- Email: [email protected]
A novel long trace profiler for synchrotron radiation optics
ARTICLE IN PRESS
Optics & Laser Technology 37 (2005) 391–396 www.elsevier.com/locate/optlastec
A novel long trace profiler for synchrotron radiation optics Zhi Lia,, Yang Zhaoa,1, Dacheng Lia, Tiqiao Xiaob, Shaojian Xiab a
State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China b Shanghai Synchrotron Radiation Facility, P.O. Box 800-204, Shanghai 201800, China Received 8 July 2002; received in revised form 18 May 2004; accepted 4 June 2004 Available online 12 August 2004
Abstract Increasing demands for accuracy in manufacturing and utilizing optics in synchrotron radiation facility require more precise surface measuring techniques. A novel profilometer design for testing aspherical optical elements, especially the optics used in synchrotron radiation, is presented, in which a phase plate is introduced so as to generate a pattern on the detector that can be easily and accurately centered. Compared with conventional long trace profilers, the optical system of this novel design is simple, therefore its potential error sources are greatly reduced. The feasibility of the method is validated in theory and experiments with a newly developed prototype. Experimental results show that this prototype works well even under an opening operation environment without temperature control and air conditioning. r 2004 Elsevier Ltd. All rights reserved. Keywords: Aspheric optics testing; Super-large surface metrology; Long trace profiler
1. Introduction Optics for synchrotrons, especially for X-ray beamlines, form a specific class of aspheres, which are different from those conventional optical components used in visible light. Because of the desired focusing properties, beamline mirrors tend to be long, and of elliptical, toroidal or other type of shapes. Such figures, however, are difficult to be tested by conventional methods [1]. As a result, various efforts have been proposed in the past decades [2–10]. Among these efforts, the long trace profiler (LTP) approach developed by Takacs and his coworkers [9–11] and shearing-scanning technique developed by Weingaertner et al. [14,15] received most of the attention. As for the latter, however, a commercial autocollimator Corresponding author. Present address. Physikalisch-Technische
Bundesanstalt, Bundesallee 100, D-38116 Brauschweig, Germany. Tel.: +49-531-5925148; fax: +49-531-5925105. E-mail address: [email protected] (Z. Li). 1 Now at Department of Civil and Material Engineering, University of Illinois at Chicago, Chicago, IL 6067, USA. 0030-3992/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2004.06.002
with super high precision and resolution (e.g. 0.01 and 0.001 arcsec, respectively) may be essential to ensure the measurement accuracy. Meanwhile, in order to reach up to the accuracy of 1 mrad (0.2 arcsec) or higher, there are still many inherent error sources in the developed LTP [11]. Recently, a novel long trace profiler (NLTP) was developed and built in Tsinghua University and has been delivered to Shanghai Synchrotron Radiation Facility (SSRF). In this paper the principle of this NLTP is described and preliminary experiments are demonstrated.
2. Advance in long trace profilometry for aspheres and complex surfaces If trivial aspects are neglected, the principles of all kinds of long trace profilers are based on an angleposition transforming system, also referring to f y lens system in some cases [16].
ARTICLE IN PRESS Z. Li et al. / Optics & Laser Technology 37 (2005) 391–396
392
2.1. Basic configuration of profilers based on f y systems As shown in Fig. 1, these profilers generally consist of three parts: (1) optical alignment subsystem which produces the datum line for shape measurement, (2) scanning component (optical head) which links the surface under test (SUT) to the datum line step by step, and (3) angle-position transforming and detecting subsystem in which a series of slope variations of the SUT could be obtained. Although autocollimator may be a convenient optical device for non-contact measurement of slopes of reflecting optical surfaces, however, up to now commercial autocollimator with high precision cannot be ordered easily. Thereby most profilers have to create their own angle-position transforming and detecting systems as shown in Fig. 1 and the detectors usually are charge coupled devices (CCDs) nowadays. It can be seen from Fig. 1 that when the slope of the detected point on the SUT deviates from the datum line by g and consequently the displacement of the image on the detector be s, we obtain g¼
