1. Field of the Invention
The present invention relates generally to phase-based imaging techniques, and in particular, to a method, apparatus, and article of manufacture for achieving non-trivial interference phase shifts using a harmonically-matched diffraction grating pair.
2. Description of the Related Art
(Note: This application references a number of different publications as indicated throughout the specification by reference numbers enclosed in brackets, e.g., [x]. A list of these different publications ordered according to these reference numbers can be found below in the section entitled “References.” Each of these publications is incorporated by reference herein.)
Full field phase based imaging techniques [B1][B2][B3][B4] are important for a wide range of applications, such as microscopy and metrology. These methods generally involve interferometry and incorporate some form of non-trivial encoding (e.g., time, space, or polarization) for phase extraction. The encoding process entails either a more complicated experimental scheme or computationally intensive post-processing. In this context, a full field interferometry scheme where the resulting interference outputs are naturally in or close to quadrature can simplify phase imaging. However, this requirement is non-trivial. In fact, the outputs of any two port non-lossy interferometer schemes, which include Michelson, Mach-Zehnder and Sagnac schemes, are constrained to be 180° shifted (trivial) by energy conservation.
In addition to the above, it may be desirable to obtain high resolution depth-resolved phase and amplitude information of turbid samples (such as tissue). Spectrometer based solutions used to provide such information provide inadequate sensitivity as a function of pathlength difference between the sample and reference arms.
To better understand the problems of the prior art, a detailed description of phase measurements, the phase of diffracted light, single grating based Michelson interferometers, and optical coherence tomography are described in detail below.
Phase Measurements
As described above, highly accurate amplitude and phase measurements of optical signals are important in many applications ranging from metrology [A1] to cell biology [A2]. Traditional phase-contrast imaging techniques such as Zernike phase [A3,A4] and Nomarski differential interference contrast (DIC) [A5] render excellent phase contrast images; however, the phase information is only qualitative in nature. To retrieve quantitative phase information in Nomarski DIC, several approaches have been proposed that include: DIC with changing shear direction [A6], phase shifting DIC [A7,A8], and non-iterative phase reconstruction methods such as half-plane Hilbert transform [A9]. In addition, Arnison et al. proposed a method that combined DIC microscopy with phase shifting, shear rotation, and Fourier phase integration to yield linear phase image of a sample [A10]. Recently, Ishiwata et al. have developed retardation-modulated DIC—a method to extract the phase component from the DIC image using two images with different retardation [A11].
Other methods for quantitative phase imaging (QPI) include digital holography [A12, A13], Hilbert phase microscopy (HPM) [A14, A15], and polarization based techniques [A16, A17]. Interference microscopy techniques based on PSI generally require recording of four interferograms with precise π/2 phase shifts of the reference field, adding complexity to the system while others can be computation extensive. HPM and digital holography are simpler as they require only one interferogram for QPI. It is to be noted that the methods depend on recording of high frequency spatial fringes for successful phase unwrapping. It may also be noted that multiport fiber based systems such as 3×3 couplers can provide non-trivial phase difference, which can be manipulated for quadrature phase measurements [A18, A19]. However, free space equivalents of a 3×3 coupler do not exist.
Phase of Diffracted Light in Shallow Gratings
The phase of transmitted/reflected and diffracted light in shallow diffraction gratings is a well understood quantity. However, for the sake of completeness, a brief account of how a diffraction grating affects the phase of light is described herein. Consider FIG. 1, which illustrates spatial phase modulation in a sinusoidal phase grating.
The complex transmittance of a sinusoidal phase grating can be expressed as:
                                          t            ⁡                          (              x              )                                =                      exp            ⁢                                                  ⁢                          {                              jα                ⁢                                                                  ⁢                cos                ⁢                                                                  ⁢                                  (                                                                                    2                        ⁢                        π                                            Λ                                        ⁢                                          (                                              x                        +                                                  x                          o                                                                    )                                                        )                                            }                                      ,                            (        1        )            where α, Λ, and xo are the amplitude of phase modulation, period, and displacement from the origin along x-direction, respectively, of the phase grating.
