- Email: [email protected]
Double grating interferometers II. Application to collimated beams
OPTICS COMMUNICATIONS
Volume 14, number 1
DOUBLE GRATING P. HARIHARAN
INTERFEROMETERS
II. APPLICATION
May 1975
TO COLLIMATED
BEAMS
and Z.S. HEGEDUS
National Measurement
Laboratory,
CSIRO, Sydney, Australia 2008
Received 17 February 1975
The application of a shearing interferometer consisting of two identical gratings to measurements of the aberrations of collimated beams is described. The theory of the system is derived, and it is shown how the shear as well as the tilt between the sheared wavefronts can be varied.
1. Introduction
Shearing interferogrom,
An improved two identical
Ronchi
gratings
interferometer
located
consisting
of
Imaging
near the focus of the
Lens .\
converging wavefront under study has been described in an earlier publication [ 11. The present communication shows how this interferometer can also be applied to tests on collimated beams.
Lens
under
2. Optical system Consider a nominally parallel, monochromatic beam from the system under test, incident normally on two identical gratings G, and G, with a sinusoidal amplitude transmittance which are separated by a distance z as shown schematically in fig. 1. Only two diffracted beams corresponding to the +l and -1 orders emerge from the first grating besides the directly transmitted beam. If the grating lines are vertical, these beams lie in the horizontal plane and make angles +0 with the directly transmitted beam, where 0 is defined by the relation sin 0 = nX,
(1)
n being the spatial frequency of the grating. When these three beams are incident on the second grating, nine beams are formed in all. Out of these, only one pair, the (1 ,O) and (OJ) beams, finally emerges at an angle 19to the directly transmitted beam. Similarly, another pair of beams, the (-1 ,O) and (0,-l) beams, emerges from the interferometer at an angle - 0.
148
Pinhole by
illuminated
Aperture
mask
Loser
Fig. 1. Schematic diagram of the optical system.
Either of these two pairs of beams can be isolated by an aperture placed at a suitable distance from the gratings. A lens placed behind this aperture forms two laterally displaced images of the exit pupil of the system under test, and a shearing interferogram of the wavefront in the plane of the exit pupil is seen in the region of overlap. Fringes of excellent visibility are obtained with this interferometer because the two interfering wavefronts have equal amplitudes. In addition, since only two beams contribute to the interference pattern, effects due to Fourier imaging [2,3] are absent, and the visibility of the fringes does not vary with the separation of the gratings.
May 1975
OPTICS COMMUNICATIONS
Volume 14, number 1
I I I
I I I
G1
G2
Fig. 2. Introduction of tilt between the sheared wavefronts in the plane of shear. Fig. 3. Introduction of tilt between the sheared wavefronts in the plane perpendicular to the direction of shear.
3. Theory It is easily seen that the lateral shear x between the two wavefronts in the plane of the exit pupil of the system under test is given by the relation x=zsinB.
(2)
This can be continuously adjusted over a wide range by varying z, the separation of the two gratings. A tilt can be introduced between the two interfering wavefronts in the plane parallel to the direction of shear by a rotation of one of the gratings about an axis parallel to the grating lines. With an aberrationfree system, this gives straight fringes across the field running perpendicular to the direction of shear. This tilt can be evaluated by assuming that the beam from the system under test is incident at a small angle P in this plane on the first grating, as shown in fig. 2. The angle 0 which the diffracted beam makes with the directly transmitted beam is then given by the relation
de = dfl[l -
(i/COS
o)].
It is also possible to introduce a tilt between the two interfering wavefronts in a plane perpendicular to the direction of shear. Nominally straight fringes are then obtained running parallel to the direction of shear, and the interpretation of the deviations of the fringes for small aberrations is quite easy. The simplest method of introducing such a tilt is by a rotation of the two gratings in their own plane in opposite directions, so that the grating lines make equal, very small angles + dy with the vertical. The principal rays of the two diffracted wavefronts now lie in planes inclined at angles f dy to the horizontal. As can be seen from fig. 3, the tilt d$ introduced between the two wavefronts is the difference of the components of the diffraction angles in the vertical plane, so that, to a first approximation,
dJ/ = 20 dy. nh = sin (0 - 0) + sin /1.
(3)
If the second grating G2 is now rotated by an additional angle do about an axis parallel to the grating lines, the change de in the angle which the diffracted beam from this grating makes with the directly transmitted beam is given by the relation
‘Osp
cos(e-p) 1 ’ which, when p is very small, is approximately
(4)
(5)
(6)
If these two adjustments are combined, both the orientation and spacing of the fringes can be varied.
4. Experimental results An interferometer of this type was set up using a pair of holographic gratings with a spatial frequency of 200 lines/mm. To obtain a brighter image, phase
149
Volume
14, number
1
OPTICS COMMUNICATIONS
Fig. 4. Interferograms of the coilimated beam from a small telescope objective obtained at two different shear settings, how a uniform field as well as fringes parallel to and perpendicular to the direction of shear can be obtained.