1 s ; 2 f
ð1Þ
where f is the back focal length of the FT lens, and the detecting surface of the detector should be on the back focal plane of the FT lens. As far as the optical scanning head is concerned, although there are some exceptions [8,9], most profilers employ penta-prisms to scan the surface of object under test [10,11,14–16]. As it is known, a penta prism of appropriate quality could optically compensate for the lack of straightness of the linear guide, i.e. it is insensitive to the movement errors of stage including pitch, yaw and roll errors. As for the datum line used for slope comparison, all profilers employ collimated laser beams (single or double beams). Moreover, since the scanning range of a profiler usually is not very long (e.g. 1 m or less), the wavefront of a collimated laser beam
Detector
F T Lens Scanning direction Alignment Laser beam(s)
x z
Optical head γ
2γ
y
s
could be represented as follows: ðx2 þ y2 Þ Uðx; yÞ ¼ A exp ; R2
ð2Þ
where R is the ‘‘1=e2 ’’ radius of the beam. 2.2. Methods to enhance the resolution of a surface scanning profiler It can be seen from Eq. (1) that in order to increase the resolution of a profiler, only two parameters could be considered, i.e. the resolution of the image’s displacement and the back focal length of the FT lens. Obviously, the lens’s focal length cannot be enlarged without limitation, since more nonlinearity would be introduced with longer focal length due to lens aberration. In addition, the range of slope detecting will be reduced when focal length is stretched. As a result, how to enhance the resolution of image displacement detecting became one of the keys to enhance the slope resolution of a profiler. For a single-beam profiler with CCD as its image readout device (e.g. in Ref. [13]) and lens aberration and some other trivial errors being omitted, the intensity distribution along xf axis of the image pattern on CCD is as follows: " # xf 2 2 4 2 2 ; ð3Þ Iðxf Þ ¼ Ap R exp 2p R lf where xf is the coordinate on the back focal plane of FT lens, l and f are the wavelengths of laser beam and the back focal length of the FT lens, respectively. Although fringe subdivision technique has been introduced to data analysis [13], it is difficult, in general, for such profiler to achieve high precision because of the characteristics of the pattern. A novel scanning profiler called pencil beam interferometer was thereby proposed by von Bieren [7], the outstanding feature of his profiler was that a doublebeam Michelson interferometer was introduced into the basic configuration of the f y lens system. As a consequence, the image pattern acquired by photoelectric reader is of interference stripe, and could be described as follows: " # xf 2 2 4 2 2 Iðxf Þ ¼ Ap R exp 2p R lf 4pd xf k Dz0 ; 1 þ cos (4) lf
f
Surface under test Fig. 1. Basic configuration of surface scanning profilometry for measuring aspheres.
where d is the lateral interval between the pairs of detecting beams, and Dz0 is the path length difference between two arms of the interferometer. A special kind of image pattern with a specified fringe number and dark line in its center could be obtained by carefully
ARTICLE IN PRESS Z. Li et al. / Optics & Laser Technology 37 (2005) 391–396
adjusting the lateral interval d and initial path difference Dz0 . In practice of image processing, the accuracy of determining the displacement of image with dark lines is always much better than that of detecting the displacement of image pattern only with peaks. Thus, higher resolution could be achieved in such types of profilers (e.g. [11]). Unfortunately, however, the measurement uncertainty of these interferential profilers is usually subject to a variation of the optical path difference arising from thermal drifts, or mechanical and environmental instabilities. Therefore a frequency-stabilized He–Ne laser and well-controlled environment are essential for running these profilers.
393
Substitute Eqs. (2) and (6) into Eq. (5), obviously function gðx; yÞ has odd symmetry with respect to x-axis, i.e. gðx; yÞ ¼ gðx; yÞ. Consequently, Gðu; vÞ is also oddly symmetric with respect to u-axis, and the pattern I f ðxf ; yf Þ must have even symmetry. Moreover we obtain I f ð0; yf Þ 0:
ð7Þ
Eq. (7) means that the center of the measurement pattern remains a dark line. Fig. 2 illustrates the general intensity distribution of the measurement pattern, and Fig. 3 shows the real image pattern on the back focal plane of the FT lens acquired with the novel long trace profile described in Section 4.
