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Wedge plate interferometry: a new dual field configuration for collimation testing
Optics & Laser Technology 30 (1998) 225±228
Wedge plate interferometry: a new dual ®eld con®guration for collimation testing J.S. Darlin, M.P. Kothiyal *, R.S. Sirohi Applied Optics Laboratory, Department Of Physics, Indian Institute of Technology, Madras-36, India Received 19 December 1997; accepted 1 June 1998
Abstract A method of deriving a dual ®eld fringe pattern from a single wedge plate for collimation testing is presented. The proposed technique uses a wedge plate and a 908 prism retrore¯ector to form two interference ®elds. With the two sets of fringes, the technique provides twofold increase in sensitivity and has its own reference. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Collimation testing; Wedge plate interferometry; Lateral shear interferometry
1. Introduction Several papers have been reported to check the collimation of the laser beams, using parallel plate and wedge plate lateral shearing interferometers [1, 2]. The plane parallel plate has been used for monitoring the collimation, but being an in®nite fringe method, it is less sensitive [1]. The wedge plate interferometer eliminates this problem. It is more sensitive than the plane parallel plate method [3], but it requires a reference ®duciary mark to check the fringe rotation. The double wedge plate method provides a two fold increase in sensitivity and has its own reference [4±6]. In this method the two wedge plates must be aligned antiparallel to each other. In the rotatable single wedge plate shearing interferometer, an accurate rotator is used for indicating a reference direction for collimation testing [7]. Gouha et al. [8] proposed a method to get the two sets of interference fringes from the single wedge plate. It uses a single wedge plate and two mirrors in a Michelson interferometer con®guration, as shown in Fig. 1. The wedge plate is so oriented that the direction of the wedge is perpendicular to the lateral shear. The beam re¯ected by the wedge plate is laterally sheared. The mirror (M1) re¯ects back this beam to * Corresponding author. Tel.: +91-44-235-1365 or +91-44-2353291; Fax: +91-44-235-0509; E-mail: [email protected]. 0030-3992/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 8 ) 0 0 0 4 1 - 3
the screen, where the corresponding interference pattern can be seen. The beam transmitted by the beam splitter is re¯ected back by mirror M2. This beam now gets sheared in re¯ection by the wedge plate and the interference pattern due to the sheared wavefronts can also be seen on the screen. If one of the two mirrors is tilted with respect to the beam, the two sets of interference patterns can be separated as seen in Fig. 1. This technique was devised to overcome the problem of aligning the two wedge plates [4±6] and the requirement of an accurate rotator [7]. For a well collimated beam, both sets of fringes are horizontal and parallel to each other. The two sets of fringes obtained in this method are separately located and the setting of two sets of fringes parallel to each other is not easy, leading to error in measurement. This separation between the two set of fringes was removed by Lee and co workers [9]. The top half of the one mirror and the lower half of the other mirror ware masked. The lower half of the shear fringe pattern from one mirror and upper half of the sheared fringes from the other mirror are thus brought side by side with a very sharp demarkation line, such that the interferometer behaves like a single shearing interferogram. The fringes in one half, act as a reference to the fringes in the other half. Parallelism of two set of fringes is a test of the laser beam collimation. In the above two methods, the re¯ected and transmitted beams from the wedge plate are folded back to the wedge plate, hence a signi®cant amount of laser beam
226
J.S. Darlin et al. / Optics & Laser Technology 30 (1998) 225±228
through and may be used for other experiments at the same time monitoring the beam collimation. 2. Modi®ed dual ®eld wedge plate lateral shear interferometer
Fig. 1. Schematic diagram [8] of a Michelson interferometer, with a single wedge plate for collimation testing.
