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Short coherence digital holography for 3D microscopy
Short coherence digital holography for 3D microscopy Giancarlo Pedrini, Staffan Schedin Institut fu¨r Technische Optik, Universita¨t Stuttgart, Pfaffenwaldring 9, D-70569 Stuttgart, Germany
Dedicated to Prof. Dr. Hans Tiziani on the occasion of his 65th birthday
Abstract: An optical system based on short coherence digital holography suitable for 3D microscopic investigations is described. An interferometer is built by using a short coherence laser (coherence length 50 micrometers). The sample is located in one arm of the interferometer, and the other arm is used as reference. The interference occurs only when the path lengths of the reference and measurement arms are matched within the coherence length of the laser. This interference pattern (hologram) is recorded on a CCD faceplate. The reference mirror position is shifted and, a sequence of holograms is recorded. These are then reconstructed numerically by using the wave propagation relation. The reconstruction contains the 3D information of the object. Experimental results showing the difference between a white light image and a short coherence hologram are presented. Key words: Digital holography – short coherence – microscopy
1. Introduction Holography was invented by Gabor [1] in an attempt to solve the problem of the aberrations of lenses used in electron microscopy. Since its invention, the technique has been used in various applications for recording large and microscopic objects [2]. In the early years of holography, reconstruction occured physically by illuminating the developed film with a reference wave. In ref. [3] is shown how the hologram can be digitalized and reconstructed numerically. In the reconstruction, the reference wave and the diffracted wavefronts are simulated. The digital reconstruction allows to obtain the amplitude and the phase of the object wavefront. Nowadays computer speed and storage capacity, and the spatial resolution of CCD sensors (CCD with more pixels and reduced pixel size) increase constantly. Hence, it is now possible to record a hologram using a CCD and evaluate it digitally in a short time. However, the resolution of the
Received 25 Mai 2001; accepted 25 July 2001. Correspondence to: G. Pedrini Fax: ++49-711-6856586 E-mail: [email protected]
Optik 112, No. 9 (2001) 427–432 ª 2001 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
sensor must be sufficient to record the fringes formed by the interference between the reference and the object wave. The maximum spatial frequency that can be recorded is limited by the pixel-size. This technique has been used to compare two wavefronts recorded at different times (digital holographic interferometry) for non-contact measurement of mechanical quantities like displacements and deformations [4]. The digital holographic technique has also been used in microscopy, see e.g. ref. [5], where a resolution of 1.4 mm has been reported by using a lensless Fourier transform arrangement. Another lensless set-up has been applied for the measurement of the shape of microscopic objects [6] and for filtering of reconstructed fields [7]. Digital reconstruction allows to focus at different depths of the object. In the investigation of microscopic objects by using a lensless arrangement the sensor has to be close to the object. As a result the numerical apertures become large and the Fresnel approximation is not valid anymore. This problem is discussed in ref. [8] where a digital reconstruction method based on the Kirchoff-Fresnel approximation is used. Another method using the Rayleigh-Sommerfeld propagation for compensation of the aberration appearing when a lensless hologram of a microscopic object is reconstructed is presented in ref. [9]. The results described in the refs. [5–9] were obtained with a lensless, off-axis arrangement where all the information necessary for the reconstruction of the wavefront is recorded on a single hologram. In ref. [10] an in-line arrangement for microscopy is described, where the phase of the wavefront in the plane of the CCD detector is determined by phase shifting. A microscope objective is used to magnify the sample and on the CCD detector are the phase and amplitude of the magnified wavefront recorded. The advantage of the holographic techniques compared with the normal microscopic imaging, is that the hologram contains all the information necessary to reconstruct the object at different positions, (e.g. focusing at a certain object plane). The disadvantage is that coherent light must be used and this produces speckle noise resulting in a degradation of the image quality. There are many inteferometric techniques that use white or partially coherent techniques for the investi0030-4026/01/112/09-427 $ 15.00/0
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Giancarlo Pedrini, Staffan Schedin, Short coherence digital holography for 3D microscopy
gation of microscopic objects. These techniques are well established and are already commonly used in research laboratories. Optical coherence tomography (OCT), can be performed using short coherence continuous light or using short light pulses. This technique allows high resolution cross-sectional tomographic imaging of internal microstructures in material and biological systems [11]. Light-in-flight digital holography for 3-dimensional shape measurements of macroscopic objects has been presented in ref. [12]. A laser emitting short light pulses (3 ps) has been used for the investigations. In this paper we will describe an arrangement based on digital holographic microscopy where a short coherence laser (coherence length 50 micrometers) is used. The interference occurs only when the path lengths of the reference and measurement arms are matched within the coherence length of the laser. This interference between the reference and the illumination produces a hologram that is recorded on a CCD faceplate. The reference mirror position is shifted and, a sequence of holograms is recorded. These holograms are then reconstructed numerically by using the wave propagation relation. The reconstruction contains the 3D information of the object.
