Improved description of the signal formation in grating generated-optical coherence tomography

Dawei Tang; A. J. Henning; Feng Gao; A. P. Gribble; Xiangqian Jiang

doi:10.1364/OE.27.033999

1. Introduction

Optical coherence tomography is a widely used low coherence interferometric technique [1,2] allowing for the measurement within a scattering medium, making it highly useful for a variety of applications including imaging of the eye [3,4], endoscopy [5], blood flow measurements [6], dermatology [7], the characterisation of hydrogels [8] and many others. There are many variations on the experimental setup and analysis methods, but the standard form of the instrument is a variation of a Michelson interferometer combined with a broadband light source.

In the following we examine the use of a tilted diffraction grating in the reference arm of such a system. The effect of such an element was described in by Zeylikovich et al [9] as introducing a continuous optical delay in the direction of the grating dispersion, following on from work by some of the same authors involving ultra short pulses [10]. OCT systems utilizing a tilted diffraction grating in the reference arm were subsequently used in [11–14]. Having constructed a system of this form and analysed the signal formation, we do not believe that this description fully captures the effect of the grating, and so present a more detailed analysis of these type of systems. Being not much more complex than a Michelson interferometer, these systems are amenable to quite a simple analysis, with the wavefront taking a simple form throughout, however the results are quite illuminating. In order to highlight the fundamentals behind the signal generation we have ignored such effects as chromatic abberations in the optical elements, imperfect collimation of the beams and misalignment of the elements. While this leads to poorer correspondence between the experimental results and the predicted results we believe that it will provide the reader with a clearer understanding of the way the system works.

The structure of the paper is as follows; in section 2 a model of the system is constructed demonstrating that the signal generated on the detector is highly dependent on the angle at which the incident light is diffracted at in the reference arm. Several numerical approximations are then made to provide a simplified expression for the signal. In section 3 the suitability of these approximations for the systems found in [11–14] are verified by comparing numerical evaluations of the equations in section 2 before and after the approximations are made. In section 4 the results from the model and experimental results are compared demonstrating that the envelope on the signal moves in the same manner in both cases.

2. Signal generation with broadband sources

The OCT systems used in [11–14] are of the form illustrated in Fig. 1, albeit in the references there is cylindrical lens in the measurement arm, and in [13,14] the lens in front of the detector is replaced by an optical zoom lens. In all these systems a tilted diffraction grating is used in the reference arm to return the incident light, a method taken from [9] where it is described as having the effect of introducing a continuous optical delay in the direction of the grating dispersion. We do not feel this description accurately captures the signal formation, especially in the cases considered in [11–14] where broadband light sources are used, and so in the following we develop a more complete description of these systems. We will demonstrate that it is the diffraction of the incident light at a wavelength dependent angle, combined with the use of a lens in front of the detector, that leads to the signal that is generated. Being only slightly more complex than a Michelson interferometer, this system is open to a particularly simple analysis which clearly illustrates the essential elements of the signal formation.

Fig. 1. A schematic of the instrument showing the path light takes when propagating (a) along the reference arm (b) along the measurement arm. The two focal points on the focal plane of the lens can be considered as sources of spherical wavefronts incident on the screen.

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Figure 1(a) highlights the path the light takes along in the reference arm, while Fig. 1(b) highlights the path along the measurement arm. For each individual wavelength, we will consider the intensity seen on the detector to be the coherent superposition of the electric field due to the light propagating in each arm. The total signal due to the broadband light source is then the incoherent superposition of the intensities obtained at each individual wavelength. Looking at Fig. 1(a), an interesting comparison can immediately be made with a Czerny-Turner spectrometer [15]. Such a spectrometer would be created if the distance $d_4$ were set to zero, with the source spectrum then being separated out into its different wavelength components across the detector.

The instruments in [11–14] use broadband light sources with a $-3$ dB spectral width of around $25$ to $30$ nm. The light is collimated before being divided into a reference beam and a measurement beam as illustrated. The reference beam is incident on a reflective diffraction grating before being incident on a lens and then a detector that is placed a significant distance beyond the focal plane of the lens.

In the collimated beam, the wavefront can be approximated at any point by a plane wave with the electric field being given by,

(1)$$E\{L\} = E_0\exp(ikL)\exp({-}i\omega t) + \textrm{c.c},$$

here $L$ is the distance along a ray in the direction the beam is propagating, $E_0$ is the magnitude of the electric field (we ignore both the reduction in magnitude that occurs at the beamsplitter, and the reduction due to only one diffraction order being collected, as it is the phase of the light that we are most interested in. Different amounts of light from the measurement and reference arm will just lead to a constant background being seen below the interference fringes), $k = 2\pi /\lambda$ is the wavenumber, $\omega$ is the angular frequency, $t$ is time, and ‘c.c’ stands for complex conjugate. The $\exp (-i\omega t)$ component will be omitted in the following, as is common practice, but it should be understood that this applies to all of the electric fields.

