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Collimation testing using a circular Dammann grating
Optics Communications 279 (2007) 1–6 www.elsevier.com/locate/optcom
Collimation testing using a circular Dammann grating Shuai Zhao *, Po Sheun Chung Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong Received 30 September 2006; received in revised form 23 May 2007; accepted 30 June 2007
Abstract A novel technique for collimation testing with a circular Dammann grating is proposed. When the beam under test is incident on a one-order circular Dammann grating with limited aperture, double-humped radial rings will be generated at the back focal plane of a focusing lens. If the beam is collimated, the separation between the two rings will reach its minimal, otherwise the two rings will be apart from each other. Therefore, the degree of collimation of the tested beam can be estimated from the separation. The principle and experimental results of the method are presented. Owing to the simplicity and low cost of the method, it is a promising method for quickly checking the collimation of a laser beam. 2007 Elsevier B.V. All rights reserved. Keywords: Collimation; Diffractive optics; Gratings; Optical testing
1. Introduction Beam collimation is always a prerequisite in optical systems, especially coherent optical systems. For example, the collimated beam is necessary in many measurement systems, and is related to the accuracy of the measurement. In free-space optical communication system, a flat beam should be emitted to achieve diffraction-limited Airy pattern at the receiver. A number of measurement methods have been proposed to check the collimation. Most of these methods fall into two main categories: self-imaging techniques and interferometric techniques. In the self-imaging techniques (Talbot interferometry), a grating replicates itself at several planes perpendicular to the propagation direction. If the second grating is placed in one of these self-imaging planes, straight parallel moire´ fringe are obtained under the illumination of collimated beam. For a noncollimated beam, these moire´ fringes rotate with respect to the reference line. Different types of gratings have been introduced into the Talbot interferometry, e.g., *
Corresponding author. Tel.: +852 2194 2683; fax: +852 2788 7741. E-mail address: [email protected] (S. Zhao).
0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.06.051
linear gratings [1], dual-field gratings [2], spiral gratings [3], and circular gratings [4]. A comparative study of collimation testing with dual-field linear gratings, spiral gratings, and evolute gratings has been reported [5]. The study shows the circular grating exhibits the highest sensitivity. In addition, the moire´ fringes can also be produced through a digital subtraction operation [6]. Collimation testing under an incoherent illumination have been carried out by use of Lau effect [7,8]. Another family of widely used methods is based on shearing interferometry. The shear can be introduced through plane parallel plate [9], single wedged plate [10–13], or double wedged plate [14– 16]. Multiple beam shearing interferometry was developed to achieve higher accuracy [17]. Besides above mentioned methods, other novel approaches have been reported such as phase conjugate interferometry using a Twyman–Green interferometer [18], cycle shearing interferometer [19,20], phase singularities [21], circular aperture sampling [22], and active materials [23]. These methods have their own pros and cons and are applicable in particular circumstances. This paper reports a simple collimation testing method with a circular Dammann grating (CDG). The unique
2
S. Zhao, P.S. Chung / Optics Communications 279 (2007) 1–6
diffraction behavior of this novel device under the noncollimated illumination will help to evaluate the degree of collimation. The most advantage of this method is its extremely simple configuration. 2. Principle of measurement Circular Dammann gratings is a kind of radially periodic phase gratings that can generate uniform-intensity impulse rings in the far field [24]. Each ring corresponds to a diffraction order. The order of a CDG is therefore defined as the number of these bright rings. The rotational symmetry of a CDG makes it not necessary to distinguish positive and negative orders. The phase modulation of the CDG can be binary, multilevel or continuous. For ease of fabrication, binary phase CDG is used in our proposed method. Under the illumination of collimated beam, a CDG with period T yields the amplitude field at the back focal plane of a lens 1 1 X n=T GðqÞ ¼ pffiffiffi cn dð1=2Þ ðq n=T Þ; p n¼1 ðn=T þ qÞ3=2
ð1Þ
which represents a series of concentric impulse rings, where q = q/kf denotes the spatial frequency, q is polar coordinate in the output plane, f is the focal length of the focusing lens, and d(1/2)(x) is the half-order derivative of the Dirac impulse d(x). The explicit expression for the fractional-order derivatives of d(x) can be found in advanced books on generalized functions such as Ref. [25]. cn are the Fourier series coefficients (or sine series coefficients) governed by Z n 2 cn ¼ ð2Þ gðrÞ sin 2p r dr; T T T where g(r) is the phase profile function of the CDG. The relative intensity of the nth pulse ring is determined by the coefficients cn. The CDG is so designed by optimizing g(r) to achieve equal intensity rings. In our proposed method, only the first diffraction order is concerned. Fig. 1 shows the phase profile of a one-order CDG. Its theoretical spectral intensities in the far field are shown in Fig. 2a. The calculated coefficients cn are given by cn ¼
2 ½1 ð1Þn : np
ð3Þ
Neglecting the terms involving higher order spectrum, Eq. (1) can then be simplified as
Fig. 2. Theoretical normalized intensity of a one-order CDG with (a) infinite aperture and (b) limited aperture.
