for implementation in a rotational-shearing interferometer.

Transcription

1 Dove prism with increased throughput for implementation in a rotational-shearing interferometer Ivan Moreno, Gonzalo Paez, and Marija Strojnik An analytical expression is derived for the tilt introduced into a wave front by a Dove prism with manufacturing errors in the prism s base angles and pyramidal angle. We found that the tilt decreases when the base angles are increased above the values of traditional design. The increase in the length aperture ratio of a prism is detrimental to the prism s performance. However, a Dove prism with a widened aperture increases throughput and maintains a manageable prism weight for implementation in a rotational shearing interferometer. Thus we propose a Dove prism designed with a widened aperture to increase throughput in a rotational shearing interferometer and with larger base angles to minimize the wave-front tilt introduced by manufacturing errors. Experimental results implemented in a rotational shearing interferometer demonstrate the feasibility of this design Optical Society of America OCIS codes: , , , , The authors are with the Centro de Investigaciones en Optica, Apdo. Postal 1-948, Leon, Guanajuato, Mexico. M. Strojnik s address is mstrojnik@aol.com. Received 5 November 2002; revised manuscript received 1 May $ Optical Society of America 1. Dove Prism as a Wave-Front Rotator The design and fabrication of a novel and optimized Dove prism arise from a need to develop instrumentation to detect faint signals or the infinitesimal deviation of a wave front from rotational symmetry in modern optical systems. A rotational shearing interferometer RSI can be used to selectively detect extrasolar planet signals 1,2 while simultaneously removing starlight. 3,4 Furthermore, a RSI has been proposed for testing the off-axis optical elements of large segmented and diluted primary mirrors that are also used for detection of faint signals and in other astronomical applications. 5 7 A Mach Zehnder interferometer with a Dove prism in each arm becomes a RSI when one Dove prism is rotated with respect to the other. Both prisms also invert the beam The difficult problem of detecting a faint planet orbiting a bright star may be modeled as two point sources: The bright star is visible as a point source on the optical axis owing to the large distance of observation. The faint, as yet invisible, planet rotates about that point. Implementation of the rotational-shearing configuration makes possible cancellation of the symmetric part of the wave front emanating from the star for any position of the planet. The planet at its off-axis position gives rise to a tilted wave front. A position in space, where an interferometer may freely rotate, is an ideal location for the search for a planet whose position about a star varies with the local year. The insensitivity of the rotationally shearing interferometer to rotationally symmetric wave fronts makes it an ideal instrument with which to detect the planet wave front and ignore the star wave front. The incorporation of the rectangular aperture proposed here does not diminish the usefulness of the instrument, as the fringe plane is scanned over the complete instrument rotation. Here we analyze the performance requirements of a Dove prism when it is used as wave-front rotator in a RSI and propose a design with a widened aperture for implementation in a rotational-shearing interferometer. The idea of widening the field of view by using a prism is not new and has been described on numerous occasions, especially for applications in Fourier-transform spectroscopy. 14,15 In rotationally shearing interferometry, two fields are created, one in the reference arm and the other in the arm with the rotated beam. Thus, in the description of a RSI that employs a rectangular entrance aperture, we find that this technique favors small-angle implementation and incorporates subaperture testing APPLIED OPTICS Vol. 42, No August 2003

2 Fig. 1. RSI implemented in a Mach Zehnder configuration, which may be used to detect extrasolar planets or to test optical systems without rotational symmetry. BSs, beam splitters. Fig. 2. A Dove prism rotates a wave front about the optical axis. When a Dove prism is rotated by an angle, the wave front is inverted and rotated by 2. Figure 1 shows a schematic layout of a RSI in a Mach Zehnder configuration. In this instrument the wave front under test is divided such that one part is rotated with respect to the other. The two wave fronts are recombined and interfere in the detection plane of the CCD camera. 16 The precision of the prism s position and the accuracy of fabricating the prism within the specified tolerances determine knowledge of the shearing angle needed to recover the wave front. 17 Dove prisms have found wide use in the traditional optical systems as the image erectors. However, their standard design results in a small aperture that may well become the limiting aperture of an optical system with its associated consequences in image distortion. 