The details of the scattering by the diffraction grating can be ignored, suffice to say that the incident light takes the form of a collimated beam, as does the scattered light, however the direction of its wavevector has been changed in accordance with the grating equation.

(2)$$D[\sin(\theta_i) +\sin(\theta_m)] = m\lambda$$

here $D$ is the period of the grating, $m$ is the diffraction order which in the following will be equal to 1, $\theta _i$ is the angle that the wavevector of the incident light makes with the normal to the grating, and $\theta _m$ is the angle that the wavevector of the diffracted light makes with the grating, as is illustrated in the inset in Fig. 2.

Fig. 2. The length of the dot dashed line (red) shown is used to calculate the phase of the wave at the point on the focal plane that the light is focused to. The shape of the wavefront about this line is also shown by the lines crossing this ray (blue). This is subsequently used to calculate the electric field on the detector plane.

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The grating is set in the Littrow configuration for a wavelength within the spectrum of the source light, $\lambda _L$. At this wavelength, $\theta _i = \theta _m = \theta _L$, which is also the angle that the normal of the grating makes with the optical axis of the system (defined by the lens in front of the detector). For other wavelengths, while $\theta _i = \theta _L$, $\theta _i \neq \theta _m$, with the reflected light propagating at an angle $\theta = \theta _m - \theta _L$ to the optical axis of the system,

(3)$$\theta = \theta_m - \theta_L = \textrm{asin}\left(\frac{(2\lambda - \lambda_L)m}{2D} \right) - \theta_L$$

The lens in front of the detector will be treated as a perfect lens, converting the flat wavefront of the diffracted beam into a portion of a spherical wave, and the sine condition will be assumed to be satisfied, meaning that light propagating at an angle $\theta$ to the optical axis is focused to a point a distance $h\{\lambda \}= f_1\sin (\theta )$ away from the optical axis where $f_1$ is the focal length of the lens, as illustrated in Fig. 2. The focal point is assumed to lie in the $(x,z)$ plane, as the wavevector of the scattered light after the diffraction grating also lies in this plane.

Just beyond the lens the electric field is viewed as a portion of a spherical wavefront converging towards a point on the focal plane, and after the focal plane as a portion of a spherical wave diverging from the focal point until it is incident on the detector. Thus the electric field on the detector can be viewed as the superposition of two spherical waves, the relative phase of which on the focal plane corresponds to the difference in phase between light propagating via the measurement arm and reference arm. Thus, by tracing a ray perpendicular to the wavefronts along each path up to the focal points after the lens, the relative phase of the light between the two spherical waves at these points is obtained, allowing the intensity recorded on the detector to be calculated. We include the justification for describing the effect of the grating in the way we do in Appendix A, though for brevity do not include it here.

In the case illustrated in Fig. 1(b), where the scattering object takes the form of a flat mirror whose normal is parallel to the wavevector of the incident light, the scattered light takes the form of a collimated beam propagating parallel to the optical axis, and thus is focused to a point that lies on the optical axis a distance of $f_1$ from the lens. The light from the reference arm is again focused to a point a distance $f_1$ from the lens but a distance $h$ away from the optical axis. The interference pattern is therefore formed by the interference of two spherical waves whose centres located on the focal plane of the lens and whose centres are separated by a distance $h\{\lambda \}$.

The intensity measured on the detector from these two spherical waves is,

(4)$$I\{\lambda\} = |E_{\textrm{ref}}\{\lambda\} + E_{\textrm{meas}}\{\lambda\}|^2$$

where

(5)$$E_{\textrm{ref}}\{\lambda,x,y\} = \frac{E_0\{\lambda\}}{\left(x-h\right)^2 + y^2 + z_0^2}\exp\left[i\left(k\sqrt{\left(x-h\right)^2 + y^2 + z_0^2 }+ \phi_{\textrm{ref}}\{\lambda\}\right)\right]$$

and

(6)$$E_{\textrm{meas}}\{\lambda,x,y\} = \frac{E_0\{\lambda\}}{x^2 + y^2 + z_0^2}\exp\left[i\left(k\sqrt{x^2 + y^2 + z_0^2 }+ \phi_{\textrm{meas}}\{\lambda\}\right)\right]$$

here $z_0$ is the distance from the focal plane to the detector, and $\phi _{\textrm {ref}}\{\lambda \} = kL_{\textrm {ref}}$, $\phi _{\textrm {meas}}\{\lambda \} = kL_{\textrm {meas}}$ are the phases of the wave after propagating down the reference and measurement arm respectively. It can be seen from Fig. 2 that the length of the path from the light source to the focal plane via the reference arm is