2. Recording of a digital hologram on a CCD sensor using short coherence light Figure 1 shows the optical set-up used for recording the digital holograms. The laser beam is focused into a monomode fibre by the lens L1. At the output of the
fibre a filtered homogeneous beam is emitted. The filtering operation was necessary in our case since the source used was a laser diode having a strong astigmatic and elliptical wavefront. The filtered beam is split into two beams by the beam splitter (BS). One beam is used as the reference and the other one for sample illumination. The illumination beam goes through a microscope objective (MO) and illuminates the sample. In order to have a parallel beam illuminating the sample, a lens (L2) focusing the beam in the back focal plane of the objective has been inserted in the arrangement. The light reflected by the object passes another time through the microscope objective and the beamsplitter and is superimposed with the reference beam which is reflected back by the mirror (M). The lens L3 in the reference collimates the beam before reflection by mirror M. If the paths of the object and reference beam have the same length, we get an interference pattern (hologram) which is recorded by the CCD sensor. We denote the reference beam with R and the wavefront produced by a point P of the sample with UP. According to the theory of partial coherence [13, 14], the intensity recorded at one point S of the CCD is given by dIðSÞ ¼ hRðt t1 Þ R*ðt t1 Þi þ hUP ðt t2 Þ UP*ðt t2 Þi þ hUP ðt t2 Þ R*ðt t1 Þi þ hUP*ðt t2 Þ Rðt t1 Þi
ð1Þ
where t1 and t2 are the times needed, by the reference and the object beam, to travel from the laser source to the point S. The sharp brackets denote the time average, * denotes the complex conjugate amplitude. It is not the purpose of this paper to give a rigorous description of the partial coherence and to explain in detail all the calculations that can be found in refs. [13– 14]. After time averaging, we can write eq. 1 in the more compact form pffiffiffiffiffi pffiffiffiffiffiffiffi dIðSÞ ¼ IR þ IUP þ 2 IR IUP gr ½ðsR sUP Þ=c : ð2Þ gr ½ðsR sUP Þ=c is here the real part of the complex degree of coherence, c is the light speed and sR sUP the path difference between the reference and object beam. The complex degree of coherence, is a function of the spectral emission of the laser source. For coherent light sources the absolute value of the complex degree of coherence is equal to 1, for completely incoherent sources it is 0 and for partially coherent sources between 0 and 1. In our particular case where a short coherence laser is used, jgj will have the value 1 for sR sUP ¼ 0. When sR sUP is larger than the coherence length this value will tends to 0. The real value of g is usually written in the form
Fig. 1. Set-up for digital holographic microscopy using a short coherent laser. L1, L2 and L3 are lenses, MO: microscope objective, M: mirror, BS: beamsplitter, DT: displacement table.
gr ½ðsR sUP Þ=c ¼ jg½ðsR sUP Þ=c j cos fa½ðsR sUP Þ=c
2pðsR sUP Þ=lÞg
ð3Þ
Giancarlo Pedrini, Staffan Schedin, Short coherence digital holography for 3D microscopy
where jg½ðsR sUP Þ=c j is responsible for the contrast of the interference fringes, a½ðsR sUP Þ=c is a phase term related to the reflectivity of the object, 2pðsR sUP Þ=lÞ is the phase due to the path difference, l is the mean wavelength of the source. Eq. 2 represents only the interference arising at the point S by superimposition between the reference and the wavefront coming from the point P of the sample. In reality the interference appearing in our experiments will be much more complicated. We need to take into account that the sample emits light from many points. Some of this light will interfere with the reference beam and produce an interference pattern when the paths are matched, some other will not. We try to keep the notation as simple as possible and we denote the intensity recorded by the CCD at a point S with P P pffiffiffiffiffi pffiffiffiffiffiffi r IðSÞ ¼ IR þ IUi þ 2 IR IUi g ½ðsR sUi Þ=c
i
i
P pffiffiffiffiffiffi pffiffiffiffiffiffi þ2 IUj IUi g ½ðsUj sUi Þ=c :
ð4Þ
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Fig. 2. Location of the points producing an interference for three path length difference (separated by 50 mm) between sample and reference beam. For the calculation a lens with 8 mm focal length and a distance of 180 mm between the lens and the CCD has been simulated.
i6¼j
The first sum term describes the intensities contribution coming to the point S from each point Pi of the object. The second sum term describes the interference between each wave coming from the sample and the reference. The third term describes the interference due to the cross correlation between the light coming from different points (Pi and Pj ) of the sample. Eq. 4 can be written even using the integral notation. Particularly interesting for us is the term describing the interference between the points coming from the sample and the reference. In order to get a good interference ðsR sUi Þ should be smaller than the coherence length lc. Only one part of the light reflected by the sample matches with the reference path and produces an interference pattern. Thus in the complex coherent term is taken into account the superposition of the different beams, it is thus a function of the object shape. By changing the reference path length sR (this is done by shifting the mirror position) we will have a change of the interference pattern, in particular others points of the sample will contribute to the interference. The first problem is to determine which points of the object will interfere with the reference. Consider fig. 1 where we denote with sL the path between the laser and the microscope objective, z is the distance between the microscope objective and one point P of the sample. The path between P and one point Q of the lens is given PQ, the thickness of the lens can be described by a quadratic function Le. The beam refracted by the lens is then directed to the point S located on the CCD. The distance between Q and S is given by QS. If sR is the path for the reference beam from the laser to the hologram plane, we can calculate the points which will contribute to the interference on the CCD simply by looking where the relation sL þ z þ PQ þ Le þ QS ¼ sR
ð5Þ
is satisfied. We consider here for simplicity a plane parallel reference beam (for the experiments we used a slightly diverging beam) and an illumination of the sample with collimated light. Figure 2 shows the location of the points satisfying eq. 2, for the case where the lens has a focal length of 8 mm, the distance between the lens and the CCD is 180 mm, the simulated diameter of the lens is 8 mm and the linear size of the CCD is 9 mm. A simple ray tracing calculation has been used to get fig. 2. The points producing interference were calculated for three match of the referenceobject path separated by 50 mm at the distance between 0.6 and 0.74 mm from the focal plane of the lens. This is a one-dimensional example which can be easily generalised to a two dimensional rotation symmetrical case. The figure shows that not the points located in a plane at a given distance z from the lens contribute to the interference but the points intersecting a paraboloid having a thickness of about 20 micrometers. The curvature and thickness of the paraboloids is a function of the lens used and the distance from the lens. Since a partially coherent (short coherent) source is used, the thickness of the paraboloid representing the points contributing to the interference in fig. 2 will increase along the z axis by the coherence length of the laser lc =2. Only the points of the object matching the paths will contribute to the interference, in order to record the information related to the full sample it is necessary to acquire a series of holograms with the reference mirror at different positions. An in-line arrangement is used to record the holograms, the phase of the wavefront at the CCD plane is obtained by phase shifting the mirror. For each scan position of the mirror, four holograms phase shifted by 0, p/2, p and 3p/2 (this is done by shifting the reference mirror using a piezo), are recorded and the phase of the object wavefront, in the
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Giancarlo Pedrini, Staffan Schedin, Short coherence digital holography for 3D microscopy
CCD plane is calculated. The method for the phase determination is a standard phase-shifting algorithm (see e.g. ref. [10]). Another possibility is to use an off-axis arrangement, in this case we do not need any phase shifting for the determination of the phase of the wavefront. However, the disadvantage of the off-axis arrangement is that the speckle size should be at least as large as three CCD pixels. In order to obtain this it is necessary to introduce an aperture in the arrangement, something that decreases the spatial resolution of the recorded wavefront. The off-axis method is very convenient in the case where all the information has to be recorded in one hologram only. This is for example the case where a pulsed digital holographic technique is used to investigate vibrations and transient events, but it is not well suited for microscopic investigation where usually a good spatial resolution is required and where we have time to apply a sequential phase shift method.
