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A dynamic zone-plate interferometer for measuring aspherical surfaces
V o l u m e 54, n u m b e r 5
OPTICS COMMUNICATIONS
1 July 1985
A DYNAMIC Z O N E - P L A T E I N T E R F E R O M E T E R FOR M E A S U R I N G A S P H E R I C A L SURFACES Nagaaki O H Y A M A , Ikuo Y A M A G U C H I , Isao I C H I M U R A , Toshio H O N D A and Jumpei T S U J I U C H I Imaging Science and Engineering Laboratory, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 227, Japan Received 2 April 1985
This paper proposes a dynamic zone-plate interferometer (DZPI) which can measure quite accurately both shapes and surface errors of aspherical mirrors or lenses. By changing the wavelength of the illuminating light in fine steps, the DZPI gives a series of interferograms in which fringes are shifted almost continuously. Therefore, all their orders can be identified even when the m a x i m u m order becomes greater than 100. From these m a n y interferograms, the shape and surface error of an optical instrument under test can be calculated with high precision by computer. A simple experiment confirms the validity of the DZPI.
1. Introduction
2. Principle of the DZPI
Optical interferometers [1] are quite useful for measuring the difference in the wavelength order between the shape of an optical surface under test and a standard surface. But conventional methods need considerable modification if the applications are to be extended: since the measurable maximum difference between surfaces is not large, it is quite difficult to obtain directly the error of aspherical surfaces. For this purpose, some special devices such as computer generated holograms (CGH) are indispensable for obtaining the deformed or aspherical standard wavefront. This paper proposed a dynamic zone-plate interferometer (DZPI) which can measure such surfaces without the need of either a CGH [2] or of a standard surface. The DZPI changes the wavelength of the illuminating light by using, for instance, a dye- or diode-laser whose wavelength can be adjusted as required by fine steps. Because the focal length of the zone plate is changed according to the wavelength shift, interference fringes are moved or scanned almost continuously. From these many interferQgrams, the shape or error of an optical surface can be calculated by computer. The fundamental principles and experimental results are shown in the following.
The optical setup of the DZPI for measuring a concave aspherical mirror is shown in fig. 1, where the arrangement of the optical components is the same as the conventional zone plate interferometer described by Smartt [3], but the wavelength of the illuminating light is changed in order to generate spherical waves with different radii. Since the on-axis zone plate is a hologram of two point sources, the reconstructed image point (+1 order) can be expressed by the fol-
0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
b t
Spectrosco[ py HM ,z.P.
•
~__.~:~rup_~ /Mirror Camera
M : Mirror
HM : Half mirror
L : Lens
Z.P. : Z o n e
plate
Fig. 1. Optical s e t u p of the DZPI.
257
Volume 54, number 5
OPTICS COMMUNICATIONS
lowing formula [4]
1/Z i = 1/Z c + / 2 ( I / Z o - 1/Zr) ,
(1)
/2 = )kr/Xc,
(2)
where Zi, Zc, Zo, and Z r are the distance from the origin to the image, reconstructing, object, and reference points, respectively, and/2 is the ratio o f the wavelength of the recording light, Xr, to that of the reconstructing light, Xc. Eq. (1) means that the wavelength shift changes the focal length, f, of the zone plate:
1/f=/2(1/Z o - 1/Zr).
(3)
In the zone-plate interferometer shown in fig. 2, the test beam is a diverging beam from R0, the +l diffracted wave, which after reflection passes through the zone plate, and becomes the zero order beam, (+1, 0). At the same time the reference beam first passes through the zone plate toward the vertex of the mirror, then is diffracted by the zone plate after reflection, so that it converges to R0, the (0, +1) beam. Light from other directions can be blocked by a pinhole stop at R 0. In general an aspherical mirror has many centers of the surface curvature corresponding to different zones, as shown in fig. 2. If one of the centers is coincident with the reference point, (0, +1), we can see some coarse fringes in the corresponding zone. With changing the wavelength, the reference point will move along the optical axis, so the zone of coarse fringes also moves. Since there is no path difference
-
Q
x
R
~
.......
