to cylinder core) and macro bends (large scale bends) (Ramsey, 1988). The effect of micro-bends on FRD is still an active field of research. On the other hand, waveguide scattering leads to the transfer of energy into the modes which have higher losses, over the length of the fiber. Due to the above two mechanisms, the energy is transferred from one mode to other, which is conspicuous in optical fibers in the form of FRD. Apart from the above mentioned mechanisms of losses, there are other parameters like poor end preparation (edge effects), non-telecentricity and the refractive index, which need to be considered in order to minimize the losses incurred due to FRD.

Instruments comprising fiber optics has now become a much promising means for measurement and research work. With the help of fiber optics in stellar spectrographs, a circular bundle of fiber can be oriented to face the star, and the other end can be used as the slit of the spectrograph. Due to the above technique, the percentage of light coming on to the instrument manifolds, and helps in better measurement. This also helps in the reduction of seeing effects and tracking error. The fibers are used predominantly now a days because of several advantages they provide us, like, small space occupation, flexibility and remote operation, and image scrambling in some cases. (Clayton, 1989)

Fiber optics can be used in applications such as sensitive telescopic instruments, like astronomical spectrographs and Integral Field Units (IFU). In our case, the work will lead to the development of an IFU for IUCAA’s Girawali Observatory (IGO) telescope. Also, the quantification of FRD is essential in making any kind of ground and space based instruments, unless it is completely fiber optics free! Quantification also allows us to decide the range of input and output f-numbers which would give us the maximum throughput. Unless and until we are sure about this range of input f-number, we can’t optimize the performance of our

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astronomical instruments. In general, this range of input f-number can be verified to be 3 – 5, and throughput in the range of 85 - 95 %.

In this paper, we present a simple method to investigate FRD, and hence quantify it. In this section, we studied about the basics behind FRD. In section 2, we will go into much detail about the investigation theory, general overview and our approach. In section 3, we present our experiment, its general organization, data reduction software, procedure adopted and precautions to be taken. In section 4, we would deal with the results and observations obtained from the experiment performed.

2. QUANTIFYING FRD AND OUR APPROACH

2.1 Investigation theory

In the previous section, we dealt with the basics needed in order to proceed further with the measurement of FRD. The theory of FRD was proposed by Gloge in 1972. Although the accurate theory of FRD (in case of randomly deformed multimode waveguide) has not been detected, the theory put forward by Gloge is used as one of the reliable theory till date. The propagation of light through fibres requires the solution of Maxwell’s equation for cylindrical symmetry, and considering the boundary conditions at the core cladding surface. Maxwell equations solution is not perfectly correct in case of fibres with imperfections. For that purpose, the coupled wave theory using an infinite set of differential equations is apt for describing any kind of waveguide imperfection (Marcuse, 1974). Gloge (1972) described the optical power in the fibre with random imperfections in the waveguide, using a single differential equation (Carrasco and Parry, 1994).

The partial equation which describes the far field distribution of optical power in a fibre of length L can be represented as,

푃/퐿 = −퐴 푃 + (퐷/) { ⁄ ( 푃 ⁄ )}, (5)

where, P = distribution of optical power, = axial angle of incidence, A = absorption coefficient and D = characterization parameter. Here D is given as:

D = 2⁄ 푑 푛 푑 , (6)

where, = wavelength of light, 푑 = fibre core diameter, and n = refractive index of the core.

The solution to the above equation (Gambling, Payne & Matsumura, 1975) gives us the following result-

P () = {1 푏퐿⁄ } exp (−(휃 + 휃 ) 4퐷퐿⁄ ) 퐼 (θ휃 / 2DL). (7)

Here,

b = 4(AD) . (8)

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And 휃 = angle of incidence of collimated input beam.

The equation (8) converts into two asymptotic form of equation depending upon two conditions, such as: (a) for << 2DL/휃 ,

P () exp (−(휃 + 휃 ) 4퐷퐿⁄ ), (9)

thus, the output will be almost a Gaussian profile with centre on optical axis.

(b) for >> 2DL/휃 ,

P () exp (−(휃 − 휃 ) 4퐷퐿⁄ ), (10)

Thus, the output will be shifted from the optical axis. Also, the Gaussian profile can be estimated by calculating the width of the output beam profile as,

= (2DL) . (11)

Thus we can say that, the parameter D can be said to be measuring means of the Gaussian profile of the output profile beam. For input beam of any aperture, we can represent the function F (휃) describing the angular flux distribution in the far field or the output profile of the beam as, (Carrasco & Parry, 1989)

퐹 (휃) = ∫ ∫ 퐺(휃 ,휑 )푃(휃, 휃′) sin(휃′)푑 휃 푑휑 . (12)

Where, sin(휃′)푑휃 = differential of solid angle, and G = input beam profile function.

