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Modified pinhole spatial filter producing a clean flat-topped beam
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Optics & Laser Technology 36 (2004) 151 – 153 www.elsevier.com/locate/optlastec
Modi$ed pinhole spatial $lter producing a clean &at-topped beam P. Hariharan∗ , Andal Narayanan Raman Research Institute, C.V. Raman Avenue, Bangalore 560 080, India Received 25 April 2003; accepted 29 July 2003
Abstract The expanded beam from a laser has a sharply peaked intensity pro$le. As a result, in many applications where uniform illumination of an extended $eld is required, it is only possible to use the central part of the beam. We show that if the pinhole normally used to spatially $lter the beam is replaced by an annular phase mask, it should be possible to obtain a clean beam providing very nearly uniform illumination over an extended area, with a minimal loss of light. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Laser beams; Beam expansion; Uniform illumination
1. Introduction The beam from a laser operating in the TEM00 mode typically has a diameter of only 1 or 2 mm and a Gaussian amplitude pro$le. To illuminate an extended area, it is necessary to expand the beam using a microscope objective with a pinhole spatial $lter placed at its focus to eliminate random di
Corresponding author. E-mail address: hariharan [email protected] (P. Hariharan).
0030-3992/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2003.07.006
noise. As a result, these devices cannot be used in an interferometer or in a holographic system. We show that by replacing the pinhole normally used to spatially $lter the beam by a simple annular phase mask, it should be possible to obtain a clean beam providing very nearly uniform illumination over an extended area, with a minimal loss of light.
2. Theoretical considerations 2.1. Pinhole with uniform transmittance The conventional spatial $lter uses a circular pinhole (radius r = 1) with a uniform amplitude transmittance which we can write as 1 for r ¡ 1; t(r) = (1) 0 for r ¿ 1; where t(r) is the amplitude transmittance at a distance r from the axis. The convergent beam incident on the pinhole has a Gaussian intensity pro$le. Accordingly, if the diameter of the pinhole is much greater than that of the beam waist, the expanded beam will also have a Gaussian pro$le. On the other hand, if the diameter of the pinhole is much smaller than the diameter of the beam waist, the incident amplitude over the pinhole can be taken to be constant. The di
152
P. Hariharan, A. Narayanan / Optics & Laser Technology 36 (2004) 151 – 153
is then given, apart from a constant of proportionality, by the integral 1 J0 (2r)2r dr; (2) A() = 0
where = (sin )= . This yields the well-known result, J1 () A() = (3) (the Airy function). In both cases, the intensity of the expanded beam drops o< rapidly as one moves away from the axis. 2.2. Pinhole mask with a radially varying amplitude transmittance If the pinhole is replaced with a mask having a radially varying complex amplitude transmittance t(r), the amplitude in the expanded beam at a radial distance from the axis is given by the integral [8] 1 A() = a(r)t(r)J0 (2r)2r dr; (4) 0
where a(r) is the complex amplitude of the beam incident on the mask. We can obtain an approximation to a &at-topped amplitude distribution by taking advantage of the observation that a reduction of the central peak, with a transfer of energy to the immediately surrounding regions, can be obtained with a very simple phase mask [9]. We consider a pinhole containing an annular phase mask with a complex amplitude transmittance t(r) where, as shown in Fig. 1, 1 for r 6 rm ; t(r) = −1 for r ¿ rm ; (5) 0 for r ¿ 1:
Fig. 2. Steps in the production of the pinhole phase mask.
Fig. 2 shows how such a mask could be produced in two steps by electron-beam etching of thin $lms deposited on a glass substrate. In the $rst step, as shown in Fig. 2(a), an opaque $lm is deposited on the substrate and a pinhole (radius r) is etched in it. In the second step, as shown in Fig. 2(b), a transparent $lm (refractive index n) of thickness d = =2(n − 1)
(6)
is deposited on the same substrate, and a concentric hole with a radius rm is etched in it. We will assume that the phase mask is placed in the focal plane of the microscope objective and the convergent beam incident on the mask has a Gaussian amplitude pro$le. We will also assume that the outer diameter of the mask (r = 1) is equal to that of the beam waist, so as to use most of the available light while ensuring a clean output beam. We can then optimize the transmittance function t(r) by evaluating the di
Fig. 1. Amplitude transmittance function for the pinhole phase mask.
