Ординатура / Офтальмология / Английские материалы / Handbook of Optical Coherence Tomography_Bouma, Tearney_2002

of buffers owns an attribute whose value can be ‘‘ping’’ or ‘‘pong.’’ While the TA task grabs incoming frames in ‘‘pong’’ buffers, the TB task computes the frames previously grabbed. When both tasks terminate their work, the B1 and B2 buffer attributes are switched, and the TA and TB tasks are repeated. Note that each task is working with the complete set of buffers (four buffers in OCM). This process needs an initialization pass that is used efficiently to reset the accumulation buffer.

However, proper operation of this double-buffering algorithm is essential to our application. For this reason we use state-of-the-art programming of this algorithm as much as possible. For example, all these critical tasks are coded in assembly language and take advantage of the internal architecture of the processor (i.e., pipeline, superscalar, and cache-optimized management).

11.4PERFORMANCE

11.4.1 Axial Resolution

We analyze in this section the ability of our optical coherence microscopes to reject light from out-of-focus planes. The depth response of our microscopes, i.e., their response when the object plane is moved out of the focal plane of the objective lens, depends on both the spectrum width of the source and the numerical aperture of the objective lenses. Without time modulation ( ¼ 0), the intensity IðzÞ received by each pixel of the CCD camera as a function of the distance z between the object and the focus plane is [23,31]

where SðkÞ is the spectrum of the light emitted by the optical source and max is related to the numerical aperture (NA) of the objective lenses by NA ¼ n sin max, n

being the refractive index of the medium.

Low Numerical Aperture

In the case of objective lenses with low numerical aperture ð max 0Þ, Fspect;NAðzÞ is reduced to

FspectðzÞ has a sinusoidal variation of period ¼ 2=k0, multiplied by an envelopespectðzÞ that also has a Gaussian shape:

322 Saint-Jalmes et al.

which is half the coherence length of the source (n is the refractive index of the

(FWHM spect 0:5 m). However, the intensity of a white lamp cannot be modulated at 50 kHz as easily as with an LED.

In conclusion, in the case of narrow-aperture systems, the short coherence length of the optical source determines the resolution in the z direction, which is typically of the order of 10 m.

High Numerical Aperture

In the case of objective lenses with high numerical aperture, the short coherence length of the light may not determine the depth response. If we suppose in the calculation a low-bandwidth optical source ð k 0Þ, Fspect;NAðzÞ is then reduced to

It is worth noting that a more rigorous analysis using Richards and Wolf’s vector theory [32] gives the same expression [Eq. (25)] obtained with the scalar approximation [33]. It is interesting to note that this expression is formally identical to the depth response of a confocal microscope [1,23]. The function FNAðzÞ exhibits oscillations that decrease with z (Fig. 16). A more explicit expression of the z response can, however, be given under the paraxial assumption (cos max 1)

Figure 16 Theoretical calculation of the depth response of an optical coherence microscope ½FNAðzÞ& as a function of the numerical aperture (NA) of the objective lenses. The coherence length of the optical source is assumed to be infinite.

From expression (27), we see that FNAðzÞ has roughly a sinusoidal variation of period modified by the factor multiplied by an envelope NAðzÞ which has a maximum at z ¼ 0 and falls off when jzj increases with oscillations like a ðsin xÞ dx function. The FWHM of the envelope NAðzÞ is approximately given by

The larger the numerical aperture of the objective lenses, the narrower the envelope. In air ðn ¼ 1Þ, with a numerical aperture NA ¼ 0:95 and a wavelength¼ 840 nm, FWHM NA 0:6 m. Thus, the effect of the high numerical aperture of the objectives can yield much better depth resolution than the effect of low coherence length of the optical source.

