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A point diffraction interferometer with random-dot filter
Optics Communications 237 (2004) 17–24 www.elsevier.com/locate/optcom
A point diffraction interferometer with random-dot filter Hideo Furuhashi a,*, Atsushi Shibata a, Yoshiyuki Uchida a, Kiyofumi Matsuda b, Chander P. Grover b a b
Department of Information Network Engineering, Aichi Institute of Technology, 1247, Yachigusa, Yakusa-cho, Toyota 4700392, Japan Optics Group, Institute for National Measurement Standards, National Research Council Canada, Ottawa, Ont., Canada K1A OR6 Received 2 December 2003; received in revised form 23 March 2004; accepted 29 March 2004
Abstract The characteristics and performance of a point diffraction interferometer with random-dot filter are analyzed in detail. Such interferometers are of simple construction and easy to align, and can be used to accurately measure optical wavefronts. The characteristics of the instrument are treated theoretically and analyzed by computer simulation. An experimental system is also constructed to confirm the theoretical results. It is found that the filter should be prepared to have a large number of dots using a high-resolution, low-granularity film such as holographic film in order to obtain a good fringe pattern with minimal noise. Ó 2004 Elsevier B.V. All rights reserved. PACS: 07.60.L; 42.25.F; 42.15.D; 42.30.K Keywords: Interferometers; Diffraction; Wave fronts; Fourier optics
1. Introduction A point diffraction interferometer (PDI) is a common-path interferometer that generates interference fringes related to the phase variation across the wavefront [1,2]. In such an interferometer, the reference beam is generated after the filter, representing a self-referencing system that is inherently stable and relatively insensitive to vibration. These instruments have a simple struc-
*
Corresponding author. Tel.: +81565488121; fax: +815654 80070. E-mail address: [email protected] (H. Furuhashi).
ture, and are useful tools for the testing and evaluation of optical elements, the analysis of phase objects, fluid flow diagnostics and so on [1–5]. The filter in a PDI has an opaque dot smaller than the airy disk. The wavefront is focused onto the dot by a Fourier transform lens, and an interference fringe pattern is obtained at the image plane. However, it is difficult to focus the wavefront on such a small dot. It has been reported that an interference fringe similar to that of a standard PDI has been obtained through the use of a filter having an arrangement of opaque dots [6]. In this paper, the theoretical aspects and advantages of a PDI with random-dot filter are discussed, and the
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.03.077
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H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
theory and effectiveness of this PDI are confirmed through simulations and experiments. Based on this discussion, the considerations for an optimal filter producing a good interference fringe are presented.
2. Theory The interference system consists of a Fourier transform lens and a random-dot filter. A wavefront on the object plane is focused on the filter, and the interference image is observed on the image plane. Here, we consider a wavefront with complex amplitude of u ¼ A exp½i/ðx; yÞ, where A is the absolute amplitude, /ðx; yÞ is the phase, and x and y are the orthogonal coordinates on the object plane. We resolve this into two components; a plane wave in the direction of the z axis (orthogonal to x and y), and all others, u ¼ A exp½i/ðx; yÞ A c, where c is the ratio of the plane wave, referring to the ratio of waves focused on an area smaller than the dot at the focal point. If each of the random dots is sufficiently small with respect to the size of the Fourier transform of the
non-plane component, and if the Fourier transform covers many spots, the non-plane component will be approximately averaged. This condition is satisfied for a wavefront with small non-plane component. The random-dot filter is also assumed to fill the entire Fourier transform plane. Under this condition, the wave on the image plane is expressed as follows: u ¼ aave f A exp ½i/ðx; yÞ A cg þ A c afoc ; ð1Þ where aave is the average amplitude transmittance of the filter, and afoc is the amplitude transmittance at the focal point on the filter. Thus, the optical intensity is given by n 2 I ¼ A2 a2ave þ c2 ðafoc aave Þ o þ 2 aave ðafoc aave Þ c cos ½/ð x; y Þ : ð2Þ An interference fringe is then obtained, with contrast define as 2a ða a Þ c ave foc ave ð3Þ V ¼ : a2ave þ c2 ðafoc aave Þ2
Fig. 1. Relationship between contrast and average amplitude transmittance. Solid line denotes the case for c ¼ 1, and dotted line indicates c ¼ 0:5.