(7)$$L_{\textrm{ref}} = d_1 + d_2 + \left(d_2 + d_3\right)/\cos(\theta) + \sqrt{f_1^2 + (a-h)^2}$$

where $a = (d_2 + d_3)\tan \theta - h$, and via the measurement arm is

(8)$$L_{\textrm{meas}} = d_1 + 2m_1 + d_3 + f_1$$

The formation of interference fringes from two spherical waves in this manner is very similar to Young’s interference experiment when using point sources instead of slits. The mathematics of this is laid out quite clearly in chapter 7 of [16] and after applying certain approximations that are detailed in appendix B, most significantly that the distance to the detector $z_0$ is much larger than $x,y$ and $h$, the intensity on the detector can be approximated by,

(9)$$I\{x\} = 2A\{\lambda\}^2\left[1 + \cos\left(\frac{(x-h/2)hk}{z_0} + \phi_{\textrm{meas}}- \phi_{\textrm{ref}}\right)\right]$$

Here $A\{\lambda \} = E_0\{\lambda \}/(x^2 + y^2 + z_0^2)$. It can be seen from Eq. (9) that under these approximations the signal on the detector for each wavelength takes the form of a sinusiod in the $x$ direction and the only variation in the signal in the $y$ direction is due to the slight variation of $A\{\lambda \}$. As both $h$ and $k$ are wavelength dependent, so is the period of the signal generated. Changing $\phi _{\textrm {ref}}$ and $\phi _{\textrm {meas}}$ does not change the period of the sinusiod, however it does change its phase.

From Eq. (9) it can be seen that the relationship between the wavelength of the signal on the detector, $\lambda _{\textrm {detector}}$, and the wavelength of the light creating it is

(10)$$\lambda_{\textrm{detector}} = \frac{z_0}{h}\lambda$$

The total signal when a broadband light source is used will be the incoherent summation of the sinusiod produced at each wavelength. If $(\phi _{\textrm {meas}}-\phi _{\textrm {ref}})$ modulo $2\pi = 0$, then the two sources have the same phase, and it can be seen that a maxima will occur on the screen at the midpoint between the two sources, with the path to this point from each source being equal, as illustrated in Fig. 3. From Eq. (9) it can be seen that at the point $x=0$ the phase of each sinusiod is given by $h^2k/(2z_0)$, and that when $(\phi _{\textrm {meas}}-\phi _{\textrm {ref}})$ modulo $2\pi \neq 0$ then the phase in the cosine term is modified by this additional amount. If the distance $m_1$ is changed by an amount $\Delta {m}$ then $\phi _{\textrm {meas}}$ is changed by an amount $2k\Delta {m}$ for each wavelength. In the next section we will examine numerically the form of the signal that is generated, and the accuracy of the solution once the approximations have been applied, showing that there is a strong peak in the signal that shifts as the location of the object is changed.

Fig. 3. This figure illustrates that, when there is no phase difference between the sources, the central point between the sources where a maxima will lie is wavelength dependent due to the dependence of the location of the second source (the blue or red dot), while the first source remains static (black dot)

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3. Numerical results

In this section we use parameters similar to the systems used in [11–14], and compare the intensity on the detector when the electric fields are given by Eqs. (5) and (6) and the intensity given by Eq. (9) where further approximations have been used. The strong correspondence between the two models implies that sufficient accuracy remains in this regime even after the approximations have been made.

The intensity will be calculated at an equally spaced set of points corresponding to a set of pixels on a detector, with the results being calculated for a set of equally spaced wavelengths which are then summed incoherently. The result using the complete form of the electric field as given by Eqs. (5) and (6) will be given by,

(11)$$I_{\textrm{complete}}\{x,y\} = \frac{1}{n+1}\sum_{j=0}^n \left[E_{\textrm{ref}}\{x,y\lambda_j\} + E_{\textrm{meas}}\{x,y\lambda_j\}\right]\left[E_{\textrm{ref}}\{x,y\lambda_j\} + E_{\textrm{meas}}\{x,y\lambda_j\}\right]^{{\ast}}.$$

where the $^\ast$ denotes the complex conjugate, and $\lambda _j = \lambda _0 + j\Delta \lambda$, $\Delta \lambda = 0.1$ nm, and $n=1000$. For simplicity it is assumed that at the points on the focal plane where the spherical waves are centred $|E_{\textrm {ref}}| = |E_{\textrm {meas}}|$. While this is unlikely to generally be the case, as the incident light is diffracted into many different orders by the grating, an imbalance between the measurement and reference beam will just lead to the visibility of the fringes being reduced.

