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Array illuminator based on self-imaging in white light
ARTICLE IN PRESS
Optics & Laser Technology 38 (2006) 636–640 www.elsevier.com/locate/optlastec
Array illuminator based on self-imaging in white light Shashi Prakash, Anupam Sharma, Santosh Rana School of Instrumentation, Devi Ahilya University, Khandwa Road, Indore-452017, India Received 7 July 2004; accepted 11 January 2005 Available online 23 February 2005
Abstract An array illuminator is a device/optical system that splits an incoming beam of light into an array of many light spots/beams. It is useful in logic systems and in optical digital computing or in switching systems to energize arrays of components such as logic gates, optically bistable devices and electro-optic modulators. Array illuminators working in white light are interesting because they do not suffer from coherent noise and yield better signal-to-noise ratio. We propose configurations for array illuminator based on Fresnel diffraction and working under white light illumination conditions. Performance evaluation of the illuminators is also undertaken and the results of investigation are reported. r 2005 Elsevier Ltd. All rights reserved. Keywords: Self-imaging; Array illuminator; White light
1. Introduction The main job of array illuminator is to transform a uniform beam into an array of uniform light spots. Many different approaches for array generation have been proposed. These are based on Fraunhofer diffraction at specially designed grating [1], phase contrast imaging [2], on Fresnel diffraction at certain gratings [3], optical coordinate transformations [4], cascades of beam splitting devices [5], pin holes, mirrors (kaleidoscopes), arrays of microlenses [6], arrays of microtelescopes, etc. A review article by Streibl [7] illustrates these devices and their fabrication techniques. Array illuminators using interferometric principles have also been suggested. Several configurations using combination of two wedge plate orientations [8], Michelsons interferometer in tandem [9] and holographic-based methods [10] have been reported. Most of the array illuminators work under coherent illumination conditions; however unless required by a particular application, incoherent methods Corresponding author. Tel.: +91 731 2462228; fax: +91 731 2470372. E-mail addresses: [email protected], [email protected] (S. Prakash).
0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.01.002
of array generation are preferred because of their high signal-to-noise ratio. Self-imaging had been used widely for applications in diverse areas [11]. Recently their application in vibration monitoring [12], collimation testing [13], displacement measurement [14], temperature measurement [15] and surface topography [16] has also been reported. In this paper, we present few novel methods of array generation using self-imaging in white light.
2. Experimental arrangement Fig. 1 shows schematic of experimental arrangement for array illuminator based on Lau effect. Halogen lamp of 150 W has been used as a extended white light source (EWLS). The extended source is condensed by a lens and illuminates the gratings G1 and G2, which are identical and are of equal periods. Grating G1 is arranged by placing two identical linear gratings each of pitch 0.4 mm in mutually perpendicular directions in tandem. G2 is also arranged such that it is formed by two gratings of equal periods in two perpendicular directions as in G1. The gratings are properly aligned
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Fig. 1. Array generation using Lau effect.
and a variable focal length system is used to focus the beam onto the phase plate of CCD. The images are acquired using a PC and a image-grabbing card, PCI1409 supplied by National Instruments, US. For image acquisition, the IMAQ software supplied by National Instruments was used and the results were displayed online via the computer monitor. The working of the interferometer can be understood in the following manner. The light emitted from a single pinhole opening in the grating G1 is spatial coherent. It produces an interference pattern at infinity after being split by grating G2. But the light emitted from different pinholes in G1 is spatially incoherent, so the interference pattern generated by all pinholes in G1 will add up in intensities. Under the conditions for self-imaging nd 21y nd 21x ; Z Oy ¼ , (1) 2l 2l where l is the average wavelength of light used, n is an integer and d 1x ¼ d 1y ; d 1x and d 1y are grating constants in the x and y directions, respectively. The maxima of all patterns coincide and the resulting pattern is of high contrast. Fig. 2(a) shows the experimental result recorded via a CCD camera at the image plane. The line profile of intensity at the CCD plane is also shown in Fig. 2(b). Fig. 3 shows the experimental arrangement for array illuminator based on Talbot effect under white light illumination. It consists of a source grating G1 and two identical gratings G2, G3. Each of the gratings G2 and G3 is a crossed grating assembled by superposing in tandem two-linear gratings of pitch 0.4 mm in mutually perpendicular direction. The grating G1 acts so as to enhance the spatial coherence of the waves emitted by the EWLS. The spherical waves after grating G1 are collimated using collimated lens, Lc. Beyond grating G2, the self-images of grating are formed at planes defined by Z Ox ¼
2d 21y 2d 21x ; Z Ty ¼ n , (2) l l where l is the wavelength of light used, n is an integer and d 1x ¼ d 1y ; d 1x and d 1y are grating periods in the x and y directions, respectively. Z Tx ; Z Ty correspond to ‘Talbot distance’. At the Talbot distance corresponding to 450 mm, the grating G3 is placed. The moire´ Z Tx ¼ n
Fig. 2. (a) Experimental result recorded via CCD camera on the experimental set up as in Fig. 1. (b) The line profile of intensity across the image, as recorded at the CCD plane.