I P'
1 July 1985
on the optical axis, the vertex of the mirror retains a zero-order fringe in spite of the wavelength shift. Accordingly, by moving the region of coarse fringes by the wavelength shift in fine steps, we can easily count the total number of fringes. At the same time the wavelength in each step should be known. F r o m these many interferograms we can calculate the shape of the mirror by computer, b y setting the wavelength as an additional parameter.
3. Fringe analysis A model for fringe analysis of the DZPI is shown in fig. 3, where O(x) represents the shape of the mirror and R(x; X) the reference sphere. The interferogram obtained at a certain wavelength X shows the difference between these at each point
d(x; X) = O(x) - R(x; X),
(4)
and the centers o f bright fringes satisfy the equation 2d(x; ?,) = nX,
(5)
where n is an integer representing the fringe order. Due to the characteristics of the common-path interferometer, both O(x) and R(x; X) can be adjusted to zero at the origin, x = 0. Then R(x; X) can be defined as
R(x; X) = D(X) - [D2(X) - x 2 ] 1/2,
(6)
/•
Z'Z,(,1.) Z.P.
0 (x) R (x,X)
Zp
(+,.o, p, ~ P,
: Equi
R, ~ R,
: Ceniers
curvature
zones
/ ........
~.
Mirror
o
!1 X
ol cufvoture
Fig. 2. Illustration of a parabolic mirror. R0-R 3 are the centers of surface curvature, and P0-P3 are the equi-curvature zones whose centers are R0-R 3, respectively. Needless to say, a practical parabolic mirror has many centers. 258
,
Fig. 3. A model for fringe analysis. As a first approximation, the difference, d(x; ~), may be considered to be producing the fringes.
Volume 54, number 5
OPTICS COMMUNICATIONS
1 July 1985
where D(?~) is the radius of the reference sphere, given by D(X) : Zp + Zi(~ ).
(7)
In eq. (7), Zp is the z-coordinate o f the zone plate and Z i is given by eq. (1). O(x) can be calculated, because all constants and the parameter,/~, appearing in eqs. (1), (6), and (7) are known and because the orders o f the fringes can be identified by observing the fringe shift as the wavelength is changed.
4. Experimental result To confirm the principle o f the DZPI, a simple experiment was carried out. The experimental setup is shown in fig. 1 with the source a frequency stabilized ring dye laser, Spectra Physics model 380D. The aspherical mirror under test is parabolic with a focal length 450 mm and aperture f/4.2. A zone plate was made by recording an in-line hologram of a point with a wavelength of 582.14 nm from a dye laser, and the focal length at this wavelength was 96.5 mm. After all optical components were properly arranged, the wavelength was changed slightly. Fig. 4 shows some of the interferograms obtained, where (a) was obtained at 582.07 nm, (b) at 579.86 nm, and (c) at 578.24 nm. Fig. 5 shows computer-generated interferograms for the same condition as the practical experiment. Note that corresponding images in figs. 4 and 5 are similar to each other. For a rough estimation, the surface curvature of the parabolic mirror along a horizontal diameter was approximated by a polynomial
O(x) = a4 x l 0 + a3x8 + a2x6 + al x4 + aox2 (mm). (8) The least-square-fit algorithm gave the coefficients as a0 =
5.58 X 10 - 4 '
a 2 = - 2 . 3 5 X 10 -14,
a 1 = - 2 . 7 0 X 10 -11, a 3 = - 2 . 3 2 X 10 -17,
a 4 = - 2 . 6 8 X 10 -20. Since the dominant coefficient in eq. (8) is a0, the shape of the mirror can be considered to be parabolic, which is in good agreement with the real shape. Fig. 6 demonstrates the simulated interferograms in case of all parameters being the same as in fig. 4 ex-
Fig. 4. Interferograms obtained at three wavelengths; (a) 582.07 nm, (b) 579.86 nm, and (c) 578.24 nm.