General Case

In the general case, the depth resolution in OCM depends on both the coherence length of the source and the numerical aperture of the objectives, the range resolution being smaller than its value due to either effect alone. For example, numerical simulations (Fig. 17) show the evolution of the depth resolution (FWHM of the depth

Figure 17 Depth resolution (FWHM of the envelope of the depth response) as a function of the numerical aperture with an 840 nm light source with 20 m coherence length in a medium of refractive index n ¼ 1:33.

envelope) as a function of the numerical aperture with an 840 nm light source with 20 m coherence length in a medium with a refractive index of 1.33. In the Michelson geometry with numerical apertures smaller than 0.2, the depth resolution is imposed by the coherence length of the source ( 7:5 m). This situation corresponds to a ‘‘pure’’ optical coherence situation, which is the case in most systems (such as our Michelson-type microscope). In the Linnik configuration, the use of a high numerical aperture objective can improve the depth resolution up to about 0:5 m.

Experimental Data

We measured the intensity depth response of our optical coherence microscopes by taking a mirror as the object and measuring the signal intensity A2SðzÞ when the mirror was moved out of the focal plane of the objective. Two numerical apertures were used: NA ¼ 0:15 and NA ¼ 0:95. The measurements are compared with theoretical calculations (Fig. 18). With NA ¼ 0:15, the depth response is determined by the coherence length of the source. The Gaussian shape response, in excellent agreement with theory, has an FWHM of 10 m, equal to half the coherence length of the source (20 m). With NA ¼ 0:95, the depth response is then determined by the numerical aperture of the objective lenses. The experimental FWHM is 0:7 m, slightly larger than the theoretical value of 0:60 m. The difference between theory and experiment is attributed to geometrical optical aberrations at the working wavelength. The depth resolution is thus better than 1 m, which means that our microscope is able to reject the light reflected by objects located less than 1 m from the focal plane. These experiments were carried out in air. In a medium with refractive index n, the resolution would be improved by a factor equal to n.

11.4.2 Lateral Resolution

If we suppose that the lateral resolution in our microscopes is limited by diffraction, the interference signal Iint for an object in the focal plane is of the form

Figure 18 Depth response of our optical coherence microscopes with a 20 m coherence length source at ¼ 840 nm and numerical apertures NA ¼ 0:15 (a) and NA ¼ 0:95 (b).

where nðx; yÞ is the point spread function of the objectives, which is the Airy function. The amplitude point spread function of our interference microscope is thus given by jhðx; yÞj2 and the intensity point spread function by jhðx; yÞj4 as it is for the confocal microscope [1]. Because of digitalization by the CCD array, the actual intensity point spread function is the square of the convolution of jhðx; yÞj2 with a square distribution of width equal to the pixel dimension in the object plane.

To measure the lateral response of our microscopes, we recorded an intensity profile along a line taken across a cleaved mirror. Two numerical apertures were used: NA ¼ 0:15 and NA ¼ 0:95. The measurements are compared with theoretical calculations (Fig. 19). With NA ¼ 0:15, the experimental 20–80% width of the intensity profile was 1:5 m, in good agreement with theory. With NA ¼ 0:95, the experimental 20–80% width of the intensity profile was 0:3 m, slightly larger than the theoretical prediction of 0:25 m. Again, the difference between theory and experiment is attributed to geometrical optical aberrations. In the calculations, values used for pixel size and wavelength were 0:15 m and 840 nm, respectively.

11.4.3 Sensitivity

We develop an estimate of the sensitivity of our system as usually defined in the OCT context, i.e., the ratio ðKminÞ 1 of the power delivered to the sample over the minimum detectable signal.

We first point out that in practice images are accumulated under illumination conditions where the CCD pixels are close to saturation. Let N0 be the number of polarized photons corresponding to one pixel incident on the objective back aperture during one image acquisition time. Assuming that the initial polarization makes a (small) angle with the S axis of the PBS, a fraction N0 sin2ð Þ reaches the reference mirror and N0 cos2ð Þ irradiates the sample. Of the latter, the objective collects a

Figure 19 Lateral resolution. Edge response of our optical coherence microscope with a 20 m coherence length source at ¼ 840 nm and numerical apertures Na ¼ 0:15 (a) and NA ¼ 0:95 (b).

The sign corresponds to the situation where the backscattered signal and the reference signal are ðþÞ in phase or ð Þ of opposite phase.

