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Image revivals in multi-mode optical fibers with periodic multiple sub-apertures
Optics Communications 326 (2014) 57–63
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Image revivals in multi-mode optical fibers with periodic multiple sub-apertures Long Wang a, Peter E. Powers a,b, Andrew Sarangan a, Joseph W. Haus a,b,n a b
Electro-optics Program, University of Dayton, 300 College Park, Dayton, OH 45469, USA Physics Department, University of Dayton, 300 College Park, Dayton, OH 45469, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 4 September 2013 Received in revised form 11 April 2014 Accepted 12 April 2014 Available online 25 April 2014
We report experiments on a multi-mode fiber-based device that reimages the input pattern after specific propagation distances. The reimaging has two propagation length scales related to the Talbot selfimaging in a periodic grating and image revival effects. We use a beam propagation method to simulate diffraction and refraction of light in the optical fiber. The details of the fiber preparation and optical experiments are described. We study the optical imaging properties using a close-packed array of subapertures placed at regular positions on a triangular lattice. We numerically analyze the propagation, diffraction and coupling characteristics of the beam oscillating inside the fiber. Our simulations identify the optimal reimaging length of the multi-mode (MM) fiber to get high fidelity image revival. Experiments are performed to validate the simulation results. & 2014 Elsevier B.V. All rights reserved.
Keywords: Image revivals Talbot imaging effects Fiber lasers Multi-mode fibers Phase-locked beamlets
1. Introduction Since its first observation in the nineteenth century, the Talbot effect has drawn renewed attention from many researchers. It is one of several basic optical phenomena that is widely discussed in the literature. Recently the Talbot effect has become a useful tool for diverse applications including optical metrology [1], optical array illumination [2], matter wave self-imaging [3] and phase locking multiple amplifiers [4,5]. Among potential wide ranging applications, the design of phase-locked fiber lasers based on the Talbot effect and a revival of the image are of primary interest in this paper. Fiber lasers have been identified as promising candidates for developing high power lasers with a high degree of coherence [6,7]. One reason is the simple geometry of optical fibers, which make alignment relatively easy and the laser cavity compact. Also, optical fibers have inherently high surface area to active volume ratios which enable efficient cooling of the fiber system. Phase-locked fiber laser designs have been theoretically and experimentally demonstrated to be candidates for the future development of high power fiber lasers [8,9]. We propose the reimaging effect in multi-mode optical fibers as a method to reinforce phase locking between multiple fiber amplifiers. In a previous publication our calculations demonstrated
n
Corresponding author. E-mail address: [email protected] (J.W. Haus).
http://dx.doi.org/10.1016/j.optcom.2014.04.022 0030-4018/& 2014 Elsevier B.V. All rights reserved.
how the use of reimaging could be useful for phasing the laser output and making the far-field laser spot coherent [4,5]. Here we report initial experiments to validate the reimaging concept using a tunable Ti:Sapphire laser whose wavelength spans the wavelength range from 740 nm to 840 nm. We find that the operation of a fiberbased device displays reimaging effects, which are related to the Talbot self-imaging and an image revival effect. The simulations are used to guide advanced experimental design of phase-lock laser sources.
2. Simulations for 37 phase-locked Gaussian beamlets The multi-mode fiber (MMF) device for phase locking multiple, high power fiber amplifiers concept is introduced and discussed in Refs. [4,5]. In an all fiber laser application the field launched into the MMF is from a bundle of tapered and fused single mode (SM) fibers. On one end of the MMF, these input SM fibers are arranged in hexagonal rings with the same nearest-neighbor separation. The beams, or beamlets, from the input SM fibers are launched into the MM fiber, diffract and interfere with one another during propagation. Here the beams from the SM fibers are assumed to be Gaussian distributed. The propagating field diffracts out of the area where the SM fibers are centered and propagates to the core/clad boundary where it is reflected back toward the center of MMF. The image revival occurs when the reflected energy reforms the image of the input beams at the center of the core; this is the image revival distance.
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If the length of the MMF is controlled to be half of the revival distance, and a partial reflector is placed on the other end of the MM fiber, then the array of beamlets will be self-imaged into the input SM fibers, provided that all the input beamlets have the same phase. A partial reflector serves as an output coupler, and the far field of the output will be a small central spot. Since the waves do not interact with one another, this process can be viewed as a one-way propagation and the reflection is simply stretched out in the same direction. In this section we model the all-fiber device with 37 input Gaussian beams with flat phase profile across each SM fiber input; the mode field diameter is assumed to be 10.4 mm. To compare our simulations to experiments we tune the input light wavelength between 740 nm and 840 nm. Also, we assume no phase and power variations between each hole in our simulations; the midpoint separation between two adjacent holes is 22 mm. The MM fiber (Thorlab's BFL37-400 fiber) used in our experiments has 400 mm core diameter, the core and cladding indices are reported to be 1.506 and 1.460, respectively and the numerical aperture is NA ¼0.37. As mentioned earlier, the wave propagation is treated as a one-way propagation. Applying the beam propagation method (BPM) the dynamic field distribution can be easily and accurately determined as the beam propagates through the MM fiber. In designing the experiment we use simulations to determine the position where the image repeats itself or at what distance the coupling efficiency is locally the highest. The normalized mode for the initial field, Ex0, is Ex0 ffi: f x0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RR jEx0 j2 dx dy
ð1Þ
The field propagating at a distance z in the fiber is denoted as Ex; the total coupling efficiency at distance z is how well it overlaps
Fig. 1. Coupling efficiency for 37 Gaussian beams input at 740 nm.