3. Reconstruction of the digital holograms recorded using short coherence light The recording method described in section 2 allows to obtain the phase and the amplitude of the wavefront at the CCD plane. From this information, it is possible to reconstruct the wavefield at a distance d from the hologram. In particular, this can be done by using the Fresnel or the Rayleigh-Sommerfeld diffraction relation. Consider at first the Fresnel diffraction. This approximation is valid only for reconstructing the wavefield when jxi xH j d and jyi yH j d, where xi and yi are co-ordinates in the plane of the reconstructed image and xH and yH are co-ordinates at the CCD plane (hologram plane). In order to use this propagation approximation, we need at first to multiply the complex amplitude at the hologram plane with the quadratic factor hðxH ; yH ; dÞ ¼ exp ðikðx2H þ y2H Þ=2dÞ
ð6Þ
where k ¼ 2p=l and l is the wavelength of the laser light source. For the reconstruction, the phase of the function h should not change more than p from one CCD pixel to the next one. This limits the minimum distance d where the wavefront can be reconstructed. Consider for example one hologram recorded on a CCD with 1024 1024 pixels, pixelsize 9 mm and a source having wavelength l ¼ 0:634 mm. Since the phase difference between two pixels located at the border of the CCD should be smaller than p, the distance d should be larger than 130 mm. At the border of the CCD the phase of the wavefront described by the function h, will change faster from one pixel to the next one compared with the change at the centre. After multiplication of the wavefront amplitude in the hologram plane by the phase factor h, we use a FFT in order to get the wavefront at the distance d.
If we like to reconstruct the wavefront at a small distance from the hologram plane we need to use the Rayleigh-Sommerfeld diffraction relation. If Uðx; y; 0Þ is the complex amplitude field at the plane d ¼ 0, and uðfx ; fy Þ its Fourier transform, the complex amplitude Uðx; y; dÞ in the plane at the distance d is given by Uðx; y; dÞ ðð 2pd ð1 l2 fx2 l2 fy2 Þ1=2 ¼ u1 uðfx ; fy Þ exp i l exp ði2pðfx x þ fy yÞÞ dfx dfy
ð7Þ
where fx and fy are the spatial frequencies, u1 is a constant. The second factor is referred to as the transfer function. Eq. (7) is solved numerically by using the Fast Fourier Transform (FFT) algorithm. In this case too there are limitation in the reconstruction. If we consider the transfer function term, we see that for large fx and fy we will have a large change in the phase from one pixel to the adjacent one. In fact the function uðfx ; fy Þ is digitalized and quantified with the same number of pixels as that used for the hologram recording. The values of the spatial frequencies are in the interval from 1=ð2DÞ to 1=ð2DÞ where D is the pixel size. If D ¼ 9 mm then fx max ¼ fy max ¼ 55 linespairs/ mm. In order to satisfy the sampling theorem, the factor in eq. 7 should not change more than p from one pixel to the adjacent. This condition is satisfied if d is smaller than 130 mm. Thus we see that we have the two reconstruction methods one for the wavefront close to the hologram and the other for the wavefront far away from the hologram.
4. Experimental results For our investigations a part of a flower was used as sample. In fig. 3a) a white light image of the sample is shown. The sample is illuminated through the microscope objective and is directly imaged on the CCD, the arrangement is the same as that shown in fig. 1, but without reference arm. In this image there are parts of the object which are in focus and other parts are not.
a)
b)
Fig. 3. Images from a part of a flower, field of view 400 400 mm2. a) White light image, b) image reconstructed from a hologram recorded with a short coherence laser.