We can also say that, we can take any input beam profile, i.e. G = G(,), and integrate it to compute the function F. After computing the function F, which describes the output profile of the beam, we can find the total energy of the beam which comes under a cone of half angle 휃. In order to calculate it; one has have to integrate the function F over the whole region coming within the half angle 휃. The equation can be written as –

N () = 2 ∫ 퐹(휃′)푑휃′ . (13)

For quantifying the FRD, the method we usually adopt is to plot this total energy N versus the output f-ratio. We can easily imagine the situation in which the value of N decreases as the output f-ratio increases.

2.2 General overview and approach

In the approach adopted in our experiment to measure the FRD, a white LED source, and ST9XEI SBIG instruments CCD camera were used. The camera had 512 512 pixels, with each pixel size being 20m. The polishing material used was Aluminium Oxide with particle size as 13.5m. The fibre used was Silica-Silica FBP broad spectrum, solarisation resistant Polymicro fibre. The NA of the fibre was 0.22 0.02. We preferred white light source, instead of a monochromatic light source, because it proves to be more practical. The length of

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the fibre was kept in the range of few meters. In total, the FRD performance of 4 fibres was investigated. All the 4 fibres differed to some extent in their end preparation, and one could easily verify this fact from the graphs obtained from the experiment. Both, unpolished and polished fibres were investigated, and the results were plotted for both. After verifying the results and its conformity as compared to the ideal profile of the fibre, the output beam profile was investigated. It was compared with two theoretical models, i.e. cos () square (Lambertian) and uniform distribution at the output. The approximate output profile was estimated at the end of the experiment, and proper curve fitted on it.

3. EXPERIMENTAL INVESTIGATION

3.1 Experimental setup

Figure 2 – Figure to explain the very basic schematic and working of the experiment.

In the experimental set up of our experiment, a pinhole of 10m size was kept in front of the white LED source. The light emanating from the pin hole would act as a point source, which would fall on to a converging lens (convex) of focal length 5cm. The light source was kept at a far enough distance from the lens, so that it could converge the input light cone to fall on the optical fibre. The light source used in the experiment can be seen in the figure 3. The size of the pinhole was varied, and different sizes were experimented with. It was observed that, the smallest pin hole would give the most appropriate results. The core of the optical fibre

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measured 50m, and thus it was very important to keep the size of the input beam as small as possible. It had to be done so because, an input beam of much wide spot on the input end of the fibre would give inappropriate results and observations. The spot should be such that, it falls very precisely on the optical axis of the optical fibre. As the spot size is very small, and very precise arrangement is needed for correct output profile, it should be given taken care of very well. As the intensity of the spot falling on the optical fibre is very less and feeble, it would give additional hindrance to the precise set up of the experiment. The optical fibre is rested on a platform, which could slide in two opposite directions. This type of sliding arrangement is necessary in order to focus the beam very precisely on the input tip of the fibre.

Figure 3 – The figure showing variable apertures, the white LED source, and the platform for the 4 optical fibres.

The platform, on which the optical fibre is kept, should be capable of moving horizontally and vertically in an ideal case. In our experiment, only vertical degree of freedom was allowed. The fibres whose measurement had to be taken, were affixed on the platform rigidly, so that they couldn’t vibrate and change their position easily.

After the light starts converging at the other face of the convex lens, an arrangement was made to affix several apertures of different sizes just after the lens. A close proximity between the lens and the aperture was necessary in order to get accurate results. The apertures were made mechanically, using hard materials, and making holes of different sizes. This variable aperture was necessary in order to vary the input f-number. As the aperture size decreased, the input f-ratio would increase and vice versa. For each fibre amongst the 4, seven input f-ratios were considered. Those were 11.67, 8.75, 7, 5.833, 5, 4.375, 3.5 for aperture size – (“a” in cm) 1.5, 2, 2.5, 3, 3.5, 4, 5 respectively. The steps taken in order to calculate input and output f-ratios are as follows:

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Figure 4 – A simple schematic to calculate input and output f-ratios. The pixel size of the CCD chip was 20m. Also, input focal length from figure 2 came to be 17.5cm. Suppose that the aperture size is = a. So the input f-ratio 푓 = 17.5/ a. The output focal length is 1.74cm. The number of pixels the output beam covered is represented as = n. Radius of the corresponding output beam would become pn. Diameter of output ring on CCD chip was 2pn. Thus, the output f-ratio 푓 = 1.74/ (2pn) = 438/ n = 1/ (2tan ()) = tan ( ).