with the initial field η1 ¼
∬ f En dx dy x0 x : ∬ jEx j2 dx dy
ð2Þ
The field Ex0(x,y) is the initial input and the Ex(x,y) is the field at distance z. The notation, Ex, indicates that the field is linearly polarized in the x-direction. The coupling efficiency for 37 Gaussian beamlets input at 740 nm from our BPM simulations is shown in Fig. 1. The wave propagation is treated as one way propagation. In Fig. 1 the local oscillations leading to a maximum coupling efficiency occur at a distance 16.348 mm. We identify this as the revival distance for the maximum overlap of the input image with the revival image. We also observe a set of partial image revivals, which locally culminates in the peak overlap with the input image. The separation between local oscillations is determined by the Talbot imaging distance, which is estimated as LT ¼πd2n/2λ ¼1.54 mm, where λ ¼740 nm is the wavelength in vacuum, d is the nearestneighbor hole separation distance (22 μm) and n is the refractive index in the core. Note, in the real all-fiber device with the partial reflector, when the length of the MM fiber matches half of 16.348 mm which is 8.174 mm, the power coupling efficiency returning to the input ports will be maximized. To estimate the revival distance based on Talbot imaging one might erroneously apply a formula: Lr ¼ πD2 n=2λ ¼ 509 nm, where D ( ¼400 μm) is the core diameter of the MM fiber, n is the core index and λ is the wavelength in air. However, this distance is about 32 greater than we observed in simulations and experiments. This estimate makes the assumption that on propagation, the reflections from the boundary of the MM fiber lead to a structure equivalent to a lattice consisting of the MM structure repeated every D. Clearly another process is responsible for the imaging. We infer that there is an angular rotation of the pattern due to excitation of many fiber modes with higher angular momentum values. The skew rays rotate about the center in an angular pattern and periodically recombine creating a partially phased image [10] at the revival distance. Fig. 2 shows a plot of the initial input beams from the tapered and fused SM fiber bundle. The intensity and phase distributions of the field after propagating at a revival distance are plotted in Fig. 3. For a length of the MM fiber taken as the revival distance, i.e. 16.34 mm, the propagated beam has maximum overlap with the input beamlets. Furthermore, a Gaussian beam-like spot is found in the far field which is in space frequency domain and the distribution along kx and ky is shown in Fig. 4. The reimaging distance as a function of the input wavelength is shown below in Fig. 5, where we determine that the revival distance decreases nearly linearly as the input wavelength increases.
Fig. 2. Isometrically distributed inputs in MMF core: (a) intensity distribution of Ex0 and (b) phase profile of Ex0.
L. Wang et al. / Optics Communications 326 (2014) 57–63
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Fig. 3. (a) The simulated intensity distribution of the field Ex at the revival distance and (b) the corresponding phase distribution.
Fig. 4. Far-field intensity distribution: (a) along kx for ky ¼0 and (b) along ky for kx ¼0.
Fig. 5. Revival distance as a function of input wavelength for input wavelength from 740 nm to 840 nm.
Fig. 7. The coupling efficiency as a function of propagation distance for 37 phaselocked flat top beamlets. The input wavelength is 740 nm.
3. Simulations for 37 phase-locked flat top beamlets
Fig. 6. An illustration of the geometry used for our multiple beam MM fiber device. Input beam impinges on the left-hand side, the beamlets are shaped by the mask with holes (37 holes in this case); the light couples directly into the MM fiber.
Designing and fabricating a multi-fiber coupler (the tapered and fused SM fiber bundle) bonded to one end of the MMF is beyond the scope of our project; hence, in our experiments we use a mask with holes fabricated on one end of a MMF to shape the input beam as shown in Fig. 6. The holes in the mask are arranged in hexagonal rings with the same nearest-neighbor separation (the same arrangement of SM fiber bundle in Section 2). By setting the reflectivity of the partial reflector to 1 and based on the discussion in Section 2, the propagation of the field inside the MMF is simply viewed as one way propagation. The mask is flood illuminated with the Ti:Sapphire laser beam, and the mask has a
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periodic arrangement of sub-apertures or holes that create beamlets that are launched into the MM fiber. In this section we present results of our simulation for flat top beamlets instead of Gaussian beamlets. The coupling efficiency versus propagation distance for flat top beams with input wavelength of 740 nm is shown in Fig. 7. Fig. 8 is an expanded view of
the coupling efficiency in Fig. 7 for the propagation distance from 9.170 mm to 16.700 mm. We identify five local maxima in that range, which are tested by experimental results later. The spatial intensity distributions corresponding to the local maxima in Fig. 8 are illustrated in Fig. 9. The maxima feature a specific spatial intensity profile; for increasing coupling efficiency the images reveal a greater organization toward the input field intensity profile at each maximum. The positions where the five local maxima occur for all wavelengths used in the experiments below are summarized in Table 1. The local coupling coefficient maxima positions as functions of wavelength are shown in Fig. 10; their positions decrease almost linearly as the input wavelength increases and are roughly inversely proportional to the wavelength. These results will be
Table 1 Simulation results: local maxima positions at different wavelengths.