GðqÞ ¼
4 1=T dð1=2Þ ðq 1=T Þ: p3=2 ð1=T þ qÞ3=2
ð4Þ
The foregoing analysis is based on the assumption that the grating is infinitely large, but a practical CDG only consists of a finite number of periods. It is, in fact, the product of an initial infinite grating with a low-pass filter, rect(r/2a), having the appropriate cut-off frequency. This means that the spectrum of such a finite grating is no longer impulsive, since the multiplication of the initial infinite grating with rect(r/2a) in the space domain implies that in the spectral domain the initial impulsive spectrum is convolved with the continuous spectrum of rect(r/2a). In this case, the Fourier spectrum will be G1 ðqÞ ¼ F ½gðrÞpðrÞ ¼ GðqÞ 4a2 jincð2aqÞ ¼ C 1 jincð1=2Þ ½2aðq 1=T Þ;
Fig. 1. Illustration of a one-order binary phase (0, p) CDG.
ð5Þ
where represents convolution, 2a is the aperture of the grating, i.e., the area illuminated by the laser beam. C1 is a constant independent on the coordinates, and jinc(q) =
S. Zhao, P.S. Chung / Optics Communications 279 (2007) 1–6
J1(pq)/2q. Eq. (5) is numerically calculated according to Grunwald–Letnikov definition to fractional derivatives ½ðtaÞ=h 1 X j a a D f ðtÞ ¼ lim ð1Þ f ðt jhÞ a t a h!0 h j j¼0 ’
½ðtaÞ=h 1 X ðaÞ xj f ðt jhÞ; ha j¼0
ð6Þ
where [x] means the integer part of x, h is time step, and the ðaÞ coefficients xj are given by the following recursive relation a þ 1 ðaÞ ðaÞ ðaÞ x0 ¼ 1; xj ¼ 1 ð7Þ xj1 ; j ¼ 1; 2; . . . j By plotting Eq. (5) in Fig. 2b, one can see that the main intensity maximum in Fig. 2a has split into two lobes owing to the limited aperture. The radial separation Dq between the two lobes is determined by the focal length and the grating aperture as shown in Fig. 3. It can be clearly seen that the separation decreases with the increase of the grating aperture, which is easily understood since the larger aperture means the grating is closer to infinite. On the contrary a lens with longer focal length will generate larger separation. If a noncollimated wave is incident on the CDG, the amplitude in the back focal plane of the focusing lens is G2 ðqÞ¼F ½wðrÞgðrÞpðrÞ
w ¼GðqÞ4a2 jincð2aqÞexp ip2 q2 k ( " 2 #)ð1=2Þ q 1 w q 1 ¼C 2 jinc 2a exp ip2 ; kf T k kf T ð8Þ
expði wk r2 Þ
where C2 is a constant and wðrÞ ¼ representing the noncollimated wave. When the beam is collimated, w is infinite. Fig. 4 numerically simulates the diffraction patterns at the back focal plane under different illumination, where f = 500 mm, T = 100 lm, and the grating aperture is 20 mm. If the separation between the two lobes is monitored, one can find that it strongly depends on the curvature radius, i.e., the degree of collimation, of the
3
illumination wave. The quantitative analysis is illustrated in Fig. 5 with the same parameters as Fig. 4. From the figure one can see that with the illumination beam deviating from collimation, the separation rapidly increases. This interesting characteristic indicates a novel way to do collimation testing in practice. The best collimation state can be achieved when the separation is minimum. 3. Experimental results The experimental setup for collimation testing is illustrated in Fig. 6. Light from a He–Ne laser with a wavelength of 632.8 nm is focused on a pinhole by a microscope objective. A precision Bi-Convex lens KBX169 supplied by the Newport Corporation, with a focal length of 250 mm, was used as collimating lens. It is mounted on a translation stage to translate it along the optical axis. The translation stage is moved by a micrometer with a resolution of 10 lm. The degree of collimation can be determined by adjusting the position of collimating lens. If the focal length of the lens is fc and the distance from the pinhole to the pupil plane of the lens is