After the converging light cone from the convex lens falls on the input end of the optical fibre, the light passes through the length of the fibre, and undergoes FRD to give us a lower f-ratio output beam profile. The fibre in between the input and the output end should be kept stress free and sharp bends free. The room in which the experiment is performed, had to be perfectly dark, and dirt on optical fibre end and CCD face needs to be avoided up to the maximum extent. Also, the output tip of the fibre should be 90 to the CCD chip. Proper care had to be taken, such that the output image is formed at exact centre of the CCD’s 2D pixel array on computer screen (watched on CCDops and DS9 software). One of the major point which needs due consideration is that, the stray radiations should be blocked completely, and the light source should be completely isolated. If the above two conditions are not taken care of, the output may contain a major percentage of photon counts (in CCDOps & DS9) originating from them.

For the experiment, in total 4 fibres were dealth with. For each fibre, 7 input f-ratios are considered. For each input f-ratio, 13 values of output data points were taken. Greater the number of data points, more accurate the graph would be. The experiment had to be repeated several number of times, with some modifications, in order to get precise and workable values. Once the output beam felled on the CCD detector, the corresponding image could be seen on the computer screen for feedback. The exposure time had to be tuned for 4 different fibres. This is because; the exposure time for each of the fibre was different, for obtaining an image with maximum S/ N ratio. The image was grabbed in AUTOGRABBING mode. Before and after taking observation for each input f-ratio of each of the fibre, the background was also measured with the same interval of exposure time. This was done in order to subtract the background image from the raw image obtained with stray radiations. All the above procedures were repeated for one of the fibres after polishing. Polishing had to be done for a minimum time interval of 90 minutes on each face. Polishing helps in better edge preparation of the fibre and thus minimizes FRD in an ideal case. Polishing can be done with materials with particle size as small as 0.5m.

Once the final image was stored inside the PC, it was analyzed using software’s like DS9. After the acquisition of satisfactory image, the procedure was stopped for that input f-ratio.

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After background subtraction using IRAF, the 3D information of the image was observed on MATLAB. It was very easy to distinguish between the output profile for several different cases in 3D. Proper operations on IRAF would give us different tables for different images and then plotting it would give the FRD characteristic curve, i.e. N versus output f-ratio. After acquiring the image, Chi Square error was also calculated between the obtained output beam profile and two kinds of distributions. It was observed that the minimum chi square error was observed in the range of NA of the fibre. Later on, it was observed that, the output beam profile was none of these. It drives a need for a new kind of theory of FRD for sources like white light LED. The one by Gloge in 1972 might not be the most successful candidate for this condition. The figure below shows us the real set up of the experiment.

Figure 5 – Few pictures in order to show the arrangement of the instruments, fibres, platform, CCD camera, a computer to acquire images grabbed, source of light, mechanical apertures, and the slide on which the platform moves.

4. RESULTS AND MEASUREMENTS

In the following images, the output profile for two different apertures are shown corresponding to their ideal profile. The one on the left hand side is the ideal one, and the one on the right hand side is experimentally obtained image.

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Figure 6 – The above figures on the right hand side shows the images with FRD for two input apertures. The image shown on the right hand side is an output spot on the CCD chip.

The 3D profile of some of the images for different input apertures are shown below.

Figure 7 – The following image shows the output image profile obtained in different conditions. The image on the upper extreme right shows us one of the bad profiles one may obtain if proper care is not given on stray and background radiations. The one on the lower extreme right shows us 3D profile of the background, with which one may have to work in practical situations. The other 4 profiles are for different apertures and are on a 512512 2D pixel array.

The following images below shows us the hat like output profile of intensity, and one of the images obtained with very large FRD.

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Figure 8 – The image on the left shows one of the ideal profile one would anticipate and on the right is one of the images with large FRD which can occur due to bad edge preparations of the fibre. Different colours of hat represent different power.

The following image shows us the profile of a much distorted image obtained due to badly adjusted exposure time.

Figure 9 – A distorted and badly adjusted image. One may have to deal with this kind of gradient profile in practical situations.

Figure 10 – Feasible contoured images for 3cm and 5cm apertures. On the right are the badly focused images. Different kinds of variation might be observed with very little deviation in the set up.

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The following figures show the FRD characteristic graph for different fibres.

Figure 11 – The images in the 1st row are for unpolished fibre 1 (left) and unpolished fibre 2 (right). The one in the 2nd row is for unpolished fibre 3 (left) and unpolished fibre 4 (right). Lastly, in the last row are the images for polished fibre 2 (left) and chi-square variation for the polished fibre 2 (right) with input aperture = 4cm, and 푓 = 4.375. The minimum chi square error was observed at around 13. It can be remarkably observed that, the graph for polished fibre 2 in the 3rd row has some irregularities for 2 input f-ratios.

The cos () and uniform distribution were investigated for two cases. In first case, the experimental graph versus theoretical was plotted for the minimum chi-square error, and in the second case, the graph was plotted for average of the ’s obtained from minimum chi square error for 7 input f-numbers. Let the value of 100 % light be at 3. It comes in the range of 0.22 0.02, which is 11.54 to 13.89. The fraction of light can be denoted as – R () = N ()/ N (3)

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Figure 12 – The diagram to illustrate the attribution process in case of cos () and uniform distribution.

