- Email: [email protected]
An interferometric method for measuring short focal length refractive lenses and diffractive lenses
1 February 1999
Optics Communications 160 Ž1999. 5–9
An interferometric method for measuring short focal length refractive lenses and diffractive lenses M. de Angelis
a,1
a b
, S. De Nicola a , P. Ferraro a,2 , A. Finizio a , G. Pierattini a , T. Hessler b,3
Istituto di Cibernetica del CNR, Õia Toiano 6, I-80072 Arco Felice, Italy Paul Scherrer Institut, Badenerstrasse 569, CH-8048 Zurich, Switzerland
Received 3 August 1998; revised 6 November 1998; accepted 29 November 1998
Abstract The possibility that an interferometric method, previously developed for measuring long focal length systems, can be equally well applied to measuring very short focal length optics and a 1 mm aperture radius diffractive lens is demonstrated. The measurement can be easily performed despite some limiting factors such as small aperture, which drop the available light intensity, and very short focal length, which magnified impurities on the optical elements, both resulting in high noise in the fringe pattern. It is easy to predict that other approaches would fail in the presence of such limiting factors. The method is based on a reflective grating interferometer and measurement of the effective focal length is performed within the limit of paraxial approximation. Focal length is obtained by the knowledge of the spatial frequency of two interferometric fringe patterns imaged by the lens under test. The spatial frequencies are evaluated by using an FFT algorithm. q 1999 Elsevier Science B.V. All rights reserved. PACS: 42.10; 42.78 Keywords: Focal length; Interferometer; FFT; Microlens
1. Introduction The capability of an FFT algorithm applied to digitised images in a reflective grating interferometer for focal length determination has been previously demonstrated w1x. In that paper the lens is illuminated by an interference pattern with a regular pitch produced by an interferometer that uses a reflective grating. The focal length is determined by measuring the fringe spatial frequency inside a fixed aperture both with and without the tested lens inserted in the set-up. The images of the interference pattern 1
E-mail: [email protected] Also at I.P.S.I.A. ‘‘G.L. Bernini’’, via Arco Mirelli 19rA, I-80122 Naples, Italy. 3 Now at Leister Technologies, Schwarzenbergstrasse 10–12, CH-6056 Kagiswill, Switzerland. 2
are digitised by a CCD camera. A one-dimensional FFT algorithm is applied to the digitised image to reconstruct the phase modulo 2p along each row of pixels in the CCD matrix. The phase is unwrapped along each row and averaged to calculate the number of fringes that fall inside the aperture of 512 pixels. This interferometric technique is an alternative to methods based on moire´ effect w2,6x. The initial work w1x consisted of only medium focal length lenses, with f between 50 mm and 1000 mm. In the present paper the extension of the method to very short focal length Ž3.7 mm - f - 30 mm. optical systems and to a diffractive microlens is demonstrated. In the analysis of traditional refractive optical systems, it has been demonstrated that moire´ techniques are more suitable for long focal length lenses and are difficult to apply to thick lenses or compound lenses with short focal length or small dimension lenses. In applying moire´ based
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 6 3 6 - 1
6
M. de Angelis et al.r Optics Communications 160 (1999) 5–9
techniques to this kind of optical system many problems may occur. When the fringe patterns are imaged before the back focal plane the dimension of the converging beam can be very small and fringe patterns are hard to see. Severe problems of noise and visibility on the fringe pattern or diffraction are also present. This happens because the moire´ technique is based on the superimposed Talbot images which show a strong degradation when moving away from the tested lens and the degradation is stronger as the focal length becomes shorter. The method proposed in this paper is an interferometric technique in which a fringe pattern is generated with a high visibility limited only by the non-zero value of the CCD sensor under dark conditions. This allows us a very good signalto-noise ratio of the fringe pattern in the whole volume where the two interfering beams overlap. A very good fringe pattern is also available in the region behind the back focal length. It is possible to place the CCD camera in this region and to achieve easily the paraxial condition together with a very good quality of the fringe pattern image. The advantages of the proposed method rely on the possibility of using the same apparatus for different focal length lenses while the paraxial condition is satisfied; in practice no changes in the set-up are required when passing from a long to a short focal length measurement. Moreover, in the moire´ technique the knowledge of the aperture size is required as explained in Ref. w2x, which is not always easy to obtain in the case of small dimension lenses. In studying diffractive microlenses, like the one we analyse in this work, the main problems are the small
aperture Ž1 mm aperture radius. which reduces the available light intensity and the diffraction structure of the lens, both resulting in high noise in the fringe pattern. The proposed interferometer has sufficient stability and visibility to be used for diffractive microlens focal length evaluation and it provides, to our knowledge, the first interferometric measurement of the focal length of a diffractive microlens. In evaluating microoptics focal lengths it is not possible to look at them as miniaturized copies of macroelements, because microlenses do not necessarily have spherical surfaces or well defined aperture stops. Therefore evaluation methods of a microoptics system focal length should give an effective focal length measurement without requiring the knowledge of the numerical aperture. Indeed the proposed method does not require the knowledge of the aperture size and it allows the measurement producing the effective focal length of the tested optics. 2. Principle of the method Fig. 1 shows the interferometric set-up. We claim that only the distance between two different positions of the lens and the fringe frequency of the interferometric pattern for each position give sufficient information for determining the effective focal length of the lens system. In the set-up an expanded and collimated He–Ne laser beam is reflected onto the measuring arrangement which consists of a reflective diffraction grating Ž1200 groovesrmm. and an adjustable mirror. The direction of the grooves of the
Fig. 1. Schematic of the experimental arrangement, G is the reflective grating and M the mirror. The two interfering beams are tilted by a small angle a . In position Ža. and Žb. two interference fringe patterns are shown without the lens inserted in the set-up Ža. and with the optical system inserted in the set-up Žb.. The straight parallel fringes are along y while a sinusoidal intensity profile is present along x. f is the focal length of the optical system under test and z c the distance between the tested optical system and the CCD camera. u is the half angle of the cone of the spherical wave emerging from the focal point.