The incoherent summation of the approximate results at each wavelength, as given by Eq. (9), is given by

(12)$$I_{\textrm{approx}} = \frac{1}{n+1}\sum_{j=0}^n 2A\{\lambda\}^2\left[1 + \cos\left(\frac{(x-h/2)hk}{z_0} + \phi_{\textrm{meas}}- \phi_{\textrm{ref}}\right)\right]$$

and in the calculations the following values were used, $d_1 = 100$ mm, $d_2 = 154$ mm, $d_3 = 20$ mm, $d_4 = 100$ mm, $f_1 = 100$ mm, $D = 1\times 10^{-3}/600$ m, and $\theta _L = \textrm {asin}(625\times 10^{9}/(2D))$ so that it is in the Littrow configuration when $\lambda = 625$ nm and $m=1$. The length of the measurement arm is $m_1 = 154 + \Delta {m}$ mm, where the shift of the mirror in the reference arm away from the initial position is given by the value of $\Delta {m}$, These parameters are chosen to be similar to those in Refs. [11–14] and which match those used in the experimental setup that is used in section 4. The intensity of the light source will be given a Gaussian form,

(13)$$S\{\lambda\} = \exp\left[\frac{-(\lambda-625\times10^9)^2}{2(5\times 10^{{-}9})^2}\right],$$

and this light is assumed to be split equally between the measurement and reference arms, with $E_0 = \sqrt {S\{\lambda \}/2}$.

The measured intensity is calculated at a set of $1001 \times 1001$ equally spaced points over the range over the range $-5$ mm < $x,y$ < $5$ mm, with each point representing a pixel on a detector. Figure 4(a) shows the results using Eq. (11) when $\Delta {m} =0$ m, while Fig. 4(b) shows the result when $\Delta {m} = 5.001\times 10^{-4}$ m, and the location of the fringes with high visibility can be seen to be highly dependent upon location of the surface in the measurement arm. The shift of $5.001\times 10^{-4}$ m was used instead of $5\times 10^{-4}$ m to highlight the fact that the form of the signal within the envelope changes, with the central peak oscillating rapidly as $\Delta {m}$ is changed. Using a shift of $5\times 10^{-4}$ m the signal appears to take the same form as that when $\Delta {m} =0$, only shifted along the $x$ axis.

Fig. 4. Part (a) shows the intensity of the signal on the detector calculated numerically when $\Delta {m} = 0$ m, while part (b) shows the result when $\Delta {m} =0.5001\times 10^{-3}$ m. Parts (c) and (d) show the results along the line of pixels $y = 600$ for parts (a) and (b) respectively, shown by the black dotted line, while the solid red line shows the results when the approximations have been made. Parts (e) and (f) show the difference between the two results in parts (c) and (d) respectively.

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Figure 4(c) shows a plot of the intensities at point along the $y=0$ axis when $\Delta {m} =0$, calculated using Eq. 12 (red line), and Eq. 11, (black dotted line), while Fig. 4(d) shows the same results when $\Delta {m} = 5.001\times 10^{-4}$. Figures 4(e) and 4(f) shows the difference between the two lines in figs. 4(c) and 4(d) respectively. It can be seen that the difference in the signal only reaches at most about $1$ percent of the peak to valley difference in the region around the centre of the envelope of the fringes.

4. Experimental verification

In this section we will look at a comparison between a set of experimental results generated by an instrument of the form shown in 1 and those predicted by Eq. (12). While the interference peak that is obtained in the experimental results is far sharper than that predicted by this model, the movement of the peak does match that predicted to a reasonable degree of accuracy. The difference in the exact form of the signal is only to be expected, as we have used a very stripped down model to represent the system; for instance the source in the model is an ideal point source which is subsequently perfectly collimated, while in the experiment the source is the end of a fibre whose core diameter is $0.4$ mm, and which will be imperfectly collimated across the spectrum by the real collimator. Effects such as these may wash out some of the interference fringes.