where the aperture effects of FT lens and related optical elements have been neglected, Gð f x ; f y Þ ¼ Ffgðx; yÞg ¼ FfUðx; yÞ Tðx; yÞg and f x0 ¼ lfs . Various phase structures can be chosen to design the phase plate as long as the displacement of the pattern it produces on the back focal plane of the FT lens can be recognized easily and precisely. Taken as an example, here a 1-D p-jump phase plate is employed, whose phase distribution is as follows: 1; 0oxo1; 1oyo1; Tðx; yÞ ¼ ð6Þ expðipÞ; otherwise:
fx
(a)
0.4
If (fx, 0)/arbitrary units
On the basis of the analyses in Section 2, we can see that the key issue for enhancing the resolution and accuracy of a profiler is to employ more stable datum line and such special kind of image pattern that its displacement could be determined easily and precisely. Fortunately, these two problems have been frequently encountered in large-scale alignment techniques for the measurement of large-scale geometrical parameters and errors such as straightness, flatness, etc. in the past several decades [18]. A novel profilometry for measuring aspherical surface is hereby proposed, which introduces a phase plate used in the diffraction alignment technique to the angle-position transforming system. Provided that a phase plate be inserted to the collimated laser beam in Fig. 1, the transmission ratio of the phase plate is Tðx; yÞ, the total optical path from the phase plate to FT lens is d and the Fresnel approximation is valid within the whole scanning range. The intensity distribution on the back focal-plane of the FT lens can be easily obtained [19]: 2 xf y I f ðxf s; yf Þ / G f x0 ; f ; ð5Þ lf lf
fy
3. New approach: surface profilometry based on an angleposition transforming system and a phase plate
0.2
0 -6000
(b)
-3000
0 fx
3000
6000
Fig. 2. Intensity distribution of the image pattern for a p-jump phase plate (simulation results). (a) 2-D image pattern. (b) Intensity distribution along f x axis.
ARTICLE IN PRESS Z. Li et al. / Optics & Laser Technology 37 (2005) 391–396
394
-25
Ref. RP
PR RP1
BS
2 Penta-prism
2 1 xf
3
CCD RP2
75
1
zf yf
2
x / pixel
Phase Plate
f
FT Lens
Optical fiber
Focal Plane
Object 1
Laser Diode
Temperature controller Current controller
175 Fig. 4. Schematic of the NLTP’s optical system: Diffraction alignment system coupled with laser diode Scanning head Angle-position transforming system.
-25
175
75 y / pixel
Fig. 3. Real image pattern in the NLTP prototype.
In the case of 2-D p-jump phase plate Tðx; yÞ ( 8 0oxo1; 0oyo1; > < 1; Tðx; yÞ ¼ 1oxo0; 1oyo0; > : expðipÞ; otherwise ð8Þ similarly we can deduce that the corresponding image pattern I f ð0; yf Þ 0 and I f ðxf ; 0Þ 0, that is, a dark cross occurs in the image pattern. Naturally it is anticipated that a new type of three dimensional profiler can hereby be developed based on this type of phase plate. It is worthwhile to point out that, although the phase plate shown in Fig. 1 is of transmissive type, a reflective phase plate can also be employed without essential difference.
4. Principle of an one-dimensional long trace profiler using a p-jump phase plate Based on the theoretical analysis in Section 3, a novel long trace profiler (NLTP) is developed. The schematic of its optical system is shown in Fig. 4. The light beam of a stabilized laser diode is collimated by an optical fiber collimating system, and then passes through a p-jump phase plate, creating a desired diffraction beam as the datum line for slope comparison. A right-angle prism ðRP1 Þ is used to rotate the diffraction beam so that it is largely parallel to the surface under test (SUT). The datum beam is then divided by a beam splitter (BS), creating two separate beams: one for measuring the slope variation of the SUT
and the other for reference. A penta-prism (PP) is employed to scan the SUT, and transmit the measuring beam downwards to the SUT. The reflected measuring beam is transmitted again by PP, BS and RP2 to the FT lens, finally creating the measuring pattern on the back focal plane of the FT lens. The reference beam is sent directly to the FT lens, and generates the reference pattern, which is used to compensate for the light source drifts. In practice, the reference pattern is usually adjusted to be located at one end of the CCD detecting area, and no interference between the reference and measuring beams would take place within the whole scanning range. A polarizer (PR) is introduced to adjust the intensity of the reference beam to be equal to that of the measurement beam. Finally, the measurement and reference patterns are collected by a CCD located at the back focal plane of the FT lens, and transferred to a computer for further evaluations. A well-manufactured penta-prism can optically compensate for such movement errors of a stage as vertical straightness error, yaw error, etc. In the event of small slope variation, although the horizontal straightness error dh, i.e. movement error of the stage perpendicular to the input beam within x z plane, will affect the position of the output beam, its influence on the measurement results can be neglected since dh is always several orders of magnitude smaller than the diameter of the diffraction beam. Therefore the NLTP could employ a standard translation slide without loss of accuracy. Another characteristic of the NLTP is that the optical path length from the phase plate to the FT lens is identical at every measurement point on the SUT. As a result, the measurement and reference patterns remain constant within the whole scanning range, and potential errors in data processing can be avoided.