M. de Angelis et al.r Optics Communications 160 (1999) 5–9
grating is perpendicular to the drawing plane in Fig. 1. One half of the illuminating beam strikes the grating directly Žbeam A. and the other half impinges the grating via the mirror Žbeam B.. The incidence angle of the illuminating beam is chosen such that the q1 and the y1 diffraction orders of beam A and beam B are respectively collinearly directed along the grating normal towards the CCD camera. Under this condition the angle between the grating and the mirror is 908. When the mirror is slightly tilted with respect to the grating normal, an interference fringe pattern is visible, which results from the superposition of the two light diffraction orders. We wish to point out that the configuration we use is based on the reflective diffraction grating as a recombining element and it allows for a non-localised interference fringe system. With careful alignment an interference pattern of fringes is generated; the interference pattern is a family of parallel straight fringes with a carrier spatial frequency corresponding to sin arl, where l s 632.8 nm is the laser wavelength and a is the angle between the two interfering diffraction orders, which can be varied by tilting the mirror. In this set-up, when no sample is inserted, the contrast of the interference pattern is limited only by the noise due to the dark current of the CCD camera. The lens under test is placed after the grating recombinating element. The interference pattern beams pass through the lens and arrive at the observation plane of a CCD camera; an array of 512 = 512 pixels Žpixel size 11 mm = 11 mm. is used to store the interferometric pattern. The two interfering beams, separated by the angle a , produce two separated point images in the back focal plane of the lens under test Žsee Ref. w1x.. Two regions in these overlapping beams are of interest: the first region is between the lens and the back focal plane and the second region is after the back focal plane. The latter starts at the point P at distance z P s drŽ drf y tan a . from the optical system where f is the focal length and d is the aperture diameter of the lens. By considering the CCD array size p Ž6 mm., it can be easily seen from Fig. 1 that the interfering pattern covers the whole CCD area if the camera is positioned, slightly off axis, at a distance z c not shorter than z P q Ž prd . z p . Substituting for z p the CCD camera must be placed at a distance z c such that zc )
dqp drf y tan a
.
7
which in our set-up results in a more strict condition than the previous one. Mounting the CCD far beyond the focal point of the tested optical system results in the light intensity available on the CCD area being low; however it is found that good quality fringe pattern images are achieved with low noise and high fringe visibility. We assume that the beam at the CCD plane is a spherical wave emerging from a point source located at the focal point, and with a cone half angle u . The amount of energy gathered by the lens is directly proportional to the area of the lens. If we neglect losses due to reflections, absorption, and so forth, the energy per unit area per unit time Ži.e. the flux density. at the CCD plane is proportional to the square of the numerical aperture NA ŽNA s sin u . of the optical system. This means that for a low numerical aperture optical system, the intensity at the positions where the CCD can be placed is low; this is the case of the microlens under test ŽNA s 0.01.. Instead for higher NA optical systems, like microscope objectives, no problems are encountered in choosing the CCD position. The limit for the largest distance z c is given by: Is
P
p w Ž z c y f . tan u x
2
) I0 ,
Ž3.
where P, assumed to be uniform over the CCD, is the energy per unit time available and I0 is the minimum acceptable flux density at the CCD. I0 is fixed by the set-up and depends on the CCD camera response: we have estimated that the dynamic response of the CCD starts to be lower than 256 grey levels when the flux density is lower than 2 =10y3 Wrm2. The original interference pattern when no lens is placed in the set-up has fringes parallel to the y direction and spatial frequency n . When the lens is placed in the beam path, the interference fringe pattern changes by an amount depending on its focal length and the distance z c from the principal plane of the lens to the CCD camera. If n 1 and n 2 are the spatial frequency of the fringe patterns corresponding to two different positions of the lens a distance ´ apart, the focal length f can be expressed as: ´ n 1n 2 fs . Ž4. n n2yn1
Ž1.