Fig. 8. Coupling efficiency for 37 phase-locked flat top beamlets input with 740 nm wavelength in the distance range from 9.170 mm to 16.70 mm along the propagation direction.
Wavelength (nm)
740
760
780
800
820
840
1st (mm) 2nd (mm) 3rd (mm) 4th (mm) 5th (mm)
10.812 12.052 13.488 14.918 16.250
10.512 11.710 13.140 14.526 15.820
10.240 11.441 12.792 14.152 15.416
9.976 11.132 12.470 13.802 15.030
9.752 10.856 12.174 13.464 14.664
9.512 10.628 11.870 13.142 14.316
Fig. 9. Simulated intensity distributions at 5 local maxima shown in Fig. 8.
L. Wang et al. / Optics Communications 326 (2014) 57–63
Fig. 10. Simulation result: local maxima's positions as functions of the input wavelength.
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Fig. 12. Photograph of the patterned fiber end after photolithography and etching, as seen under the microscope.
the fiber, we deposited a thin film of chromium on the fiber end and photoresist was spin coated over the metal film. After the mask was aligned to the center of the MMF the pattern was transferred to the photoresist using photolithography. After developing the photoresist holes were etched in the chromium film. The microscopic photograph of the fiber end after completing the photolithography and etching the pattern into the chromium is shown in Fig. 12. The holes are separated by 22 μm and their diameter is 10.4 μm. Fig. 11. The experimental set-up used to validate the reimaging concept. The image depth inside the device sample is changed by moving the lens labeled Object.
compared to our experimental results described in the following section.
4. Experimental procedure In this section a sample is fabricated and tested to validate the simulation results. The experimental configuration to image the patterns is schematically presented in Fig. 11. Theoretically, the optimal length of the fiber device is half of the revival distance. However, it is easy to verify the simulation results by fixing the length of our fiber segment and imaging the planes inside the fiber onto the CCD. The mode-locked Ti:Sapphire laser is tunable from 740 nm to 840 nm. The laser has a narrow linewidth around each wavelength. The output of the Ti:Sapphire laser is coupled into an “endless single-mode” photonic-crystal fiber (PCF). It was used to ensure that the illumination pattern on the input plane of the device remains constant as the laser is tuned. 4.1. Fiber end patterning To fabricate the array of holes for our sample the following process was developed. First a large hole was drilled into a glass plate to hold the fiber; the hole size was slightly larger than the fiber diameter ( 425 mm). To help prevent chipping, the glass slide was attached to a sacrificial slide using wax. This helped prevent breakage of the glass plate as it was drilled. A smooth edge along the hole rim was achieved when the SiC drill bit ran at a low speed and was oil cooled. Finally, the fiber was glued into the hole using an epoxy. This process was followed for both ends of the MM fiber. A spacer was epoxied to the two glass plates to support the entire structure. Second the fiber ends were polished flush with the surface of the glass plate to make the surfaces appear seamless between the glass plate and the fiber end. In the third step, applied to just one end of
4.2. Microscope imaging In this section we assume that the objective lens is a thin lens with focal length f. By moving the objective lens towards and away, we image selected planes inside the fiber (opposite end from the hole pattern end) and capture the image onto the CCD; captured sample images are shown in Fig. 13. The beam waist (focal plane or image plane) shifts faster than the translation of the objective lens, given by Δd2 ¼ nc Δd1 ;
ð3Þ
where Δd1 is the translation of the objective lens from a given location, and Δd2 is the change in location of the image plane. This equation reveals that if the objective lens is moved closer to the fiber, the focal image plane will move nc times faster away from the surface of the fiber. To know the precise location of the imaging plane, we first measured the MM fiber length. This measurement was accomplished by measuring the distance between the four corners of the sample block that holds the fiber. Since the MM fiber is placed in the center of the block, we assume that the MM fiber length is the average of four numbers, which yields 16,700 7125 μm. The depths that the microscope can image into the un-patterned end of the fiber are limited by the travel of the microscope's objective lens and ultimately by placing the objective lens on the end face of the MM fiber. With our setup we are able to image from the end-face of the MM fiber to a depth of 7530 μm. Equivalently, we are able to image from 9.170 to 16.7 mm away from the patterned surface. 4.3. Experimental results For the input beams with wavelength of 740 nm, the intensity distributions for the 2nd, 3rd, 4th and 5th local maxima are shown in Fig. 13. The corresponding pictures from the simulation are shown in Fig. 9. In order to calculate the location of the experimental local maxima, we should translate the shift of objective lens into the shift of the image plane as discussed in Section 4.2. For example, in Fig. 13(a) we translate the objective lens by 3.200 mm. Recalling that the
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L. Wang et al. / Optics Communications 326 (2014) 57–63
distance change of the focal plane in the fiber is nc times larger than the distance change of the object lens, we arrive at a focal plane distance of 4.819 mm (1.506 3.200) away from the fiber output surface which means that the focal plane is at 11.881 mm (16.700– 4.819) along the light propagation direction. Following the same rule we have found that the third local maximum occurs at 13.206 mm, fourth local maximum at 14.682 mm and the fifth local maximum at 16.158 mm. The local maxima for all the wavelengths tested are shown in Table 2. Note that there is no data for first local maximum in Table 2 since the coupling efficiency between the first local maximum and the input pattern is too low. A comparison between Fig. 14 and Fig. 10 shows that the experimental results closely follow the simulation results. The positional differences between simulation results and experimental results are at most a few hundred microns. For instance, by comparison of the slopes of each curve we determine that for the second local maximum the experiment result is within 3.44% of the simulation result. The relative errors for local maxima are reasonable compared to those of the relative error of the MM fiber length (125/16700¼ 0.75%) and also there are a number of other error sources that are discussed later in Section 5. The trend in each case closely corresponds to our simulations results, which validates the process we used in our simulations.