is fed back to the source and can in¯uence the behaviour of the light source. These methods have the additional drawback that when the collimation tester is placed in the test beam, the laser beam cannot be used for subsequent stages of experiments. Thus the collimation of the laser beam cannot be continuously monitored. Here we present a new con®guration, using a retrore¯ector and a single wedge plate for collimation testing, which is self-referencing as in the above methods and has the following advantages. First, the amount of optical feed back to the source is very small (10.16%). Second, the interferometer uses only about 12% of the incident light. The rest of the light is transmitted
Fig. 2 shows the schematic of the new con®guration. Two sheared wavefronts with a tilt between them are produced in re¯ection, as explained in connection with Fig. 1. A major fraction of the incident beam is transmitted through and can be used elsewhere. The re¯ected beam is incident on the retrore¯ector. The upperpart of the front surface of the retro re¯ector is mirrored as shown in Fig. 3. The fringe pattern before it is incident on the retrore¯ector will appear in general as in Fig. 4(a). The top portion of beam is re¯ected back without any alteration as in the top half of Fig. 4(b). The lower half of the beam is inverted about the roof of the prism and is re¯ected back. This re¯ected fringe pattern will appear as shown in the lower half of Fig. 4(b). Thus, two set of interferograms can be derived from the single ®eld after re¯ection from the retrore¯ector. It can be seen, that when the fringe pattern produced by the wedge plate due to the change in the curvature of the test beam rotates, the fringe in the two halves of the ®eld will rotate in opposite directions. This property gives doubled sensitivity and self-referencing. As mentioned earlier, the wedge direction is perpendicular to the shear direction in the wedge plate interferometer for collimation testing. The fringes are parallel to the shear direction for collimated beam in this case. The roof axis of the retrore¯ector in the present case is parallel to the wedge direction. Hence
Fig. 2. Schematic diagram of the new con®guration of a wedge plate interferometer for collimation testing.
J.S. Darlin et al. / Optics & Laser Technology 30 (1998) 225±228
227
fringes in the two ®elds is given by a 2 tanÿ1
2DDx : b
5
For a collimated beam, D = 0 and the fringes in the two ®elds are parallel. For a noncollimated beam D$0 and the fringes in the two ®elds enclose the angle between them. When D takes negative values, the fringes in the two ®elds are inclined in opposite direction to that when D is positive. Thus, it is easy to tell whether the point source is inside or outside the focus, according to the direction of the fringe orientation. The angle a is twice as large, as that obtained in the single wedge plate method. The smallest displacement Df of the collimator lens, that can be detected is Fig. 3. Schematic of a 908 prism, used as a retrore¯ector in a new con®guration for collimation testing. The top half of the front face of the prism is mirrored.
the fringe pattern re¯ected by the front surface of the retrore¯ector, as well as the retro re¯ected fringe pattern, will be parallel to the shear axis; that is, the fringes in the two ®elds will be parallel to each other. For a noncollimated beam the fringes will be found to rotate in the opposite directions. Thus, we have both referencing and improved sensitivity. The defocusing error in wavefront is represented by W x, y D x 2 y2 ,
1
where D represents the defocusing factor. The defocusing factor is a function of the radius of the curvature R of the wavefront, which is given by D
1 Df 2, 2R 2f
2
where f is focal-length of the collimating tens and Df its deviation from the collimation position. The fringe equations in the two halves in Fig. 4(b) can be written as 2DxDx bg nl,
3
ÿ2DxDx gb nl,
4
where b is the tilt between the wavefront and Dx is the amount of shear. The ÿve sign in Eq. (4) is due to the reversal of the coordinate on retro re¯ection. These equations show that the fringes in the two ®elds are straight and inclined at equal angle from the horizontal axis in the opposite direction. Since the two sets of fringes rotate in opposite direction the sensitivity of the modi®ed interferometer is twice than that of the single wedge plate method. The angle between the
jDf j
f 2 ba : 2Dx
6
Here a is the smallest angle, between the fringes of the two ®elds, that can be detected. The sensitivity of the collimation test can be controlled by proper choice of b and Dx. 3. Experimental veri®cation and results An interferometer is setup based on the optical con®guration, shown in Fig. 2. A HeNe laser is used in this experiment as a light source. A laser beam is focused by means of a microscope objective at the focal-point of a collimating lens. A pin hole is placed at the focus of the microscope objective, to serve as a spatial ®lter. The divergent beam, so obtained, was incident on the collimating lens of focal-length 0.25 m. The collimating lens was mounted on a translation stage, in order to translate the lens along the optic axis, to introduce a known defocus. Two re¯ected beams from the front and back surfaces of the wedge plate travel normal to the incident beam direction, which are split into two ®elds, as explained earlier, by the retro re¯ector. The wedge direction of the wedge
Fig. 4. Wedge plate interference pattern (a) before incidence on the prism (b) after re¯ection from the prism.