A further lower limit to the distance z c is given by requiring paraxial approximation. In the optical arrangement the CCD array size p corresponds to the effective optical aperture; in order to satisfy the paraxial approximation Žwhich is taken to require angles of less than 10y2 ., the CCD camera must be placed beyond the focal point at distance z c such that p - 10y2 , Ž2. 2Ž zc y f .
3. Results In the experimental arrangement the fringe frequency n is easily changed by rotating the mirror M and can be suitably set. The optical systems under test in our laboratory are microscope objectives and collimating lenses for diode lasers. The distance between the two different positions of the tested lens is ´ s 15 mm. The error in the whole number of fringes in the CCD aperture is assumed
8
M. de Angelis et al.r Optics Communications 160 (1999) 5–9
to be 0.002, which corresponds to an error of 3 =10y4 mmy1 in the fringe spatial frequency. As can be noted in Fig. 1 the main problem in the small focal length measurement is to achieve the largest fringe spatial frequency compatible with the small diameter of the lens systems under test. Moreover looking at the expression of f given by Eq. Ž5. it is clear that the error in the focal length is mainly due to the difference n 2 y n 1 between the spatial frequencies. It means that a large excursion of ´ compatible with the superposition of the two interfering beams after the lens is desirable. The measurement of each quantity is performed with the same procedure; the fringe patterns are digitised with a 8-bit grey scale, corresponding to 256 levels. A one-dimensional FFT algorithm is applied to each row of 512 pixels to obtain a power spectra of spatial frequencies. Let us assume a fringe pattern intensity distribution I Ž x , y . s a Ž x , y . q b Ž x , y . cos Ž 2pn x . ,
Ž5.
where aŽ x, y . and bŽ x, y . are the background intensity and the local contrast for the fringe pattern and the spatial frequency is given by n . In order to determine the fringe spatial frequency n of the fringe pattern, we apply the one-dimensional fast Fourier algorithm by Fourier transforming with respect to x each row y of the digitised fringe pattern intensity distribution I Ž x, y .. This spectrum is characterised by a contribution of the low frequency background illumination, which is centered around the zero frequency. Since spatial changes of bŽ x, y . are slow compared to spatial frequency n , the function spectrum is characterised by two peaks at frequency n and yn . The low and the negative frequencies are bandpass filtered and the filtered spectrum is inverse Fourier transformed in order to obtain the complex map Ž1r2 . b Ž x, y . expŽ i f Ž x, y ... The value of phase f s 2pn x wrapped in the range between yp rad and p rad is calculated for each point Ž x, y . of the fringe pattern by using a computer subroutine. From the wrapped phase the unwrapped phase value f can be computed by using a standard unwrapping procedure Žsee Refs. w3,4x and references therein. and the frequency value can be determined from the simple relationship n s D fr2p p where D f is the phase change of a given row of the interferometric pattern between pixel 1 and pixel p of the CCD camera. In the absence of aberrations the phase varies linearly along each row of the interferogram and the unwrapped phase can be fitted to a straight line. Finally the D f values obtained on each row have been averaged over 100 rows of the examinated pattern. Thus, from the knowledge of n and from Eq. Ž5. we can determine the focal length. Measurements have been performed on three achromatic microscope objectives; one is a 4 = with numerical aperture of 0.12 and nominal focal length of 30.8 mm ŽMelles–Griot mod. 04 OAS 002., the second is a 10 = with numerical aperture of 0.25 and nominal focal length of 16.9 mm ŽMelles–Griot mod. 04 OAS 010., the third is
Table 1 Results of the focal length measurements of microscope objectives, diode laser collimating lenses and the diffractive microlens Tested optics
Nom. f Žmm.
Meas. f Žmm.
d fr f
Micr. objective 4= Micr. objective 10= Micr. objective 40= Aspheric lens Collimating lens Microlens
30.8 16.9 4.5 11 3.7 100.0
30.82 16.88 4.48 11.17 3.66 100
0.2% 0.2% 0.2% 0.3% 0.3% - 3%
a Zeiss objective 40 = with numerical aperture of 0.65 and focal length of 4.5 mm. Other optics tested were collimating lenses for diode lasers: we have tested an aspheric lens of nominal focal length of 11.0 mm and numerical aperture of 0.25 mm ŽThorlabs mod. C220TM-B. and a GRIN lens with focal length of 3.7 mm and numerical aperture of 0.450 ŽMelles-Griot mod. 06 GLC 009.. In Table 1 the characteristics of the tested optical systems are summarised, together with their nominal focal length, the measured focal length with the FFT interferometric method and the estimated error measurement. Typical frequency values of the spatial fringes are n s 20 mmy1, and between 2 mmy1 and 12 mmy1 for n 1 and n 2 . If we take into account Eqs. Ž2. and Ž4. which give the limit for the range of measurability of focal length f and assume that the distance between the camera and the optical system is z c s 600 mm Žwhich is the right distance for satisfying the paraxial condition of Eq. Ž4.. and the aperture diameter of the optics is of the order of 10 mm, the measurable focal length f cannot be smaller than 1.7 mm and larger than 300 mm Žfor larger focal length the more practical set-up is to place the CCD camera before the back focal plane as explained in Ref. w1x..