5. Discussion and summary To fabricate and use an all-fiber device to phase-lock multiple fiber amplifiers there are significant additional challenges still to be faced. Multiple single mode fibers must be fused together and tapered to fit into the core area of a multi-mode fiber. In the experiments a Gaussian beam was expanded to cover the mask to get the initial 37 inputs which means the inputs are not Gaussian beams but flat top beams in each sub-aperture. Our simulations show that the coupling efficiency of Gaussian input beamlets is higher than that of flat top beamlets input. Also, when the center of the mask is misaligned with the center of the MMF, as shown in Fig. 15, the coupling efficiency is greatly reduced. In Fig. 15 the fiber end diameter is 430 mm including cladding. Another red
Table 2 Experimental results: local maxima positions at different wavelengths. Wavelength (nm)
740
760
780
800
820
840
2nd (mm) 3rd (mm) 4th (mm) 5th (mm)
11.881 13.206 14.682 16.158
11.580 12.754 14.290 15.706
11.354 12.4230 13.778 15.254
10.977 12.062 13.507 14.833
10.751 11.851 13.176 14.622
10.525 11.489 12.784 14.230
Fig. 13. Experimental intensity distribution for 740 nm input wavelength: (a) 2nd local maximum; (b) 3rd local maximum; (c) 4th local maximum and (d) 5th local maximum.
L. Wang et al. / Optics Communications 326 (2014) 57–63
Fig. 14. Experimental results: local maxima's positions as functions of the input wavelength. The inset formulas are best fits to the curves with y values in microns and the x values in nm.
Fig. 15. Photograph with annotations that indicate the mask misalignment from the fiber's center.
circle with diameter of 430 mm is added on the fiber end showing that the center of the pattern is off the center of the fiber by about 13 mm which greatly reduces the coupling efficiency, as seen by comparing Fig. 16 with Fig. 7. As shown in the experimental set-up, the Gaussian beam from the PCF is simply expanded to cover the whole mask which indicates that power and phase variations across the mask are unavoidable in our experiment. The maximum phase variation can be estimated on the edge of the pattern by using the equation: exp (ikr2/(2R)), where k is wave number for 740 nm wavelength, r is the radius of the hole pattern (100 um) and R is the radius of curvature of Gaussian beam on the mask. The separation between the mask and the PCF is about z ¼10 cm and the beam waist of the Gaussian beam from the PCF is around w0 ¼30 μm. Thus R can be calculated by using equation: R¼ z(1 þ (z0/z)2), where z0 ¼ πw20 l. By combining all the numbers, the phase difference between the center and the edge of the pattern is about 0.1349π, which is about
63
Fig. 16. Coupling efficiency for BFL37-400 with 13 mm misaligned 37 flat top beamlets input.
6.75% of 2π with 740 nm input. Our simulations including the phase variations of the field across the mask produce results that differ by only a small amount from the results shown in Fig. 16. Combining phase variations and power variations in each beamlet the coupling efficiency along the propagation will be further reduced from that of the flap top assumption. The separation between adjacent inputs should also be optimized for the wavelength of interest. 22 mm separation between sub-apertures was chosen based on 632.8 nm wavelength. To optimize the coupling efficiency in 740–840 nm wavelength range, the separation should be adjusted and the length of the fiber should be finely controlled. Future design with optimization of the fiber mode size and separation for a specific MM fiber can greatly improve the efficiency. In our numerical simulations using 6 beamlets, efficiencies better than 30% were achieved and we should be able to greatly improve this result with a comprehensive design process. In conclusion, in this paper we have studied self-imaging revivals using a periodic array of sub-apertures in a multi-mode optical fiber. For our geometry there are two length scales in our simulations and experiments, one is the revival length which occurs on the scale of cm and the other is the Talbot imaging distance within the array, which is on the scale of mm. We corroborated the existence and scale of both lengths through simulations and experiments and the wavelength dependence of the results.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
A.W. Lohmann, D.E. Silva, Opt. Commun. 2 (1971) 413. A.W. Lohmann, J.A. Thomas, Appl. Opt. 29 (1990) 4437. S. Nowak, C. Kurtsiefer, T. Pfau, C. David, Opt. Lett. 22 (1997) 1430. R. Zhou, Q. Zhan, P.E. Powers, B. Ibarra-Escamilla, J.W. Haus, Proc. SPIE 7839 (2010) 78390M-1. R. Zhou, Q. Zhan, P.E. Powers, B. Ibarra-Escamilla, J.W. Haus, J. Eur. Opt. Soc. 7 (2012) 12012. Y. Jeong, J. Sahu, D. Payne, J. Nilsson, Opt. Express 12 (2004) 6088. A. Galvanauskas, Opt. Photonics News 15 (2004) 42. Y. Zhou, L. Liu, C. Etson, Y. Abranyos, A. Padilla, Y.C. Chen, Appl. Phys. Lett. 84 (2004) 3025. B. He, et al., Appl. Phys. Lett. 92 (2008) 251115. M. Hautakorpi, M. Kaivola, Appl. Opt. 45 (2006) 6388.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Image revivals in multi-mode optical fibers with periodic multiple sub-apertures Long Wang a, Peter E. Powers a,b, Andrew Sarangan a, Joseph W. Haus a,b,n a b