Fig. 3. Array generation using Talbot effect in white light.
pattern resulting because of misalignment between different gratings has to be removed to get the array illumination at the image plane. Fig. 4(a) shows image recorded at image plane using CCD camera. The line profile of intensity at the CCD plane is also shown in Fig. 4(b). The problem of fringe intensity profile in case of self-imaging phenomenon using incoherent light has been addressed by Patorski in series of papers [17,18]. Since the array illuminators are based on principle of
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s1 and d 1 are the slit width and period of G1; z is the self-imaging distance, n d 22 =l: Here n, d 2 and l are an integer, period and wavelength of light, respectively. Hence the opening ratio of the second function becomes nðs1 =d 1 Þ: Having established that the opening ratios of the two functions are convolved, the results of the extensive analysis presented in the referenced paper [19] can directly be applied to yield the parameters of the trapezoidal profile. As per the analysis, the width of the base of the trapezoid is proportional to the sum of the numerators in the opening ratios of the gratings at the image plane. The width of the upper parallel arm is proportional to the difference between the numerators in the grating opening ratios. The amplitude of the trapezoid function (i.e. the maximum intensity) is that intensity which emerges from the grating with smaller opening ratio in the absence of the other grating. The slopes of the profile elements will be identical for all opening ratio combinations and equal to 45 : Throughout the discussion, it has to be taken into consideration that trapezoid parameters be normalized to grating period, and the grating period be normalized to unity. The analysis can similarly be extended to understand the variation of pitch parameters on the intensity line profile of the Talbot effect-based array illuminator. Fig. 4. (a) Experimental result recorded via CCD camera on the experimental set up as in Fig. 3. (b) The line profile of intensity across the image, as recorded at the CCD plane.
self-imaging in incoherent light, the applicability of the theory is direct. Mathematically, in the case of Laubased array illuminator, the intensity of the doubly diffracted Fresnel field is given by convolving the selfimage generated by point source with the actual function of intensity distribution of source grating. For binary type of gratings used in the setup, the convolution operation results in general in a trapezoidal fringe profile. This is clearly visible in the experimental results shown in Figs. 2(b) and 4(b). The parameters of the trapezoidal waveform (such as width of the base and parallel arm, height and the slopes of profile elements) depend on the opening ratios of the two binary functions (Ronchi-type gratings) being convolved, at the image plane. The opening ratios are defined as the ratio of slit width to the grating period ðs=dÞ: Since we are investigating the fringe profiles in the in-registry planes coinciding with the self-image planes of grating, G2, the opening ratio of the rectangular function describing the self-image intensity pattern is equal to the opening ratio of G2 ðs2 =d 2 Þ: On the other hand, the width of slit image of source grating in observation plane is given by ðs1 =d 1 Þ ðl=d 2 Þ z; where
3. Results and discussion Various parameters to describe the performance of an array illuminator were discussed in detail by Streibl et al. [6]. To determine the quality of array generated, we evaluated parameters describing their performance. Splitting ratio for both types of illuminators is determined to be 99 99: The homogeneity in the intensity for both the array illuminators has been found to be good. However, white-light Talbot array spots are more homogenous than Lau-based array illuminator. Compression ratio is the quotient of bright area of the spot pattern and the whole area of the elementary cell of the spot pattern. Hence, the compression ratio is used to indicate separation between the array spots. It is determined to be 5 in case of Talbot-based illuminator and 4 in case of Lau-based illuminator. Background suppression ratio is defined as the ratio between spot intensity and the background stray light present in the illuminator. Hence background suppression ratio measures the noise of array illuminator in bright and dark areas of the array. In nutshell, background suppression ratio is direct measure of image contrast. Very low values of background suppression ratio indicate poor contrast and high values indicate good image contrast among various array elements. It is determined to be 2.08 in Lau-based illuminator and 4.25 in Talbot-based system. Other parameters such as splitting ratio,
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4. Conclusions In summary, communication presents configurations for array illuminator based on self-imaging and working in white light. The performance characteristics of the Talbot effect-based array illuminator are better than Lau effect-based array illuminator. However, the array illuminators devised using amplitude gratings do not offer high efficiency. Hence configurations for achieving high efficiency in the array illuminators are also suggested. Fig. 5. Schematic for typical Lau-based efficient array illuminator using phase gratings.