4. Measurement on a microlens We also performed measurements on a diffractive microlens. For such lenses, aperture diffraction effect can be of considerable importance. A suitable parameter for describing the aperture diffraction effects is the Fresnel number N of the lens which is defined as: Ns
a2
lf
,
Ž6.
where f is the geometric optical focal length, l the wavelength and a s dr2 the lens aperture radius. The tested microlens was a diffractive microlens available in our laboratory; it was manufactured at the Paul Scherrer Institute ŽZurich. w5x. It is a continuous relief lens fabricated by direct laser writing into photoresist. It is designed for wavelength l s 632.8 nm, has aperture radius of 1.0060 mm, focal length of 100 mm and Fresnel number 16. The
M. de Angelis et al.r Optics Communications 160 (1999) 5–9
9
microlens inserted in the interferometer, the CCD is placed in the region where a straight fringe pattern of about 4 Žor 8. fringes is visible. In Fig. 2 a typical image of the two interfering beams is shown. The CCD camera is placed in such a way as to detect only the central part of the image where the fringes are straight and the condition for the paraxial approximation is satisfied. The microlens under test has a very long focal length for its aperture, which forces us to choose a rather large displacement ´ of the CCD. A shorter focal length microoptic Ž f of the order of 1 mm., which is more common, would have been easier to measure resulting in a more compact set-up. In fact for this microlens the limit on the measurability of the focal length f imposed by Eqs. Ž2. and Ž4. limit f between 1 mm and 950 mm for the chosen value of z c . Strong diffraction rings are visible on the interferometric pattern and introduce noise on the straight fringe pattern. Because of this noise the statistical analysis of the microlens data has been performed on a larger number of row data. We have applied the linear least square fitting procedure on 500 rows of the fringe pattern and computed the mean value of the fringe spatial frequency and the standard deviation. The relative error of the focal length of the tested microlens has been evaluated to be less than 3%.
Fig. 2. Ža. Fringe pattern at the CCD camera while a microlens is inserted in the set-up at a distance of 1250 mm from the CCD. Žb. Image of the same microlens at a shorter distance, where the two interfering beams are totally visible. Strong diffraction rings are present which are caused by the microlens.
microlens is part of a matrix of microlenses which are all on the same support which is a 5 = 5 cm2 photoresist on a glass plate; the distance between the microlenses is 0.5 mm. In the set-up for microlens analysis an adjustable iris diaphragm was mounted in the beam because we wanted to analyse only one lens at a time. Moreover, in the optical set-up, before the lens under test we have mounted a recollimating optical system. In this way we have reduced the diameter of the beam to 2 mm before the microlens sample and we have obtained a fringe pattern of about 50 fringes when no test optics are inserted in the interferometer. The recollimating optical system was needed because a low intensity was available at the CCD plane because of the small microlens aperture; a more powerful laser or a less expanded collimated laser beam would have avoided the recollimating system. The distance between the lens and the closer position of the CCD camera is 1250 mm. We chose to displace the CCD camera rather than the optics under test; the distance between the two different positions of the CCD camera is ´ s 800 mm. With the
5. Conclusion We have demonstrated the possibility that an interferometric method can be applied for measuring very short focal length refractive optics and a 1 mm aperture radius diffractive lens. The focal length is determined by measuring the spatial frequency of two interferometric fringe patterns imaged by the lens under test and the spatial frequencies are evaluated by using an FFT algorithm. The relative error of the f measurement is less than 0.3% for the tested refractive lenses while it is less than 3% for the diffractive microlens.
References w1x M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, Optics Comm. 136 Ž1996. 370. w2x E. Keren, K.M. Kreske, O. Kafri, Appl. Optics 27 Ž1988. 1383. w3x T. Kreis, in: P.K. Rastogi ŽEd.., Holographic Interferometry, Springer, Berlin, 1994, pp. 184–185. w4x M. Taked, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72 Ž1982. 156. w5x T. Hessler, M. Rossi, in: S. Martellucci, A. Chester ŽEds.., Diffractive Optics and Optical Microsystems, Plenum, New York, 1997, pp. 139–148. w6x D.C. Su, C.W. Chang, Optics Comm. 78 Ž1990. 118.