18,19 Another issue to be considered is the polarization change produced by the rotation of a beam in one arm of an interferometer with respect to that in the other. 20,21 The polarization problem may lead to reduced contrast of the fringe patterns in the interference plane. 22 A Dove prism is usually four times longer than its aperture diameter. Therefore its aperture size is constrained by its weight. To overcome this limitation we propose a shortened Dove prism with a rectangular aperture to increase throughput in the interferometer. Novel arrays of cemented Dove prisms have been proposed for image rotation to satisfy similar requirements. 23 However, their performance is not adequate for the RSI: The process of cementing the prism array results in some misalignment between individual prisms. In this paper, first we describe the desired characteristics of the Dove prism to perform wave-front rotation for its implementation in the RSI. Then in Section 2 we analyze the fabrication tolerances for optimal performance of the Dove prism. First, the tilt introduced into the wave front as a result of manufacturing errors in the base angles of prism is analyzed. 24 Next, the wave-front tilt introduced when a wave traverses a prism that incorporates pyramidal error is also analyzed. Finally in Section 2, we discuss some optomechanical limitations that result from the large ratio of length to aperture diameter. They lead us to propose in Sections 3 and 4 a short but wider design for a Dove prism with higher base angles. We summarize our findings in Section Manufacturing Errors in a Dove Prism A Dove prism is often used to rotate an image about an optical axis, as illustrated in Fig. 2. The angle of wave-front rotation is twice the angle of mechanical rotation of the Dove prism. An image can also be rotated with a flat mirror, but the line of sight will deviate. Refraction of a beam of light at the entrance and the exit faces of a Dove prism permits rotation of the wave front without deviation of the line of sight in a perfect prism devoid of manufacturing errors. Other devices, 25,26 such as K mirrors and Pechan prisms, may be used to accomplish the same objective when the beam is internally reflected an odd number of times. There a beam of light is internally reflected three and five times, respectively. 27 In terms of manufacturing and alignment requirements, the Dove prism remains the most popular component because of its one-piece fabrication and the absence of cementing and alignment steps. Hence we analyze the Dove prism and its manufacturing requirements to rotate the wave front with high precision and knowledge of the rotation. A Dove prism has the attractive property of conserving the direction of the optical axis. However, the presence of manufacturing errors results in the introduction of tilt and lateral displacement into a wave front, as illustrated in Fig. 3. A. Error in Base Angles Figure 4 shows base angles 1 and 2 and their manufacturing errors. Manufacturing errors 1 and 2 in the input and the output base angles introduce tilt into the exiting wave front that is seen to increase with the error in the base angles, as illustrated in Fig. 5. Errors in the first and second base angles introduce approximately the same amount of tilt. From Fig. 5 we note that fabrication of Dove prisms 1 August 2003 Vol. 42, No. 22 APPLIED OPTICS 4515

3 Fig. 3. A Dove prism with manufacturing errors introduces tilt and displacement into the rotated wave front. The tilt introduced into the wave front is, and d is the lateral displacement at z o. Fig. 6. A Dove prism suffers pyramidal error when angle p is different from zero. This angle represents a slight inclination of the normal to the reflecting and refracting surfaces with respect to the vertical plane of symmetry. with base angles larger than in the traditional design results in reduction of the tilt that arises from manufacturing errors. The amount of possible improvement is limited because the tilt introduced by a prism with base angles approaching 90 is only 17% less than that introduced by a prism with base angles of 65. This improvement does not justify an extremely long prism, limiting the base angles to a maximum feasible value of 65. When errors 1 and 2 are equal, the output wave front does not exhibit angular deviation; however, it deviates laterally. The analytical expression Fig. 4. Error in base angles. The prism with manufacturing errors 1 and 2 introduces tilt into a wave front. for tilt introduced into the wave front as a result of errors in the base angles is 2 arcos n sin 1 2 arcsin 1 n cos 1. (1) Here n is the index of refraction; 1 and 2 are the angular errors in the base angles of the input and output faces of prism. Errors in the first and second base angles introduce approximately the same amount of tilt, as one may observe by examining Eq. 1. In the case of fused silica, the prism is shortest when the base angles are 31 n , nm. For this reason we evaluated a set of prisms with base angles of 35, 