Acknowledgement compression ratio, homogeneity, and efficiency though do not appear directly related to image contrast, but poor contrast shall severely degrade the performance of the illuminator. This shall induce errors, which may eventually undermine the experimental evaluation of these parameters. The attractive features of the illuminators include the ease of fabrication of gratings and their low cost. However as we have realized illuminators using amplitude gratings, the efficiency of the illuminator is not high. The signal-to-noise ratio and the efficiency of the array illuminator based on Talbot effect under white-light illumination can be improved by choosing phase gratings instead of amplitude gratings and modifying the distance between two gratings suitably, based on theory and reasoning put forward by Lohmann [3]. The signal-tonoise ratio and efficiency in case of Lau-based illuminator can also be improved considerably by using phase gratings instead of amplitude gratings, but using laser as source. Schematic for typical Lau-based efficient array illuminator is illustrated in Fig. 5. The collimated laser light illuminates a crossed phase grating, which produces high contrast Fresnel diffraction pattern that is periodic and has binary irradiance distribution. This high contrast binary irradiance distribution is converted to incoherent source with periodic irradiance using a rotating scatter plate, which destroys the coherence between any point over the Fresnel diffraction pattern. Since the mutual coherence of waves beyond the rotating reflector has been destroyed, the results at image plane will not be afflicted with the coherence noise. The conditions of ‘self-imaging’ are satisfied if another crossed phase grating is placed at distance ‘Z 0 ’ between the scatter plate and the grating G2 such that [20] Z 0 ¼ 2ðE þ n0 Þ
d2 , l
(3) 3
where E ¼ 0; 1; 2; . . . ; n0 ¼ 14; 4 ; l is the wavelength of light used and d the pitch of the grating. In this case, we obtain the array illuminator of very high efficiency at the focal plane of the lens.
This research was performed under the research project [Grant no. 03(0938)/02/EMR-II] funded by Council of Scientific and Industrial Research (CSIR), New Delhi. Authors gratefully acknowledge the financial assistance provided by CSIR, New Delhi. Authors are also grateful to the referee’s suggestions/comments towards improving the manuscript.