150
May 1975
showing
OPTICS COMMUNICATIONS
Volume 14. number 1
were used, but non-linear effects were kept to a minimum by limiting the phase modulation in these gratings so that their diffraction efficiency in the first order was approximately 25%; under these conditions, the irradiance of the higher orders was negligible. Typical interferograms of the collimated beam from a small telescope objective obtained at two different values of the shear are presented in fig. 4. These show how the direction and magnitude of the tilt between the sheared wavefronts can be conveniently varied.
gratings
5. Modified optical system A more compact version of this interferometer for use only with collimated beams can be set up with one grating and a plane mirror, as shown schematically in fig. 5. In this, the shearing interferogram is formed by the (1,O) beam, which is diffracted on its first passage through the grating and then reflected from the mirror, and the (0,l) beam, which is initially reflected back on its own path by the mirror and then diffracted by the grating. Effectively, the image of the grating in the mirror functions as the second grating. As before, the separation of the grating and the mirror is varied to change the shear, and tilt can be introduced between the two sheared wavefronts in the plane of shear by a rotation of the grating about an axis parallel to the grating lines. A rotation of the grating in its own plane now has no effect on the tilt. However, tilt can be introduced between the two sheared wavefronts in the vertical plane by a rotation of the grating about an
May 1975
axis in its own plane perpendicular
to the grating lines. For a rotation of the grating by a small angle da, the tilt d$ introduced between the sheared wavefronts is given by the relation d$ = 2(1 - cos 0) da.
(7)
This adjustment also serves to eliminate interference patterns due to parasitic diffracted beams formed by reflection at the grating. For small apertures, a tilt of the grating which introduces a few fringes parallel to the direction of shear is usually adequate to ensure that these beams fall outside the field of observation.
6. Conclusions These experiments show that the double-grating interferometer is a versatile instrument which should have considerable practical utility for tests on collimated beams as well as converging beams. Besides its simplicity and ease of operation, this interferometer has several other advantages. Since interference takes place between two diffracted wavefronts of the same amplitude, the visibility of the fringes is close to unity. The shear can be continuously adjusted over a wide range, and it is possible to work at small shears without problems due to overlapping images from different diffraction orders. In addition, a tilt of variable magnitude can be introduced in any direction between the two sheared wavefronts to give reference fringes with any orientation and spacing.
Acknowledgements The authors thank Dr. W.H. Steel for helpful discussions .
References
Grating
Mirror
Fig. 5. Modified optical system using only one grating.
[ 11 P. Hariharan, W.H. Steel and J.C. Wyant, Opt. Commun. 11 (1974) 317. [2] S. Yokozeki and T. Suzuki, Appl. Opt. 10 (1971) 1575. [3] A.W. Lohmann and D.E. Silva, Opt. Commun. 2 (1971) 413.
151
Volume 14, number 1
DOUBLE GRATING P. HARIHARAN
INTERFEROMETERS
II. APPLICATION
May 1975
TO COLLIMATED
BEAMS
and Z.S. HEGEDUS
National Measurement
Laboratory,
CSIRO, Sydney, Australia 2008
Received 17 February 1975
The application of a shearing interferometer consisting of two identical gratings to measurements of the aberrations of collimated beams is described. The theory of the system is derived, and it is shown how the shear as well as the tilt between the sheared wavefronts can be varied.
1. Introduction
Shearing interferogrom,
An improved two identical
Ronchi
gratings
interferometer
located
consisting
of
Imaging
near the focus of the
Lens .\
converging wavefront under study has been described in an earlier publication [ 11. The present communication shows how this interferometer can also be applied to tests on collimated beams.
Lens
under
2. Optical system Consider a nominally parallel, monochromatic beam from the system under test, incident normally on two identical gratings G, and G, with a sinusoidal amplitude transmittance which are separated by a distance z as shown schematically in fig. 1. Only two diffracted beams corresponding to the +l and -1 orders emerge from the first grating besides the directly transmitted beam. If the grating lines are vertical, these beams lie in the horizontal plane and make angles +0 with the directly transmitted beam, where 0 is defined by the relation sin 0 = nX,
(1)
n being the spatial frequency of the grating. When these three beams are incident on the second grating, nine beams are formed in all. Out of these, only one pair, the (1 ,O) and (OJ) beams, finally emerges at an angle 19to the directly transmitted beam. Similarly, another pair of beams, the (-1 ,O) and (0,-l) beams, emerges from the interferometer at an angle - 0.
148
Pinhole by
illuminated
Aperture
mask
Loser
Fig. 1. Schematic diagram of the optical system.
Either of these two pairs of beams can be isolated by an aperture placed at a suitable distance from the gratings. A lens placed behind this aperture forms two laterally displaced images of the exit pupil of the system under test, and a shearing interferogram of the wavefront in the plane of the exit pupil is seen in the region of overlap. Fringes of excellent visibility are obtained with this interferometer because the two interfering wavefronts have equal amplitudes. In addition, since only two beams contribute to the interference pattern, effects due to Fourier imaging [2,3] are absent, and the visibility of the fringes does not vary with the separation of the gratings.
May 1975
OPTICS COMMUNICATIONS
Volume 14, number 1
I I I
I I I
G1
G2
Fig. 2. Introduction of tilt between the sheared wavefronts in the plane of shear. Fig. 3. Introduction of tilt between the sheared wavefronts in the plane perpendicular to the direction of shear.