Optics Communications 160 Ž1999. 5–9
An interferometric method for measuring short focal length refractive lenses and diffractive lenses M. de Angelis
a,1
a b
, S. De Nicola a , P. Ferraro a,2 , A. Finizio a , G. Pierattini a , T. Hessler b,3
Istituto di Cibernetica del CNR, Õia Toiano 6, I-80072 Arco Felice, Italy Paul Scherrer Institut, Badenerstrasse 569, CH-8048 Zurich, Switzerland
Received 3 August 1998; revised 6 November 1998; accepted 29 November 1998
Abstract The possibility that an interferometric method, previously developed for measuring long focal length systems, can be equally well applied to measuring very short focal length optics and a 1 mm aperture radius diffractive lens is demonstrated. The measurement can be easily performed despite some limiting factors such as small aperture, which drop the available light intensity, and very short focal length, which magnified impurities on the optical elements, both resulting in high noise in the fringe pattern. It is easy to predict that other approaches would fail in the presence of such limiting factors. The method is based on a reflective grating interferometer and measurement of the effective focal length is performed within the limit of paraxial approximation. Focal length is obtained by the knowledge of the spatial frequency of two interferometric fringe patterns imaged by the lens under test. The spatial frequencies are evaluated by using an FFT algorithm. q 1999 Elsevier Science B.V. All rights reserved. PACS: 42.10; 42.78 Keywords: Focal length; Interferometer; FFT; Microlens
1. Introduction The capability of an FFT algorithm applied to digitised images in a reflective grating interferometer for focal length determination has been previously demonstrated w1x. In that paper the lens is illuminated by an interference pattern with a regular pitch produced by an interferometer that uses a reflective grating. The focal length is determined by measuring the fringe spatial frequency inside a fixed aperture both with and without the tested lens inserted in the set-up. The images of the interference pattern 1
E-mail: [email protected] Also at I.P.S.I.A. ‘‘G.L. Bernini’’, via Arco Mirelli 19rA, I-80122 Naples, Italy. 3 Now at Leister Technologies, Schwarzenbergstrasse 10–12, CH-6056 Kagiswill, Switzerland. 2
are digitised by a CCD camera. A one-dimensional FFT algorithm is applied to the digitised image to reconstruct the phase modulo 2p along each row of pixels in the CCD matrix. The phase is unwrapped along each row and averaged to calculate the number of fringes that fall inside the aperture of 512 pixels. This interferometric technique is an alternative to methods based on moire´ effect w2,6x. The initial work w1x consisted of only medium focal length lenses, with f between 50 mm and 1000 mm. In the present paper the extension of the method to very short focal length Ž3.7 mm - f - 30 mm. optical systems and to a diffractive microlens is demonstrated. In the analysis of traditional refractive optical systems, it has been demonstrated that moire´ techniques are more suitable for long focal length lenses and are difficult to apply to thick lenses or compound lenses with short focal length or small dimension lenses. In applying moire´ based
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 6 3 6 - 1
6
M. de Angelis et al.r Optics Communications 160 (1999) 5–9
techniques to this kind of optical system many problems may occur. When the fringe patterns are imaged before the back focal plane the dimension of the converging beam can be very small and fringe patterns are hard to see. Severe problems of noise and visibility on the fringe pattern or diffraction are also present. This happens because the moire´ technique is based on the superimposed Talbot images which show a strong degradation when moving away from the tested lens and the degradation is stronger as the focal length becomes shorter. The method proposed in this paper is an interferometric technique in which a fringe pattern is generated with a high visibility limited only by the non-zero value of the CCD sensor under dark conditions. This allows us a very good signalto-noise ratio of the fringe pattern in the whole volume where the two interfering beams overlap. A very good fringe pattern is also available in the region behind the back focal length. It is possible to place the CCD camera in this region and to achieve easily the paraxial condition together with a very good quality of the fringe pattern image. The advantages of the proposed method rely on the possibility of using the same apparatus for different focal length lenses while the paraxial condition is satisfied; in practice no changes in the set-up are required when passing from a long to a short focal length measurement. Moreover, in the moire´ technique the knowledge of the aperture size is required as explained in Ref. w2x, which is not always easy to obtain in the case of small dimension lenses. In studying diffractive microlenses, like the one we analyse in this work, the main problems are the small
aperture Ž1 mm aperture radius. which reduces the available light intensity and the diffraction structure of the lens, both resulting in high noise in the fringe pattern. The proposed interferometer has sufficient stability and visibility to be used for diffractive microlens focal length evaluation and it provides, to our knowledge, the first interferometric measurement of the focal length of a diffractive microlens. In evaluating microoptics focal lengths it is not possible to look at them as miniaturized copies of macroelements, because microlenses do not necessarily have spherical surfaces or well defined aperture stops. Therefore evaluation methods of a microoptics system focal length should give an effective focal length measurement without requiring the knowledge of the numerical aperture. Indeed the proposed method does not require the knowledge of the aperture size and it allows the measurement producing the effective focal length of the tested optics. 2. Principle of the method Fig. 1 shows the interferometric set-up. We claim that only the distance between two different positions of the lens and the fringe frequency of the interferometric pattern for each position give sufficient information for determining the effective focal length of the lens system. In the set-up an expanded and collimated He–Ne laser beam is reflected onto the measuring arrangement which consists of a reflective diffraction grating Ž1200 groovesrmm. and an adjustable mirror. The direction of the grooves of the
Fig. 1. Schematic of the experimental arrangement, G is the reflective grating and M the mirror. The two interfering beams are tilted by a small angle a . In position Ža. and Žb. two interference fringe patterns are shown without the lens inserted in the set-up Ža. and with the optical system inserted in the set-up Žb.. The straight parallel fringes are along y while a sinusoidal intensity profile is present along x. f is the focal length of the optical system under test and z c the distance between the tested optical system and the CCD camera. u is the half angle of the cone of the spherical wave emerging from the focal point.