ARTICLE IN PRESS Z. Li et al. / Optics & Laser Technology 37 (2005) 391–396 1000
s /rad
500 0 -500 -1000 0
50
100
150
200 x /mm
250
300
350
400
50
100
150
200 x /mm
250
300
350
400
2 1 ∆s /rad
The reference beam can compensate not only for the collimated input beam’s pointing instability, but also for the relative vibration between the collimated beam and the phase plate. Suppose that, after one scanning procedure, a series of relative displacements of the measurement and reference patterns are obtained, and denoted as fsxi g and frxi g, respectively. In the case of a 90 -Porro prism being employed as reference optics, the slope variation of the SUT should be 1 sxi þ rxi S¼ ; ð9Þ 2 f
395
0 -1 -2
where i is the serial number of scanning point.
0
Fig. 6. Experimental results of the prototype of NLTP on measuring a long narrow mirror, where Ds ¼ s s.
5. Experimental testing
prototype is
A prototype of the NLTP (shown in Fig. 5) has been built, which consists of the following four parts: a diffraction alignment subsystem coupled with laser diode, scanning head, positioning subsystem, an angleposition transforming and detecting subsystem. The basic structural parameters of the prototype are: the FT lens focal length is 400 mm, the maximum scanning length is 370 mm, the diameter of the collimated beam is about 1.5 mm, and the slope measurement range is 8.0 mrad. CCD camera used in the prototype is a Pulnix TM1040, whose pixel size is 9:0 9:0 mm, and the total pixel number is 1024 1024. With curve fitting and data interpolating technique, we obtain the resolution of the measurement/reference pattern’s displacement dsx o0:1 mm, consequently, the slope resolution of the
dS ¼
Fig. 5. Prototype of the NLTP used for measuring aspherical optics: Surface under test Angle-position transforming system Scanning head Positioning system Diffraction alignment system.
dsx o0:25 mrad: f
As a preliminary experimental investigation of the performance of this prototype when it operated in a laboratory circumstance, a flat mirror with a length of 400 mm was measured. Generally it takes the prototype about 200 s to scan the whole range of the workpiece. Six scans were carried out sequentially, and the experimental results are shown in Fig. 6. The standard deviation of the slope measurement is 0:51 mrad.
6. Summary In this paper, a novel long trace profiler for large aspherical optics, especially those optical components used in synchrotron radiation, has been described. Compared with traditional surface profilers, interferential long trace profilers in particular, one piece of phase plate is introduced to replace the complicated interferometer, the optical system of this NLTP is rather concise, consequently its potential error sources are greatly reduced, which leads naturally to an improved measurement precision. Meanwhile, similar to traditional profilers, measurement accuracy of the NLTP is subject to its working environment. In a laboratory with thermal gradients and air turbulence, NLTP’s measurement errors tend to increase. A stable holder for the mirror under test is necessary in the event of long-term measurement. Nonlinearity would be introduced into the measurement results when the camera used in the system is not positioned exactly at the focal plane of the FT lens, and/ or its pixels are not uniform. All these factors will be
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carefully considered in the further investigation of this novel long trace profiler. References [1] Sostero G, Cocco D, Qian SN. Metrological challenges of synchrotron radiation optics. Proc SPIE 1999;3739:310–6. [2] Malacara D. Optical shop testing, 2nd ed. New York: Wiley; 1992. [3] Yun Z, Liu Z, Li Y. Aspheric surface testing with a liquid compensatory interferometer. Opt Eng 1998;37(4):1364–7. [4] Fairman PS, Ward BK, Oreb BF, Farrant DI, Gilliand Y, Freund CH, Leistner AJ, Seckold JA, Walsh CJ. 300-mm-aperture phaseshifting Fizeau interferometer. Opt Eng 1999;38(8):1371–80. [5] Garcia-Torales G, Paez G, Strojnik M. Simulations and experimental results with a vectorial shearing interferometer. Opt Eng 2001;40(5):767–73. [6] Brinkmann S, Schreiner R, Dresel T, Schwider J. Interferometric testing of plane and cylindrical workpieces with computergenerated holograms. Opt Eng 1998;37(9):2506–11. [7] von Bieren K. Pencil beam interferometer for aspherical optical surfaces. Proc SPIE 1982;343:101–8. [8] Takacs PZ, Qian SN, Colbert J. Design of a long trace surface profiler. Proc SPIE 1987;749:59–64.