Giancarlo Pedrini, Staffan Schedin, Short coherence digital holography for 3D microscopy
The defocused parts usually disturb the investigation of the sample. The microscope objective used was of magnification 20X and a numerical aperture of 0.50. The distance between the sample and the CCD was 190 mm. The CCD used had 1024 1024 pixels and the pixel size was 9 mm. The field-of-view in fig. 3a) was 400 400 mm2. An example of a reconstructed hologram recorded with a short coherence diode laser (wavelength 634 nm, coherence length 50 mm, laser power 3 mW, only 0.5 mW are emitted at the fiber output) is shown in fig. 3b). The CCD records 4 phase shifted interferograms (the phase shift is done by a piezoelectric element) of a sample (slightly defocused compared with the white light image of fig. 3a)) and the complex amplitude of the wavefront at the CCD plane is calculated. By digital reconstruction of the wavefront we can focus or defocus this image. A focused image of the sample was digitally reconstructed at the distance d ¼ 13 mm by using the Rayleigh-Sommerfeld propagation relation. By using the lens formula relation we can calculate that this corresponds to a focusing by 40 mm at the sample plane. As already discussed before, the interference appears only for the points of the samples which match with the reference, thus in fig. 3b) we see that only the part of the object around a certain plane (paraboloid) z is reconstructed. The two arrows between the figures, show that the two small filament that can be seen focused in fig. 3b) corresponds to those appearing defocused in the white
a)
b)
c)
d)
Fig. 4. Images of a part of a flower recorded by different position of the reference mirror. The difference between the mirror position from one image to the next one is 20 mm.
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light image. There are not other defocused parts which contribute as in fig. 3a), where focused and defocused parts are present at the same time. In order to get a 3D information it is necessary to shift the reference mirror located on the displacement table (DT), this is in practice a scan along the depth of the sample. Figure 4 shows an example where holograms are recorded at 4 different mirror positions. The holograms are then reconstructed such that the parts which produce the interference are brought into focus. In this case the sample was defocused quite a lot, and the reconstructed focused image appears at a distance of d ¼ 150 mm, the Fresnel approximation has been used to reconstruct the hologram. The shifts of the mirror from one image to the other was 20 micrometers. Figures a) and b) show the same filament, in b), we have a better match between the filament and the reference thus the filament appears brigther. At a short distance from this filament (along the scanning direction), we find other as seen in the figs. c) and d). In d) is possible to see that there is still a part of the filament present in c). This is explained by the fact that the point of the sample producing interference are located on paraboloids (see fig. 2) with a thickness of about 20 mm. Furthermore the coherence of the laser is lc ¼ 50 mm, these increase the thickness of contributing point clouds by lc/2.
5. Discussion and conclusion The digital holographic system described allows the digital recording and reconstruction of microscopic objects. The reconstruction shown were obtained by using the Fresnel and the Rayleigh-Sommerfeld diffraction formula. The quality of the results is the same for the two methods. Compared with the imaging using a white light source, we have the advantage that only the part of the object located at a certain position (paraboloid with a certain thickness) is reconstructed and the light coming from other planes does not disturb. Another advantage is the possibility of digital focusing. The example shown are quite simple, but the method can of course be used to perform investigations inside a tissue or a liquid. In this case the possibility of discrimination of different parts of a tissue or particles in a liquid located at different planes will be of great importance. The method is based on the interference and is thus more sensitive compared with an incoherent imaging, few photons coming from the object, already produce an interference detectable by the CCD and allow the reconstruction. The disadvantage is that the optical arrangement is more complex compared with the simple white light imaging, we need a short coherence light source, a reference beam and a piezoelectric element to translate the mirror. The digital reconstruction involves FFT calculations that take some seconds with out PC. There is of course the possibility to calculate the FFT in a few milliseconds by using special signal processing equipment.
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Compared with the classical OCT the digital holographic technique has some advantages. OCT is usually performed with low numerical aperture focusing because it is desirable to have a large depth of field and use low coherence interferometry to achieve axial resolution [11]. However, low numerical aperture means low transversal spatial resolution. If classical OCT is performed with high numerical aperture, a focusing is necessary. The holographic method presented here use a high numerical aperture objective and do not need any focusing by moving the objective, the focusing is done digitally. Compared with the recording with a coherent source, we have here only the interference between the reference and light coming from some parts of the sample. We tryed to record holograms with a long coherence laser (Nd-YAG) and the arrangement shown in figure 1, but we did not get good results, this because the light reflected by the microscope objective interferes with the reference and with the light reflected by the sample, producing a dramatical degradation of the quality of the reconstructed images. The arrangement using a tilted reference (instead as a temporal phase step method) has been investigated as well, in this case the size of the speckle need to be larger and this involves a reduction of the spatial resolution. This is well suited when there is no time to apply a temporal phase shift, e.g. when the sample to be investigated contains particles or cells moving with high speed. The arrangement shown in fig. 1 use a microscope objective but it is of course possible to record a short coherence hologram by using a lensless arrangement. If a high numerical aperture is needed, the CCD should come close to the sample and the illumination of the object has to be chosen in order to have only a small part illuminated. This because a large illumination will produce high spatial frequencies which cannot be resoved by the CCD.
Acknowledgements. We gratefully acknowledge the financial support of the DFG. S. Schedin would like to acknowledge the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), for financial support.