45, 55, and 65. B. Pyramidal Error Pyramidal angle p, illustrated in Fig. 6, represents another manufacturing error. Basically, there is a wedge between the plane of the first last refracting surface and the plane of the base bottom or reflecting surface. Another way to say this is that the normal to the refracting surface and the normal to the reflecting surface do not lie in the same plane of symmetry of the Dove prism. The pyramidal error also introduces tilt into the wave front. Pyramidal error introduces tilt but in a plane perpendicular to that which arises as a result of the base angle error. The tilt is 2 p n 2 cos sin cos. (2) Fig. 5. Dove prisms display smaller tilt for the same base angle errors when they are designed with large values of base angle fused silica, n , nm. Here n and p are the index of refraction and the pyramidal angle, respectively. Tilt is plotted in Fig. 7 as a function of pyramidal angle p and with base angle as a parameter. Again, we note that Dove prisms with larger base angles introduce smaller tilt. Thus Dove prisms with base angles larger than those of traditional designs and with the same manufacturing error in the pyramidal angle generate a reduced amount of tilt. Nevertheless, the 4516 APPLIED OPTICS Vol. 42, No August 2003

4 Fig. 7. Dove prisms display smaller tilt for the pyramidal angle errors when they are designed with large values of base angle fused silica, n , nm. tilt introduced by the prism with base angles approaching 90 is 50% smaller than that introduced by a prism with base angles of 65. This appreciable improvement still does not justify designing an extremely long prism; it limits the base angles to a reasonable value of 65. The only difference between the wave-front tilt introduced by manufacturing errors in the base angles and that in the pyramidal angle is that they lie in orthogonal planes, as illustrated in Fig. 8, which is based on calculation of the deviations by use of exact vector formulation. The total tilt total introduced into the wave front as a result of manufacturing errors is the geometric sum of the tilts introduced by error in base angles base and that by pyramidal angle pyramid : total pyramid 2 base (3) Expression 3 might be interpreted to mean that manufacturing tolerances in the base angles and in the pyramidal angle must be specified such that the two introduce the same amount of tilt, pyramid base. The error in the prism length is of secondary importance, because the wave front is just laterally displaced. With proper alignment this lateral displacement may be compensated for. Fig. 8. Exact ray trace using vector formulation, showing that tilts introduced by pyramidal-angle error and base-angle error are along two orthogonal directions. Total tilt is obtained by vector addition. Fig. 9. Prism length L decreases with index of refraction. The length of the Dove prism is shorter by 23% when the prism is made from a material with a refractive index of 1.9 Optical glass LaSFN9 than the length of fused silica with n When the wave front passes through a rotating imperfect Dove prism, its center describes a circle in the image plane at z o. While the wave front rotates by twice the angle of the prism rotation, the wave front s center rotates by the same angle. The implementation of an imperfect Dove prism in the RSI has detrimental consequences for interpretation of interferograms: The additional tilt increases the fringe density in an uncontrolled manner. C. Alignment of a Dove Prism Previous study of manufacturing errors in base angles and in pyramidal angles suggests that Dove prisms with larger base angles are less sensitive to fabrication errors. However, the length of the Dove prism increases for the same aperture diameter when the base angles are made larger. A longer Dove prism is more sensitive to misalignment, resulting in lateral displacement of a wave front: d n 1 L. (4) n Here d,, L, and n are the lateral displacement, the angular misalignment, the prism length, and the index of refraction, respectively. A short prism is preferable not just because it will weigh less but also for alignment considerations. A commonly used fused silica has a relatively small index of refraction and a desirably small coefficient of thermal expansion. Figure 9 shows the length of a Dove prism versus its index of refraction. Making a Dove prism with materials of high index of refraction reduces the length aperture ratio. The smallest thermal expansion coefficient of optical glass with a high index of refraction is approximately ten times higher than that of fused silica. A traditional Dove prism is by 23% shorter when it is made from a material with an index of refraction of 1.9 Optical glass LaSFN9, than when it is made from fused 1 August 2003 Vol. 42, No. 22 APPLIED OPTICS 4517