Electro-optics Program, University of Dayton, 300 College Park, Dayton, OH 45469, USA Physics Department, University of Dayton, 300 College Park, Dayton, OH 45469, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 4 September 2013 Received in revised form 11 April 2014 Accepted 12 April 2014 Available online 25 April 2014
We report experiments on a multi-mode fiber-based device that reimages the input pattern after specific propagation distances. The reimaging has two propagation length scales related to the Talbot selfimaging in a periodic grating and image revival effects. We use a beam propagation method to simulate diffraction and refraction of light in the optical fiber. The details of the fiber preparation and optical experiments are described. We study the optical imaging properties using a close-packed array of subapertures placed at regular positions on a triangular lattice. We numerically analyze the propagation, diffraction and coupling characteristics of the beam oscillating inside the fiber. Our simulations identify the optimal reimaging length of the multi-mode (MM) fiber to get high fidelity image revival. Experiments are performed to validate the simulation results. & 2014 Elsevier B.V. All rights reserved.
Keywords: Image revivals Talbot imaging effects Fiber lasers Multi-mode fibers Phase-locked beamlets
1. Introduction Since its first observation in the nineteenth century, the Talbot effect has drawn renewed attention from many researchers. It is one of several basic optical phenomena that is widely discussed in the literature. Recently the Talbot effect has become a useful tool for diverse applications including optical metrology [1], optical array illumination [2], matter wave self-imaging [3] and phase locking multiple amplifiers [4,5]. Among potential wide ranging applications, the design of phase-locked fiber lasers based on the Talbot effect and a revival of the image are of primary interest in this paper. Fiber lasers have been identified as promising candidates for developing high power lasers with a high degree of coherence [6,7]. One reason is the simple geometry of optical fibers, which make alignment relatively easy and the laser cavity compact. Also, optical fibers have inherently high surface area to active volume ratios which enable efficient cooling of the fiber system. Phase-locked fiber laser designs have been theoretically and experimentally demonstrated to be candidates for the future development of high power fiber lasers [8,9]. We propose the reimaging effect in multi-mode optical fibers as a method to reinforce phase locking between multiple fiber amplifiers. In a previous publication our calculations demonstrated
n
Corresponding author. E-mail address: [email protected] (J.W. Haus).
http://dx.doi.org/10.1016/j.optcom.2014.04.022 0030-4018/& 2014 Elsevier B.V. All rights reserved.
how the use of reimaging could be useful for phasing the laser output and making the far-field laser spot coherent [4,5]. Here we report initial experiments to validate the reimaging concept using a tunable Ti:Sapphire laser whose wavelength spans the wavelength range from 740 nm to 840 nm. We find that the operation of a fiberbased device displays reimaging effects, which are related to the Talbot self-imaging and an image revival effect. The simulations are used to guide advanced experimental design of phase-lock laser sources.
2. Simulations for 37 phase-locked Gaussian beamlets The multi-mode fiber (MMF) device for phase locking multiple, high power fiber amplifiers concept is introduced and discussed in Refs. [4,5]. In an all fiber laser application the field launched into the MMF is from a bundle of tapered and fused single mode (SM) fibers. On one end of the MMF, these input SM fibers are arranged in hexagonal rings with the same nearest-neighbor separation. The beams, or beamlets, from the input SM fibers are launched into the MM fiber, diffract and interfere with one another during propagation. Here the beams from the SM fibers are assumed to be Gaussian distributed. The propagating field diffracts out of the area where the SM fibers are centered and propagates to the core/clad boundary where it is reflected back toward the center of MMF. The image revival occurs when the reflected energy reforms the image of the input beams at the center of the core; this is the image revival distance.