z, the wave front of the beam emerging from the lens is expressed by k ik 1 1 2 wðrÞ ¼ exp i r2 ¼ exp r : ð9Þ w 2 fc z A CDG is placed in front of a high-quality Newport Bi-Convex KBX178 lens with a focal length of 500 mm. The images at the focal plane are recorded by a Mintron MTV-1802CB CCD sensor. The pixel size is 8.05 lm · 8.05 lm. The CDG in the experiment was produced with wet etching technique. Firstly, a positive photoresist AZ5206E was spun onto K9 glass. After exposure and development, the latent pattern in the resist was transferred to the glass by wet etching. Finally, the remaining chromium was stripped off to get the final sample. The refractive index of glass is about 1.5164 at wavelength of 632.8 nm, the thickness corresponding to the phase difference p is equal to 613 nm. To keep consistent with the numerical simulation, the designed pitch of the one-order CDG is also 100 lm, and the total aperture is 20 mm. The diffraction patterns recorded by the CCD camera are shown in Fig. 7. To make the separation evident in the figure, the defocusing was chosen as 0.75 mm. The splitting of the rings can be distinguished clearly by naked eyes in Fig. 7b. The measured peak-to-peak distance was seven pixels, which is in good agreement with the numerical simulations. The experimental results fully demonstrated the validity of our proposed method. 4. Discussion
Fig. 3. Dependence of the radial separation on grating aperture and focal length, where N is the number of the periods.
As shown in Fig. 5, the radial separation almost does not change with the degree of noncollimation in a small region around 1/w = 0. Therefore, the accuracy obtainable in the collimation test can be determined by the smallest detectable deviation from perfect collimation, which is
4
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Fig. 4. Numerical simulated diffraction patterns with different (a) 1/w = 0, (b) 1/w = 8.03 · 109 lm1, (c) 1/w = 16.13 · 109 lm1, where f = 500 mm, T = 100 lm and the grating aperture was 20 mm.
measured from the separation of the two rings in the proposed technique. According to Eq. (8), besides the amount of noncollimation, the separation also depends on the focal length of the focusing lens and the grating aperture, which determine the precision of this method. For a given noncollimated wave front the dependence of the separation on focal length is shown in Fig. 8, where 1/w = 4.01 · 109 lm1, which is equivalent to a focal shift of 0.5 mm for a collimating lens with a focal length of 250 mm in Eq. (9). The grating aperture is 20 mm and period is 100 lm. From Fig. 8, it can be clearly seen that the separation is linearly proportional to the focal length. The relation between the separation and the grating aperture is illustrated in Fig. 9, where the focal length is
assumed to be 500 mm and other parameters are the same with what is used in Fig. 8. As shown in Fig. 9, when the grating aperture is smaller than a certain value, the separation widens as the aperture decreases. However, when the aperture exceeds this value, the separation increases with the aperture. In the proposed method, toward improving the measurement accuracy, we expect that the separation under collimated illumination is as small as possible. At the same time, the large separation is desired under noncollimated illumination. In Fig. 9, before the turning point the smaller aperture can generate the larger separation. However, in this condition, the separation under collimated illumination is also large as seen in Fig. 3, which means the difference between collimated and noncollimated
S. Zhao, P.S. Chung / Optics Communications 279 (2007) 1–6
5
Fig. 5. Radial separation between the two rings under different illumination.