5 Fig. 10. Throughput is increased by use of a widened aperture, without increasing the prism length. In a traditional prism with a square aperture, aperture side A is equal to side D. silica The length aperture ratio is 4.5 when the Dove prism is made from fused silica. Dove prisms with large aperture diameters are necessary for implementation in RSIs to detect the faint signals carried on wave fronts. 28 A useful prism diameter could be 10 cm. When it is fabricated from optical glass of high index of refraction, a prism will have a minimum length of 35 cm with a 45 base angle, resulting in an unacceptable length. 3. Dove Prisms with Rectangular Apertures The analytical expression for design of a Dove prism 9 is a function of side D only. Increasing side A does not change the prism s length. Figure 10, likewise, demonstrates that increasing side A does not increase the length of prism: L D sin 2 1 n2 cos (5) n 2 cos sin Therefore we suggest employing a Dove prism with a widened aperture to control the prism s length and to increase throughput. The wave front can be rotated with this widened prism that rotates a rectangular portion of the beam in one arm of the interferometer. A complete circular pupil may be examined by rotation of both prisms simultaneously, as indicated in Fig. 11. This proposal makes sense especially for a RSI used for planet detection, because the shear will be implemented for small angles. When a RSI is used Fig. 12. A RSI in a Mach Zehnder configuration that incorporates a rectangular aperture may be used to test a complete circular wave front by rotation of the Dove prisms at the same time. Subapertures are tested sequentially by rotation of one prism by and the other by in increments of 2 sin 1 D A 2. BSs, beam splitters. for testing the optical components, the subapertures may be tested sequentially by rotation of one prism by and the other by in increments of 2 sin 1 D A 2. This interferometric concept is illustrated in Fig Rotational Shearing Interferometer Incorporating Dove Prisms with Rectangular Apertures We favor building Dove prisms with larger base angles for implementation of rotational shearing in interferometers, considering the importance of minimizing the wave-front tilt for such applications. Recently several Dove prisms with the parameters listed in Table 1 were manufactured in our optical shop. For the purpose of demonstration, rather than as the optimized design, we chose prism parameters that fitted within the largest blank of glass that was available in the shop. Even with these relatively small prisms, we can assess their feasibility in a rotational shearing interferometer. With these tolerances, the worst-case performance of the Dove prism is as indicated in Table 2. We incorporated two prisms into an existing Mach Zehnder interferometer to convert the interferometer into a RSI. It is estimated that 12 prism rotations are needed to cover a circular fringe field with the given parameters. When the prisms are incorporated into a rotating RSI, the scanning of the field is continuous, so subaperture testing is no longer an issue. We cannot detect wave fronts with smaller Table 1. Representative Prism Parameters and Their Tolerances Parameter Value Tolerance Fig. 11. A short, widened prism rotates a rectangular entrance pupil. A wave front with large diameter may be tested by rotation of the prism by a scanning angle. Prism width A 80 mm 0.1 mm Prism height D 15 mm 0.1 mm Base angles 1, arc sec Pyramidal angle p 0 20 arc sec 4518 APPLIED OPTICS Vol. 42, No August 2003

6 Table 2. Worst-Case Performance of Dove Prism Source of Tilt Value arc sec Error in base angle 20 arcsec 10.8 Error in pyramidal angle 20 arcsec 8.0 Total tilt in both directions T 13.4 tilt than the maximum tilt error of 13 arc sec that has been predicted and limited by the fabrication errors. Some improvements may be possible when the tilt is well characterized and understood. At least for some prism orientations, this tilt number will be sufficiently well known that it may be compensated for in the testing operations. Figure 13 shows a photo of two Dove prisms, fabricated in optical glass SFN64 with index of refraction nm and a rectangular aperture of dimensions D 15 mm 0.1 mm, A 80 mm 0.1 mm, and L 87 mm 0.1 mm. Its base angle is These prisms are 8.7 cm long and weigh kg each; these values are appreciably smaller than those estimated for the conventional Dove prisms with equivalent throughput, facilitating the mount design and the incorporation of these prisms in a compact optical system. A photo of the RSI in the Mach Zehnder configuration, with the shear incorporated by a rotated Dove prism with the rectangular aperture, is shown in Fig. 14. The lens in the lower left corner collimates the incident wave front. A Dove prism rotated by 10 is seen in its rotary holder at the left; a compensatory prism is seen at the right mounted parallel to the table top. We designed and fabricated the rotary mount to control the prism rotation for shear implementation. A white card in the holder at the top of the photo is the image plane where a CCD camera is placed to record the interferograms. In this setup we are testing an achromatic lens f 300 mm, d 42 mm placed at a distance from the laser source that is equal to the