References [1] Gabor D: A new microscopic principle. Nature 161 (1948) 777–778 [2] Collier RJ, Burckhard CB, Lin LH: Optical Holography. Academic Press, New York 1971 [3] Yaroslavsky LP, Merzyalov NS: Methods of Digital Holography. Consultant Bureau, New York 1980 [4] Pedrini G, Tiziani H, Zou Y: Digital Double Pulse-TVHolography. Opt. Laser Eng. 26 (1997) 199–219 [5] Hassad W, Cullen D, Solem J, Longworth J, McPherson A, Boyer K, Rhodes C: Fourier-transform holographic microscope. Appl. Opt. 31 (1992) 4973–4978 [6] Pedrini G, Fro¨ning Ph, Tiziani H, Mendoza Santoyo F: Shape measurement of microscopic structures using digital holograms. Opt. Commun. 164 (1999) 257–268 [7] Pedrini G, Schedin S, Tiziani HJ: Spatial filtering in digital holographic microscopy. J. Mod. Opt. 47 (2000) 1447– 1454. [8] Takaki Y, Ohzu H: Fast numerical reconstruction techniques for high-resolution hybrid holographic microscopy. Appl. Opt. 38 (1999) 2204–2211 [9] Pedrini G, Schedin S, Tiziani HJ: Aberration compensation in digital holographic reconstruction of microscopic objects. J. Mod. Opt. (2001) [10] Zhang T, Yamaguchi I: Three-dimensional microscopy with phase-shifting digital holography. Opt. Lett. 23 (1998) 1221–1223 [11] Fujimoto JG, Drexler W, Morgner U, Ka¨rtner F, Ippen E: Optical Coherence Tomography: High Resolution Imaging using Echoes of Light. Opt. Photon. News 11 (2000) 24–31 [12] Nilsson B, Carlsson TE: Direct three-dimensional shape measurement by digital light-in-flight holography. Appl. Opt. 37 (1998) 7954–7959 [13] Born M, Wolf E: Principles of Optics. Pergamon Press, fifth edition, Oxford 1975 [14] Goodman JW: Introduction of Fourier optics. McGrawHill International Editions, Second Edition, New York 1996
Dedicated to Prof. Dr. Hans Tiziani on the occasion of his 65th birthday
Abstract: An optical system based on short coherence digital holography suitable for 3D microscopic investigations is described. An interferometer is built by using a short coherence laser (coherence length 50 micrometers). The sample is located in one arm of the interferometer, and the other arm is used as reference. The interference occurs only when the path lengths of the reference and measurement arms are matched within the coherence length of the laser. This interference pattern (hologram) is recorded on a CCD faceplate. The reference mirror position is shifted and, a sequence of holograms is recorded. These are then reconstructed numerically by using the wave propagation relation. The reconstruction contains the 3D information of the object. Experimental results showing the difference between a white light image and a short coherence hologram are presented. Key words: Digital holography – short coherence – microscopy
1. Introduction Holography was invented by Gabor [1] in an attempt to solve the problem of the aberrations of lenses used in electron microscopy. Since its invention, the technique has been used in various applications for recording large and microscopic objects [2]. In the early years of holography, reconstruction occured physically by illuminating the developed film with a reference wave. In ref. [3] is shown how the hologram can be digitalized and reconstructed numerically. In the reconstruction, the reference wave and the diffracted wavefronts are simulated. The digital reconstruction allows to obtain the amplitude and the phase of the object wavefront. Nowadays computer speed and storage capacity, and the spatial resolution of CCD sensors (CCD with more pixels and reduced pixel size) increase constantly. Hence, it is now possible to record a hologram using a CCD and evaluate it digitally in a short time. However, the resolution of the
Received 25 Mai 2001; accepted 25 July 2001. Correspondence to: G. Pedrini Fax: ++49-711-6856586 E-mail: [email protected]
Optik 112, No. 9 (2001) 427–432 ª 2001 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
sensor must be sufficient to record the fringes formed by the interference between the reference and the object wave. The maximum spatial frequency that can be recorded is limited by the pixel-size. This technique has been used to compare two wavefronts recorded at different times (digital holographic interferometry) for non-contact measurement of mechanical quantities like displacements and deformations [4]. The digital holographic technique has also been used in microscopy, see e.g. ref. [5], where a resolution of 1.4 mm has been reported by using a lensless Fourier transform arrangement. Another lensless set-up has been applied for the measurement of the shape of microscopic objects [6] and for filtering of reconstructed fields [7]. Digital reconstruction allows to focus at different depths of the object. In the investigation of microscopic objects by using a lensless arrangement the sensor has to be close to the object. As a result the numerical apertures become large and the Fresnel approximation is not valid anymore. This problem is discussed in ref. [8] where a digital reconstruction method based on the Kirchoff-Fresnel approximation is used. Another method using the Rayleigh-Sommerfeld propagation for compensation of the aberration appearing when a lensless hologram of a microscopic object is reconstructed is presented in ref. [9]. The results described in the refs. [5–9] were obtained with a lensless, off-axis arrangement where all the information necessary for the reconstruction of the wavefront is recorded on a single hologram. In ref. [10] an in-line arrangement for microscopy is described, where the phase of the wavefront in the plane of the CCD detector is determined by phase shifting. A microscope objective is used to magnify the sample and on the CCD detector are the phase and amplitude of the magnified wavefront recorded. The advantage of the holographic techniques compared with the normal microscopic imaging, is that the hologram contains all the information necessary to reconstruct the object at different positions, (e.g. focusing at a certain object plane). The disadvantage is that coherent light must be used and this produces speckle noise resulting in a degradation of the image quality. There are many inteferometric techniques that use white or partially coherent techniques for the investi0030-4026/01/112/09-427 $ 15.00/0
428
Giancarlo Pedrini, Staffan Schedin, Short coherence digital holography for 3D microscopy
gation of microscopic objects. These techniques are well established and are already commonly used in research laboratories. Optical coherence tomography (OCT), can be performed using short coherence continuous light or using short light pulses. This technique allows high resolution cross-sectional tomographic imaging of internal microstructures in material and biological systems [11]. Light-in-flight digital holography for 3-dimensional shape measurements of macroscopic objects has been presented in ref. [12]. A laser emitting short light pulses (3 ps) has been used for the investigations. In this paper we will describe an arrangement based on digital holographic microscopy where a short coherence laser (coherence length 50 micrometers) is used. The interference occurs only when the path lengths of the reference and measurement arms are matched within the coherence length of the laser. This interference between the reference and the illumination produces a hologram that is recorded on a CCD faceplate. The reference mirror position is shifted and, a sequence of holograms is recorded. These holograms are then reconstructed numerically by using the wave propagation relation. The reconstruction contains the 3D information of the object.