Fig. 6. Schematic of the experimental arrangement for collimation testing.
conditions is inconspicuous. Therefore, a small aperture is useless for the precision enhancement. On the contrary, after the turning point in Fig. 9, the separation under noncollimated illumination increases with the grating size, whereas the trend is opposite under collimated illumination. It means the difference between them will be enlarged with the grating size. Therefore, we arrive at a conclusion that if permitted, the grating size should be as large as possible to increase the measurement accuracy. If we assume that the smallest distinguishable peak-topeak separation is four pixels in the image recorded by our CCD camera, according to the numerical calculation in Fig. 5, the separation at 1/w = 4.01 · 109 mm1 is 39.3 lm, which can be resolved. Therefore, the defocusing of 0.5 mm can be detected in our experiment. Many commercial CCD or CMOS cameras have the pixel smaller than 6 lm. If the grating size can be enlarged to 40 mm, the numerical analysis shows that the accuracy can reach 0.15 mm for a collimating lens with a focal length of 250 mm. According to Eq. (9), it is equivalent to that the measurement precision of wave front height in the entire aperture of the incident wave on the grating is about 0.76k. If the beam under test is relatively small in diameter, the grating aperture under illumination will be small too. In this case, the grating period should be decreased correspondingly. As illustrated above, compared with large grating, the measurement sensitivity will be slightly lower, which is a limitation of this method.
Fig. 7. Experimental images captured at the focal plane (a) in focus and (b) inside focus with a defocusing of 0.75 mm.
Fig. 8. Separation between the two rings corresponding to different focal lengths, where 1/w = 4.01 · 109 lm1.
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of collimation can be quickly checked by measuring the separation. It should be noted that the method we have reported in this paper is not a substitute for those welldeveloped techniques. It needs more components compared with the techniques using shear plates or Talbot gratings. It is a supplemental method suitable for collimation detection, especially for the beam with large aperture. In this condition, it has a precision comparable with other methods. The judgment to collimation can also be carried out automatically by the computer if a simple algorithm is programmed. All these advantages offer this method the potential to be embedded and integrated into other optical systems. Fig. 9. Dependence of separation on the grating aperture.
Acknowledgement The possible aberrations of the focusing lens should be considered. Here we take third-order spherical aberration as an example to illustrate its effect on the measurement. The amplitude distribution in the observation plane is given by r4 G3 ðqÞ ¼ F wðrÞgðrÞpðrÞ exp iAsph k 3 f w ¼ C 3 GðqÞ jincð2aqÞ exp ip2 q2 k 4 r F exp iAsph k 3 ; ð10Þ f where C3 is a constant. The dimensionless coefficient Asph quantitates the spherical aberration, which is a function of the shape of the lens surfaces and the wave front curvature incident onto the lens. Numerical calculation of Eq. (10) shows that under the collimated illumination, the position where the separation is minimal is not the paraxial focus of the lens any longer with presence of the spherical aberration. It has a displacement DZf from the paraxial focus given by 2 wb ; ð11Þ DZ f ¼ 2Asph f f where wb is the radius of the illumination beam. Therefore, this position must be accurately located and put the detector in it before the measurement. 5. Conclusion It has been shown that the separation of the two rings in the focal plane is definitely related to the wave front height of an incident noncollimated wave. Therefore, the degree