M. de Angelis et al.r Optics Communications 160 (1999) 5–9
grating is perpendicular to the drawing plane in Fig. 1. One half of the illuminating beam strikes the grating directly Žbeam A. and the other half impinges the grating via the mirror Žbeam B.. The incidence angle of the illuminating beam is chosen such that the q1 and the y1 diffraction orders of beam A and beam B are respectively collinearly directed along the grating normal towards the CCD camera. Under this condition the angle between the grating and the mirror is 908. When the mirror is slightly tilted with respect to the grating normal, an interference fringe pattern is visible, which results from the superposition of the two light diffraction orders. We wish to point out that the configuration we use is based on the reflective diffraction grating as a recombining element and it allows for a non-localised interference fringe system. With careful alignment an interference pattern of fringes is generated; the interference pattern is a family of parallel straight fringes with a carrier spatial frequency corresponding to sin arl, where l s 632.8 nm is the laser wavelength and a is the angle between the two interfering diffraction orders, which can be varied by tilting the mirror. In this set-up, when no sample is inserted, the contrast of the interference pattern is limited only by the noise due to the dark current of the CCD camera. The lens under test is placed after the grating recombinating element. The interference pattern beams pass through the lens and arrive at the observation plane of a CCD camera; an array of 512 = 512 pixels Žpixel size 11 mm = 11 mm. is used to store the interferometric pattern. The two interfering beams, separated by the angle a , produce two separated point images in the back focal plane of the lens under test Žsee Ref. w1x.. Two regions in these overlapping beams are of interest: the first region is between the lens and the back focal plane and the second region is after the back focal plane. The latter starts at the point P at distance z P s drŽ drf y tan a . from the optical system where f is the focal length and d is the aperture diameter of the lens. By considering the CCD array size p Ž6 mm., it can be easily seen from Fig. 1 that the interfering pattern covers the whole CCD area if the camera is positioned, slightly off axis, at a distance z c not shorter than z P q Ž prd . z p . Substituting for z p the CCD camera must be placed at a distance z c such that zc )
dqp drf y tan a
.
7
which in our set-up results in a more strict condition than the previous one. Mounting the CCD far beyond the focal point of the tested optical system results in the light intensity available on the CCD area being low; however it is found that good quality fringe pattern images are achieved with low noise and high fringe visibility. We assume that the beam at the CCD plane is a spherical wave emerging from a point source located at the focal point, and with a cone half angle u . The amount of energy gathered by the lens is directly proportional to the area of the lens. If we neglect losses due to reflections, absorption, and so forth, the energy per unit area per unit time Ži.e. the flux density. at the CCD plane is proportional to the square of the numerical aperture NA ŽNA s sin u . of the optical system. This means that for a low numerical aperture optical system, the intensity at the positions where the CCD can be placed is low; this is the case of the microlens under test ŽNA s 0.01.. Instead for higher NA optical systems, like microscope objectives, no problems are encountered in choosing the CCD position. The limit for the largest distance z c is given by: Is
P
p w Ž z c y f . tan u x
2
) I0 ,
Ž3.
where P, assumed to be uniform over the CCD, is the energy per unit time available and I0 is the minimum acceptable flux density at the CCD. I0 is fixed by the set-up and depends on the CCD camera response: we have estimated that the dynamic response of the CCD starts to be lower than 256 grey levels when the flux density is lower than 2 =10y3 Wrm2. The original interference pattern when no lens is placed in the set-up has fringes parallel to the y direction and spatial frequency n . When the lens is placed in the beam path, the interference fringe pattern changes by an amount depending on its focal length and the distance z c from the principal plane of the lens to the CCD camera. If n 1 and n 2 are the spatial frequency of the fringe patterns corresponding to two different positions of the lens a distance ´ apart, the focal length f can be expressed as: ´ n 1n 2 fs . Ž4. n n2yn1
Ž1.