Fig. 3 shows the normalized amplitude distribution A() in the expanded beam from a pinhole with uniform transmittance (rm = 1:0), along with the amplitude distribution for pinholes incorporating annular phase masks with values of rm of 0.80 and 0.75. As can be seen, with a pinhole incorporating an annular phase mask having a value of rm = 0:75, it is possible to obtain an almost &at-topped beam in which the amplitude at a distance from the axis corresponding to a value of = 0:62 is the same as that at the centre ( = 0). Over this region, the variation in the value of the amplitude, with respect to its average value, is only around ±10%. It is also
P. Hariharan, A. Narayanan / Optics & Laser Technology 36 (2004) 151 – 153
153
A minor problem is the need to match the diameter of the pinhole mask to that of the beam waist. This problem can be solved quite easily by inserting a weak concave lens in the beam incident on the microscope objective and adjusting its distance from the microscope objective so as to optimize the intensity distribution in the expanded beam. References
Fig. 3. Normalized amplitude distribution in the expanded beam for a pinhole with uniform transmittance (rm = 1:0) and for pinholes incorporating annular phase masks with values of rm of 0.80 and 0.75.
possible to obtain a &at-topped beam exhibiting a negligible variation in amplitude for values of up to 0.34 with an annular phase mask having a value of rm = 0:8. This improvement in uniformity is obtained with a minimal loss of light since, with a phase mask, no light is lost by absorption. Most of the light in the central part of the beam is redistributed in the adjacent regions and there is only a small increase in the amount of light di
[1] Hariharan P. Optical holography. Cambridge: Cambridge University Press; 1996. p.74 –5. [2] Chang SP, Kuo J-M, Lee Y-P, Ling K-J. Transformation of Gaussian to coherent uniform beams by inverse-Gaussian transmittance $lters. Appl Opt 1998;37:747–52. [3] Xie C, Gupta R, Metcalf H. Beam pro$le &attener for Gaussian beams. Opt Lett 1993;18:173–5. [4] Kasinski JJ, Burnham RL. Near-di
Optics & Laser Technology 36 (2004) 151 – 153 www.elsevier.com/locate/optlastec
Modi$ed pinhole spatial $lter producing a clean &at-topped beam P. Hariharan∗ , Andal Narayanan Raman Research Institute, C.V. Raman Avenue, Bangalore 560 080, India Received 25 April 2003; accepted 29 July 2003
Abstract The expanded beam from a laser has a sharply peaked intensity pro$le. As a result, in many applications where uniform illumination of an extended $eld is required, it is only possible to use the central part of the beam. We show that if the pinhole normally used to spatially $lter the beam is replaced by an annular phase mask, it should be possible to obtain a clean beam providing very nearly uniform illumination over an extended area, with a minimal loss of light. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Laser beams; Beam expansion; Uniform illumination
1. Introduction The beam from a laser operating in the TEM00 mode typically has a diameter of only 1 or 2 mm and a Gaussian amplitude pro$le. To illuminate an extended area, it is necessary to expand the beam using a microscope objective with a pinhole spatial $lter placed at its focus to eliminate random di
Corresponding author. E-mail address: hariharan [email protected] (P. Hariharan).
0030-3992/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2003.07.006
noise. As a result, these devices cannot be used in an interferometer or in a holographic system. We show that by replacing the pinhole normally used to spatially $lter the beam by a simple annular phase mask, it should be possible to obtain a clean beam providing very nearly uniform illumination over an extended area, with a minimal loss of light.