Defining ξ(xo)=2πxo/Λ, Eq. (1) can be rewritten in the form:
                                          t            ⁡                          (              x              )                                =                                    ∑                              m                =                                  -                  ∞                                            ∞                        ⁢                                                            J                  m                                ⁡                                  (                  α                  )                                            ⁢              exp              ⁢                                                          ⁢                              {                                  j                  ⁢                                                                          ⁢                                      m                    ⁡                                          (                                                                                                                                  2                              ⁢                              π                                                        Λ                                                    ⁢                          x                                                +                                                  ξ                          ⁡                                                      (                                                          x                              o                                                        )                                                                          +                                                  π                          2                                                                    )                                                                      }                                                    ,                            (        2        )            where Jm(α) is mth order Bessel function of the first kind. Using the identity J−m(α)=(−1)mJm(α), the relative phase of the mth diffracted order with respect to the zeroth order is given by:
                              ϕ          ⁡                      (                          x              o                        )                          =                  {                                                                                          m                    ⁢                                          {                                                                        ξ                          ⁡                                                      (                                                          x                              o                                                        )                                                                          +                                                  π                          2                                                                    }                                                        ,                                                                              m                  ≥                  1                                                                                                                                              m                                                        ⁢                                                            {                                                                        -                                                      ξ                            ⁡                                                          (                                                              x                                0                                                            )                                                                                                      +                                                  π                          2                                                                    }                                        .                                                                                                m                  ≤                                      -                    1                                                                                                          (        3        )            
It may be noted that for shallow phase gratings, Eq. (3) holds regardless of the grating profile [A24].
Single Grating Based Michelson Interferometer
Diffraction gratings can be used as beam splitters in different interferometric designs. As described above, the diffracted light in diffraction gratings acquires a unique phase with respect to the undiffracted light. Moreover, this distinct phase φ(xo) can be adjusted by translating the diffraction grating in x-direction [see Eq. (3)]. However, the phase shifts between different output ports of single grating-based Michelson/Mach-Zehnder interferometers are only trivial in nature. To better understand this phenomenon and the operation of the invention's harmonically-related gratings-based interferometer, a simpler system is described herein—a Michelson interferometer based on a single shallow diffraction grating G1 [see FIG. 2(a,b)].
FIG. 2.(a,b) illustrates a schematic of a Michelson interferometer using a single shallow diffraction grating, G1. FIG. 2(a) illustrates the transmitted sample and diffracted reference beams with path lengths d1 and d2, respectively. FIG. 2(b) shows dashed lines representative of coincident sample and reference beam at output ports I, II, and III of the interferometer. FIG. 2(c) shows phase shift of the diffracted beam with respect to the undiffracted beam during the first diffraction. FIGS. 2(d,e) show phase shifts of the diffracted sample and reference beams, respectively, during the second diffraction. x1 is the actuation of grating G1 along the x-direction for the experiment. Mi: ith Mirror; BS: Beam splitter.
In the arrangement shown in FIG. 2(a), a collimated beam from a laser source is directed at normal incidence at G1, which acts as a beam splitter during the first diffraction. Only two beams are considered, i.e., the zeroth order and a first order diffracted beam which form the sample and reference arms, respectively, of the Michelson interferometer. It is to be noted that the grating period Λ1 can be chosen so that only the zeroth and the first order diffracted beams exist. Mirror M1 is shown as the sample whereas M2 represents the reference mirror. The returning sample and reference beams reach the grating G1 and undergo a second diffraction. This time, the grating acts both as a beam splitter as well as a combiner, since it splits and combines the incoming sample and reference beams at the three output ports I, II, and III of the interferometer. The coincident reference and sample beams at the three ports are shown as dashed lines [see FIG. 2(b)]. A beam splitter (BS) is used to separate the output beam at port II from the input beam.
In the context of FIGS. 2(c)-(e), which illustrate the phase of the diffracted beams with respect to the undiffracted beams during the two passes, the total electric field at port I of the single grating-based interferometer can be written as:E1=E1exp{i(2kd1−ξ1(x1)+π/2)}→Field comp. from the sample arm+E2exp{i(2kd2+ξ1(x1)+π/2)}, →Field comp. from the reference arm  (4)where E1 and E2 are the amplitudes of the field components reaching port I from the sample and reference arms, respectively. k is the optical wave vector, and the parameters d1 and d2 correspond to the path lengths of the sample and reference arms, respectively. Moreover, ξ1(x1)=2πx1/Λ1, where x1 is the displacement of the grating G1. Therefore, using EIEI*, the interference signal at port I of the interferometer can be written as:iI=2A1 cos {2k(d2−d1)+2ξ1(x1)},  (5a)
Similarly, the interference signals at ports II and III of the interferometer can be written as:iII=2A2 cos {2k(d2−d1)+2ξ1(x1)+π},  (5b)iIII=2A3 cos {2k(d2−d1)+2ξ1(x1)+π}.  (5c)respectively. The parameters Ai, i=1, 2, 3 are the amplitudes of the interference signals at ports I, II, and III, respectively, and depend upon the diffraction efficiency of the grating G1. It is clear from Eq. (5) that ports II and III of a single shallow grating-based Michelson interferometer are in phase whereas the port I of the interferometer is 180° out of phase with respect to the other two output ports. This geometry is therefore unsuitable for extracting quadrature signals.