The authors acknowledge the support of CERG (No. 112805) of the Research Grants Council in Hong Kong. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
D.E. Silva, Appl. Opt. 10 (1971) 1980. M.P. Kothiyal, R.S. Sirohi, Appl. Opt. 26 (1987) 4056. C.W. Chang, D.C. Su, Opt. Lett. 16 (1991) 1783. C. Shakher, S. Prakash, D. Nand, R. Kumar, Appl. Opt. 40 (2001) 1175. K.V. Sriram, M.P. Kothiyal, R.S. Sirohi, Appl. Opt. 33 (1994) 7258. R. Torroba, N. Bolognini, M. Tebaldi, A. Tagliaferri, Opt. Commun. 201 (2002) 283. K.V. Avudainayagam, S. Chitralekha, J. Mod. Opt. 44 (1997) 175. S. Rana, S. Prakash, Appl. Opt. 45 (2006) 3213. M.V.R.K. Murty, Appl. Opt. 3 (1964) 531. D. Malacara, Optical shopping testing, Wiley, New York, 1992, p. 123. D.Y. Xu, K.J. Rosenbruch, Opt. Eng. 30 (1991) 391. H. Zhang, M.J. Lalor, D.R. Burton, Opt. Eng. 36 (1997) 3318. Y.W. Lee, Opt. Eng. 32 (1993) 2837. R.S. Sirohi, M.P. Kothiyal, Appl. Opt. 26 (1987) 4054. K.V. Sriram, P. Senthilkumaran, M.P. Kothiyal, R.S. Sirohi, Appl. Opt. 32 (1993) 4199. J. Choi, G.M. Perera, M.D. Aggarwal, R.P. Shukla, M.V. Mantravadi, Appl. Opt. 34 (1995) 3628. K. Matsuda, J.W. O’Byrne, C.J.R. Sheppard, S. Rehman, T. Eiju, Opt. Commun. 194 (2001) 1. W.L. Howes, Appl. Opt. 25 (1986) 473. T.D. Henning, J.L. Carlsten, Appl. Opt. 31 (1992) 1199. J.S. Darlin, M.P. Kothiyal, R.S. Sirohi, J. Modern Opt. 44 (1997) 929. P. Senthilkumaran, Appl. Opt. 42 (2003) 6314. M. Zhang, L. Liu, L. Wan, Z. Luan, Appl. Opt. 44 (2005) 2705. A. Anand, V.K. Chhaniwal, C.S. Narayanamurthy, Opt. Eng. 42 (2003) 1108. S. Zhao, P.S. Chung, Opt. Lett. 31 (2006) 2387. I.M. Gelfand, G.R. shilovGeneralized Functions, vol. 1, Academic, San Diego, CA, 1964.
Collimation testing using a circular Dammann grating Shuai Zhao *, Po Sheun Chung Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong Received 30 September 2006; received in revised form 23 May 2007; accepted 30 June 2007
Abstract A novel technique for collimation testing with a circular Dammann grating is proposed. When the beam under test is incident on a one-order circular Dammann grating with limited aperture, double-humped radial rings will be generated at the back focal plane of a focusing lens. If the beam is collimated, the separation between the two rings will reach its minimal, otherwise the two rings will be apart from each other. Therefore, the degree of collimation of the tested beam can be estimated from the separation. The principle and experimental results of the method are presented. Owing to the simplicity and low cost of the method, it is a promising method for quickly checking the collimation of a laser beam. 2007 Elsevier B.V. All rights reserved. Keywords: Collimation; Diffractive optics; Gratings; Optical testing
1. Introduction Beam collimation is always a prerequisite in optical systems, especially coherent optical systems. For example, the collimated beam is necessary in many measurement systems, and is related to the accuracy of the measurement. In free-space optical communication system, a flat beam should be emitted to achieve diffraction-limited Airy pattern at the receiver. A number of measurement methods have been proposed to check the collimation. Most of these methods fall into two main categories: self-imaging techniques and interferometric techniques. In the self-imaging techniques (Talbot interferometry), a grating replicates itself at several planes perpendicular to the propagation direction. If the second grating is placed in one of these self-imaging planes, straight parallel moire´ fringe are obtained under the illumination of collimated beam. For a noncollimated beam, these moire´ fringes rotate with respect to the reference line. Different types of gratings have been introduced into the Talbot interferometry, e.g., *
Corresponding author. Tel.: +852 2194 2683; fax: +852 2788 7741. E-mail address: [email protected] (S. Zhao).