References [1] Lee WH. High efficiency multiple beam gratings. Appl Opt 1979; 18:2152–8. [2] Glu¨ckstad J, Mogensen PC. Reconfigurable ternary-phase array illuminator based on the generalised phase contrast method. Opt Commun 2000;173:169–75. [3] Lohmann AW. An array illuminator based on Talbot effect. Optik 1988;79:41–5. [4] Bryngdahl O. Geometrical transformations in optics. J Opt Soc Am 1974;64:1092–8. [5] Kubota T, Takeda M. Array illuminator using grating couplers. Opt Lett 1989;14:651–2. [6] Streibl N, Nalscher U, Jahns J, Walker SJ. Array generation with lenslet arrays. Appl Opt 1991;30:2739–42. [7] Streibl N. Beam shaping with optical array generation. J Mod Opt 1989;38:1559–73. [8] Senthilkumaran P, Sirohi RS. Michelson interferometer in tendom for array generation. Opt Commun 1994;105:158–60. [9] Senthilkumaran P, Sriram KV, Kothiyal MP, Sirohi RS. Array generation using double wedge plate interferometer. J Mod Opt 1994;41:481–9. [10] Bhattacharya S, Senthilkumaran P, Darlin JS, Kothiyal MP, Sirohi RS. Holographic array illuminators using tendom Michelson interferometers; fabrication and analysis. J Mod Opt 1995;42: 2275–83. [11] Patorski K. The self imaging phenomenon and its applications. In: Wolf E, editor. Progress in optics XXVII. Netherland: Elsevier; 1989. p. 1–110. [12] Prakash S, Upadhyay S, Shakher C. Real time out-of-plane vibration measurement using Talbot interferometry. Opt Lasers Eng 2004;34:251–9. [13] Shakher C, Prakash S, Dayanand X, Kumar R. Collimation testing with circular gratings. Appl Opt 2001;40:1175–9. [14] Spagnolo GS, Amrosini D, Paoletti D. Displacement measurement using the Talbot effect with Ronchi grating. J Opt A: Pure Appl Opt 2002;4:5376–80.
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[15] Thakur M, Angra SK, Shakher C. Measurement of temperature profile of the 2-D slot burner using Lau phase interferometry with linear gratings. Opt Eng 2003;42:85–91. [16] Mirza S, Faridi MS, Kumar R, Vyas AL, Shakher C. Surface topography of steps and dips by phase shifting Talbot interferometric technique. In: Proceedings of the international conference on laser applications and optical metrology, New Delhi; 1–4 December 2003. p. 418–21.
[17] Patorski K. Incoherent superposition of multiple self-imaging Lau effect and moire´ fringe explanation. Opt Acta 1983;30:745–58. [18] Patorski K. Incoherent superposition of multiple self-imaging under plane wave-front illumination. Appl Opt 1986;25:2396–403. [19] Patorski K, Yokozeki S, Suzuki T. Moire´ profile prediction using Fourier series formalism. Jap J Appl Phys 1976;15:443–56. [20] Barreiro JC, Andres P, Ojeda-Castaneda J. Lau effect with only phase gratings. Opt Commun 1989;73:106–10.
Optics & Laser Technology 38 (2006) 636–640 www.elsevier.com/locate/optlastec
Array illuminator based on self-imaging in white light Shashi Prakash, Anupam Sharma, Santosh Rana School of Instrumentation, Devi Ahilya University, Khandwa Road, Indore-452017, India Received 7 July 2004; accepted 11 January 2005 Available online 23 February 2005
Abstract An array illuminator is a device/optical system that splits an incoming beam of light into an array of many light spots/beams. It is useful in logic systems and in optical digital computing or in switching systems to energize arrays of components such as logic gates, optically bistable devices and electro-optic modulators. Array illuminators working in white light are interesting because they do not suffer from coherent noise and yield better signal-to-noise ratio. We propose configurations for array illuminator based on Fresnel diffraction and working under white light illumination conditions. Performance evaluation of the illuminators is also undertaken and the results of investigation are reported. r 2005 Elsevier Ltd. All rights reserved. Keywords: Self-imaging; Array illuminator; White light
1. Introduction The main job of array illuminator is to transform a uniform beam into an array of uniform light spots. Many different approaches for array generation have been proposed. These are based on Fraunhofer diffraction at specially designed grating [1], phase contrast imaging [2], on Fresnel diffraction at certain gratings [3], optical coordinate transformations [4], cascades of beam splitting devices [5], pin holes, mirrors (kaleidoscopes), arrays of microlenses [6], arrays of microtelescopes, etc. A review article by Streibl [7] illustrates these devices and their fabrication techniques. Array illuminators using interferometric principles have also been suggested. Several configurations using combination of two wedge plate orientations [8], Michelsons interferometer in tandem [9] and holographic-based methods [10] have been reported. Most of the array illuminators work under coherent illumination conditions; however unless required by a particular application, incoherent methods Corresponding author. Tel.: +91 731 2462228; fax: +91 731 2470372. E-mail addresses: [email protected], [email protected] (S. Prakash).