To corroborate the above discussion, an experimental setup may be constructed using a collimated beam (1/e2 diameter≈1 mm) from a HeNe laser (λ=633 nm) and a shallow 600 grooves/mm blazed transmission grating (Thorlabs, Inc., GT25-06V). The reference mirror is mounted on a voice coil to modulate the reference arm. Heterodyne interference signals are acquired at the three output ports using New Focus photodetectors (model 2001) and a 16-bit analog-to-digital converter (National Instruments, model PXI-6120). The grating is mounted on a computer-controlled piezo actuator (25.5 nm/V) in order to measure the phase shifts between different ports of the interferometer for various positions of the grating. For each position of the grating, DC components are removed from the acquired heterodyne signals at the three ports; the resulting interference signals, represented by Eqs. 5(a)-(c), are subsequently processed to determine the phase shifts between the output ports.
FIG. 3 illustrates the measured phase shifts between different ports of a single grating-based Michelson interferometer versus grating displacement (x1) along the x-direction. More specifically, FIG. 3 shows the measured phase shifts between different output ports of the interferometer versus grating displacement over 3.5 microns (˜2 grating periods) along the x1 direction specified in FIGS. 2(a,b). As expected, ports II and III are in phase whereas port I is ˜180° out of phase with respect to the ports II and III, indicating that a single shallow diffraction grating-based Michelson interferometer cannot provide but trivial phase shifts between different output ports. Although, p-polarized light can be used in the above described reported experiment (see results in FIG. 3), similar results may be observed for a s-polarized light.
Optical Coherence Tomography
Embodiments of the invention relate generally to low coherence interferometry systems and methods, and more particularly to optical coherence tomography (OCT)—a method for obtaining high resolution depth-resolved phase and amplitude information of turbid samples (such as tissue). The reported methods for depth-resolved imaging include time-domain (TD) [C1, C2] and spectral domain (SD) [C3-C6] OCT (which includes both spectrometer as well as swept-source based OCT systems). The later technique, i.e., SDOCT, offers increased sensitivity [C7-C9] compared with the earlier method (TDOCT). This increased sensitivity can be translated into a higher OCT scan acquisition rate [C10-C13], decrease in the light fluence level, greater depth penetration, or to boost the sensitivity of the various functional OCT methods [C14-C17].
The SDOCT scheme is very similar to that of a typical TDOCT scheme except that a moving reference mirror is immobilized and the detector is replaced by a low-loss spectrometer. The spectrometer is used to record the spectral variation (also known as spectral interferogram or spectrogram) of the detected signal. Moreover, the depth resolved structural information is retrieved by performing a Fourier transform of the acquired spectrogram. Since a typical SDOCT system measures only the real part of the complex cross-spectral density, the Fourier transform of the real valued spectrum yields redundant information for positive and negative spatial frequencies corresponding to positive and negative path length differences between sample and reference arms. To keep the mirrored representations in the two adjacent Fourier half spaces from mixing with each other, the reference arm delay is typically kept slightly shorter than the relative distance of the first sample interface. Thus, only half of the Fourier space can be used to retrieve the sample structural information. This ambiguity can be resolved by acquiring (or reconstructing) the complex signal, resulting in a signal space that is double in range.
In spectrometer-based SDOCT systems, the sensitivity [also known as signal-to-noise-ratio (SNR)] is not the same through the whole depth scan range. Instead, it drops as the path length difference between reference and sample arms increases. This SNR drop-off can be as serious as ˜20 dB over the scan range of 2-3 mm. The ability to record or reconstruct complex signal can therefore significantly improve the performance of spectrometer-based SDOCT systems by increasing the effective or overall depth scan range.
The most common method for complex signal acquisition involves acquisition of phase-shifted interferograms. Several different approaches have been proposed and demonstrated for reference beam phase-shifting, which include piezo-driven reference mirror [C18, C19], electro-optic modulator [C20], and acousto-optic modulators [C21]. Typically, three or more spectral interferograms shifted by π/2 are recorded in a serial fashion in the above methods. Another approach involves the use of multiport fiber-based systems such as 3×3 couplers, which inherently exhibit non-trivial phase difference between the different ports, for complex signal acquisition [C22-C24]. An advantage of the above technique is that it can simultaneously acquire the phase-shifted signals without any optical or electronic modulation as in the previously mentioned OCT systems. Simultaneous acquisition of phase-shifted spectral interferograms is also important as it keeps the reconstructed full-range image from corrupting due to the sub-wavelength sample motion or interferometric drift. It may be noted, however, that the free space equivalents of a 3×3 coupler do not exist.