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J.S. Darlin et al. / Optics & Laser Technology 30 (1998) 225±228
Fig. 5. Interferograms obtained from the modi®ed dual ®eld single wedge plate collimation tester showing the change in fringe orientation when the point source is (a) inside (b) at and (c) outside the focus.
plate is kept vertical, so that it is parallel to the roof of the prism and the tilt between the beams in the two parts are unaected. The height of the prism is adjusted to split the beam into two equal parts. The lateral shear between the interfering beams is 0.005 m and the diameter of the test beam is 0.025 m. The fringe spacing at the collimated position of the lens is equal to 0.004 m. Fig. 5(a)±(c) show fringe patterns, obtained inside, at and outside the collimation setting of the lens for Df =2100 10 ÿ 6 m. The minimum shift of the lens on either side of the collimation setting, for which decollimation can just be detected by the observing the lack of parallelism between the fringes, is 30 10 ÿ 6 m. The quality of fringes obtained, can be considerably improved by using better quality optics. A wedge plate with an angle of 20 arcsec, 0.05 m diameter and 0.007 m thickness was used in the experiment. The roof of the right angle prism is seen in the lower part of the interferogram. However it does not aect the sensitivity of the collimation tester. 4. Conclusions We have presented a simple method for producing a dual ®eld interferogram, derived from the single wedge
plate interferometer, for collimation testing. The proposed technique has a self-reference and its sensitivity is twice that of a single wedge plate method. Further, the transmitted laser beam through the wedge plate can be used for subsequent applications.
References [1] Murty MVRK. The use of single plane parallel plate as a lateral shearing interferometer with a visible laser gas source. Appl Opt 1964;3:531±4. [2] Riley ME, Gusinow MA. Laser beam divergence utilizing a lateral shearing interferometer. Appl Opt 1977;16:2753±6. [3] Malacara D. Optical Shop Testing, 2nd edn. New York: Wiley, 1992. [4] Sirohi RS, Kothiyal MP. Double wedge plate shearing interferometer for collimation testing. Appl Opt 1987;26:4054±7. [5] Kothiyal MP, Sirohi RS, Rosenbruch KJ. Improved techniques of collimation testing. Opt Laser Technol 1988;20:139±44. [6] Sriram KV, Kothiyal MP, Sirohi RS. Self-referencing collimation testing techniques. Opt Eng 1993;32:94±100. [7] Xu DY, Rosenbruch KJ. Rotatable single wedge plate shearing interference technique for collimation testing. Opt Eng 1991;30:391±6. [8] Gouha U, Mingshan Z, Jingbin Z. Improved wedge-plate shearing interferometric technique for a collimation test. Appl Opt 1992;31:4363±4. [9] Lee YW, Cho HM, Lee IW. Half aperture shearing interferometer for collimation testing. Opt Eng 1993;32:2837±40.
Wedge plate interferometry: a new dual ®eld con®guration for collimation testing J.S. Darlin, M.P. Kothiyal *, R.S. Sirohi Applied Optics Laboratory, Department Of Physics, Indian Institute of Technology, Madras-36, India Received 19 December 1997; accepted 1 June 1998
Abstract A method of deriving a dual ®eld fringe pattern from a single wedge plate for collimation testing is presented. The proposed technique uses a wedge plate and a 908 prism retrore¯ector to form two interference ®elds. With the two sets of fringes, the technique provides twofold increase in sensitivity and has its own reference. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Collimation testing; Wedge plate interferometry; Lateral shear interferometry
1. Introduction Several papers have been reported to check the collimation of the laser beams, using parallel plate and wedge plate lateral shearing interferometers [1, 2]. The plane parallel plate has been used for monitoring the collimation, but being an in®nite fringe method, it is less sensitive [1]. The wedge plate interferometer eliminates this problem. It is more sensitive than the plane parallel plate method [3], but it requires a reference ®duciary mark to check the fringe rotation. The double wedge plate method provides a two fold increase in sensitivity and has its own reference [4±6]. In this method the two wedge plates must be aligned antiparallel to each other. In the rotatable single wedge plate shearing interferometer, an accurate rotator is used for indicating a reference direction for collimation testing [7]. Gouha et al. [8] proposed a method to get the two sets of interference fringes from the single wedge plate. It uses a single wedge plate and two mirrors in a Michelson interferometer con®guration, as shown in Fig. 1. The wedge plate is so oriented that the direction of the wedge is perpendicular to the lateral shear. The beam re¯ected by the wedge plate is laterally sheared. The mirror (M1) re¯ects back this beam to * Corresponding author. Tel.: +91-44-235-1365 or +91-44-2353291; Fax: +91-44-235-0509; E-mail: [email protected]. 0030-3992/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 8 ) 0 0 0 4 1 - 3
the screen, where the corresponding interference pattern can be seen. The beam transmitted by the beam splitter is re¯ected back by mirror M2. This beam now gets sheared in re¯ection by the wedge plate and the interference pattern due to the sheared wavefronts can also be seen on the screen. If one of the two mirrors is tilted with respect to the beam, the two sets of interference patterns can be separated as seen in Fig. 1. This technique was devised to overcome the problem of aligning the two wedge plates [4±6] and the requirement of an accurate rotator [7]. For a well collimated beam, both sets of fringes are horizontal and parallel to each other. The two sets of fringes obtained in this method are separately located and the setting of two sets of fringes parallel to each other is not easy, leading to error in measurement. This separation between the two set of fringes was removed by Lee and co workers [9]. The top half of the one mirror and the lower half of the other mirror ware masked. The lower half of the shear fringe pattern from one mirror and upper half of the sheared fringes from the other mirror are thus brought side by side with a very sharp demarkation line, such that the interferometer behaves like a single shearing interferogram. The fringes in one half, act as a reference to the fringes in the other half. Parallelism of two set of fringes is a test of the laser beam collimation. In the above two methods, the re¯ected and transmitted beams from the wedge plate are folded back to the wedge plate, hence a signi®cant amount of laser beam
226
J.S. Darlin et al. / Optics & Laser Technology 30 (1998) 225±228
through and may be used for other experiments at the same time monitoring the beam collimation. 2. Modi®ed dual ®eld wedge plate lateral shear interferometer
Fig. 1. Schematic diagram [8] of a Michelson interferometer, with a single wedge plate for collimation testing.