[9] Takacs PZ, Qian SN. Surface profile interferometer. US Patent No. 4,884,697, 1989. [10] Qian SN, Jark W, Takacs PZ. The penta-prism LTP: a long-traceprofiler with stationary optical head and moving penta prism. Rev Sci Instrum 1995;66(3):2562–9. [11] Qian SN, Sostero G, Takacs PZ. Precision calibration and systematic error reduction in the long trace profiler. Opt Eng 2000;39(1):304–10. [13] Yu JC, Xu JQ, Zhang XJ, Sun XF. Surface contour measurement using an optical scanner. Proc SPIE 1995;2536:192–9. [14] Weingaertner I, Schulz M, Elster C. Novel scanning technique for ultraprecise measurement of topography. Proc SPIE 1999;3782: 306–17. [15] Weingaertner I, Schulz M. Novel scanning technique for ultraprecise measurement of slope and topography of flats. and complex surfaces, Proc SPIE 1999;3739:274–82. [16] Irick SC, McKinney WR. Advancements in one-dimensional profiling with a long-trace profiler. Proc SPIE 1992;1720: 162–8. [18] Hao Q, Li DC, Wang YT. High-accuracy long distance alignment using single-mode optical fiber and phase plate. Opt Laser Technol 2002;34(4):287–92. [19] Goodman JW. Introduction to Fourier optics. San Francisco: McGraw-Hill; 1968.
Optics & Laser Technology 37 (2005) 391–396 www.elsevier.com/locate/optlastec
A novel long trace profiler for synchrotron radiation optics Zhi Lia,, Yang Zhaoa,1, Dacheng Lia, Tiqiao Xiaob, Shaojian Xiab a
State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China b Shanghai Synchrotron Radiation Facility, P.O. Box 800-204, Shanghai 201800, China Received 8 July 2002; received in revised form 18 May 2004; accepted 4 June 2004 Available online 12 August 2004
Abstract Increasing demands for accuracy in manufacturing and utilizing optics in synchrotron radiation facility require more precise surface measuring techniques. A novel profilometer design for testing aspherical optical elements, especially the optics used in synchrotron radiation, is presented, in which a phase plate is introduced so as to generate a pattern on the detector that can be easily and accurately centered. Compared with conventional long trace profilers, the optical system of this novel design is simple, therefore its potential error sources are greatly reduced. The feasibility of the method is validated in theory and experiments with a newly developed prototype. Experimental results show that this prototype works well even under an opening operation environment without temperature control and air conditioning. r 2004 Elsevier Ltd. All rights reserved. Keywords: Aspheric optics testing; Super-large surface metrology; Long trace profiler
1. Introduction Optics for synchrotrons, especially for X-ray beamlines, form a specific class of aspheres, which are different from those conventional optical components used in visible light. Because of the desired focusing properties, beamline mirrors tend to be long, and of elliptical, toroidal or other type of shapes. Such figures, however, are difficult to be tested by conventional methods [1]. As a result, various efforts have been proposed in the past decades [2–10]. Among these efforts, the long trace profiler (LTP) approach developed by Takacs and his coworkers [9–11] and shearing-scanning technique developed by Weingaertner et al. [14,15] received most of the attention. As for the latter, however, a commercial autocollimator Corresponding author. Present address. Physikalisch-Technische
Bundesanstalt, Bundesallee 100, D-38116 Brauschweig, Germany. Tel.: +49-531-5925148; fax: +49-531-5925105. E-mail address: [email protected] (Z. Li). 1 Now at Department of Civil and Material Engineering, University of Illinois at Chicago, Chicago, IL 6067, USA. 0030-3992/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2004.06.002
with super high precision and resolution (e.g. 0.01 and 0.001 arcsec, respectively) may be essential to ensure the measurement accuracy. Meanwhile, in order to reach up to the accuracy of 1 mrad (0.2 arcsec) or higher, there are still many inherent error sources in the developed LTP [11]. Recently, a novel long trace profiler (NLTP) was developed and built in Tsinghua University and has been delivered to Shanghai Synchrotron Radiation Facility (SSRF). In this paper the principle of this NLTP is described and preliminary experiments are demonstrated.
