Width of the apodization area in the case of diffractive optical elements with variable efficiency

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1 Width of the apodization area in the case of diffractive optical elements with variable efficiency Tomasz Osuch 1, Zbigniew Jaroszewicz 1,, Andrzej Kołodziejczyk 3 1 National Institute of Telecommunications, Laboratory of Electrical, Electronic and Optoelectronic Metrology, Szachowa 1, Warsaw, Poland Institute of Applied Optics, Department of Physical Optics, Kamionkowska 18, Warsaw, Poland 3 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, Warsaw, Poland ABSTRACT We present simulations of averaged intensity of light behind apodized phase masks. Two types of apodization profile were assumed: Gaussian and tanh. In reality, because of limitations of electron-beam exposure system used for phase mask fabrication, we simulated phase masks with eight values of step height. For comparison, the averaged intensity distributions behind ideal phase masks with variable intensity were also calculated. Simulations and description of intensity distribution perturbations due to phase jumps in real apodized phase masks were performed. Keywords: diffractive optics, apodization 1. INTRODUCTION In the case of diffractive optical elements the implementation of apodizing function by a local change of their diffraction efficiency was proposed recently. There were described various methods for achieving this goal: the duty ratio s change of the binary amplitude grating [1],[], the phase step s splitting of the binary phase grating [3] and the gradual transformation of the step-like kinoform into its conjugate across the apodization region [4]. However, in the majority of cases the apodization function was varying along the length of the elements periods, what means that the contribution of every period to the given point within the apodization range was identical. Quite different situation can be encountered in the case, when the apodization function runs perpendicularly to the periods length. Such situations takes place in the case of apodized phase masks designed for the exposure of apodized fiber Bragg gratings (FBG). Since contribution of every period will be then different, it could be expected that the width of the apodization range should be significantly greater than the periods width. In the present contribution there are made numerical evaluations of the width of the apodization area vs. the period of the grating, which allows determining the admissible minimal width of the apodization range. The spectral characteristics of uniform fiber Bragg gratings exhibit large sidelobes around the main peak of reflection. To minimize these sidelobes it is necessary to decrease modulation of refractive index in the fiber core at the ends of the gratings [5],[6]. It is usually achieved by use of phase masks with variable diffraction efficiency in the fabrication process, which are binary phase gratings with variable height of the phase steps. The main advantage of this method is based on maintenance the effective index of refraction constant along the Bragg grating s length and prevents broadening of their spectral characteristics simultaneously [7]. Design of apodized phase masks for fiber Bragg gratings fabrication consists of the following steps: - choosing an apodization profile. Typical and most often applied apodization profiles are: Gaussian, cosine, raised cosine and tanh [6]. - determining of phase distribution along mask using formula, which describes diffraction intensities I(ϕ) of n±1 orders versus phase step height ϕ [8] Photon Management II, edited by John T. Sheridan, Frank Wyrowski, Proc. of SPIE Vol. 6187, 61871G, (006) X/06/$15 doi: / Proc. of SPIE Vol G-1 Downloaded from SPIE Digital Library on 14 Mar 011 to Terms of Use:

2 iϕ n πn ϕ ( ϕ) A exp( iπn)exp sin c cos I n (1) - determining of step height distribution of the phase mask [9] ϕ ϕ d π λ d () π ( n 1) ϕ where d is step height of the phase mask which guarantee minimization of 0 diffraction order and maximization of ±1 diffraction orders according to the following formula: d λ ( n ) 1 (3) and λ is laser wavelength used for exposure, n the index of refraction of the phase mask material for wavelength λ [10]. Ideal phase mask has continuous change of diffraction efficiency, i.e. each period has slightly different step height corresponding to the apodization profile 1. However, real apodized phase mask must consist of some subsections, which have appropriate step heights, and therefore its profile have staircase shape because of technological limitations imposed by some of existing facilities for electron beam lithography. Discontinuity in diffraction efficiency (step heights) where subsections join, contribute to perturbations of intensity distribution behind the phase mask.. SIMULATIONS Our simulations are based on non-paraxial scalar diffraction theory [11] and numerical approach proposed in [1],[13]. All procedures were written in Matlab. Taking into account geometry of the system for fabrication apodized FBG using phase mask and mechanism of theirs production (creation of periodic modulation of refractive index in fiber core due to place the fiber in the diffractive field of the phase mask), it was assumed that diffractive field I av (z), which takes part in FBG formation is an arithmetic mean of distributions I i (z) in planes with the range of core diameter. Therefore I av (z) can be expressed as follows where: I av 1 N + 1 N ( z) I ( z) i N i d zi z f + i (5) N (4) and: I i (z) simulation s result of intensity distribution in distance z i from phase mask z f distance between optical fiber and phase mask d fiber core diameter N+1 number of planes (within the precincts of core diameter) in which the intensity distribution is calculated. 1 In reality ideal apodized phase mask should have continuous change of diffraction efficiency and infinitesimal differences of the step heights between the following periods. Proc. of SPIE Vol G- Downloaded from SPIE Digital Library on 14 Mar 011 to Terms of Use:

3 In our simulations parameters in accordance with real setup geometry (figure 1) were assumed. Therefore the following values were established: d9 µm (standard single-mode optical fiber), phase mask period Λ PM 1060 nm and incident laser wavelength λ 44 nm. Moreover, in the case of simulation averaged distribution of intensity, 1 number of planes and z f 150µm were assumed. z x laser beam phase mask z f d optical fiber Figure 1. Setup for apodized fiber Bragg gratings fabrication using phase mask with variable diffraction efficiency. Firstly averaged distribution intensities were obtained I av for ideal and real (consists of eight step heights) apodization profiles with Gaussian and tanh(3,3) shapes. These profiles are described by the following formulas: Gaussian: y( x) exp( x ) L L for x ; (6) tanh(a,b): y ( x) B 1+ tanh A 1 1 x L B x L 1+ tanh A 1 L for for x x L 0; L ;L (7) where: L - phase mask length; A, B - parameters [6]. In our case A3, and B3 were assumed. Ideal apodization profiles (formulas (6) and (7)) and their staircase equivalents (real profiles) are shown on figure. In simulations eight values of step height were assumed because of limitation of electron-beam exposure system ZBA-0, which will be used in further experiments. According to the Talbot effect in the case of pure phase diffractive optical elements (for example phase mask), it is well known that the intensity is constant in plane described by the following formula z 0 Λ v λ PM (8) where v - integer and Λ PM phase mask period [14]. Therefore simulations of intensity distribution were prepared for propagations distances which satisfy formula (8), because in these planes perturbations are most visible. It was also taken into consideration that in the region of optical fiber self-image planes should be calculated using following equation z 0 Λ vn λ PM (9) Proc. of SPIE Vol G-3 Downloaded from SPIE Digital Library on 14 Mar 011 to Terms of Use:

4 where n is an index of refraction in glass. Because of indices of refraction in fiber core n co and cladding n cl slightly differs, it was assumed that nn cl n co. a C 0 a 0 a -D0 N a -D a Na E 0 z Phase mask length Figure. Ideal and real (staircase) apodization profiles: Gaussian and tanh(3,3). Moreover, simulations of averaged intensity distribution versus propagation distance were performed. In this case parts of the total distribution intensities (where diffraction efficiency changes abruptly) are shown on figure 1. Characteristic of width of perturbation area with the distance between phase mask and fiber was also determined (figure 13). In this work we mainly focused on perturbation of apodization profile, therefore it is sufficient to observe envelope of distribution intensity. In reality, results of our simulations are periodical changes of intensities (see inset in figure 5), and period of the intensity distribution is half of the phase mask period. However for the sake of large number of phase mask periods ( ), only envelopes of intensity distributions can be noticeable. 3. RESULTS With our point of view only upper part of averaged distribution intensity (between values 1 and ) is interesting (figure 3). This part of characteristic corresponds to the apodization profile. Whereas lower part of averaged intensity distribution is connected with 0 diffraction order, and can be omitted in further figures Averaged intensity distribution behind the phase masks Figures 4 7 show averaged intensity of ideal and real phase masks. o Figure 3. Averaged intensity distribution I av behind the ideal phase mask with Gaussian profile - total characteristic. Proc. of SPIE Vol G-4 Downloaded from SPIE Digital Library on 14 Mar 011 to Terms of Use:

5 8 Figure 4. Averaged intensity distribution I av behind the ideal phase mask with Gaussian profile - interesting part of total characteristic (envelope of averaged intensity distribution) Figure 5. Averaged intensity distribution I av behind the real phase mask with Gaussian profile. Figure 6. Averaged intensity distribution I av behind the ideal phase mask with tanh(3,3) profile. Figure 7. Averaged intensity distribution I av behind the real phase mask with tanh(3,3) profile. Proc. of SPIE Vol G-5 Downloaded from SPIE Digital Library on 14 Mar 011 to Terms of Use:

6 Averaged intensities of ideal phase masks (Gaussian and tanh(3,3)) do not exhibit perturbations because of very little changes in step heights in adjacent periods. Quite different situations can be observed in the case of real phase masks. Here deformations of averaged intensity are observed caused by discontinuity in staircase apodization profiles. Despite of this disadvantage, overall apodization profiles are maintained even for only eight level profile (eight step heights). 3.. Intensity distribution of the phase masks in Talbot plane Below (figures 8-11), there are results of simulations of intensities behind phase masks (real and ideal) in plane z 0, where intensity distribution is constant. U 4 8 lb Phase mask length [mm] Figure 8. Intensity distribution behind the ideal phase mask with Gaussian profile in plane z 0. lisniunil U Phase mask length [mm] Figure 9. Intensity distribution behind the real phase mask with Gaussian profile in plane z 0. - U 4 8 lb Phase mask length [mm] Figure 10. Intensity distribution behind the ideal phase mask with tanh(3,3) profile in plane z 0. U 4 8 Phase mask length [mm] Figure 11. Intensity distribution behind the real phase mask with tanh(3,3) profile in plane z Proc. of SPIE Vol G-6 Downloaded from SPIE Digital Library on 14 Mar 011 to Terms of Use:

7 Intensity distributions in plane z 0 behind ideal phase masks with continuous changes of diffraction efficiency (figures 8 and 10) manifest very small distortions because of differences between the following step heights in apodization profile are not infinitesimal. Perturbations are especially visible in steep parts of apodization profiles. In the case of intensity distributions of real phase masks, levels of perturbation are considerably larger. Moreover, on figure 11 it should be noted, that perturbations, originated from stepped changes in apodization profile overlap, because of too small width of the apodization areas (size of subsections) Averaged intensity distribution versus propagation distance Figures 1, 13 shows averaged intensity distribution and perturbations spread out respectively vs. propagation distance. 1!,iifl 1.75 a) Length [mm] b) Length [mm] c) Length [mm] Length [mm] d) Figure 1. Averaged intensity distribution versus propagation distance z for: a) z5z T, b) z10z T, c) z15z T, d) z0z T, where z T is Talbot distance and z T z 0 (see equations (8) and (9)). V Co 14 1 xl 0 V C C z o Co 63 Co C Multiplication factor of Talbot distance Figure 13. Characteristic of perturbations spread out versus distance between phase mask and optical fiber. Proc. of SPIE Vol G-7 Downloaded from SPIE Digital Library on 14 Mar 011 to Terms of Use:

8 According to figure 1 it was noticed that the width of the perturbation area is proportional to the propagation distance (distance between phase mask and observation plane). Spread out of disturbance depending on propagation distance is shown on figure 13, where distances between characteristic points of perturbation were determined on the basis of plots of averaged intensity distributions similar to these from figure 1. Together with increase of perturbation area it can be observed that amplitude of disturbances decreased. Dependence between width of perturbation area and propagation distance is linear. 4. CONCLUSIONS In the case of apodized phase masks, each departure from ideal, continuous profile is source of perturbation in distribution intensity. Large changes of diffraction efficiency (step heights) produce perturbations, which spreads out across distribution intensity (along x axis figure 1) versus propagation distance. Moreover, when widths of apodization areas are too small, adjacent perturbations overlap and distribution of intensity is then distorted. Each inhomogeneity can be treated as source of spherical wave, which superimposes on ideal structure. Additionally, in connection with propagation of spherical wave phenomenon and on the basis of our simulations, amplitude of this perturbation decreases and disturbances spreads out linear versus propagation distance. Take into consideration above conclusions, during the design of phase masks with variable diffraction efficiency it is necessary to remember about assurance of minimal width of apodization area, in order to avoidance of situations, when perturbations overlap. Propagation distance should also be taken into account, and as small as possible distance between phase mask and optical fiber should be achieved during fiber Bragg gratings fabrication. Shape of averaged intensity distribution is slightly different than apodization profile (especially in the case of Gaussian function), because of dependence If(ϕ) (see equation 1) for ±1 diffraction orders is nonlinear. 5. ACKNOWLEDGEMENT We acknowledge the support of the NEMO (Network of Excellence on Micro-Optics), in which National Institute of Telecommunications and Warsaw University of Technology participate. REFERENCES 1. N. Château, D. Phalippou, and P. Chavel, "A method for splitting a gaussian laser beam into two coherent uniform beams", Optical Communications 88, 33-36, S.Yu. Popov and A.T. Friberg, "Apodization of generalized axicons to produce uniform axial line images", Pure Appl. Opt. 7, , S.Yu. Popov, A.T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, "Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography", Opt. Commun. 154, , Z. Jaroszewicz, A.T. Friberg, and S.Yu. Popov: "Kinoform apodization", J. Mod. Opt. 47, , A. Othonos, K. Kalli, Fiber Bragg Gratings. Fundamentals and Applications in Telecommunications and Sensing, Artech House Optoelectronics Library, R. Kashyap, Fiber Bragg Gratings, Academic Press, J. Albert, K.O. Hill, B. Malo, S. Thériault, F. Bilodeau, D.C. Johnson and L.E. Erickson Apodisation of spectral response of fibre Bragg gratings using a phase mask with variable diffraction efficiency, Electronics Letters Vol.31 No.3, -3 (1995). 8. Z. Jaroszewicz, A. Kołodziejczyk, A. Kowalik, R. Restrepo Determination of the step height of the binary phase grating from its Fresnel images Optik 111, 07-10, T. Osuch, Z. Jaroszewicz, Apodized diffractive optical elements for fiber Bragg gratings fabrication, in European Optical Society Topical Meeting on Diffractive Optics, 3 September - 7 September 005, Warsaw, Poland. 10. X.Liu, J.S. Aitchison, R.M. De La Rue, S. Thoms, L. Zhang, J.A.R. Williams and I. Bennion Electron beam lithography of phase mask gratins for near field holographic production of optical fibre gratings, Microelectronic Engineering 35, , J.W. Goodman, Introduction to Fourier optics, McGraw-Hill, New York, Proc. of SPIE Vol G-8 Downloaded from SPIE Digital Library on 14 Mar 011 to Terms of Use:

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