259
Volume 54, n u m b e r 5
OPTICS COMMUNICATIONS
Fig. 5. Computer-simulated interferograms. All parameters are the same as those of the practical experiment; (a), (b), and (c) correspond to fig. 4. 260
1 July 1985
Fig. 6. Computer-simulated interferograms obtained for a parabolic mirror of f/3; (a) shows the case where the center of the paraxial curvature is coincident with that of the reference sphere at 582.14 n m ; (b) and (c) show the cases for 574.14 n m and 567.15 n m , respectively. Note that the region of the coarse fringes shifts toward the edge.
Volume 54, number 5
OPTICS COMMUNICATIONS
1 July 1985
cept that the mirror is f/3. In fig. 6 (a) the center of paraxial curvature is coincident with the virtual point from which the test beam diverges. The closer to the edge of the mirror, the finer the fringes and the lower the contrast become, so the number o f fringes cannot be counted by any means. By changing the wavelength from 582.15 run to 574.14 nm, the area of coarse fringes is shifted toward the edge as shown in fig. 6 (b), and fig. 6 (c) shows the case of 567.15 nm. Remember that all fringes will move almost continuously if the wavelength is changed in b y fine steps, so the order of each fringe can be easily identified.
In the optical system, it is only when passing through the zone plate that the object and reference waves take different paths, so the substrate o f the zone plate must be optically flat. Otherwise, an additional phase variation will occur as a systematic error, which we could compensate by using a computer. In the experiment it was confirmed that all fringes moved continuously according to the wavelength shift as expected by the theory. More detailed discussions and experimental results obtained by the fringe scanning method, including error compensation, will be reported in a later paper.
5. Conclusion
References
The dynamic zone-plate interferometer (DZPI) proposed can measure both shapes and surface errors of aspherical mirrors or lens surfaces without using a standard surface or a CGH. With the aid o f the wavelength shift, the DZPI has succeeded in not only realizing a common-path fringe scanning but also widening the measurable range enough to cover aspherical surfaces.
[1 ] W.H. Steel, Interferometry (University Press, Cambridge, 1967). [2 ] H. Tanigawa, K. Nakajima and S. Mastuura, Optica Acta 27 (1980) 1327. [3] R.N. Smartt, Appl. Optics 13 (1974) 1093. [4] R.W. Meier, J. Opt. Soc. Am. 55 (1965) 987.
261
OPTICS COMMUNICATIONS
1 July 1985
A DYNAMIC Z O N E - P L A T E I N T E R F E R O M E T E R FOR M E A S U R I N G A S P H E R I C A L SURFACES Nagaaki O H Y A M A , Ikuo Y A M A G U C H I , Isao I C H I M U R A , Toshio H O N D A and Jumpei T S U J I U C H I Imaging Science and Engineering Laboratory, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 227, Japan Received 2 April 1985
This paper proposes a dynamic zone-plate interferometer (DZPI) which can measure quite accurately both shapes and surface errors of aspherical mirrors or lenses. By changing the wavelength of the illuminating light in fine steps, the DZPI gives a series of interferograms in which fringes are shifted almost continuously. Therefore, all their orders can be identified even when the m a x i m u m order becomes greater than 100. From these m a n y interferograms, the shape and surface error of an optical instrument under test can be calculated with high precision by computer. A simple experiment confirms the validity of the DZPI.
1. Introduction
2. Principle of the DZPI
Optical interferometers [1] are quite useful for measuring the difference in the wavelength order between the shape of an optical surface under test and a standard surface. But conventional methods need considerable modification if the applications are to be extended: since the measurable maximum difference between surfaces is not large, it is quite difficult to obtain directly the error of aspherical surfaces. For this purpose, some special devices such as computer generated holograms (CGH) are indispensable for obtaining the deformed or aspherical standard wavefront. This paper proposed a dynamic zone-plate interferometer (DZPI) which can measure such surfaces without the need of either a CGH [2] or of a standard surface. The DZPI changes the wavelength of the illuminating light by using, for instance, a dye- or diode-laser whose wavelength can be adjusted as required by fine steps. Because the focal length of the zone plate is changed according to the wavelength shift, interference fringes are moved or scanned almost continuously. From these many interferQgrams, the shape or error of an optical surface can be calculated by computer. The fundamental principles and experimental results are shown in the following.