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If the length of the MMF is controlled to be half of the revival distance, and a partial reflector is placed on the other end of the MM fiber, then the array of beamlets will be self-imaged into the input SM fibers, provided that all the input beamlets have the same phase. A partial reflector serves as an output coupler, and the far field of the output will be a small central spot. Since the waves do not interact with one another, this process can be viewed as a one-way propagation and the reflection is simply stretched out in the same direction. In this section we model the all-fiber device with 37 input Gaussian beams with flat phase profile across each SM fiber input; the mode field diameter is assumed to be 10.4 mm. To compare our simulations to experiments we tune the input light wavelength between 740 nm and 840 nm. Also, we assume no phase and power variations between each hole in our simulations; the midpoint separation between two adjacent holes is 22 mm. The MM fiber (Thorlab's BFL37-400 fiber) used in our experiments has 400 mm core diameter, the core and cladding indices are reported to be 1.506 and 1.460, respectively and the numerical aperture is NA ¼0.37. As mentioned earlier, the wave propagation is treated as a one-way propagation. Applying the beam propagation method (BPM) the dynamic field distribution can be easily and accurately determined as the beam propagates through the MM fiber. In designing the experiment we use simulations to determine the position where the image repeats itself or at what distance the coupling efficiency is locally the highest. The normalized mode for the initial field, Ex0, is Ex0 ffi: f x0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RR jEx0 j2 dx dy
ð1Þ
The field propagating at a distance z in the fiber is denoted as Ex; the total coupling efficiency at distance z is how well it overlaps
Fig. 1. Coupling efficiency for 37 Gaussian beams input at 740 nm.
with the initial field η1 ¼
∬ f En dx dy x0 x : ∬ jEx j2 dx dy
ð2Þ
The field Ex0(x,y) is the initial input and the Ex(x,y) is the field at distance z. The notation, Ex, indicates that the field is linearly polarized in the x-direction. The coupling efficiency for 37 Gaussian beamlets input at 740 nm from our BPM simulations is shown in Fig. 1. The wave propagation is treated as one way propagation. In Fig. 1 the local oscillations leading to a maximum coupling efficiency occur at a distance 16.348 mm. We identify this as the revival distance for the maximum overlap of the input image with the revival image. We also observe a set of partial image revivals, which locally culminates in the peak overlap with the input image. The separation between local oscillations is determined by the Talbot imaging distance, which is estimated as LT ¼πd2n/2λ ¼1.54 mm, where λ ¼740 nm is the wavelength in vacuum, d is the nearestneighbor hole separation distance (22 μm) and n is the refractive index in the core. Note, in the real all-fiber device with the partial reflector, when the length of the MM fiber matches half of 16.348 mm which is 8.174 mm, the power coupling efficiency returning to the input ports will be maximized. To estimate the revival distance based on Talbot imaging one might erroneously apply a formula: Lr ¼ πD2 n=2λ ¼ 509 nm, where D ( ¼400 μm) is the core diameter of the MM fiber, n is the core index and λ is the wavelength in air. However, this distance is about 32 greater than we observed in simulations and experiments. This estimate makes the assumption that on propagation, the reflections from the boundary of the MM fiber lead to a structure equivalent to a lattice consisting of the MM structure repeated every D. Clearly another process is responsible for the imaging. We infer that there is an angular rotation of the pattern due to excitation of many fiber modes with higher angular momentum values. The skew rays rotate about the center in an angular pattern and periodically recombine creating a partially phased image [10] at the revival distance. Fig. 2 shows a plot of the initial input beams from the tapered and fused SM fiber bundle. The intensity and phase distributions of the field after propagating at a revival distance are plotted in Fig. 3. For a length of the MM fiber taken as the revival distance, i.e. 16.34 mm, the propagated beam has maximum overlap with the input beamlets. Furthermore, a Gaussian beam-like spot is found in the far field which is in space frequency domain and the distribution along kx and ky is shown in Fig. 4. The reimaging distance as a function of the input wavelength is shown below in Fig. 5, where we determine that the revival distance decreases nearly linearly as the input wavelength increases.
Fig. 2. Isometrically distributed inputs in MMF core: (a) intensity distribution of Ex0 and (b) phase profile of Ex0.
L. Wang et al. / Optics Communications 326 (2014) 57–63
59
Fig. 3. (a) The simulated intensity distribution of the field Ex at the revival distance and (b) the corresponding phase distribution.
Fig. 4. Far-field intensity distribution: (a) along kx for ky ¼0 and (b) along ky for kx ¼0.
Fig. 5. Revival distance as a function of input wavelength for input wavelength from 740 nm to 840 nm.
Fig. 7. The coupling efficiency as a function of propagation distance for 37 phaselocked flat top beamlets. The input wavelength is 740 nm.
3. Simulations for 37 phase-locked flat top beamlets
Fig. 6. An illustration of the geometry used for our multiple beam MM fiber device. Input beam impinges on the left-hand side, the beamlets are shaped by the mask with holes (37 holes in this case); the light couples directly into the MM fiber.
Designing and fabricating a multi-fiber coupler (the tapered and fused SM fiber bundle) bonded to one end of the MMF is beyond the scope of our project; hence, in our experiments we use a mask with holes fabricated on one end of a MMF to shape the input beam as shown in Fig. 6. The holes in the mask are arranged in hexagonal rings with the same nearest-neighbor separation (the same arrangement of SM fiber bundle in Section 2). By setting the reflectivity of the partial reflector to 1 and based on the discussion in Section 2, the propagation of the field inside the MMF is simply viewed as one way propagation. The mask is flood illuminated with the Ti:Sapphire laser beam, and the mask has a
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periodic arrangement of sub-apertures or holes that create beamlets that are launched into the MM fiber. In this section we present results of our simulation for flat top beamlets instead of Gaussian beamlets. The coupling efficiency versus propagation distance for flat top beams with input wavelength of 740 nm is shown in Fig. 7. Fig. 8 is an expanded view of
the coupling efficiency in Fig. 7 for the propagation distance from 9.170 mm to 16.700 mm. We identify five local maxima in that range, which are tested by experimental results later. The spatial intensity distributions corresponding to the local maxima in Fig. 8 are illustrated in Fig. 9. The maxima feature a specific spatial intensity profile; for increasing coupling efficiency the images reveal a greater organization toward the input field intensity profile at each maximum. The positions where the five local maxima occur for all wavelengths used in the experiments below are summarized in Table 1. The local coupling coefficient maxima positions as functions of wavelength are shown in Fig. 10; their positions decrease almost linearly as the input wavelength increases and are roughly inversely proportional to the wavelength. These results will be
Table 1 Simulation results: local maxima positions at different wavelengths.