H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
The maximum contrast (1) is obtained under the conditionl ð1 þ cÞaave ¼ cafoc :
ð4Þ
19
Fig. 1 shows the relationship between the contrast and average amplitude transmittance aave . If the ratio of the plane wave is nearly equal to 1, the maximum contrast is obtained at aave ¼ 0.5 or
Fig. 2. Relationship between fringe amplitude and amplitude transmittance at the focal point. Solid line denotes the case for c ¼ 1, and dotted line indicates c ¼ 0:5.
Fig. 3. Sample waveform for computer simulation.
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H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
Fig. 4. Results of simulation for: (a) afoc ¼ 1, (b) afoc ¼ 0:2 and (c) afoc ¼ 0. (1) Image. (2) Optical intensity along the center line.
H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
afoc ¼ 0. If the ratio is not equal to 1, the maximum contrast is obtained at aave ¼ afoc =ð1 þ cÞ. Since the amplitudes afoc and aave are positive values, there is no other solution. The amplitude of the interference fringe is given by 2jaave ðafoc aave Þ cj. Fig. 2 shows the relationship between the fringe amplitudes and the amplitude transmittance at the focal point afoc . A large amplitude is obtained for afoc ¼ 0 or 1, representing the case in which the film absorbs all or none of the wavefront at the focal point. A larger amplitude of the interference fringe is obtained in the case where jafoc aave j is large. The amplitude decreases with a decrease in the average amplitude transmittance. A good interference fringe is therefore obtained when the wavefront corresponds to a large ratio of the plane wave. This theory can be confirmed through simulation. Consider a triangular wave (20 mm 20 mm 3 lm in height) on a plane wave (100 mm 100 mm) as shown in Fig. 3. The ratio of the plane wave in this case is about 96%. The optical Fourier transform is calculated for an optical wavelength of 633 nm and a lens with a focal length of 150 mm in a system with a filter consisting of dots of dimension 1 lm 1 lm. The amplitude transmittance of the dots is 0%, that in the filter region around the dots is 100%, and the filter is assumed to consist of 50% dots by area to give an average amplitude transmittance of 50%. The Fourier transformed image is multiplied by the filter pattern, and then the inverse Fourier transform is obtained. Specific cases of afoc are described below. (i) afoc ¼ 1. In this case, the wavefront is not focused on a dot, and the maximum contrast is obtained.
21
Fig. 4(a-1) shows the image obtained by this simulation, and Fig. 4(a-2) shows the optical intensity along the center line. An interference fringe pattern is obtained, conforming to Eq. (2). The background intensity is quite high, and the contrast is approximately 1. (ii) afoc ¼ 0:2. In this case, the wavefront is partially focused on a dot. Fig. 4(b-1) shows the image obtained by this simulation, and Fig. 4(b-2) shows the optical intensity along the center line. An interference fringe pattern is also observed in this case. However, the amplitude of the interference fringe is low. The contrast is approximately 0.85. (iii) afoc ¼ 0. In this case, the wavefront is focused fully on a dot, giving the maximum contrast. Fig. 4(c-1) shows the image obtained by simulation, and Fig. 4(b-2) shows the optical intensity along the center line. An interference fringe pattern is obtained, conforming to Eq. (2). The contrast is approximately 1, and the fringe is 180° out of phase compared with the fringe in case (i). These simulation results confirm the theory well.
3. Experimental Fig. 5 shows the experimental setup. A collimated beam produced by a He–Ne laser (k ¼ 633 nm, output power 3.6 mW) was used to illuminate a phase object. The transmitted beam was focused on the random-dot filter by Fourier transform lens L with focal length of 160 mm and diameter of 40 mm. The interference pattern at the image plane was then observed using a charge-coupled device (CCD) camera (640 400 pixels).
