Abstract: This study presents a new technique to measure the phase difference between two diffracted fields: The field diffracted by a reference object and the field diffracted by an unknown object. For that we use the interferential technique of Young slits. We measured the phase difference between the diffracted fields of two rods of resin. And knowing the phase of the diffracted field of the reference object helps deducing the phase of the field diffracted by the sample. This setup is simple and it is very strong in the presence of disturbances because both objects are illuminated with the same incident beam. Moreover, this technique allows us measuring the phase of the diffracted field on a wide range of angle so that a high resolution of the image can be obtained.
INTRODUCTION
There has been a considerable interest these last few years in the development of new optical imaging systems that are able to give the three dimensional optical properties of a sample, as encoded in the spatial variations of the permittivity, at the nanoscale (Carney et al., 2004; Sentenac et al., 2006; Liu et al., 2007). Potential applications range across multiple fields in life and material sciences.
Standard optical microscopes do not provide quantitative information on the sample permittivity. Crucial information is lost when only the intensity of the diffracted field is detected. To overcome this difficulty, it has been proposed to measure the amplitude and the phase of the diffracted field in a conventional far field microscope and to increase the resolution of the imager, the sample is illuminated under various incident angles (Lauer, 2002). An inversion algorithm is then used to form the image from the multiple dataset (Belkebir and Sentenac, 2003). This approach, known as Optical Diffraction Tomography (ODT) has stirred a wealth of research in the last five years and different setups, adapted to biological applications (Lauer, 2002; Debailleul et al., 2008) or to surface imaging (Alexandrov et al., 2006; Mico et al., 2006) were proposed.
Herein, we present a setup which allowed us to measure the phase of the diffracted field on a wide range of angle. So far, in the experimental configuration used to measure the phase of the diffracted field, the underlying principle is interference between a probe beam and a reference beam (Lauer, 2002; Destouches et al., 2001). We use here only one beam in this set up.
SETUP
The interferential technique of Young slits is composed of two slits
in the same plane. These two slits are lit by the same incident beam.
The waves diffracted by these two slits interfere to give the interference
figure of young slits. In the far field, there is an interference between
the fields diffracted in the same direction. The angle of observation,
θd, defined as the angle between the diffracted wave vector
| Fig. 1: | Setup diagram, O is the sample deposited on the fine glass plate P1 and M is the reference object deposited on the fine glass plate P2 |
For the other interferential systems, i.e., the Michelson interferometer, the angle of observation ranges from –45° to 45° at the best, when using a cube as a beam splitter.
There are two things to consider in this setup: the reference object
and the sample. Let us denote
The reference object and the sample we use in this experiment are rods of resin, with rectangular cross-sections, deposited on a glass substrate. The relative permittivity of this resin is εr = 2.66. The incident plan and the polarization of the illuminating wave are perpendicular and parallel to the invariant axis of the rods, respectively.
The emitted light at 633 nm by a 30 mW Helium-Neon laser, illuminates the reference object and the sample. The reference object is translated along the x-axis direction by an electro optic modulator, so that the reference object does not leave the illuminated beam. With a photodetector placed on a moving arm which turns around the sample, the intensity Ird of the interference between the diffracted field of the sample and the diffracted field of the reference object is measured for different angles. The moving arm is 60 cm long so that we work in the far field region.
We assume that the illuminated wave
| (1) |
where, Ai and Δφ are, respectively the amplitude and the phase of the illuminating field.
The diffracted field of the sample is:
| (2) |
where,
| (3) |
where,
Assuming that
| (4) |
where,
| (5) |
We assume that the translation is proportional to the tension: d = kV so,
| (6) |
The signal s (θd, θinc) is evaluated:
| (7) |
| (8) |
We determine k and then we calculate the signal:
| (9) |
With both signals s (θd, θinc) and scal
(θd, θinc), we estimate
RESULTS AND DISCUSSION
Figure 6: and 7 show the results
when the reference object and the sample are illuminated by the incident
beam with the angle of incident θinc = 37°.
| Fig. 2: | Intensity of the interference between the diffracted field of the sample and the diffracted field of the reference object for θinc = 0 and θd = 10° |
| Fig. 3: | Intensity of the interference between the diffracted field of the sample and the diffracted field of the reference object for θinc = 0 and θd = 60° |
| Fig. 4: | Experimental signal s (θd, θinc) and calculated signal scal (θd, θinc) for θd = 10° and θinc = 0° |
| Fig. 5: | Experimental signal |
| Fig. 6: | Experimental signal s (θd, θinc) and calculated signal scal (θd, θinc) for θd = 10° and θinc = 37° |
| Fig. 7: | Experimental signal |
When we compare the measured signal with the calculated signal (Eq. 9) with the good phase shift for this direction, we note a very good agreement between the two signals.
The intensity of the interference between the diffracted field of the sample and the diffracted field of the reference object for θd = 60° is not very noisy. This allows measurements on a wide range of angle for a given direction of incidence.
Comparing with the other tomography techniques, this technique presents two main advantages. Firstly it uses only one beam, not two and secondly it allows measures on a wide range of angle for a given direction of incidence.
A number of directions of the illuminating wave are used and the corresponding
intensities of the interferences are measured. Following this, the corresponding
phases
CONCLUSION
A simple setup based on Young slits technique has been built. It has allowed making measurements of difference phase of the diffracted fields without using lenses. These measurements have been made on a wide rang of angle [–80°, 80°]. That helps to have a high resolution of the imager. Moreover, this setup is very strong in presence of disturbances.
With the different results, it is experimentally demonstrated that it is possible to measure the phase of the diffracted field. A further study is projected in order to improve this technique by motoring the movement of the detector.
ACKNOWLEDGMENT
We thank Dr. André Kra Koffi, teacher of English for specific purposes who helped us to write this study in English.