3. Theory It is easily seen that the lateral shear x between the two wavefronts in the plane of the exit pupil of the system under test is given by the relation x=zsinB.
(2)
This can be continuously adjusted over a wide range by varying z, the separation of the two gratings. A tilt can be introduced between the two interfering wavefronts in the plane parallel to the direction of shear by a rotation of one of the gratings about an axis parallel to the grating lines. With an aberrationfree system, this gives straight fringes across the field running perpendicular to the direction of shear. This tilt can be evaluated by assuming that the beam from the system under test is incident at a small angle P in this plane on the first grating, as shown in fig. 2. The angle 0 which the diffracted beam makes with the directly transmitted beam is then given by the relation
de = dfl[l -
(i/COS
o)].
It is also possible to introduce a tilt between the two interfering wavefronts in a plane perpendicular to the direction of shear. Nominally straight fringes are then obtained running parallel to the direction of shear, and the interpretation of the deviations of the fringes for small aberrations is quite easy. The simplest method of introducing such a tilt is by a rotation of the two gratings in their own plane in opposite directions, so that the grating lines make equal, very small angles + dy with the vertical. The principal rays of the two diffracted wavefronts now lie in planes inclined at angles f dy to the horizontal. As can be seen from fig. 3, the tilt d$ introduced between the two wavefronts is the difference of the components of the diffraction angles in the vertical plane, so that, to a first approximation,
dJ/ = 20 dy. nh = sin (0 - 0) + sin /1.
(3)
If the second grating G2 is now rotated by an additional angle do about an axis parallel to the grating lines, the change de in the angle which the diffracted beam from this grating makes with the directly transmitted beam is given by the relation
‘Osp
cos(e-p) 1 ’ which, when p is very small, is approximately
(4)
(5)
(6)
If these two adjustments are combined, both the orientation and spacing of the fringes can be varied.
4. Experimental results An interferometer of this type was set up using a pair of holographic gratings with a spatial frequency of 200 lines/mm. To obtain a brighter image, phase
149
Volume
14, number
1
OPTICS COMMUNICATIONS
Fig. 4. Interferograms of the coilimated beam from a small telescope objective obtained at two different shear settings, how a uniform field as well as fringes parallel to and perpendicular to the direction of shear can be obtained.
150
May 1975
showing
OPTICS COMMUNICATIONS
Volume 14. number 1
were used, but non-linear effects were kept to a minimum by limiting the phase modulation in these gratings so that their diffraction efficiency in the first order was approximately 25%; under these conditions, the irradiance of the higher orders was negligible. Typical interferograms of the collimated beam from a small telescope objective obtained at two different values of the shear are presented in fig. 4. These show how the direction and magnitude of the tilt between the sheared wavefronts can be conveniently varied.
gratings
5. Modified optical system A more compact version of this interferometer for use only with collimated beams can be set up with one grating and a plane mirror, as shown schematically in fig. 5. In this, the shearing interferogram is formed by the (1,O) beam, which is diffracted on its first passage through the grating and then reflected from the mirror, and the (0,l) beam, which is initially reflected back on its own path by the mirror and then diffracted by the grating. Effectively, the image of the grating in the mirror functions as the second grating. As before, the separation of the grating and the mirror is varied to change the shear, and tilt can be introduced between the two sheared wavefronts in the plane of shear by a rotation of the grating about an axis parallel to the grating lines. A rotation of the grating in its own plane now has no effect on the tilt. However, tilt can be introduced between the two sheared wavefronts in the vertical plane by a rotation of the grating about an
May 1975
axis in its own plane perpendicular
to the grating lines. For a rotation of the grating by a small angle da, the tilt d$ introduced between the sheared wavefronts is given by the relation d$ = 2(1 - cos 0) da.
(7)
This adjustment also serves to eliminate interference patterns due to parasitic diffracted beams formed by reflection at the grating. For small apertures, a tilt of the grating which introduces a few fringes parallel to the direction of shear is usually adequate to ensure that these beams fall outside the field of observation.
6. Conclusions These experiments show that the double-grating interferometer is a versatile instrument which should have considerable practical utility for tests on collimated beams as well as converging beams. Besides its simplicity and ease of operation, this interferometer has several other advantages. Since interference takes place between two diffracted wavefronts of the same amplitude, the visibility of the fringes is close to unity. The shear can be continuously adjusted over a wide range, and it is possible to work at small shears without problems due to overlapping images from different diffraction orders. In addition, a tilt of variable magnitude can be introduced in any direction between the two sheared wavefronts to give reference fringes with any orientation and spacing.
Acknowledgements The authors thank Dr. W.H. Steel for helpful discussions .
References
Grating
Mirror
Fig. 5. Modified optical system using only one grating.
[ 11 P. Hariharan, W.H. Steel and J.C. Wyant, Opt. Commun. 11 (1974) 317. [2] S. Yokozeki and T. Suzuki, Appl. Opt. 10 (1971) 1575. [3] A.W. Lohmann and D.E. Silva, Opt. Commun. 2 (1971) 413.
151