Fig. 5. Experimental setup. L: Fourier transform lens (focal length 160 mm, diameter 40 mm). Filter: random-dot filter. CCD: CCD camera (640 400 pixels).
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H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
The random-dot filter was an exposed AGFA 8E75 HD holographic film. Slavich PFG-01 holographic film, which has similar characteristics to AGFA 8E75 HD, was also used as AGFA 8E75 HD has been discontinued. These films have a grain size of 40 nm and resolving power of 3000 lines/mm. The amplitude transmittance of the dots
Fig. 6. (a) Image obtained without the filter. (b) Fringe pattern obtained using the random-dot filter. (c) Differential phase contrast image.
was below 10%, and that around the dots was approximately 90%. The average amplitude transmittance was about 0.6. The phase objects were thin glass wedges with a refractive index of 1.5, diameter of 2.5 mm, thickness of 0.3 mm, and wedge angle of approximately 103 rad. The area of the wavefront, defined by the diameter of the Fourier transform lens, was 1250 mm2 , corresponding to a ratio of the plane wave of 99.6%. The contrast of the PDI fringe was calculated theoretically using Eq. (3) and exceeded 89%. The diameter of the Fourier transform of the non-
Fig. 7. PDI image obtained using the random-dot filter with (a) a dot at the focal point, and (b) no dot at the focal point.
H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
plane component on the focal plane, calculated from diffraction theory, was about 100 lm.
4. Results Fig. 6(b) shows the interference fringe pattern obtained using this system, and the image obtained without the filter is shown in Fig. 6(a). An inter-
23
ference fringe with good contrast was obtained using the random-dot filter. Fig. 6(c) shows the differential phase contrast image obtained for the same object. The variable decrease in thickness at the edge can be clearly seen in both the interference fringe and the differential phase contrast image. The fringe contrast was more than 0.9. Fig. 7 shows the interference fringes of a different sample with and without a dot at the focal
Table 1 Characteristics of films used in experiments and RMS granularity of corresponding PDI images Film
Resolution (contrast 1000:1)
Photographic sensitivity
RMS granularity (aperture 48 lm)
RMS granularity of PDI image (aperture 2 2 pixels)
Slavich PFG-01 Fuji Film NEOPAN-F Kodak TMAX-100 Kodak TMAX-P3200
3000 lines/mm 150 lines/mm 200 lines/mm 125 lines/mm
80 lJ/cm2 (ISO103 ) ISO32 ISO100 ISO3200
NA (grain size: 40 nm) 7 8 18
4.6 6.0 5.6 14.5
Fig. 8. PDI images obtained using four types of film: (a) Slavich PFG-01, (b) Fuji Film NEOPAN-F, (c) Kodak TMAX-100, and (d) Kodak TMAX-P3200.
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point. As shown in the theoretical treatment and simulation, the phases of the interference fringe are shifted by 180°. Thus, a PDI fringe is obtained not only when the wavefront is focused on a dot, but also when the wavefront is focused between dots. Furthermore, as the film contains many dots, alignment is very easy. However, some noise was observed in the interference fringe pattern, attributed to the granularity and resolution of the film. In the theoretical discussion, the dots were approximated as being only relatively small, and the intensity of the transparent wave was averaged. However, the Fourier-transformed spot is in fact very small and is not averaged adequately by the photographic film. Therefore, the interference fringes were compared using various films, the characteristics of which are listed in Table 1. Fig. 8 shows the PDI images obtained with four different filters. The best result was obtained with the filter prepared using Slavich PFG-01 holographic film. This film has very high resolution and excellent granularity. In contrast, much noise was observed when using the Kodak TMAX-P3200 film, which has low resolution and poor granularity. Table 1 also lists the root mean square (RMS) granularities of the PDI images obtained for a plane wave. The PDI image captures the overlap of waves generated by the dots. When there are many dots in one spot, many waves are generated and the averaging is adequate. However, if the dots are coarse, as on the TMAX-P3200 filter, the averaging is inadequate and noise appears. Increasing the number of dots reduces noise, but also reduces the transmittance and intensity of fringes. It is therefore important to use a filter with a large number of dots prepared from high-resolution, low-granularity film. In this respect, holographic films are more suitable than standard photographic films.