is fed back to the source and can in¯uence the behaviour of the light source. These methods have the additional drawback that when the collimation tester is placed in the test beam, the laser beam cannot be used for subsequent stages of experiments. Thus the collimation of the laser beam cannot be continuously monitored. Here we present a new con®guration, using a retrore¯ector and a single wedge plate for collimation testing, which is self-referencing as in the above methods and has the following advantages. First, the amount of optical feed back to the source is very small (10.16%). Second, the interferometer uses only about 12% of the incident light. The rest of the light is transmitted
Fig. 2 shows the schematic of the new con®guration. Two sheared wavefronts with a tilt between them are produced in re¯ection, as explained in connection with Fig. 1. A major fraction of the incident beam is transmitted through and can be used elsewhere. The re¯ected beam is incident on the retrore¯ector. The upperpart of the front surface of the retro re¯ector is mirrored as shown in Fig. 3. The fringe pattern before it is incident on the retrore¯ector will appear in general as in Fig. 4(a). The top portion of beam is re¯ected back without any alteration as in the top half of Fig. 4(b). The lower half of the beam is inverted about the roof of the prism and is re¯ected back. This re¯ected fringe pattern will appear as shown in the lower half of Fig. 4(b). Thus, two set of interferograms can be derived from the single ®eld after re¯ection from the retrore¯ector. It can be seen, that when the fringe pattern produced by the wedge plate due to the change in the curvature of the test beam rotates, the fringe in the two halves of the ®eld will rotate in opposite directions. This property gives doubled sensitivity and self-referencing. As mentioned earlier, the wedge direction is perpendicular to the shear direction in the wedge plate interferometer for collimation testing. The fringes are parallel to the shear direction for collimated beam in this case. The roof axis of the retrore¯ector in the present case is parallel to the wedge direction. Hence
Fig. 2. Schematic diagram of the new con®guration of a wedge plate interferometer for collimation testing.
J.S. Darlin et al. / Optics & Laser Technology 30 (1998) 225±228
227
fringes in the two ®elds is given by a 2 tanÿ1
2DDx : b
5
For a collimated beam, D = 0 and the fringes in the two ®elds are parallel. For a noncollimated beam D$0 and the fringes in the two ®elds enclose the angle between them. When D takes negative values, the fringes in the two ®elds are inclined in opposite direction to that when D is positive. Thus, it is easy to tell whether the point source is inside or outside the focus, according to the direction of the fringe orientation. The angle a is twice as large, as that obtained in the single wedge plate method. The smallest displacement Df of the collimator lens, that can be detected is Fig. 3. Schematic of a 908 prism, used as a retrore¯ector in a new con®guration for collimation testing. The top half of the front face of the prism is mirrored.