0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.06.051
linear gratings [1], dual-field gratings [2], spiral gratings [3], and circular gratings [4]. A comparative study of collimation testing with dual-field linear gratings, spiral gratings, and evolute gratings has been reported [5]. The study shows the circular grating exhibits the highest sensitivity. In addition, the moire´ fringes can also be produced through a digital subtraction operation [6]. Collimation testing under an incoherent illumination have been carried out by use of Lau effect [7,8]. Another family of widely used methods is based on shearing interferometry. The shear can be introduced through plane parallel plate [9], single wedged plate [10–13], or double wedged plate [14– 16]. Multiple beam shearing interferometry was developed to achieve higher accuracy [17]. Besides above mentioned methods, other novel approaches have been reported such as phase conjugate interferometry using a Twyman–Green interferometer [18], cycle shearing interferometer [19,20], phase singularities [21], circular aperture sampling [22], and active materials [23]. These methods have their own pros and cons and are applicable in particular circumstances. This paper reports a simple collimation testing method with a circular Dammann grating (CDG). The unique
2
S. Zhao, P.S. Chung / Optics Communications 279 (2007) 1–6
diffraction behavior of this novel device under the noncollimated illumination will help to evaluate the degree of collimation. The most advantage of this method is its extremely simple configuration. 2. Principle of measurement Circular Dammann gratings is a kind of radially periodic phase gratings that can generate uniform-intensity impulse rings in the far field [24]. Each ring corresponds to a diffraction order. The order of a CDG is therefore defined as the number of these bright rings. The rotational symmetry of a CDG makes it not necessary to distinguish positive and negative orders. The phase modulation of the CDG can be binary, multilevel or continuous. For ease of fabrication, binary phase CDG is used in our proposed method. Under the illumination of collimated beam, a CDG with period T yields the amplitude field at the back focal plane of a lens 1 1 X n=T GðqÞ ¼ pffiffiffi cn dð1=2Þ ðq n=T Þ; p n¼1 ðn=T þ qÞ3=2
ð1Þ
which represents a series of concentric impulse rings, where q = q/kf denotes the spatial frequency, q is polar coordinate in the output plane, f is the focal length of the focusing lens, and d(1/2)(x) is the half-order derivative of the Dirac impulse d(x). The explicit expression for the fractional-order derivatives of d(x) can be found in advanced books on generalized functions such as Ref. [25]. cn are the Fourier series coefficients (or sine series coefficients) governed by Z n 2 cn ¼ ð2Þ gðrÞ sin 2p r dr; T T T where g(r) is the phase profile function of the CDG. The relative intensity of the nth pulse ring is determined by the coefficients cn. The CDG is so designed by optimizing g(r) to achieve equal intensity rings. In our proposed method, only the first diffraction order is concerned. Fig. 1 shows the phase profile of a one-order CDG. Its theoretical spectral intensities in the far field are shown in Fig. 2a. The calculated coefficients cn are given by cn ¼
2 ½1 ð1Þn : np
ð3Þ
Neglecting the terms involving higher order spectrum, Eq. (1) can then be simplified as
Fig. 2. Theoretical normalized intensity of a one-order CDG with (a) infinite aperture and (b) limited aperture.
GðqÞ ¼
4 1=T dð1=2Þ ðq 1=T Þ: p3=2 ð1=T þ qÞ3=2
ð4Þ
The foregoing analysis is based on the assumption that the grating is infinitely large, but a practical CDG only consists of a finite number of periods. It is, in fact, the product of an initial infinite grating with a low-pass filter, rect(r/2a), having the appropriate cut-off frequency. This means that the spectrum of such a finite grating is no longer impulsive, since the multiplication of the initial infinite grating with rect(r/2a) in the space domain implies that in the spectral domain the initial impulsive spectrum is convolved with the continuous spectrum of rect(r/2a). In this case, the Fourier spectrum will be G1 ðqÞ ¼ F ½gðrÞpðrÞ ¼ GðqÞ 4a2 jincð2aqÞ ¼ C 1 jincð1=2Þ ½2aðq 1=T Þ;
Fig. 1. Illustration of a one-order binary phase (0, p) CDG.