A further lower limit to the distance z c is given by requiring paraxial approximation. In the optical arrangement the CCD array size p corresponds to the effective optical aperture; in order to satisfy the paraxial approximation Žwhich is taken to require angles of less than 10y2 ., the CCD camera must be placed beyond the focal point at distance z c such that p - 10y2 , Ž2. 2Ž zc y f .
3. Results In the experimental arrangement the fringe frequency n is easily changed by rotating the mirror M and can be suitably set. The optical systems under test in our laboratory are microscope objectives and collimating lenses for diode lasers. The distance between the two different positions of the tested lens is ´ s 15 mm. The error in the whole number of fringes in the CCD aperture is assumed
8
M. de Angelis et al.r Optics Communications 160 (1999) 5–9
to be 0.002, which corresponds to an error of 3 =10y4 mmy1 in the fringe spatial frequency. As can be noted in Fig. 1 the main problem in the small focal length measurement is to achieve the largest fringe spatial frequency compatible with the small diameter of the lens systems under test. Moreover looking at the expression of f given by Eq. Ž5. it is clear that the error in the focal length is mainly due to the difference n 2 y n 1 between the spatial frequencies. It means that a large excursion of ´ compatible with the superposition of the two interfering beams after the lens is desirable. The measurement of each quantity is performed with the same procedure; the fringe patterns are digitised with a 8-bit grey scale, corresponding to 256 levels. A one-dimensional FFT algorithm is applied to each row of 512 pixels to obtain a power spectra of spatial frequencies. Let us assume a fringe pattern intensity distribution I Ž x , y . s a Ž x , y . q b Ž x , y . cos Ž 2pn x . ,
Ž5.
where aŽ x, y . and bŽ x, y . are the background intensity and the local contrast for the fringe pattern and the spatial frequency is given by n . In order to determine the fringe spatial frequency n of the fringe pattern, we apply the one-dimensional fast Fourier algorithm by Fourier transforming with respect to x each row y of the digitised fringe pattern intensity distribution I Ž x, y .. This spectrum is characterised by a contribution of the low frequency background illumination, which is centered around the zero frequency. Since spatial changes of bŽ x, y . are slow compared to spatial frequency n , the function spectrum is characterised by two peaks at frequency n and yn . The low and the negative frequencies are bandpass filtered and the filtered spectrum is inverse Fourier transformed in order to obtain the complex map Ž1r2 . b Ž x, y . expŽ i f Ž x, y ... The value of phase f s 2pn x wrapped in the range between yp rad and p rad is calculated for each point Ž x, y . of the fringe pattern by using a computer subroutine. From the wrapped phase the unwrapped phase value f can be computed by using a standard unwrapping procedure Žsee Refs. w3,4x and references therein. and the frequency value can be determined from the simple relationship n s D fr2p p where D f is the phase change of a given row of the interferometric pattern between pixel 1 and pixel p of the CCD camera. In the absence of aberrations the phase varies linearly along each row of the interferogram and the unwrapped phase can be fitted to a straight line. Finally the D f values obtained on each row have been averaged over 100 rows of the examinated pattern. Thus, from the knowledge of n and from Eq. Ž5. we can determine the focal length. Measurements have been performed on three achromatic microscope objectives; one is a 4 = with numerical aperture of 0.12 and nominal focal length of 30.8 mm ŽMelles–Griot mod. 04 OAS 002., the second is a 10 = with numerical aperture of 0.25 and nominal focal length of 16.9 mm ŽMelles–Griot mod. 04 OAS 010., the third is
Table 1 Results of the focal length measurements of microscope objectives, diode laser collimating lenses and the diffractive microlens Tested optics
Nom. f Žmm.
Meas. f Žmm.
d fr f
Micr. objective 4= Micr. objective 10= Micr. objective 40= Aspheric lens Collimating lens Microlens
30.8 16.9 4.5 11 3.7 100.0
30.82 16.88 4.48 11.17 3.66 100
0.2% 0.2% 0.2% 0.3% 0.3% - 3%
a Zeiss objective 40 = with numerical aperture of 0.65 and focal length of 4.5 mm. Other optics tested were collimating lenses for diode lasers: we have tested an aspheric lens of nominal focal length of 11.0 mm and numerical aperture of 0.25 mm ŽThorlabs mod. C220TM-B. and a GRIN lens with focal length of 3.7 mm and numerical aperture of 0.450 ŽMelles-Griot mod. 06 GLC 009.. In Table 1 the characteristics of the tested optical systems are summarised, together with their nominal focal length, the measured focal length with the FFT interferometric method and the estimated error measurement. Typical frequency values of the spatial fringes are n s 20 mmy1, and between 2 mmy1 and 12 mmy1 for n 1 and n 2 . If we take into account Eqs. Ž2. and Ž4. which give the limit for the range of measurability of focal length f and assume that the distance between the camera and the optical system is z c s 600 mm Žwhich is the right distance for satisfying the paraxial condition of Eq. Ž4.. and the aperture diameter of the optics is of the order of 10 mm, the measurable focal length f cannot be smaller than 1.7 mm and larger than 300 mm Žfor larger focal length the more practical set-up is to place the CCD camera before the back focal plane as explained in Ref. w1x..