5. Conclusion The characteristics of a point diffraction interferometer with a random-dot filter were discussed theoretically and demonstrated through computer simulation. The analysis showed that an interference fringe is obtained when the optical wave is focused on either a dot or the surrounding film, and this finding was confirmed in experiments. PDI images were obtained using four different types of film for preparation of the filter. A good fringe was obtained using a high-resolution lowgranularity film. This system is of simple construction and is easy to align, and can be used to accurately and quickly measure optical wavefronts. The system is expected to be useful in many applications such as the testing and evaluation of optical elements, the analysis of phase objects, and fluid flow diagnostics, and is particularly suitable for the measurement of wavefronts when the Fourier transform of the non-plane component is larger than approximately 10 lm and the ratio of the plane wave component is large. These results were obtained using holographic film with a resolution of 3000 lines/mm: with the use of higher-resolution film, this system will be applicable for wavefront analysis on even smaller scales. References [1] R. Smartt, W. Steel, Jpn. J. Appl. Phys. Suppl. 14-1 (1975) 351. [2] R.N. Smartt, in: G.W. Hopkins (Ed.), Interferometry, Proceedings of the SPIE, vol. 192, 1979, p. 35. [3] W. Bachalo, M. Houser, Opt. Eng. 24 (1985) 455. [4] M. Giglio, U. Perini, E. Paginini, Opt. Eng. 27 (1988) 197. [5] Measurement of gas-phase temperatures in flames with a point-diffraction interferometerAppl. Opt. 40 (2001) 4816– 4823. [6] C.P. Grover, Opt. Commun. 13 (1975) 335.
A point diffraction interferometer with random-dot filter Hideo Furuhashi a,*, Atsushi Shibata a, Yoshiyuki Uchida a, Kiyofumi Matsuda b, Chander P. Grover b a b
Department of Information Network Engineering, Aichi Institute of Technology, 1247, Yachigusa, Yakusa-cho, Toyota 4700392, Japan Optics Group, Institute for National Measurement Standards, National Research Council Canada, Ottawa, Ont., Canada K1A OR6 Received 2 December 2003; received in revised form 23 March 2004; accepted 29 March 2004
Abstract The characteristics and performance of a point diffraction interferometer with random-dot filter are analyzed in detail. Such interferometers are of simple construction and easy to align, and can be used to accurately measure optical wavefronts. The characteristics of the instrument are treated theoretically and analyzed by computer simulation. An experimental system is also constructed to confirm the theoretical results. It is found that the filter should be prepared to have a large number of dots using a high-resolution, low-granularity film such as holographic film in order to obtain a good fringe pattern with minimal noise. Ó 2004 Elsevier B.V. All rights reserved. PACS: 07.60.L; 42.25.F; 42.15.D; 42.30.K Keywords: Interferometers; Diffraction; Wave fronts; Fourier optics
1. Introduction A point diffraction interferometer (PDI) is a common-path interferometer that generates interference fringes related to the phase variation across the wavefront [1,2]. In such an interferometer, the reference beam is generated after the filter, representing a self-referencing system that is inherently stable and relatively insensitive to vibration. These instruments have a simple struc-
*
Corresponding author. Tel.: +81565488121; fax: +815654 80070. E-mail address: [email protected] (H. Furuhashi).
ture, and are useful tools for the testing and evaluation of optical elements, the analysis of phase objects, fluid flow diagnostics and so on [1–5]. The filter in a PDI has an opaque dot smaller than the airy disk. The wavefront is focused onto the dot by a Fourier transform lens, and an interference fringe pattern is obtained at the image plane. However, it is difficult to focus the wavefront on such a small dot. It has been reported that an interference fringe similar to that of a standard PDI has been obtained through the use of a filter having an arrangement of opaque dots [6]. In this paper, the theoretical aspects and advantages of a PDI with random-dot filter are discussed, and the
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.03.077
18
H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
theory and effectiveness of this PDI are confirmed through simulations and experiments. Based on this discussion, the considerations for an optimal filter producing a good interference fringe are presented.