ð5Þ
where represents convolution, 2a is the aperture of the grating, i.e., the area illuminated by the laser beam. C1 is a constant independent on the coordinates, and jinc(q) =
S. Zhao, P.S. Chung / Optics Communications 279 (2007) 1–6
J1(pq)/2q. Eq. (5) is numerically calculated according to Grunwald–Letnikov definition to fractional derivatives ½ðtaÞ=h 1 X j a a D f ðtÞ ¼ lim ð1Þ f ðt jhÞ a t a h!0 h j j¼0 ’
½ðtaÞ=h 1 X ðaÞ xj f ðt jhÞ; ha j¼0
ð6Þ
where [x] means the integer part of x, h is time step, and the ðaÞ coefficients xj are given by the following recursive relation a þ 1 ðaÞ ðaÞ ðaÞ x0 ¼ 1; xj ¼ 1 ð7Þ xj1 ; j ¼ 1; 2; . . . j By plotting Eq. (5) in Fig. 2b, one can see that the main intensity maximum in Fig. 2a has split into two lobes owing to the limited aperture. The radial separation Dq between the two lobes is determined by the focal length and the grating aperture as shown in Fig. 3. It can be clearly seen that the separation decreases with the increase of the grating aperture, which is easily understood since the larger aperture means the grating is closer to infinite. On the contrary a lens with longer focal length will generate larger separation. If a noncollimated wave is incident on the CDG, the amplitude in the back focal plane of the focusing lens is G2 ðqÞ¼F ½wðrÞgðrÞpðrÞ
w ¼GðqÞ4a2 jincð2aqÞexp ip2 q2 k ( " 2 #)ð1=2Þ q 1 w q 1 ¼C 2 jinc 2a exp ip2 ; kf T k kf T ð8Þ
expði wk r2 Þ
where C2 is a constant and wðrÞ ¼ representing the noncollimated wave. When the beam is collimated, w is infinite. Fig. 4 numerically simulates the diffraction patterns at the back focal plane under different illumination, where f = 500 mm, T = 100 lm, and the grating aperture is 20 mm. If the separation between the two lobes is monitored, one can find that it strongly depends on the curvature radius, i.e., the degree of collimation, of the
3
illumination wave. The quantitative analysis is illustrated in Fig. 5 with the same parameters as Fig. 4. From the figure one can see that with the illumination beam deviating from collimation, the separation rapidly increases. This interesting characteristic indicates a novel way to do collimation testing in practice. The best collimation state can be achieved when the separation is minimum. 3. Experimental results The experimental setup for collimation testing is illustrated in Fig. 6. Light from a He–Ne laser with a wavelength of 632.8 nm is focused on a pinhole by a microscope objective. A precision Bi-Convex lens KBX169 supplied by the Newport Corporation, with a focal length of 250 mm, was used as collimating lens. It is mounted on a translation stage to translate it along the optical axis. The translation stage is moved by a micrometer with a resolution of 10 lm. The degree of collimation can be determined by adjusting the position of collimating lens. If the focal length of the lens is fc and the distance from the pinhole to the pupil plane of the lens is z, the wave front of the beam emerging from the lens is expressed by k ik 1 1 2 wðrÞ ¼ exp i r2 ¼ exp r : ð9Þ w 2 fc z A CDG is placed in front of a high-quality Newport Bi-Convex KBX178 lens with a focal length of 500 mm. The images at the focal plane are recorded by a Mintron MTV-1802CB CCD sensor. The pixel size is 8.05 lm · 8.05 lm. The CDG in the experiment was produced with wet etching technique. Firstly, a positive photoresist AZ5206E was spun onto K9 glass. After exposure and development, the latent pattern in the resist was transferred to the glass by wet etching. Finally, the remaining chromium was stripped off to get the final sample. The refractive index of glass is about 1.5164 at wavelength of 632.8 nm, the thickness corresponding to the phase difference p is equal to 613 nm. To keep consistent with the numerical simulation, the designed pitch of the one-order CDG is also 100 lm, and the total aperture is 20 mm. The diffraction patterns recorded by the CCD camera are shown in Fig. 7. To make the separation evident in the figure, the defocusing was chosen as 0.75 mm. The splitting of the rings can be distinguished clearly by naked eyes in Fig. 7b. The measured peak-to-peak distance was seven pixels, which is in good agreement with the numerical simulations. The experimental results fully demonstrated the validity of our proposed method. 4. Discussion
Fig. 3. Dependence of the radial separation on grating aperture and focal length, where N is the number of the periods.
As shown in Fig. 5, the radial separation almost does not change with the degree of noncollimation in a small region around 1/w = 0. Therefore, the accuracy obtainable in the collimation test can be determined by the smallest detectable deviation from perfect collimation, which is
4
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Fig. 4. Numerical simulated diffraction patterns with different (a) 1/w = 0, (b) 1/w = 8.03 · 109 lm1, (c) 1/w = 16.13 · 109 lm1, where f = 500 mm, T = 100 lm and the grating aperture was 20 mm.