2. Recording of a digital hologram on a CCD sensor using short coherence light Figure 1 shows the optical set-up used for recording the digital holograms. The laser beam is focused into a monomode fibre by the lens L1. At the output of the
fibre a filtered homogeneous beam is emitted. The filtering operation was necessary in our case since the source used was a laser diode having a strong astigmatic and elliptical wavefront. The filtered beam is split into two beams by the beam splitter (BS). One beam is used as the reference and the other one for sample illumination. The illumination beam goes through a microscope objective (MO) and illuminates the sample. In order to have a parallel beam illuminating the sample, a lens (L2) focusing the beam in the back focal plane of the objective has been inserted in the arrangement. The light reflected by the object passes another time through the microscope objective and the beamsplitter and is superimposed with the reference beam which is reflected back by the mirror (M). The lens L3 in the reference collimates the beam before reflection by mirror M. If the paths of the object and reference beam have the same length, we get an interference pattern (hologram) which is recorded by the CCD sensor. We denote the reference beam with R and the wavefront produced by a point P of the sample with UP. According to the theory of partial coherence [13, 14], the intensity recorded at one point S of the CCD is given by dIðSÞ ¼ hRðt t1 Þ R*ðt t1 Þi þ hUP ðt t2 Þ UP*ðt t2 Þi þ hUP ðt t2 Þ R*ðt t1 Þi þ hUP*ðt t2 Þ Rðt t1 Þi
ð1Þ
where t1 and t2 are the times needed, by the reference and the object beam, to travel from the laser source to the point S. The sharp brackets denote the time average, * denotes the complex conjugate amplitude. It is not the purpose of this paper to give a rigorous description of the partial coherence and to explain in detail all the calculations that can be found in refs. [13– 14]. After time averaging, we can write eq. 1 in the more compact form pffiffiffiffiffi pffiffiffiffiffiffiffi dIðSÞ ¼ IR þ IUP þ 2 IR IUP gr ½ðsR sUP Þ=c : ð2Þ gr ½ðsR sUP Þ=c is here the real part of the complex degree of coherence, c is the light speed and sR sUP the path difference between the reference and object beam. The complex degree of coherence, is a function of the spectral emission of the laser source. For coherent light sources the absolute value of the complex degree of coherence is equal to 1, for completely incoherent sources it is 0 and for partially coherent sources between 0 and 1. In our particular case where a short coherence laser is used, jgj will have the value 1 for sR sUP ¼ 0. When sR sUP is larger than the coherence length this value will tends to 0. The real value of g is usually written in the form
Fig. 1. Set-up for digital holographic microscopy using a short coherent laser. L1, L2 and L3 are lenses, MO: microscope objective, M: mirror, BS: beamsplitter, DT: displacement table.
gr ½ðsR sUP Þ=c ¼ jg½ðsR sUP Þ=c j cos fa½ðsR sUP Þ=c
2pðsR sUP Þ=lÞg
ð3Þ
Giancarlo Pedrini, Staffan Schedin, Short coherence digital holography for 3D microscopy
where jg½ðsR sUP Þ=c j is responsible for the contrast of the interference fringes, a½ðsR sUP Þ=c is a phase term related to the reflectivity of the object, 2pðsR sUP Þ=lÞ is the phase due to the path difference, l is the mean wavelength of the source. Eq. 2 represents only the interference arising at the point S by superimposition between the reference and the wavefront coming from the point P of the sample. In reality the interference appearing in our experiments will be much more complicated. We need to take into account that the sample emits light from many points. Some of this light will interfere with the reference beam and produce an interference pattern when the paths are matched, some other will not. We try to keep the notation as simple as possible and we denote the intensity recorded by the CCD at a point S with P P pffiffiffiffiffi pffiffiffiffiffiffi r IðSÞ ¼ IR þ IUi þ 2 IR IUi g ½ðsR sUi Þ=c
i
i
P pffiffiffiffiffiffi pffiffiffiffiffiffi þ2 IUj IUi g ½ðsUj sUi Þ=c :
ð4Þ
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Fig. 2. Location of the points producing an interference for three path length difference (separated by 50 mm) between sample and reference beam. For the calculation a lens with 8 mm focal length and a distance of 180 mm between the lens and the CCD has been simulated.