2. Theory The interference system consists of a Fourier transform lens and a random-dot filter. A wavefront on the object plane is focused on the filter, and the interference image is observed on the image plane. Here, we consider a wavefront with complex amplitude of u ¼ A exp½i/ðx; yÞ, where A is the absolute amplitude, /ðx; yÞ is the phase, and x and y are the orthogonal coordinates on the object plane. We resolve this into two components; a plane wave in the direction of the z axis (orthogonal to x and y), and all others, u ¼ A exp½i/ðx; yÞ A c, where c is the ratio of the plane wave, referring to the ratio of waves focused on an area smaller than the dot at the focal point. If each of the random dots is sufficiently small with respect to the size of the Fourier transform of the
non-plane component, and if the Fourier transform covers many spots, the non-plane component will be approximately averaged. This condition is satisfied for a wavefront with small non-plane component. The random-dot filter is also assumed to fill the entire Fourier transform plane. Under this condition, the wave on the image plane is expressed as follows: u ¼ aave f A exp ½i/ðx; yÞ A cg þ A c afoc ; ð1Þ where aave is the average amplitude transmittance of the filter, and afoc is the amplitude transmittance at the focal point on the filter. Thus, the optical intensity is given by n 2 I ¼ A2 a2ave þ c2 ðafoc aave Þ o þ 2 aave ðafoc aave Þ c cos ½/ð x; y Þ : ð2Þ An interference fringe is then obtained, with contrast define as 2a ða a Þ c ave foc ave ð3Þ V ¼ : a2ave þ c2 ðafoc aave Þ2
Fig. 1. Relationship between contrast and average amplitude transmittance. Solid line denotes the case for c ¼ 1, and dotted line indicates c ¼ 0:5.
H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
The maximum contrast (1) is obtained under the conditionl ð1 þ cÞaave ¼ cafoc :
ð4Þ
19
Fig. 1 shows the relationship between the contrast and average amplitude transmittance aave . If the ratio of the plane wave is nearly equal to 1, the maximum contrast is obtained at aave ¼ 0.5 or
Fig. 2. Relationship between fringe amplitude and amplitude transmittance at the focal point. Solid line denotes the case for c ¼ 1, and dotted line indicates c ¼ 0:5.
Fig. 3. Sample waveform for computer simulation.
20
H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
Fig. 4. Results of simulation for: (a) afoc ¼ 1, (b) afoc ¼ 0:2 and (c) afoc ¼ 0. (1) Image. (2) Optical intensity along the center line.
H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
afoc ¼ 0. If the ratio is not equal to 1, the maximum contrast is obtained at aave ¼ afoc =ð1 þ cÞ. Since the amplitudes afoc and aave are positive values, there is no other solution. The amplitude of the interference fringe is given by 2jaave ðafoc aave Þ cj. Fig. 2 shows the relationship between the fringe amplitudes and the amplitude transmittance at the focal point afoc . A large amplitude is obtained for afoc ¼ 0 or 1, representing the case in which the film absorbs all or none of the wavefront at the focal point. A larger amplitude of the interference fringe is obtained in the case where jafoc aave j is large. The amplitude decreases with a decrease in the average amplitude transmittance. A good interference fringe is therefore obtained when the wavefront corresponds to a large ratio of the plane wave. This theory can be confirmed through simulation. Consider a triangular wave (20 mm 20 mm 3 lm in height) on a plane wave (100 mm 100 mm) as shown in Fig. 3. The ratio of the plane wave in this case is about 96%. The optical Fourier transform is calculated for an optical wavelength of 633 nm and a lens with a focal length of 150 mm in a system with a filter consisting of dots of dimension 1 lm 1 lm. The amplitude transmittance of the dots is 0%, that in the filter region around the dots is 100%, and the filter is assumed to consist of 50% dots by area to give an average amplitude transmittance of 50%. The Fourier transformed image is multiplied by the filter pattern, and then the inverse Fourier transform is obtained. Specific cases of afoc are described below. (i) afoc ¼ 1. In this case, the wavefront is not focused on a dot, and the maximum contrast is obtained.