measured from the separation of the two rings in the proposed technique. According to Eq. (8), besides the amount of noncollimation, the separation also depends on the focal length of the focusing lens and the grating aperture, which determine the precision of this method. For a given noncollimated wave front the dependence of the separation on focal length is shown in Fig. 8, where 1/w = 4.01 · 109 lm1, which is equivalent to a focal shift of 0.5 mm for a collimating lens with a focal length of 250 mm in Eq. (9). The grating aperture is 20 mm and period is 100 lm. From Fig. 8, it can be clearly seen that the separation is linearly proportional to the focal length. The relation between the separation and the grating aperture is illustrated in Fig. 9, where the focal length is
assumed to be 500 mm and other parameters are the same with what is used in Fig. 8. As shown in Fig. 9, when the grating aperture is smaller than a certain value, the separation widens as the aperture decreases. However, when the aperture exceeds this value, the separation increases with the aperture. In the proposed method, toward improving the measurement accuracy, we expect that the separation under collimated illumination is as small as possible. At the same time, the large separation is desired under noncollimated illumination. In Fig. 9, before the turning point the smaller aperture can generate the larger separation. However, in this condition, the separation under collimated illumination is also large as seen in Fig. 3, which means the difference between collimated and noncollimated
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Fig. 5. Radial separation between the two rings under different illumination.
Fig. 6. Schematic of the experimental arrangement for collimation testing.
conditions is inconspicuous. Therefore, a small aperture is useless for the precision enhancement. On the contrary, after the turning point in Fig. 9, the separation under noncollimated illumination increases with the grating size, whereas the trend is opposite under collimated illumination. It means the difference between them will be enlarged with the grating size. Therefore, we arrive at a conclusion that if permitted, the grating size should be as large as possible to increase the measurement accuracy. If we assume that the smallest distinguishable peak-topeak separation is four pixels in the image recorded by our CCD camera, according to the numerical calculation in Fig. 5, the separation at 1/w = 4.01 · 109 mm1 is 39.3 lm, which can be resolved. Therefore, the defocusing of 0.5 mm can be detected in our experiment. Many commercial CCD or CMOS cameras have the pixel smaller than 6 lm. If the grating size can be enlarged to 40 mm, the numerical analysis shows that the accuracy can reach 0.15 mm for a collimating lens with a focal length of 250 mm. According to Eq. (9), it is equivalent to that the measurement precision of wave front height in the entire aperture of the incident wave on the grating is about 0.76k. If the beam under test is relatively small in diameter, the grating aperture under illumination will be small too. In this case, the grating period should be decreased correspondingly. As illustrated above, compared with large grating, the measurement sensitivity will be slightly lower, which is a limitation of this method.
Fig. 7. Experimental images captured at the focal plane (a) in focus and (b) inside focus with a defocusing of 0.75 mm.
Fig. 8. Separation between the two rings corresponding to different focal lengths, where 1/w = 4.01 · 109 lm1.
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of collimation can be quickly checked by measuring the separation. It should be noted that the method we have reported in this paper is not a substitute for those welldeveloped techniques. It needs more components compared with the techniques using shear plates or Talbot gratings. It is a supplemental method suitable for collimation detection, especially for the beam with large aperture. In this condition, it has a precision comparable with other methods. The judgment to collimation can also be carried out automatically by the computer if a simple algorithm is programmed. All these advantages offer this method the potential to be embedded and integrated into other optical systems. Fig. 9. Dependence of separation on the grating aperture.
Acknowledgement The possible aberrations of the focusing lens should be considered. Here we take third-order spherical aberration as an example to illustrate its effect on the measurement. The amplitude distribution in the observation plane is given by r4 G3 ðqÞ ¼ F wðrÞgðrÞpðrÞ exp iAsph k 3 f w ¼ C 3 GðqÞ jincð2aqÞ exp ip2 q2 k 4 r F exp iAsph k 3 ; ð10Þ f where C3 is a constant. The dimensionless coefficient Asph quantitates the spherical aberration, which is a function of the shape of the lens surfaces and the wave front curvature incident onto the lens. Numerical calculation of Eq. (10) shows that under the collimated illumination, the position where the separation is minimal is not the paraxial focus of the lens any longer with presence of the spherical aberration. It has a displacement DZf from the paraxial focus given by 2 wb ; ð11Þ DZ f ¼ 2Asph f f where wb is the radius of the illumination beam. Therefore, this position must be accurately located and put the detector in it before the measurement. 5. Conclusion It has been shown that the separation of the two rings in the focal plane is definitely related to the wave front height of an incident noncollimated wave. Therefore, the degree
The authors acknowledge the support of CERG (No. 112805) of the Research Grants Council in Hong Kong. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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