Which gives R () for cos () distribution as –

R () =∫ 퐾(cos (휃)) 푑휃 ∫ 퐾(푐표푠(휃)) 푑휃3 . (14)

And, R () for Uniform distribution as –

R () = / 3. (15)

Few graphs which are obtained after plotting a particular input f-ratio = 4.375 versus cos () and uniform distribution can be drawn. Unpolished and polished fibre 2 was considered along with the above two distributions. Some of the plots are shown below –

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Figure 13 – Graphs for Unpolished versus cos (휃), unpolished versus uniform, polished versus cos (휃), polished versus uniform.( to be read from top left and then clockwise). All the graphs of f-ratio = 4.375 are plotted versus average 휃 value (average of the sum of the minimum 휃 corresponding to minimum chi square values for each input f-ratios) as discussed earlier. The plots for f-ratio = 4.375 versus minimum 휃 values are almost similar to their average counterpart.

A simple approach was adopted for fitting a function on the graph of fibre 2 and input f-ratio = 4.375. The fitting graph proved to be a sum of three Gaussian profiles. The fitted curve has been drawn below –

Figure 14 – The graphs fitted would give the chi square estimate to be = 0.005868, for the set of data points adopted. The function was obtained as - F(x) = a1*exp (-((x-b1)/c1) ^2) + a2*exp (-((x-b2)/c2) ^2) + a3*exp (-((x-b3)/c3) ^2). F(x) is equivalent to the ratio of variable and total enegy or R(). F(x) is a sum of three gaussian profiles with RMSE = 0.0766 and CSE = 0.005868. The coefficients as calculated with 95% confidence bounds. The values of parameter are as follows – a1= 11.65, b1 = 3.473, c1 = 2.151, a2 = -1.264 e+04, b2 = 5.109, c2 = 8.966, a3 = 1.27e+04, b3 = 5.077, c3 = 8.989.

5. SUMMARY AND DISCUSSION

To summarize, it can be said that, polished and unpolished fibre didn’t give much quantifiable differences. This averts sign towards the need of better polishing with smaller particle size material and for a much longer duration. From our results and observations, it can be inferred that, FRD for higher input f-ratio beam is considerable. The FRD characteristic graphs suggest that, input f-ratio in the range of 3 to 5 is best for the present and future instruments, and it can give us 80 to 90 % throughput easily. The not too small scale bending and larger core diameter would decrease the FRD. One of the main point of discussion which emerges from the paper is that, neither the cos (휃) nor the uniform distribution output profile correctly fits with the output FRD characteristic graph. This point towards the fact that a better theoretical model for white light input source is needed. In future, the theoretical model can be readily be correlated with the observations. The values

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obtained are in conformation with the inbuilt distinct parameters of the fibre, such as NA, which is provided by the manufacturer.

It can further be said that, measurement of FRD characteristics and its modification is essential for the development of fibre optic applications in instrumentation. Future works will be mainly concentrated on re-polishing of the edge of the fibre and finding its conformity with the IFU to be established in IUCAA’s Girawali Observatory.

ACKNOWLEDGEMENTS

Research opportunity from IUCAA, in the form of Vacation Student’s programme, was crucial in the progression of the work. Other than that, we thank Mr. Siddartha S Verma and Mr. Mudit Shrivastava for their support in carrying out the project.

REFERENCES

Gloge, D., 1972, Optical Power Flow in Mulitimode Fibers, Bell Syst. Tech. J., Volume 51, 1767

Carrasco, Esperanza, Parry, Ian R., 1994, A method for determining the focal ratio degradation of optical fibres for astronomy, Royal Astronomical Society, NASA Astrophysics Data Systems

Ramsey, L.W., 1988, Focal ratio degradation in Optical Fibres of Astronomical interest, Astronomical Society of the Pacific, NASA Astrophysics Data Systems

Crause, Lisa, Bershady, Matthew, Buckley David, 2008, Investigation of focal ratio degradation in optical fibres for astronomical instrumentation, SPIE Vol. 7014 70146C-1

Clayton, C.A., 1989, The implications of image scrambling and focal ration degradation in fibre optics on the design of astronomical instrumentation, European Southern Observatory, NASA Astrophysics Data Systems

Marcuse, D., 1972, Derivation of Coupled Power Equations, B.S.T.J., Volume 51

Oliveira, A.C., de Oliveira, L.S., dos Santos, J.B., 2004, Studying focal ratio degradation of optical fibres with a core size of 50 m for astronomy, Royal Astronomical Society, NASA Astrophysics Data System

Aslund, Mattias L., Canning John, 2008, Air-clad fibres for astronomical instrumentation: focal-ratio degradation, Springer Science + Business Media B.V. 2008