21
Fig. 4(a-1) shows the image obtained by this simulation, and Fig. 4(a-2) shows the optical intensity along the center line. An interference fringe pattern is obtained, conforming to Eq. (2). The background intensity is quite high, and the contrast is approximately 1. (ii) afoc ¼ 0:2. In this case, the wavefront is partially focused on a dot. Fig. 4(b-1) shows the image obtained by this simulation, and Fig. 4(b-2) shows the optical intensity along the center line. An interference fringe pattern is also observed in this case. However, the amplitude of the interference fringe is low. The contrast is approximately 0.85. (iii) afoc ¼ 0. In this case, the wavefront is focused fully on a dot, giving the maximum contrast. Fig. 4(c-1) shows the image obtained by simulation, and Fig. 4(b-2) shows the optical intensity along the center line. An interference fringe pattern is obtained, conforming to Eq. (2). The contrast is approximately 1, and the fringe is 180° out of phase compared with the fringe in case (i). These simulation results confirm the theory well.
3. Experimental Fig. 5 shows the experimental setup. A collimated beam produced by a He–Ne laser (k ¼ 633 nm, output power 3.6 mW) was used to illuminate a phase object. The transmitted beam was focused on the random-dot filter by Fourier transform lens L with focal length of 160 mm and diameter of 40 mm. The interference pattern at the image plane was then observed using a charge-coupled device (CCD) camera (640 400 pixels).
Fig. 5. Experimental setup. L: Fourier transform lens (focal length 160 mm, diameter 40 mm). Filter: random-dot filter. CCD: CCD camera (640 400 pixels).
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H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
The random-dot filter was an exposed AGFA 8E75 HD holographic film. Slavich PFG-01 holographic film, which has similar characteristics to AGFA 8E75 HD, was also used as AGFA 8E75 HD has been discontinued. These films have a grain size of 40 nm and resolving power of 3000 lines/mm. The amplitude transmittance of the dots
Fig. 6. (a) Image obtained without the filter. (b) Fringe pattern obtained using the random-dot filter. (c) Differential phase contrast image.
was below 10%, and that around the dots was approximately 90%. The average amplitude transmittance was about 0.6. The phase objects were thin glass wedges with a refractive index of 1.5, diameter of 2.5 mm, thickness of 0.3 mm, and wedge angle of approximately 103 rad. The area of the wavefront, defined by the diameter of the Fourier transform lens, was 1250 mm2 , corresponding to a ratio of the plane wave of 99.6%. The contrast of the PDI fringe was calculated theoretically using Eq. (3) and exceeded 89%. The diameter of the Fourier transform of the non-
Fig. 7. PDI image obtained using the random-dot filter with (a) a dot at the focal point, and (b) no dot at the focal point.
H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
plane component on the focal plane, calculated from diffraction theory, was about 100 lm.