The optical setup of the DZPI for measuring a concave aspherical mirror is shown in fig. 1, where the arrangement of the optical components is the same as the conventional zone plate interferometer described by Smartt [3], but the wavelength of the illuminating light is changed in order to generate spherical waves with different radii. Since the on-axis zone plate is a hologram of two point sources, the reconstructed image point (+1 order) can be expressed by the fol-
0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
b t
Spectrosco[ py HM ,z.P.
•
~__.~:~rup_~ /Mirror Camera
M : Mirror
HM : Half mirror
L : Lens
Z.P. : Z o n e
plate
Fig. 1. Optical s e t u p of the DZPI.
257
Volume 54, number 5
OPTICS COMMUNICATIONS
lowing formula [4]
1/Z i = 1/Z c + / 2 ( I / Z o - 1/Zr) ,
(1)
/2 = )kr/Xc,
(2)
where Zi, Zc, Zo, and Z r are the distance from the origin to the image, reconstructing, object, and reference points, respectively, and/2 is the ratio o f the wavelength of the recording light, Xr, to that of the reconstructing light, Xc. Eq. (1) means that the wavelength shift changes the focal length, f, of the zone plate:
1/f=/2(1/Z o - 1/Zr).
(3)
In the zone-plate interferometer shown in fig. 2, the test beam is a diverging beam from R0, the +l diffracted wave, which after reflection passes through the zone plate, and becomes the zero order beam, (+1, 0). At the same time the reference beam first passes through the zone plate toward the vertex of the mirror, then is diffracted by the zone plate after reflection, so that it converges to R0, the (0, +1) beam. Light from other directions can be blocked by a pinhole stop at R 0. In general an aspherical mirror has many centers of the surface curvature corresponding to different zones, as shown in fig. 2. If one of the centers is coincident with the reference point, (0, +1), we can see some coarse fringes in the corresponding zone. With changing the wavelength, the reference point will move along the optical axis, so the zone of coarse fringes also moves. Since there is no path difference
-
Q
x
R
~
.......
I P'
1 July 1985
on the optical axis, the vertex of the mirror retains a zero-order fringe in spite of the wavelength shift. Accordingly, by moving the region of coarse fringes by the wavelength shift in fine steps, we can easily count the total number of fringes. At the same time the wavelength in each step should be known. F r o m these many interferograms we can calculate the shape of the mirror by computer, b y setting the wavelength as an additional parameter.
3. Fringe analysis A model for fringe analysis of the DZPI is shown in fig. 3, where O(x) represents the shape of the mirror and R(x; X) the reference sphere. The interferogram obtained at a certain wavelength X shows the difference between these at each point
d(x; X) = O(x) - R(x; X),
(4)
and the centers o f bright fringes satisfy the equation 2d(x; ?,) = nX,
(5)
where n is an integer representing the fringe order. Due to the characteristics of the common-path interferometer, both O(x) and R(x; X) can be adjusted to zero at the origin, x = 0. Then R(x; X) can be defined as
R(x; X) = D(X) - [D2(X) - x 2 ] 1/2,
(6)
/•
Z'Z,(,1.) Z.P.
0 (x) R (x,X)
Zp
(+,.o, p, ~ P,
: Equi
R, ~ R,
: Ceniers
curvature
zones
/ ........
~.
Mirror
o
!1 X
ol cufvoture
Fig. 2. Illustration of a parabolic mirror. R0-R 3 are the centers of surface curvature, and P0-P3 are the equi-curvature zones whose centers are R0-R 3, respectively. Needless to say, a practical parabolic mirror has many centers. 258
,
Fig. 3. A model for fringe analysis. As a first approximation, the difference, d(x; ~), may be considered to be producing the fringes.
Volume 54, number 5
OPTICS COMMUNICATIONS
1 July 1985
where D(?~) is the radius of the reference sphere, given by D(X) : Zp + Zi(~ ).
(7)
In eq. (7), Zp is the z-coordinate o f the zone plate and Z i is given by eq. (1). O(x) can be calculated, because all constants and the parameter,/~, appearing in eqs. (1), (6), and (7) are known and because the orders o f the fringes can be identified by observing the fringe shift as the wavelength is changed.