4. Measurement on a microlens We also performed measurements on a diffractive microlens. For such lenses, aperture diffraction effect can be of considerable importance. A suitable parameter for describing the aperture diffraction effects is the Fresnel number N of the lens which is defined as: Ns
a2
lf
,
Ž6.
where f is the geometric optical focal length, l the wavelength and a s dr2 the lens aperture radius. The tested microlens was a diffractive microlens available in our laboratory; it was manufactured at the Paul Scherrer Institute ŽZurich. w5x. It is a continuous relief lens fabricated by direct laser writing into photoresist. It is designed for wavelength l s 632.8 nm, has aperture radius of 1.0060 mm, focal length of 100 mm and Fresnel number 16. The
M. de Angelis et al.r Optics Communications 160 (1999) 5–9
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microlens inserted in the interferometer, the CCD is placed in the region where a straight fringe pattern of about 4 Žor 8. fringes is visible. In Fig. 2 a typical image of the two interfering beams is shown. The CCD camera is placed in such a way as to detect only the central part of the image where the fringes are straight and the condition for the paraxial approximation is satisfied. The microlens under test has a very long focal length for its aperture, which forces us to choose a rather large displacement ´ of the CCD. A shorter focal length microoptic Ž f of the order of 1 mm., which is more common, would have been easier to measure resulting in a more compact set-up. In fact for this microlens the limit on the measurability of the focal length f imposed by Eqs. Ž2. and Ž4. limit f between 1 mm and 950 mm for the chosen value of z c . Strong diffraction rings are visible on the interferometric pattern and introduce noise on the straight fringe pattern. Because of this noise the statistical analysis of the microlens data has been performed on a larger number of row data. We have applied the linear least square fitting procedure on 500 rows of the fringe pattern and computed the mean value of the fringe spatial frequency and the standard deviation. The relative error of the focal length of the tested microlens has been evaluated to be less than 3%.
Fig. 2. Ža. Fringe pattern at the CCD camera while a microlens is inserted in the set-up at a distance of 1250 mm from the CCD. Žb. Image of the same microlens at a shorter distance, where the two interfering beams are totally visible. Strong diffraction rings are present which are caused by the microlens.
microlens is part of a matrix of microlenses which are all on the same support which is a 5 = 5 cm2 photoresist on a glass plate; the distance between the microlenses is 0.5 mm. In the set-up for microlens analysis an adjustable iris diaphragm was mounted in the beam because we wanted to analyse only one lens at a time. Moreover, in the optical set-up, before the lens under test we have mounted a recollimating optical system. In this way we have reduced the diameter of the beam to 2 mm before the microlens sample and we have obtained a fringe pattern of about 50 fringes when no test optics are inserted in the interferometer. The recollimating optical system was needed because a low intensity was available at the CCD plane because of the small microlens aperture; a more powerful laser or a less expanded collimated laser beam would have avoided the recollimating system. The distance between the lens and the closer position of the CCD camera is 1250 mm. We chose to displace the CCD camera rather than the optics under test; the distance between the two different positions of the CCD camera is ´ s 800 mm. With the
5. Conclusion We have demonstrated the possibility that an interferometric method can be applied for measuring very short focal length refractive optics and a 1 mm aperture radius diffractive lens. The focal length is determined by measuring the spatial frequency of two interferometric fringe patterns imaged by the lens under test and the spatial frequencies are evaluated by using an FFT algorithm. The relative error of the f measurement is less than 0.3% for the tested refractive lenses while it is less than 3% for the diffractive microlens.
References w1x M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, Optics Comm. 136 Ž1996. 370. w2x E. Keren, K.M. Kreske, O. Kafri, Appl. Optics 27 Ž1988. 1383. w3x T. Kreis, in: P.K. Rastogi ŽEd.., Holographic Interferometry, Springer, Berlin, 1994, pp. 184–185. w4x M. Taked, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72 Ž1982. 156. w5x T. Hessler, M. Rossi, in: S. Martellucci, A. Chester ŽEds.., Diffractive Optics and Optical Microsystems, Plenum, New York, 1997, pp. 139–148. w6x D.C. Su, C.W. Chang, Optics Comm. 78 Ž1990. 118.