lens s focal length. The experimental setup also employs a Fig. 14. RSI in the Mach Zehnder configuration, with the shear incorporated by a rotated Dove prism with the rectangular aperture. In this setup we are testing an achromatic lens f 300 mm, d 42 mm located at a distance from the laser source equal to the lens s focal length and employing a microscope objective and a spatial filter not shown. Fig. 15. Interferogram recorded with the CCD camera at the image plane in Fig. 14, with a shearing angle of 20 Dove prism rotated by 10. Nearly straight tilt fringes indicate that the lens has some combination of astigmatism and coma. The shape of the image field depends on the Dove prism s aperture and on the magnitude of the shear angle. Fig. 13. Two Dove prisms fabricated with optical glass SFN64 and index of refraction nm, with a rectangular aperture of dimensions D mm, A mm, and L mm. The prism base angle is He Ne laser, a microscope objective, and a spatial filter, which are not shown in the photo that presents only the key components. A typical interferogram recorded with the CCD camera at the image plane, with a shearing angle of 20, appears in Fig. 15. Nearly straight tilt fringes indicate that the lens has some combination of astigmatism and coma. 16 The shape of the fringe field depends on the Dove prism aperture and decreases in size with the magnitude of the shear angle up to 90. To perform additional subaperture testing to cover the aperture area completely, one has to adjust the orientations of Dove prisms in both arms. 5. Summary We have presented analytical expressions for the total tilt introduced into a wave front by a Dove prism 1 August 2003 Vol. 42, No. 22 APPLIED OPTICS 4519

7 when the prism is manufactured with errors in its base angles and in its pyramidal angle. We found that the tilt that arises from manufacturing errors decreases in a Dove prism as the base angles increase, favoring designs with base angles larger than the traditional value of 45. Considering the importance of minimizing wave-front tilt for testing asymmetrical optical systems or detecting faint asymmetrical radiometric signals, we favor building Dove prisms with larger base angles for implementation in rotational shearing interferometers. Similarly, fabricating a Dove prism with a high index of refraction reduces the prism s length aperture ratio, but not enough for our requirements of low weight. Therefore we recommend designing a short Dove prism with a wide aperture to reduce the prism s length and weight and to increase the throughput in the RSI. By subaperture testing, large-diameter wave fronts can be tested with the proposed Dove prism. In such a test the wave front exhibits a controllable amount of tilt caused by residual manufacturing errors of the prism. We designed a Dove prism with large base angles to reduce the wave-front tilt caused by manufacturing errors and with a wide aperture to increase throughput in a RSI. In our optical shop we fabricated several Dove prisms with parameters suitable for incorporation of the prisms into an existing Mach Zehnder interferometer to convert that interferometer into a RSI. We designed and fabricated a rotary mount to control prism rotation for shear implementation. The RSI performed as predicted, with an irregular field of fringe pattern that depended on the prism aperture and the shear angle. The instantaneous field of view of the RSI is not critical for our long-term application of planet detection. The astronomical community is pursuing observations of smaller, fainter, and more-distant objects. 29,30 The traditional telescope with a single aperture of increasingly larger diameter is often traded against interferometric imaging and object detection in which a single aperture is replaced by a number of separate, smaller apertures to detect specific spatial frequencies. 31 With the current design, fabrication, and demonstration we have taken the first steps to demonstrate the feasibility of the rotational shearing interferometer as an instrument that may replace some traditional astronomical instruments. The authors express their gratitude to the Consejo Nacional de Ciencia y Tecnología Mexico for funding the research reported in this paper. References 1. M. S. Scholl, Signal generated by an extra-solar-system planet detected by a rotating rotationally shearing interferometer, J. Opt. Soc. Am. A 13, M. S. Scholl and G. 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8 28. M. Strojnik and G. Paez, Radiometry, in Handbook of Optical Engineering, D. Malacara and B. Thompson, eds. Marcel Dekker, New York, 2001, Chap. 18, pp G. Paez, M. Strojnik, and J. Garcia-Marquez, On performance evaluation of future telescopes, J. Mod. Opt. to be published. 30. D. M. Alloin and J. M. Mariotti, Diffraction Limited Imaging with Very Large Telescopes, NATO ASI Ser. C G. Paez and M. Strojnik, Telescopes, in Handbook of Optical Engineering, D. Malacara and B. Thompson, eds. Marcel Dekker, New York, 2001, Chap. 8, pp August 2003 Vol. 42, No. 22 APPLIED OPTICS 4521