the fringe pattern re¯ected by the front surface of the retrore¯ector, as well as the retro re¯ected fringe pattern, will be parallel to the shear axis; that is, the fringes in the two ®elds will be parallel to each other. For a noncollimated beam the fringes will be found to rotate in the opposite directions. Thus, we have both referencing and improved sensitivity. The defocusing error in wavefront is represented by W x, y D x 2 y2 ,
1
where D represents the defocusing factor. The defocusing factor is a function of the radius of the curvature R of the wavefront, which is given by D
1 Df 2, 2R 2f
2
where f is focal-length of the collimating tens and Df its deviation from the collimation position. The fringe equations in the two halves in Fig. 4(b) can be written as 2DxDx bg nl,
3
ÿ2DxDx gb nl,
4
where b is the tilt between the wavefront and Dx is the amount of shear. The ÿve sign in Eq. (4) is due to the reversal of the coordinate on retro re¯ection. These equations show that the fringes in the two ®elds are straight and inclined at equal angle from the horizontal axis in the opposite direction. Since the two sets of fringes rotate in opposite direction the sensitivity of the modi®ed interferometer is twice than that of the single wedge plate method. The angle between the
jDf j
f 2 ba : 2Dx
6
Here a is the smallest angle, between the fringes of the two ®elds, that can be detected. The sensitivity of the collimation test can be controlled by proper choice of b and Dx. 3. Experimental veri®cation and results An interferometer is setup based on the optical con®guration, shown in Fig. 2. A HeNe laser is used in this experiment as a light source. A laser beam is focused by means of a microscope objective at the focal-point of a collimating lens. A pin hole is placed at the focus of the microscope objective, to serve as a spatial ®lter. The divergent beam, so obtained, was incident on the collimating lens of focal-length 0.25 m. The collimating lens was mounted on a translation stage, in order to translate the lens along the optic axis, to introduce a known defocus. Two re¯ected beams from the front and back surfaces of the wedge plate travel normal to the incident beam direction, which are split into two ®elds, as explained earlier, by the retro re¯ector. The wedge direction of the wedge
Fig. 4. Wedge plate interference pattern (a) before incidence on the prism (b) after re¯ection from the prism.
228
J.S. Darlin et al. / Optics & Laser Technology 30 (1998) 225±228
Fig. 5. Interferograms obtained from the modi®ed dual ®eld single wedge plate collimation tester showing the change in fringe orientation when the point source is (a) inside (b) at and (c) outside the focus.
plate is kept vertical, so that it is parallel to the roof of the prism and the tilt between the beams in the two parts are unaected. The height of the prism is adjusted to split the beam into two equal parts. The lateral shear between the interfering beams is 0.005 m and the diameter of the test beam is 0.025 m. The fringe spacing at the collimated position of the lens is equal to 0.004 m. Fig. 5(a)±(c) show fringe patterns, obtained inside, at and outside the collimation setting of the lens for Df =2100 10 ÿ 6 m. The minimum shift of the lens on either side of the collimation setting, for which decollimation can just be detected by the observing the lack of parallelism between the fringes, is 30 10 ÿ 6 m. The quality of fringes obtained, can be considerably improved by using better quality optics. A wedge plate with an angle of 20 arcsec, 0.05 m diameter and 0.007 m thickness was used in the experiment. The roof of the right angle prism is seen in the lower part of the interferogram. However it does not aect the sensitivity of the collimation tester. 4. Conclusions We have presented a simple method for producing a dual ®eld interferogram, derived from the single wedge
plate interferometer, for collimation testing. The proposed technique has a self-reference and its sensitivity is twice that of a single wedge plate method. Further, the transmitted laser beam through the wedge plate can be used for subsequent applications.
References [1] Murty MVRK. The use of single plane parallel plate as a lateral shearing interferometer with a visible laser gas source. Appl Opt 1964;3:531±4. [2] Riley ME, Gusinow MA. Laser beam divergence utilizing a lateral shearing interferometer. Appl Opt 1977;16:2753±6. [3] Malacara D. Optical Shop Testing, 2nd edn. New York: Wiley, 1992. [4] Sirohi RS, Kothiyal MP. Double wedge plate shearing interferometer for collimation testing. Appl Opt 1987;26:4054±7. [5] Kothiyal MP, Sirohi RS, Rosenbruch KJ. Improved techniques of collimation testing. Opt Laser Technol 1988;20:139±44. [6] Sriram KV, Kothiyal MP, Sirohi RS. Self-referencing collimation testing techniques. Opt Eng 1993;32:94±100. [7] Xu DY, Rosenbruch KJ. Rotatable single wedge plate shearing interference technique for collimation testing. Opt Eng 1991;30:391±6. [8] Gouha U, Mingshan Z, Jingbin Z. Improved wedge-plate shearing interferometric technique for a collimation test. Appl Opt 1992;31:4363±4. [9] Lee YW, Cho HM, Lee IW. Half aperture shearing interferometer for collimation testing. Opt Eng 1993;32:2837±40.