0030-3992/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2005.01.002
of array generation are preferred because of their high signal-to-noise ratio. Self-imaging had been used widely for applications in diverse areas [11]. Recently their application in vibration monitoring [12], collimation testing [13], displacement measurement [14], temperature measurement [15] and surface topography [16] has also been reported. In this paper, we present few novel methods of array generation using self-imaging in white light.
2. Experimental arrangement Fig. 1 shows schematic of experimental arrangement for array illuminator based on Lau effect. Halogen lamp of 150 W has been used as a extended white light source (EWLS). The extended source is condensed by a lens and illuminates the gratings G1 and G2, which are identical and are of equal periods. Grating G1 is arranged by placing two identical linear gratings each of pitch 0.4 mm in mutually perpendicular directions in tandem. G2 is also arranged such that it is formed by two gratings of equal periods in two perpendicular directions as in G1. The gratings are properly aligned
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Fig. 1. Array generation using Lau effect.
and a variable focal length system is used to focus the beam onto the phase plate of CCD. The images are acquired using a PC and a image-grabbing card, PCI1409 supplied by National Instruments, US. For image acquisition, the IMAQ software supplied by National Instruments was used and the results were displayed online via the computer monitor. The working of the interferometer can be understood in the following manner. The light emitted from a single pinhole opening in the grating G1 is spatial coherent. It produces an interference pattern at infinity after being split by grating G2. But the light emitted from different pinholes in G1 is spatially incoherent, so the interference pattern generated by all pinholes in G1 will add up in intensities. Under the conditions for self-imaging nd 21y nd 21x ; Z Oy ¼ , (1) 2l 2l where l is the average wavelength of light used, n is an integer and d 1x ¼ d 1y ; d 1x and d 1y are grating constants in the x and y directions, respectively. The maxima of all patterns coincide and the resulting pattern is of high contrast. Fig. 2(a) shows the experimental result recorded via a CCD camera at the image plane. The line profile of intensity at the CCD plane is also shown in Fig. 2(b). Fig. 3 shows the experimental arrangement for array illuminator based on Talbot effect under white light illumination. It consists of a source grating G1 and two identical gratings G2, G3. Each of the gratings G2 and G3 is a crossed grating assembled by superposing in tandem two-linear gratings of pitch 0.4 mm in mutually perpendicular direction. The grating G1 acts so as to enhance the spatial coherence of the waves emitted by the EWLS. The spherical waves after grating G1 are collimated using collimated lens, Lc. Beyond grating G2, the self-images of grating are formed at planes defined by Z Ox ¼
2d 21y 2d 21x ; Z Ty ¼ n , (2) l l where l is the wavelength of light used, n is an integer and d 1x ¼ d 1y ; d 1x and d 1y are grating periods in the x and y directions, respectively. Z Tx ; Z Ty correspond to ‘Talbot distance’. At the Talbot distance corresponding to 450 mm, the grating G3 is placed. The moire´ Z Tx ¼ n
Fig. 2. (a) Experimental result recorded via CCD camera on the experimental set up as in Fig. 1. (b) The line profile of intensity across the image, as recorded at the CCD plane.
Fig. 3. Array generation using Talbot effect in white light.
pattern resulting because of misalignment between different gratings has to be removed to get the array illumination at the image plane. Fig. 4(a) shows image recorded at image plane using CCD camera. The line profile of intensity at the CCD plane is also shown in Fig. 4(b). The problem of fringe intensity profile in case of self-imaging phenomenon using incoherent light has been addressed by Patorski in series of papers [17,18]. Since the array illuminators are based on principle of
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s1 and d 1 are the slit width and period of G1; z is the self-imaging distance, n d 22 =l: Here n, d 2 and l are an integer, period and wavelength of light, respectively. Hence the opening ratio of the second function becomes nðs1 =d 1 Þ: Having established that the opening ratios of the two functions are convolved, the results of the extensive analysis presented in the referenced paper [19] can directly be applied to yield the parameters of the trapezoidal profile. As per the analysis, the width of the base of the trapezoid is proportional to the sum of the numerators in the opening ratios of the gratings at the image plane. The width of the upper parallel arm is proportional to the difference between the numerators in the grating opening ratios. The amplitude of the trapezoid function (i.e. the maximum intensity) is that intensity which emerges from the grating with smaller opening ratio in the absence of the other grating. The slopes of the profile elements will be identical for all opening ratio combinations and equal to 45 : Throughout the discussion, it has to be taken into consideration that trapezoid parameters be normalized to grating period, and the grating period be normalized to unity. The analysis can similarly be extended to understand the variation of pitch parameters on the intensity line profile of the Talbot effect-based array illuminator. Fig. 4. (a) Experimental result recorded via CCD camera on the experimental set up as in Fig. 3. (b) The line profile of intensity across the image, as recorded at the CCD plane.