i6¼j
The first sum term describes the intensities contribution coming to the point S from each point Pi of the object. The second sum term describes the interference between each wave coming from the sample and the reference. The third term describes the interference due to the cross correlation between the light coming from different points (Pi and Pj ) of the sample. Eq. 4 can be written even using the integral notation. Particularly interesting for us is the term describing the interference between the points coming from the sample and the reference. In order to get a good interference ðsR sUi Þ should be smaller than the coherence length lc. Only one part of the light reflected by the sample matches with the reference path and produces an interference pattern. Thus in the complex coherent term is taken into account the superposition of the different beams, it is thus a function of the object shape. By changing the reference path length sR (this is done by shifting the mirror position) we will have a change of the interference pattern, in particular others points of the sample will contribute to the interference. The first problem is to determine which points of the object will interfere with the reference. Consider fig. 1 where we denote with sL the path between the laser and the microscope objective, z is the distance between the microscope objective and one point P of the sample. The path between P and one point Q of the lens is given PQ, the thickness of the lens can be described by a quadratic function Le. The beam refracted by the lens is then directed to the point S located on the CCD. The distance between Q and S is given by QS. If sR is the path for the reference beam from the laser to the hologram plane, we can calculate the points which will contribute to the interference on the CCD simply by looking where the relation sL þ z þ PQ þ Le þ QS ¼ sR
ð5Þ
is satisfied. We consider here for simplicity a plane parallel reference beam (for the experiments we used a slightly diverging beam) and an illumination of the sample with collimated light. Figure 2 shows the location of the points satisfying eq. 2, for the case where the lens has a focal length of 8 mm, the distance between the lens and the CCD is 180 mm, the simulated diameter of the lens is 8 mm and the linear size of the CCD is 9 mm. A simple ray tracing calculation has been used to get fig. 2. The points producing interference were calculated for three match of the referenceobject path separated by 50 mm at the distance between 0.6 and 0.74 mm from the focal plane of the lens. This is a one-dimensional example which can be easily generalised to a two dimensional rotation symmetrical case. The figure shows that not the points located in a plane at a given distance z from the lens contribute to the interference but the points intersecting a paraboloid having a thickness of about 20 micrometers. The curvature and thickness of the paraboloids is a function of the lens used and the distance from the lens. Since a partially coherent (short coherent) source is used, the thickness of the paraboloid representing the points contributing to the interference in fig. 2 will increase along the z axis by the coherence length of the laser lc =2. Only the points of the object matching the paths will contribute to the interference, in order to record the information related to the full sample it is necessary to acquire a series of holograms with the reference mirror at different positions. An in-line arrangement is used to record the holograms, the phase of the wavefront at the CCD plane is obtained by phase shifting the mirror. For each scan position of the mirror, four holograms phase shifted by 0, p/2, p and 3p/2 (this is done by shifting the reference mirror using a piezo), are recorded and the phase of the object wavefront, in the
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Giancarlo Pedrini, Staffan Schedin, Short coherence digital holography for 3D microscopy
CCD plane is calculated. The method for the phase determination is a standard phase-shifting algorithm (see e.g. ref. [10]). Another possibility is to use an off-axis arrangement, in this case we do not need any phase shifting for the determination of the phase of the wavefront. However, the disadvantage of the off-axis arrangement is that the speckle size should be at least as large as three CCD pixels. In order to obtain this it is necessary to introduce an aperture in the arrangement, something that decreases the spatial resolution of the recorded wavefront. The off-axis method is very convenient in the case where all the information has to be recorded in one hologram only. This is for example the case where a pulsed digital holographic technique is used to investigate vibrations and transient events, but it is not well suited for microscopic investigation where usually a good spatial resolution is required and where we have time to apply a sequential phase shift method.
3. Reconstruction of the digital holograms recorded using short coherence light The recording method described in section 2 allows to obtain the phase and the amplitude of the wavefront at the CCD plane. From this information, it is possible to reconstruct the wavefield at a distance d from the hologram. In particular, this can be done by using the Fresnel or the Rayleigh-Sommerfeld diffraction relation. Consider at first the Fresnel diffraction. This approximation is valid only for reconstructing the wavefield when jxi xH j d and jyi yH j d, where xi and yi are co-ordinates in the plane of the reconstructed image and xH and yH are co-ordinates at the CCD plane (hologram plane). In order to use this propagation approximation, we need at first to multiply the complex amplitude at the hologram plane with the quadratic factor hðxH ; yH ; dÞ ¼ exp ðikðx2H þ y2H Þ=2dÞ
ð6Þ
where k ¼ 2p=l and l is the wavelength of the laser light source. For the reconstruction, the phase of the function h should not change more than p from one CCD pixel to the next one. This limits the minimum distance d where the wavefront can be reconstructed. Consider for example one hologram recorded on a CCD with 1024 1024 pixels, pixelsize 9 mm and a source having wavelength l ¼ 0:634 mm. Since the phase difference between two pixels located at the border of the CCD should be smaller than p, the distance d should be larger than 130 mm. At the border of the CCD the phase of the wavefront described by the function h, will change faster from one pixel to the next one compared with the change at the centre. After multiplication of the wavefront amplitude in the hologram plane by the phase factor h, we use a FFT in order to get the wavefront at the distance d.
If we like to reconstruct the wavefront at a small distance from the hologram plane we need to use the Rayleigh-Sommerfeld diffraction relation. If Uðx; y; 0Þ is the complex amplitude field at the plane d ¼ 0, and uðfx ; fy Þ its Fourier transform, the complex amplitude Uðx; y; dÞ in the plane at the distance d is given by Uðx; y; dÞ ðð 2pd ð1 l2 fx2 l2 fy2 Þ1=2 ¼ u1 uðfx ; fy Þ exp i l exp ði2pðfx x þ fy yÞÞ dfx dfy
ð7Þ
where fx and fy are the spatial frequencies, u1 is a constant. The second factor is referred to as the transfer function. Eq. (7) is solved numerically by using the Fast Fourier Transform (FFT) algorithm. In this case too there are limitation in the reconstruction. If we consider the transfer function term, we see that for large fx and fy we will have a large change in the phase from one pixel to the adjacent one. In fact the function uðfx ; fy Þ is digitalized and quantified with the same number of pixels as that used for the hologram recording. The values of the spatial frequencies are in the interval from 1=ð2DÞ to 1=ð2DÞ where D is the pixel size. If D ¼ 9 mm then fx max ¼ fy max ¼ 55 linespairs/ mm. In order to satisfy the sampling theorem, the factor in eq. 7 should not change more than p from one pixel to the adjacent. This condition is satisfied if d is smaller than 130 mm. Thus we see that we have the two reconstruction methods one for the wavefront close to the hologram and the other for the wavefront far away from the hologram.