2. Advance in long trace profilometry for aspheres and complex surfaces If trivial aspects are neglected, the principles of all kinds of long trace profilers are based on an angleposition transforming system, also referring to f y lens system in some cases [16].
ARTICLE IN PRESS Z. Li et al. / Optics & Laser Technology 37 (2005) 391–396
392
2.1. Basic configuration of profilers based on f y systems As shown in Fig. 1, these profilers generally consist of three parts: (1) optical alignment subsystem which produces the datum line for shape measurement, (2) scanning component (optical head) which links the surface under test (SUT) to the datum line step by step, and (3) angle-position transforming and detecting subsystem in which a series of slope variations of the SUT could be obtained. Although autocollimator may be a convenient optical device for non-contact measurement of slopes of reflecting optical surfaces, however, up to now commercial autocollimator with high precision cannot be ordered easily. Thereby most profilers have to create their own angle-position transforming and detecting systems as shown in Fig. 1 and the detectors usually are charge coupled devices (CCDs) nowadays. It can be seen from Fig. 1 that when the slope of the detected point on the SUT deviates from the datum line by g and consequently the displacement of the image on the detector be s, we obtain g¼
1 s ; 2 f
ð1Þ
where f is the back focal length of the FT lens, and the detecting surface of the detector should be on the back focal plane of the FT lens. As far as the optical scanning head is concerned, although there are some exceptions [8,9], most profilers employ penta-prisms to scan the surface of object under test [10,11,14–16]. As it is known, a penta prism of appropriate quality could optically compensate for the lack of straightness of the linear guide, i.e. it is insensitive to the movement errors of stage including pitch, yaw and roll errors. As for the datum line used for slope comparison, all profilers employ collimated laser beams (single or double beams). Moreover, since the scanning range of a profiler usually is not very long (e.g. 1 m or less), the wavefront of a collimated laser beam
Detector
F T Lens Scanning direction Alignment Laser beam(s)
x z
Optical head γ
2γ
y
s
could be represented as follows: ðx2 þ y2 Þ Uðx; yÞ ¼ A exp ; R2
ð2Þ
where R is the ‘‘1=e2 ’’ radius of the beam. 2.2. Methods to enhance the resolution of a surface scanning profiler It can be seen from Eq. (1) that in order to increase the resolution of a profiler, only two parameters could be considered, i.e. the resolution of the image’s displacement and the back focal length of the FT lens. Obviously, the lens’s focal length cannot be enlarged without limitation, since more nonlinearity would be introduced with longer focal length due to lens aberration. In addition, the range of slope detecting will be reduced when focal length is stretched. As a result, how to enhance the resolution of image displacement detecting became one of the keys to enhance the slope resolution of a profiler. For a single-beam profiler with CCD as its image readout device (e.g. in Ref. [13]) and lens aberration and some other trivial errors being omitted, the intensity distribution along xf axis of the image pattern on CCD is as follows: " # xf 2 2 4 2 2 ; ð3Þ Iðxf Þ ¼ Ap R exp 2p R lf where xf is the coordinate on the back focal plane of FT lens, l and f are the wavelengths of laser beam and the back focal length of the FT lens, respectively. Although fringe subdivision technique has been introduced to data analysis [13], it is difficult, in general, for such profiler to achieve high precision because of the characteristics of the pattern. A novel scanning profiler called pencil beam interferometer was thereby proposed by von Bieren [7], the outstanding feature of his profiler was that a doublebeam Michelson interferometer was introduced into the basic configuration of the f y lens system. As a consequence, the image pattern acquired by photoelectric reader is of interference stripe, and could be described as follows: " # xf 2 2 4 2 2 Iðxf Þ ¼ Ap R exp 2p R lf 4pd xf k Dz0 ; 1 þ cos (4) lf
f
Surface under test Fig. 1. Basic configuration of surface scanning profilometry for measuring aspheres.
where d is the lateral interval between the pairs of detecting beams, and Dz0 is the path length difference between two arms of the interferometer. A special kind of image pattern with a specified fringe number and dark line in its center could be obtained by carefully
ARTICLE IN PRESS Z. Li et al. / Optics & Laser Technology 37 (2005) 391–396
adjusting the lateral interval d and initial path difference Dz0 . In practice of image processing, the accuracy of determining the displacement of image with dark lines is always much better than that of detecting the displacement of image pattern only with peaks. Thus, higher resolution could be achieved in such types of profilers (e.g. [11]). Unfortunately, however, the measurement uncertainty of these interferential profilers is usually subject to a variation of the optical path difference arising from thermal drifts, or mechanical and environmental instabilities. Therefore a frequency-stabilized He–Ne laser and well-controlled environment are essential for running these profilers.