Fig. 8. Coupling efficiency for 37 phase-locked flat top beamlets input with 740 nm wavelength in the distance range from 9.170 mm to 16.70 mm along the propagation direction.
Wavelength (nm)
740
760
780
800
820
840
1st (mm) 2nd (mm) 3rd (mm) 4th (mm) 5th (mm)
10.812 12.052 13.488 14.918 16.250
10.512 11.710 13.140 14.526 15.820
10.240 11.441 12.792 14.152 15.416
9.976 11.132 12.470 13.802 15.030
9.752 10.856 12.174 13.464 14.664
9.512 10.628 11.870 13.142 14.316
Fig. 9. Simulated intensity distributions at 5 local maxima shown in Fig. 8.
L. Wang et al. / Optics Communications 326 (2014) 57–63
Fig. 10. Simulation result: local maxima's positions as functions of the input wavelength.
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Fig. 12. Photograph of the patterned fiber end after photolithography and etching, as seen under the microscope.
the fiber, we deposited a thin film of chromium on the fiber end and photoresist was spin coated over the metal film. After the mask was aligned to the center of the MMF the pattern was transferred to the photoresist using photolithography. After developing the photoresist holes were etched in the chromium film. The microscopic photograph of the fiber end after completing the photolithography and etching the pattern into the chromium is shown in Fig. 12. The holes are separated by 22 μm and their diameter is 10.4 μm. Fig. 11. The experimental set-up used to validate the reimaging concept. The image depth inside the device sample is changed by moving the lens labeled Object.
compared to our experimental results described in the following section.
4. Experimental procedure In this section a sample is fabricated and tested to validate the simulation results. The experimental configuration to image the patterns is schematically presented in Fig. 11. Theoretically, the optimal length of the fiber device is half of the revival distance. However, it is easy to verify the simulation results by fixing the length of our fiber segment and imaging the planes inside the fiber onto the CCD. The mode-locked Ti:Sapphire laser is tunable from 740 nm to 840 nm. The laser has a narrow linewidth around each wavelength. The output of the Ti:Sapphire laser is coupled into an “endless single-mode” photonic-crystal fiber (PCF). It was used to ensure that the illumination pattern on the input plane of the device remains constant as the laser is tuned. 4.1. Fiber end patterning To fabricate the array of holes for our sample the following process was developed. First a large hole was drilled into a glass plate to hold the fiber; the hole size was slightly larger than the fiber diameter ( 425 mm). To help prevent chipping, the glass slide was attached to a sacrificial slide using wax. This helped prevent breakage of the glass plate as it was drilled. A smooth edge along the hole rim was achieved when the SiC drill bit ran at a low speed and was oil cooled. Finally, the fiber was glued into the hole using an epoxy. This process was followed for both ends of the MM fiber. A spacer was epoxied to the two glass plates to support the entire structure. Second the fiber ends were polished flush with the surface of the glass plate to make the surfaces appear seamless between the glass plate and the fiber end. In the third step, applied to just one end of
4.2. Microscope imaging In this section we assume that the objective lens is a thin lens with focal length f. By moving the objective lens towards and away, we image selected planes inside the fiber (opposite end from the hole pattern end) and capture the image onto the CCD; captured sample images are shown in Fig. 13. The beam waist (focal plane or image plane) shifts faster than the translation of the objective lens, given by Δd2 ¼ nc Δd1 ;
ð3Þ
where Δd1 is the translation of the objective lens from a given location, and Δd2 is the change in location of the image plane. This equation reveals that if the objective lens is moved closer to the fiber, the focal image plane will move nc times faster away from the surface of the fiber. To know the precise location of the imaging plane, we first measured the MM fiber length. This measurement was accomplished by measuring the distance between the four corners of the sample block that holds the fiber. Since the MM fiber is placed in the center of the block, we assume that the MM fiber length is the average of four numbers, which yields 16,700 7125 μm. The depths that the microscope can image into the un-patterned end of the fiber are limited by the travel of the microscope's objective lens and ultimately by placing the objective lens on the end face of the MM fiber. With our setup we are able to image from the end-face of the MM fiber to a depth of 7530 μm. Equivalently, we are able to image from 9.170 to 16.7 mm away from the patterned surface. 4.3. Experimental results For the input beams with wavelength of 740 nm, the intensity distributions for the 2nd, 3rd, 4th and 5th local maxima are shown in Fig. 13. The corresponding pictures from the simulation are shown in Fig. 9. In order to calculate the location of the experimental local maxima, we should translate the shift of objective lens into the shift of the image plane as discussed in Section 4.2. For example, in Fig. 13(a) we translate the objective lens by 3.200 mm. Recalling that the
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distance change of the focal plane in the fiber is nc times larger than the distance change of the object lens, we arrive at a focal plane distance of 4.819 mm (1.506 3.200) away from the fiber output surface which means that the focal plane is at 11.881 mm (16.700– 4.819) along the light propagation direction. Following the same rule we have found that the third local maximum occurs at 13.206 mm, fourth local maximum at 14.682 mm and the fifth local maximum at 16.158 mm. The local maxima for all the wavelengths tested are shown in Table 2. Note that there is no data for first local maximum in Table 2 since the coupling efficiency between the first local maximum and the input pattern is too low. A comparison between Fig. 14 and Fig. 10 shows that the experimental results closely follow the simulation results. The positional differences between simulation results and experimental results are at most a few hundred microns. For instance, by comparison of the slopes of each curve we determine that for the second local maximum the experiment result is within 3.44% of the simulation result. The relative errors for local maxima are reasonable compared to those of the relative error of the MM fiber length (125/16700¼ 0.75%) and also there are a number of other error sources that are discussed later in Section 5. The trend in each case closely corresponds to our simulations results, which validates the process we used in our simulations.