4. Experimental results For our investigations a part of a flower was used as sample. In fig. 3a) a white light image of the sample is shown. The sample is illuminated through the microscope objective and is directly imaged on the CCD, the arrangement is the same as that shown in fig. 1, but without reference arm. In this image there are parts of the object which are in focus and other parts are not.
a)
b)
Fig. 3. Images from a part of a flower, field of view 400 400 mm2. a) White light image, b) image reconstructed from a hologram recorded with a short coherence laser.
Giancarlo Pedrini, Staffan Schedin, Short coherence digital holography for 3D microscopy
The defocused parts usually disturb the investigation of the sample. The microscope objective used was of magnification 20X and a numerical aperture of 0.50. The distance between the sample and the CCD was 190 mm. The CCD used had 1024 1024 pixels and the pixel size was 9 mm. The field-of-view in fig. 3a) was 400 400 mm2. An example of a reconstructed hologram recorded with a short coherence diode laser (wavelength 634 nm, coherence length 50 mm, laser power 3 mW, only 0.5 mW are emitted at the fiber output) is shown in fig. 3b). The CCD records 4 phase shifted interferograms (the phase shift is done by a piezoelectric element) of a sample (slightly defocused compared with the white light image of fig. 3a)) and the complex amplitude of the wavefront at the CCD plane is calculated. By digital reconstruction of the wavefront we can focus or defocus this image. A focused image of the sample was digitally reconstructed at the distance d ¼ 13 mm by using the Rayleigh-Sommerfeld propagation relation. By using the lens formula relation we can calculate that this corresponds to a focusing by 40 mm at the sample plane. As already discussed before, the interference appears only for the points of the samples which match with the reference, thus in fig. 3b) we see that only the part of the object around a certain plane (paraboloid) z is reconstructed. The two arrows between the figures, show that the two small filament that can be seen focused in fig. 3b) corresponds to those appearing defocused in the white
a)
b)
c)
d)
Fig. 4. Images of a part of a flower recorded by different position of the reference mirror. The difference between the mirror position from one image to the next one is 20 mm.
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light image. There are not other defocused parts which contribute as in fig. 3a), where focused and defocused parts are present at the same time. In order to get a 3D information it is necessary to shift the reference mirror located on the displacement table (DT), this is in practice a scan along the depth of the sample. Figure 4 shows an example where holograms are recorded at 4 different mirror positions. The holograms are then reconstructed such that the parts which produce the interference are brought into focus. In this case the sample was defocused quite a lot, and the reconstructed focused image appears at a distance of d ¼ 150 mm, the Fresnel approximation has been used to reconstruct the hologram. The shifts of the mirror from one image to the other was 20 micrometers. Figures a) and b) show the same filament, in b), we have a better match between the filament and the reference thus the filament appears brigther. At a short distance from this filament (along the scanning direction), we find other as seen in the figs. c) and d). In d) is possible to see that there is still a part of the filament present in c). This is explained by the fact that the point of the sample producing interference are located on paraboloids (see fig. 2) with a thickness of about 20 mm. Furthermore the coherence of the laser is lc ¼ 50 mm, these increase the thickness of contributing point clouds by lc/2.
5. Discussion and conclusion The digital holographic system described allows the digital recording and reconstruction of microscopic objects. The reconstruction shown were obtained by using the Fresnel and the Rayleigh-Sommerfeld diffraction formula. The quality of the results is the same for the two methods. Compared with the imaging using a white light source, we have the advantage that only the part of the object located at a certain position (paraboloid with a certain thickness) is reconstructed and the light coming from other planes does not disturb. Another advantage is the possibility of digital focusing. The example shown are quite simple, but the method can of course be used to perform investigations inside a tissue or a liquid. In this case the possibility of discrimination of different parts of a tissue or particles in a liquid located at different planes will be of great importance. The method is based on the interference and is thus more sensitive compared with an incoherent imaging, few photons coming from the object, already produce an interference detectable by the CCD and allow the reconstruction. The disadvantage is that the optical arrangement is more complex compared with the simple white light imaging, we need a short coherence light source, a reference beam and a piezoelectric element to translate the mirror. The digital reconstruction involves FFT calculations that take some seconds with out PC. There is of course the possibility to calculate the FFT in a few milliseconds by using special signal processing equipment.
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Giancarlo Pedrini, Staffan Schedin, Short coherence digital holography for 3D microscopy
Compared with the classical OCT the digital holographic technique has some advantages. OCT is usually performed with low numerical aperture focusing because it is desirable to have a large depth of field and use low coherence interferometry to achieve axial resolution [11]. However, low numerical aperture means low transversal spatial resolution. If classical OCT is performed with high numerical aperture, a focusing is necessary. The holographic method presented here use a high numerical aperture objective and do not need any focusing by moving the objective, the focusing is done digitally. Compared with the recording with a coherent source, we have here only the interference between the reference and light coming from some parts of the sample. We tryed to record holograms with a long coherence laser (Nd-YAG) and the arrangement shown in figure 1, but we did not get good results, this because the light reflected by the microscope objective interferes with the reference and with the light reflected by the sample, producing a dramatical degradation of the quality of the reconstructed images. The arrangement using a tilted reference (instead as a temporal phase step method) has been investigated as well, in this case the size of the speckle need to be larger and this involves a reduction of the spatial resolution. This is well suited when there is no time to apply a temporal phase shift, e.g. when the sample to be investigated contains particles or cells moving with high speed. The arrangement shown in fig. 1 use a microscope objective but it is of course possible to record a short coherence hologram by using a lensless arrangement. If a high numerical aperture is needed, the CCD should come close to the sample and the illumination of the object has to be chosen in order to have only a small part illuminated. This because a large illumination will produce high spatial frequencies which cannot be resoved by the CCD.
Acknowledgements. We gratefully acknowledge the financial support of the DFG. S. Schedin would like to acknowledge the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), for financial support.
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