4. Results Fig. 6(b) shows the interference fringe pattern obtained using this system, and the image obtained without the filter is shown in Fig. 6(a). An inter-
23
ference fringe with good contrast was obtained using the random-dot filter. Fig. 6(c) shows the differential phase contrast image obtained for the same object. The variable decrease in thickness at the edge can be clearly seen in both the interference fringe and the differential phase contrast image. The fringe contrast was more than 0.9. Fig. 7 shows the interference fringes of a different sample with and without a dot at the focal
Table 1 Characteristics of films used in experiments and RMS granularity of corresponding PDI images Film
Resolution (contrast 1000:1)
Photographic sensitivity
RMS granularity (aperture 48 lm)
RMS granularity of PDI image (aperture 2 2 pixels)
Slavich PFG-01 Fuji Film NEOPAN-F Kodak TMAX-100 Kodak TMAX-P3200
3000 lines/mm 150 lines/mm 200 lines/mm 125 lines/mm
80 lJ/cm2 (ISO103 ) ISO32 ISO100 ISO3200
NA (grain size: 40 nm) 7 8 18
4.6 6.0 5.6 14.5
Fig. 8. PDI images obtained using four types of film: (a) Slavich PFG-01, (b) Fuji Film NEOPAN-F, (c) Kodak TMAX-100, and (d) Kodak TMAX-P3200.
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H. Furuhashi et al. / Optics Communications 237 (2004) 17–24
point. As shown in the theoretical treatment and simulation, the phases of the interference fringe are shifted by 180°. Thus, a PDI fringe is obtained not only when the wavefront is focused on a dot, but also when the wavefront is focused between dots. Furthermore, as the film contains many dots, alignment is very easy. However, some noise was observed in the interference fringe pattern, attributed to the granularity and resolution of the film. In the theoretical discussion, the dots were approximated as being only relatively small, and the intensity of the transparent wave was averaged. However, the Fourier-transformed spot is in fact very small and is not averaged adequately by the photographic film. Therefore, the interference fringes were compared using various films, the characteristics of which are listed in Table 1. Fig. 8 shows the PDI images obtained with four different filters. The best result was obtained with the filter prepared using Slavich PFG-01 holographic film. This film has very high resolution and excellent granularity. In contrast, much noise was observed when using the Kodak TMAX-P3200 film, which has low resolution and poor granularity. Table 1 also lists the root mean square (RMS) granularities of the PDI images obtained for a plane wave. The PDI image captures the overlap of waves generated by the dots. When there are many dots in one spot, many waves are generated and the averaging is adequate. However, if the dots are coarse, as on the TMAX-P3200 filter, the averaging is inadequate and noise appears. Increasing the number of dots reduces noise, but also reduces the transmittance and intensity of fringes. It is therefore important to use a filter with a large number of dots prepared from high-resolution, low-granularity film. In this respect, holographic films are more suitable than standard photographic films.
5. Conclusion The characteristics of a point diffraction interferometer with a random-dot filter were discussed theoretically and demonstrated through computer simulation. The analysis showed that an interference fringe is obtained when the optical wave is focused on either a dot or the surrounding film, and this finding was confirmed in experiments. PDI images were obtained using four different types of film for preparation of the filter. A good fringe was obtained using a high-resolution lowgranularity film. This system is of simple construction and is easy to align, and can be used to accurately and quickly measure optical wavefronts. The system is expected to be useful in many applications such as the testing and evaluation of optical elements, the analysis of phase objects, and fluid flow diagnostics, and is particularly suitable for the measurement of wavefronts when the Fourier transform of the non-plane component is larger than approximately 10 lm and the ratio of the plane wave component is large. These results were obtained using holographic film with a resolution of 3000 lines/mm: with the use of higher-resolution film, this system will be applicable for wavefront analysis on even smaller scales. References [1] R. Smartt, W. Steel, Jpn. J. Appl. Phys. Suppl. 14-1 (1975) 351. [2] R.N. Smartt, in: G.W. Hopkins (Ed.), Interferometry, Proceedings of the SPIE, vol. 192, 1979, p. 35. [3] W. Bachalo, M. Houser, Opt. Eng. 24 (1985) 455. [4] M. Giglio, U. Perini, E. Paginini, Opt. Eng. 27 (1988) 197. [5] Measurement of gas-phase temperatures in flames with a point-diffraction interferometerAppl. Opt. 40 (2001) 4816– 4823. [6] C.P. Grover, Opt. Commun. 13 (1975) 335.