393
Substitute Eqs. (2) and (6) into Eq. (5), obviously function gðx; yÞ has odd symmetry with respect to x-axis, i.e. gðx; yÞ ¼ gðx; yÞ. Consequently, Gðu; vÞ is also oddly symmetric with respect to u-axis, and the pattern I f ðxf ; yf Þ must have even symmetry. Moreover we obtain I f ð0; yf Þ 0:
ð7Þ
Eq. (7) means that the center of the measurement pattern remains a dark line. Fig. 2 illustrates the general intensity distribution of the measurement pattern, and Fig. 3 shows the real image pattern on the back focal plane of the FT lens acquired with the novel long trace profile described in Section 4.
where the aperture effects of FT lens and related optical elements have been neglected, Gð f x ; f y Þ ¼ Ffgðx; yÞg ¼ FfUðx; yÞ Tðx; yÞg and f x0 ¼ lfs . Various phase structures can be chosen to design the phase plate as long as the displacement of the pattern it produces on the back focal plane of the FT lens can be recognized easily and precisely. Taken as an example, here a 1-D p-jump phase plate is employed, whose phase distribution is as follows: 1; 0oxo1; 1oyo1; Tðx; yÞ ¼ ð6Þ expðipÞ; otherwise:
fx
(a)
0.4
If (fx, 0)/arbitrary units
On the basis of the analyses in Section 2, we can see that the key issue for enhancing the resolution and accuracy of a profiler is to employ more stable datum line and such special kind of image pattern that its displacement could be determined easily and precisely. Fortunately, these two problems have been frequently encountered in large-scale alignment techniques for the measurement of large-scale geometrical parameters and errors such as straightness, flatness, etc. in the past several decades [18]. A novel profilometry for measuring aspherical surface is hereby proposed, which introduces a phase plate used in the diffraction alignment technique to the angle-position transforming system. Provided that a phase plate be inserted to the collimated laser beam in Fig. 1, the transmission ratio of the phase plate is Tðx; yÞ, the total optical path from the phase plate to FT lens is d and the Fresnel approximation is valid within the whole scanning range. The intensity distribution on the back focal-plane of the FT lens can be easily obtained [19]: 2 xf y I f ðxf s; yf Þ / G f x0 ; f ; ð5Þ lf lf
fy
3. New approach: surface profilometry based on an angleposition transforming system and a phase plate
0.2
0 -6000
(b)
-3000
0 fx
3000
6000
Fig. 2. Intensity distribution of the image pattern for a p-jump phase plate (simulation results). (a) 2-D image pattern. (b) Intensity distribution along f x axis.
ARTICLE IN PRESS Z. Li et al. / Optics & Laser Technology 37 (2005) 391–396
394
-25
Ref. RP
PR RP1
BS
2 Penta-prism
2 1 xf
3
CCD RP2
75
1
zf yf
2
x / pixel
Phase Plate
f
FT Lens
Optical fiber
Focal Plane
Object 1
Laser Diode
Temperature controller Current controller
175 Fig. 4. Schematic of the NLTP’s optical system: Diffraction alignment system coupled with laser diode Scanning head Angle-position transforming system.
-25
175
75 y / pixel
Fig. 3. Real image pattern in the NLTP prototype.
In the case of 2-D p-jump phase plate Tðx; yÞ ( 8 0oxo1; 0oyo1; > < 1; Tðx; yÞ ¼ 1oxo0; 1oyo0; > : expðipÞ; otherwise ð8Þ similarly we can deduce that the corresponding image pattern I f ð0; yf Þ 0 and I f ðxf ; 0Þ 0, that is, a dark cross occurs in the image pattern. Naturally it is anticipated that a new type of three dimensional profiler can hereby be developed based on this type of phase plate. It is worthwhile to point out that, although the phase plate shown in Fig. 1 is of transmissive type, a reflective phase plate can also be employed without essential difference.