Thus, the modulated signal S that is processed by our parallel lock-in technique and the average flux S0 can be expressed as

Assuming that sin2ð Þ Ki (the input polarization is oriented so that the noise level is not dominated by the incoherent backscattered light), in the case of shot noise

limited signal detection, and if S0 sin2ð ÞN0 ¼ Ssat (where Ssat is the saturation level of the CCD pixel), the signal-to-noise ratio for N accumulated image quad-

ruplets can be expressed as

In expression (33) the parameter sin2ð Þ is in general inversely proportional to the power delivered to the sample. As a matter of fact, it describes the interferometer balance setting that enables one to almost reach the saturation level of the sensor with the signal returning from the reference arm.

Taking Ssat ¼ 2 105 and ¼ 10 , a 1 s record of a full image at the maximum speed of the camera (200 images/s) enables one to detect a minimum signal of relative

magnitude Kmin 7:5 10, corresponding to a sensitivity of 91 dB. We point out that there is a trade-off between sensitivity and detection bandwidth, so that under the same conditions an 81 dB sensitivity is obtained within 10 1 s of recording time.

In the case of single quadrature images, the signal-to-noise ratio for N accu-

A 1 s record under the conditions specified above would then be characterized by a sensitivity of 97 dB.

We confirmed this estimate by measuring the noise level in featureless images acquired under illumination conditions where the CCD sensor was almost saturated.

Finally, we point out that it is assumed in the above calculation that sin2 Ki, meaning that the noise does not mainly originate from the incoherent backscattered light. This was typically the case in our experiments with a 0.25 NA objective and a polarizer setting corresponding to 10 . However, if an objective lens with higher numerical aperture is used, might be increased, depending on the overall backscattering of the sample.

11.4.4 Temporal Resolution and SNR Issues

We saw in the previous section that a sensitivity of 90 dB can be achieved in optimal conditions with an exposure time of about 1 s.

In this derivation we have supposed that the signal backscattered from the structure was stationary on the time scale of our measurement. We can now raise the question of the behavior of such a signal (and of its time average) if the medium under study is quickly changing with time (for example, if the signal originates from scatterers undergoing Brownian or some physiological motion).

In a ‘‘classical’’ OCT system the measurement time per voxel is on the order of a few microseconds and the sample can generally be considered to be static at this time scale. However, line-to-line evolutions (a few milliseconds to a few seconds) usually have to be taken into account in the processing software that reconstructs the final image.

In a full-field approach using a CCD camera, the voxel acquisition time is on the order of a few milliseconds, and, most of the time, averaging over about 1 s is necessary to obtain an acceptable SNR.

If we average coherently (with respect to the reference signal) the signals from moving scatterers over 10–100 images, rapid changes in the optical phase of the backscattered wave may wipe out any contrast present in the interference signals. For instance, if the optical phase changes by 2 during the acquisition time, the contrast is zero.

So one may think of first computing the amplitude of the interference signal from the four unprocessed images necessary to get the signal, then averaging this value over several acquisitions. Such incoherent averaging is obviously valid only if the variance of the noise is smaller than the signal amplitude (otherwise noise amplitude rather than signal is averaged). Only in this latter case can signal-to-noise ratio and thus the image quality of the moving sample be improved by averaging.

Figure 21 Cross-sectional intensity images at different depths from a multilayer silicon integrated circuit, obtained with our Linnik-type optical coherence microscope. The distance between successive images is 0:1 m in the z direction. Each image corresponds to a field of 35 m 35 m. The numerical aperture of the objective lenses was NA ¼ 0:95.

11.5.2 Integrated Circuit Inspection

The unceasing advances in optical microlithography and processing technology require more and more powerful optical imaging instruments for characterization and measurement. For this purpose, our high resolution Linnik-type interference microscope is a useful tool. Cross-sectional intensity (A2S) images from a multilayer silicon integrated circuit are shown in Fig. 21. Objective lenses with 0.95 numerical aperture were used, which yields 0:7 m depth resolution and 0:3 m lateral resolution. Two consecutive images are separated by 0:1 m in the z direction. Note that offfocus features are suppressed, whereas they are not in classical microscopy (Fig. 22).

11.5.3 Phase Images and Profilometry

As mentioned before, optical coherence microscopes can provide the optical phase, which is proportional (modulo =2) to the height between the surface of the object