4. Experimental result To confirm the principle o f the DZPI, a simple experiment was carried out. The experimental setup is shown in fig. 1 with the source a frequency stabilized ring dye laser, Spectra Physics model 380D. The aspherical mirror under test is parabolic with a focal length 450 mm and aperture f/4.2. A zone plate was made by recording an in-line hologram of a point with a wavelength of 582.14 nm from a dye laser, and the focal length at this wavelength was 96.5 mm. After all optical components were properly arranged, the wavelength was changed slightly. Fig. 4 shows some of the interferograms obtained, where (a) was obtained at 582.07 nm, (b) at 579.86 nm, and (c) at 578.24 nm. Fig. 5 shows computer-generated interferograms for the same condition as the practical experiment. Note that corresponding images in figs. 4 and 5 are similar to each other. For a rough estimation, the surface curvature of the parabolic mirror along a horizontal diameter was approximated by a polynomial
O(x) = a4 x l 0 + a3x8 + a2x6 + al x4 + aox2 (mm). (8) The least-square-fit algorithm gave the coefficients as a0 =
5.58 X 10 - 4 '
a 2 = - 2 . 3 5 X 10 -14,
a 1 = - 2 . 7 0 X 10 -11, a 3 = - 2 . 3 2 X 10 -17,
a 4 = - 2 . 6 8 X 10 -20. Since the dominant coefficient in eq. (8) is a0, the shape of the mirror can be considered to be parabolic, which is in good agreement with the real shape. Fig. 6 demonstrates the simulated interferograms in case of all parameters being the same as in fig. 4 ex-
Fig. 4. Interferograms obtained at three wavelengths; (a) 582.07 nm, (b) 579.86 nm, and (c) 578.24 nm.
259
Volume 54, n u m b e r 5
OPTICS COMMUNICATIONS
Fig. 5. Computer-simulated interferograms. All parameters are the same as those of the practical experiment; (a), (b), and (c) correspond to fig. 4. 260
1 July 1985
Fig. 6. Computer-simulated interferograms obtained for a parabolic mirror of f/3; (a) shows the case where the center of the paraxial curvature is coincident with that of the reference sphere at 582.14 n m ; (b) and (c) show the cases for 574.14 n m and 567.15 n m , respectively. Note that the region of the coarse fringes shifts toward the edge.
Volume 54, number 5
OPTICS COMMUNICATIONS
1 July 1985
cept that the mirror is f/3. In fig. 6 (a) the center of paraxial curvature is coincident with the virtual point from which the test beam diverges. The closer to the edge of the mirror, the finer the fringes and the lower the contrast become, so the number o f fringes cannot be counted by any means. By changing the wavelength from 582.15 run to 574.14 nm, the area of coarse fringes is shifted toward the edge as shown in fig. 6 (b), and fig. 6 (c) shows the case of 567.15 nm. Remember that all fringes will move almost continuously if the wavelength is changed in b y fine steps, so the order of each fringe can be easily identified.
In the optical system, it is only when passing through the zone plate that the object and reference waves take different paths, so the substrate o f the zone plate must be optically flat. Otherwise, an additional phase variation will occur as a systematic error, which we could compensate by using a computer. In the experiment it was confirmed that all fringes moved continuously according to the wavelength shift as expected by the theory. More detailed discussions and experimental results obtained by the fringe scanning method, including error compensation, will be reported in a later paper.
5. Conclusion
References
The dynamic zone-plate interferometer (DZPI) proposed can measure both shapes and surface errors of aspherical mirrors or lens surfaces without using a standard surface or a CGH. With the aid o f the wavelength shift, the DZPI has succeeded in not only realizing a common-path fringe scanning but also widening the measurable range enough to cover aspherical surfaces.
[1 ] W.H. Steel, Interferometry (University Press, Cambridge, 1967). [2 ] H. Tanigawa, K. Nakajima and S. Mastuura, Optica Acta 27 (1980) 1327. [3] R.N. Smartt, Appl. Optics 13 (1974) 1093. [4] R.W. Meier, J. Opt. Soc. Am. 55 (1965) 987.
261