5. Discussion and summary To fabricate and use an all-fiber device to phase-lock multiple fiber amplifiers there are significant additional challenges still to be faced. Multiple single mode fibers must be fused together and tapered to fit into the core area of a multi-mode fiber. In the experiments a Gaussian beam was expanded to cover the mask to get the initial 37 inputs which means the inputs are not Gaussian beams but flat top beams in each sub-aperture. Our simulations show that the coupling efficiency of Gaussian input beamlets is higher than that of flat top beamlets input. Also, when the center of the mask is misaligned with the center of the MMF, as shown in Fig. 15, the coupling efficiency is greatly reduced. In Fig. 15 the fiber end diameter is 430 mm including cladding. Another red
Table 2 Experimental results: local maxima positions at different wavelengths. Wavelength (nm)
740
760
780
800
820
840
2nd (mm) 3rd (mm) 4th (mm) 5th (mm)
11.881 13.206 14.682 16.158
11.580 12.754 14.290 15.706
11.354 12.4230 13.778 15.254
10.977 12.062 13.507 14.833
10.751 11.851 13.176 14.622
10.525 11.489 12.784 14.230
Fig. 13. Experimental intensity distribution for 740 nm input wavelength: (a) 2nd local maximum; (b) 3rd local maximum; (c) 4th local maximum and (d) 5th local maximum.
L. Wang et al. / Optics Communications 326 (2014) 57–63
Fig. 14. Experimental results: local maxima's positions as functions of the input wavelength. The inset formulas are best fits to the curves with y values in microns and the x values in nm.
Fig. 15. Photograph with annotations that indicate the mask misalignment from the fiber's center.
circle with diameter of 430 mm is added on the fiber end showing that the center of the pattern is off the center of the fiber by about 13 mm which greatly reduces the coupling efficiency, as seen by comparing Fig. 16 with Fig. 7. As shown in the experimental set-up, the Gaussian beam from the PCF is simply expanded to cover the whole mask which indicates that power and phase variations across the mask are unavoidable in our experiment. The maximum phase variation can be estimated on the edge of the pattern by using the equation: exp (ikr2/(2R)), where k is wave number for 740 nm wavelength, r is the radius of the hole pattern (100 um) and R is the radius of curvature of Gaussian beam on the mask. The separation between the mask and the PCF is about z ¼10 cm and the beam waist of the Gaussian beam from the PCF is around w0 ¼30 μm. Thus R can be calculated by using equation: R¼ z(1 þ (z0/z)2), where z0 ¼ πw20 l. By combining all the numbers, the phase difference between the center and the edge of the pattern is about 0.1349π, which is about
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Fig. 16. Coupling efficiency for BFL37-400 with 13 mm misaligned 37 flat top beamlets input.
6.75% of 2π with 740 nm input. Our simulations including the phase variations of the field across the mask produce results that differ by only a small amount from the results shown in Fig. 16. Combining phase variations and power variations in each beamlet the coupling efficiency along the propagation will be further reduced from that of the flap top assumption. The separation between adjacent inputs should also be optimized for the wavelength of interest. 22 mm separation between sub-apertures was chosen based on 632.8 nm wavelength. To optimize the coupling efficiency in 740–840 nm wavelength range, the separation should be adjusted and the length of the fiber should be finely controlled. Future design with optimization of the fiber mode size and separation for a specific MM fiber can greatly improve the efficiency. In our numerical simulations using 6 beamlets, efficiencies better than 30% were achieved and we should be able to greatly improve this result with a comprehensive design process. In conclusion, in this paper we have studied self-imaging revivals using a periodic array of sub-apertures in a multi-mode optical fiber. For our geometry there are two length scales in our simulations and experiments, one is the revival length which occurs on the scale of cm and the other is the Talbot imaging distance within the array, which is on the scale of mm. We corroborated the existence and scale of both lengths through simulations and experiments and the wavelength dependence of the results.
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