self-imaging in incoherent light, the applicability of the theory is direct. Mathematically, in the case of Laubased array illuminator, the intensity of the doubly diffracted Fresnel field is given by convolving the selfimage generated by point source with the actual function of intensity distribution of source grating. For binary type of gratings used in the setup, the convolution operation results in general in a trapezoidal fringe profile. This is clearly visible in the experimental results shown in Figs. 2(b) and 4(b). The parameters of the trapezoidal waveform (such as width of the base and parallel arm, height and the slopes of profile elements) depend on the opening ratios of the two binary functions (Ronchi-type gratings) being convolved, at the image plane. The opening ratios are defined as the ratio of slit width to the grating period ðs=dÞ: Since we are investigating the fringe profiles in the in-registry planes coinciding with the self-image planes of grating, G2, the opening ratio of the rectangular function describing the self-image intensity pattern is equal to the opening ratio of G2 ðs2 =d 2 Þ: On the other hand, the width of slit image of source grating in observation plane is given by ðs1 =d 1 Þ ðl=d 2 Þ z; where
3. Results and discussion Various parameters to describe the performance of an array illuminator were discussed in detail by Streibl et al. [6]. To determine the quality of array generated, we evaluated parameters describing their performance. Splitting ratio for both types of illuminators is determined to be 99 99: The homogeneity in the intensity for both the array illuminators has been found to be good. However, white-light Talbot array spots are more homogenous than Lau-based array illuminator. Compression ratio is the quotient of bright area of the spot pattern and the whole area of the elementary cell of the spot pattern. Hence, the compression ratio is used to indicate separation between the array spots. It is determined to be 5 in case of Talbot-based illuminator and 4 in case of Lau-based illuminator. Background suppression ratio is defined as the ratio between spot intensity and the background stray light present in the illuminator. Hence background suppression ratio measures the noise of array illuminator in bright and dark areas of the array. In nutshell, background suppression ratio is direct measure of image contrast. Very low values of background suppression ratio indicate poor contrast and high values indicate good image contrast among various array elements. It is determined to be 2.08 in Lau-based illuminator and 4.25 in Talbot-based system. Other parameters such as splitting ratio,
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4. Conclusions In summary, communication presents configurations for array illuminator based on self-imaging and working in white light. The performance characteristics of the Talbot effect-based array illuminator are better than Lau effect-based array illuminator. However, the array illuminators devised using amplitude gratings do not offer high efficiency. Hence configurations for achieving high efficiency in the array illuminators are also suggested. Fig. 5. Schematic for typical Lau-based efficient array illuminator using phase gratings.
Acknowledgement compression ratio, homogeneity, and efficiency though do not appear directly related to image contrast, but poor contrast shall severely degrade the performance of the illuminator. This shall induce errors, which may eventually undermine the experimental evaluation of these parameters. The attractive features of the illuminators include the ease of fabrication of gratings and their low cost. However as we have realized illuminators using amplitude gratings, the efficiency of the illuminator is not high. The signal-to-noise ratio and the efficiency of the array illuminator based on Talbot effect under white-light illumination can be improved by choosing phase gratings instead of amplitude gratings and modifying the distance between two gratings suitably, based on theory and reasoning put forward by Lohmann [3]. The signal-tonoise ratio and efficiency in case of Lau-based illuminator can also be improved considerably by using phase gratings instead of amplitude gratings, but using laser as source. Schematic for typical Lau-based efficient array illuminator is illustrated in Fig. 5. The collimated laser light illuminates a crossed phase grating, which produces high contrast Fresnel diffraction pattern that is periodic and has binary irradiance distribution. This high contrast binary irradiance distribution is converted to incoherent source with periodic irradiance using a rotating scatter plate, which destroys the coherence between any point over the Fresnel diffraction pattern. Since the mutual coherence of waves beyond the rotating reflector has been destroyed, the results at image plane will not be afflicted with the coherence noise. The conditions of ‘self-imaging’ are satisfied if another crossed phase grating is placed at distance ‘Z 0 ’ between the scatter plate and the grating G2 such that [20] Z 0 ¼ 2ðE þ n0 Þ
d2 , l
(3) 3
where E ¼ 0; 1; 2; . . . ; n0 ¼ 14; 4 ; l is the wavelength of light used and d the pitch of the grating. In this case, we obtain the array illuminator of very high efficiency at the focal plane of the lens.