2. Theoretical considerations 2.1. Pinhole with uniform transmittance The conventional spatial $lter uses a circular pinhole (radius r = 1) with a uniform amplitude transmittance which we can write as 1 for r ¡ 1; t(r) = (1) 0 for r ¿ 1; where t(r) is the amplitude transmittance at a distance r from the axis. The convergent beam incident on the pinhole has a Gaussian intensity pro$le. Accordingly, if the diameter of the pinhole is much greater than that of the beam waist, the expanded beam will also have a Gaussian pro$le. On the other hand, if the diameter of the pinhole is much smaller than the diameter of the beam waist, the incident amplitude over the pinhole can be taken to be constant. The di
152
P. Hariharan, A. Narayanan / Optics & Laser Technology 36 (2004) 151 – 153
is then given, apart from a constant of proportionality, by the integral 1 J0 (2r)2r dr; (2) A() = 0
where = (sin )= . This yields the well-known result, J1 () A() = (3) (the Airy function). In both cases, the intensity of the expanded beam drops o< rapidly as one moves away from the axis. 2.2. Pinhole mask with a radially varying amplitude transmittance If the pinhole is replaced with a mask having a radially varying complex amplitude transmittance t(r), the amplitude in the expanded beam at a radial distance from the axis is given by the integral [8] 1 A() = a(r)t(r)J0 (2r)2r dr; (4) 0
where a(r) is the complex amplitude of the beam incident on the mask. We can obtain an approximation to a &at-topped amplitude distribution by taking advantage of the observation that a reduction of the central peak, with a transfer of energy to the immediately surrounding regions, can be obtained with a very simple phase mask [9]. We consider a pinhole containing an annular phase mask with a complex amplitude transmittance t(r) where, as shown in Fig. 1, 1 for r 6 rm ; t(r) = −1 for r ¿ rm ; (5) 0 for r ¿ 1:
Fig. 2. Steps in the production of the pinhole phase mask.
Fig. 2 shows how such a mask could be produced in two steps by electron-beam etching of thin $lms deposited on a glass substrate. In the $rst step, as shown in Fig. 2(a), an opaque $lm is deposited on the substrate and a pinhole (radius r) is etched in it. In the second step, as shown in Fig. 2(b), a transparent $lm (refractive index n) of thickness d = =2(n − 1)
(6)
is deposited on the same substrate, and a concentric hole with a radius rm is etched in it. We will assume that the phase mask is placed in the focal plane of the microscope objective and the convergent beam incident on the mask has a Gaussian amplitude pro$le. We will also assume that the outer diameter of the mask (r = 1) is equal to that of the beam waist, so as to use most of the available light while ensuring a clean output beam. We can then optimize the transmittance function t(r) by evaluating the di
Fig. 1. Amplitude transmittance function for the pinhole phase mask.
Fig. 3 shows the normalized amplitude distribution A() in the expanded beam from a pinhole with uniform transmittance (rm = 1:0), along with the amplitude distribution for pinholes incorporating annular phase masks with values of rm of 0.80 and 0.75. As can be seen, with a pinhole incorporating an annular phase mask having a value of rm = 0:75, it is possible to obtain an almost &at-topped beam in which the amplitude at a distance from the axis corresponding to a value of = 0:62 is the same as that at the centre ( = 0). Over this region, the variation in the value of the amplitude, with respect to its average value, is only around ±10%. It is also
P. Hariharan, A. Narayanan / Optics & Laser Technology 36 (2004) 151 – 153
153
A minor problem is the need to match the diameter of the pinhole mask to that of the beam waist. This problem can be solved quite easily by inserting a weak concave lens in the beam incident on the microscope objective and adjusting its distance from the microscope objective so as to optimize the intensity distribution in the expanded beam. References
Fig. 3. Normalized amplitude distribution in the expanded beam for a pinhole with uniform transmittance (rm = 1:0) and for pinholes incorporating annular phase masks with values of rm of 0.80 and 0.75.
possible to obtain a &at-topped beam exhibiting a negligible variation in amplitude for values of up to 0.34 with an annular phase mask having a value of rm = 0:8. This improvement in uniformity is obtained with a minimal loss of light since, with a phase mask, no light is lost by absorption. Most of the light in the central part of the beam is redistributed in the adjacent regions and there is only a small increase in the amount of light di
[1] Hariharan P. Optical holography. Cambridge: Cambridge University Press; 1996. p.74 –5. [2] Chang SP, Kuo J-M, Lee Y-P, Ling K-J. Transformation of Gaussian to coherent uniform beams by inverse-Gaussian transmittance $lters. Appl Opt 1998;37:747–52. [3] Xie C, Gupta R, Metcalf H. Beam pro$le &attener for Gaussian beams. Opt Lett 1993;18:173–5. [4] Kasinski JJ, Burnham RL. Near-di