4. Principle of an one-dimensional long trace profiler using a p-jump phase plate Based on the theoretical analysis in Section 3, a novel long trace profiler (NLTP) is developed. The schematic of its optical system is shown in Fig. 4. The light beam of a stabilized laser diode is collimated by an optical fiber collimating system, and then passes through a p-jump phase plate, creating a desired diffraction beam as the datum line for slope comparison. A right-angle prism ðRP1 Þ is used to rotate the diffraction beam so that it is largely parallel to the surface under test (SUT). The datum beam is then divided by a beam splitter (BS), creating two separate beams: one for measuring the slope variation of the SUT
and the other for reference. A penta-prism (PP) is employed to scan the SUT, and transmit the measuring beam downwards to the SUT. The reflected measuring beam is transmitted again by PP, BS and RP2 to the FT lens, finally creating the measuring pattern on the back focal plane of the FT lens. The reference beam is sent directly to the FT lens, and generates the reference pattern, which is used to compensate for the light source drifts. In practice, the reference pattern is usually adjusted to be located at one end of the CCD detecting area, and no interference between the reference and measuring beams would take place within the whole scanning range. A polarizer (PR) is introduced to adjust the intensity of the reference beam to be equal to that of the measurement beam. Finally, the measurement and reference patterns are collected by a CCD located at the back focal plane of the FT lens, and transferred to a computer for further evaluations. A well-manufactured penta-prism can optically compensate for such movement errors of a stage as vertical straightness error, yaw error, etc. In the event of small slope variation, although the horizontal straightness error dh, i.e. movement error of the stage perpendicular to the input beam within x z plane, will affect the position of the output beam, its influence on the measurement results can be neglected since dh is always several orders of magnitude smaller than the diameter of the diffraction beam. Therefore the NLTP could employ a standard translation slide without loss of accuracy. Another characteristic of the NLTP is that the optical path length from the phase plate to the FT lens is identical at every measurement point on the SUT. As a result, the measurement and reference patterns remain constant within the whole scanning range, and potential errors in data processing can be avoided.
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The reference beam can compensate not only for the collimated input beam’s pointing instability, but also for the relative vibration between the collimated beam and the phase plate. Suppose that, after one scanning procedure, a series of relative displacements of the measurement and reference patterns are obtained, and denoted as fsxi g and frxi g, respectively. In the case of a 90 -Porro prism being employed as reference optics, the slope variation of the SUT should be 1 sxi þ rxi S¼ ; ð9Þ 2 f
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Fig. 6. Experimental results of the prototype of NLTP on measuring a long narrow mirror, where Ds ¼ s s.
5. Experimental testing
prototype is
A prototype of the NLTP (shown in Fig. 5) has been built, which consists of the following four parts: a diffraction alignment subsystem coupled with laser diode, scanning head, positioning subsystem, an angleposition transforming and detecting subsystem. The basic structural parameters of the prototype are: the FT lens focal length is 400 mm, the maximum scanning length is 370 mm, the diameter of the collimated beam is about 1.5 mm, and the slope measurement range is 8.0 mrad. CCD camera used in the prototype is a Pulnix TM1040, whose pixel size is 9:0 9:0 mm, and the total pixel number is 1024 1024. With curve fitting and data interpolating technique, we obtain the resolution of the measurement/reference pattern’s displacement dsx o0:1 mm, consequently, the slope resolution of the
dS ¼
Fig. 5. Prototype of the NLTP used for measuring aspherical optics: Surface under test Angle-position transforming system Scanning head Positioning system Diffraction alignment system.
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As a preliminary experimental investigation of the performance of this prototype when it operated in a laboratory circumstance, a flat mirror with a length of 400 mm was measured. Generally it takes the prototype about 200 s to scan the whole range of the workpiece. Six scans were carried out sequentially, and the experimental results are shown in Fig. 6. The standard deviation of the slope measurement is 0:51 mrad.
6. Summary In this paper, a novel long trace profiler for large aspherical optics, especially those optical components used in synchrotron radiation, has been described. Compared with traditional surface profilers, interferential long trace profilers in particular, one piece of phase plate is introduced to replace the complicated interferometer, the optical system of this NLTP is rather concise, consequently its potential error sources are greatly reduced, which leads naturally to an improved measurement precision. Meanwhile, similar to traditional profilers, measurement accuracy of the NLTP is subject to its working environment. In a laboratory with thermal gradients and air turbulence, NLTP’s measurement errors tend to increase. A stable holder for the mirror under test is necessary in the event of long-term measurement. Nonlinearity would be introduced into the measurement results when the camera used in the system is not positioned exactly at the focal plane of the FT lens, and/ or its pixels are not uniform. All these factors will be
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