This research was performed under the research project [Grant no. 03(0938)/02/EMR-II] funded by Council of Scientific and Industrial Research (CSIR), New Delhi. Authors gratefully acknowledge the financial assistance provided by CSIR, New Delhi. Authors are also grateful to the referee’s suggestions/comments towards improving the manuscript.
References [1] Lee WH. High efficiency multiple beam gratings. Appl Opt 1979; 18:2152–8. [2] Glu¨ckstad J, Mogensen PC. Reconfigurable ternary-phase array illuminator based on the generalised phase contrast method. Opt Commun 2000;173:169–75. [3] Lohmann AW. An array illuminator based on Talbot effect. Optik 1988;79:41–5. [4] Bryngdahl O. Geometrical transformations in optics. J Opt Soc Am 1974;64:1092–8. [5] Kubota T, Takeda M. Array illuminator using grating couplers. Opt Lett 1989;14:651–2. [6] Streibl N, Nalscher U, Jahns J, Walker SJ. Array generation with lenslet arrays. Appl Opt 1991;30:2739–42. [7] Streibl N. Beam shaping with optical array generation. J Mod Opt 1989;38:1559–73. [8] Senthilkumaran P, Sirohi RS. Michelson interferometer in tendom for array generation. Opt Commun 1994;105:158–60. [9] Senthilkumaran P, Sriram KV, Kothiyal MP, Sirohi RS. Array generation using double wedge plate interferometer. J Mod Opt 1994;41:481–9. [10] Bhattacharya S, Senthilkumaran P, Darlin JS, Kothiyal MP, Sirohi RS. Holographic array illuminators using tendom Michelson interferometers; fabrication and analysis. J Mod Opt 1995;42: 2275–83. [11] Patorski K. The self imaging phenomenon and its applications. In: Wolf E, editor. Progress in optics XXVII. Netherland: Elsevier; 1989. p. 1–110. [12] Prakash S, Upadhyay S, Shakher C. Real time out-of-plane vibration measurement using Talbot interferometry. Opt Lasers Eng 2004;34:251–9. [13] Shakher C, Prakash S, Dayanand X, Kumar R. Collimation testing with circular gratings. Appl Opt 2001;40:1175–9. [14] Spagnolo GS, Amrosini D, Paoletti D. Displacement measurement using the Talbot effect with Ronchi grating. J Opt A: Pure Appl Opt 2002;4:5376–80.
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[15] Thakur M, Angra SK, Shakher C. Measurement of temperature profile of the 2-D slot burner using Lau phase interferometry with linear gratings. Opt Eng 2003;42:85–91. [16] Mirza S, Faridi MS, Kumar R, Vyas AL, Shakher C. Surface topography of steps and dips by phase shifting Talbot interferometric technique. In: Proceedings of the international conference on laser applications and optical metrology, New Delhi; 1–4 December 2003. p. 418–21.
[17] Patorski K. Incoherent superposition of multiple self-imaging Lau effect and moire´ fringe explanation. Opt Acta 1983;30:745–58. [18] Patorski K. Incoherent superposition of multiple self-imaging under plane wave-front illumination. Appl Opt 1986;25:2396–403. [19] Patorski K, Yokozeki S, Suzuki T. Moire´ profile prediction using Fourier series formalism. Jap J Appl Phys 1976;15:443–56. [20] Barreiro JC, Andres P, Ojeda-Castaneda J. Lau effect with only phase gratings. Opt Commun 1989;73:106–10.