The experimental setup consists of light from an LED light source (Thorlabs, M625F2), being coupled into a multimode fibre (Thorlabs M48L01, core diameter 400 µm) before being collimated by a triplet collimator (Thorlabs TC25FC-633) and split using a non-polarising 50:50 beamsplitter. The grating in the reference arm is a ruled reflective diffraction grating with 600 lines/mm (Thorlabs GR25-0605). In the measurement arm a mirror was mounted on a manual translation stage allowing it to be moved along the optical axis of the system. The light from both the reference and measurement arm is incident on a lens with a focal length of $100$ mm, and the detector is a further $154$ mm after this. A Lumenera camera, Lw235M, was used as the detector which has 1616 by 1216 pixels that are $4.4$ µm square, giving a total detector size of 7.11 mm by 5.35 mm.

Looking at Fig. 5(a) it can be seen that there is a bright vertical line around the point $x= 180$. This interference signal moves across the detector as the mirror is moved along the optical axis. In Fig. 5(b) the mirror has been moved along the axis by $750$ µm and the interference signal has moved across the detector, being seen around the point $x = 1245$. Figures 5(c) and 5(d) show line plots along the pixel line $y = 600$. While in both cases here the interference is constructive giving a peak that is higher than the background signal, at other points the interference is destructive leading to a lower intensity than the background. Variation such as this was seen in the model as can be seen in the plots shown in figs. 5(e) and 5(f). In the numerical simulation the results for 1001 equally spaced wavelengths between 628.5 nm and 636.5 nm at used, with the grating being in Littrow configuration for light at a wavelength of 632.5 nm. The plot in Fig. 5(e) was generated when $\Delta {m} = 0.41321x10^{-3}$ m, meaning the centre of the envelope on the signal is around the pixel $x=180$, while Fig. 5(f) shows the result when $\Delta {m}$ has been reduced by $750$ µm at which point it can be seen that the centre on the envelope on the signal is around $x=1350$.

Fig. 5. Part (a) and (b) shows the intensity seen on the camera before and after a 750 µm shift of the mirror along the measurement arm, while parts (c) and (d) show the intensity along the pixel line $y=600$ for parts (a) and (b) respectively. Parts (e) and (f) show the results for a simulation where in part (e) the centre of the envelope on the interference fringes is set close to the location of the peak in part (c), and part (f) shows the signal after a shift of 750 µm of the object in the measurement arm. It can be seen that, while the envelope is far wider, the shift in the interference signal along the $x$ axis is close to that seen experimentally.

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5. Discussion and conclusions

Work in previous papers has described the effect of placing a tilted diffraction grating in the reference arm of an interferometer as applying a continuous optical delay in the direction of the grating dispersion. This appears to be a great simplification and does not clearly describe its effect, especially when a broadband light source is used, and as such we have carried out a detailed examination of the signal formation. While we have restricted ourselves to a very simple scatterer in the measurement arm, this still allows us to examine the effect of such a reference signal. It is also enlightening to make a comparison between the path the light takes via the reference arm and a Czerny-Turner spectrometer with a detector shifted back from the focal plane.

It is clearly highly significant that light of different wavelengths is diffracted at different angles by the grating, which leads to it being focused to different points on the focal plane of the lens in front of the detector. This gives a wavelength dependent reference path, which when combined with the light from the measurement arm leading to the formation of an envelope on the interference fringes which moves as the scattering object is moved along the measurement arm, giving a signal that can be used to track its location. However, unlike the signal created in an instrument such as a white light interferometer, the shape that the interference signal takes within this envelope varies rapidly as the object moves. While the experimental results give a sharper peak in the interference signal than is predicted by the simulation, the signal does move in the manner predicted. The differences between the experimental signal and that predicted by the model can be attributed to numerous factors that the simple model does not accurately include, such as the large size of the light source (a $0.4$ mm core to the multimode optical fiber) combined with an imperfect collimator will lead the light at each wavelength to be focused to a larger region than the sharp focal spot that is predicted by the model. Such an effect may lead to the interference fringes being somewhat washed out. Imperfect spacing of the elements, chromatic dispersion, and any tilt on the scattering measurement object will also lead to changes in the signal. However, there is sufficient correspondence to imply that the model is capturing the essential nature of the system.

We note that when the object in the measurement arm is replaced by something with more complex features than used here, then the light falling on the screen will no longer take the form of a single spherical wavefront diverging from a point on the optical axis, however the light from the reference arm that it interferes with will remain exactly the same as in the case here. It is likely that the complex form of the reference signal will mean that the interpretation of signals generated on the detector will be fairly difficult, making it hard to get results that can be accurately assessed in the general case.

Appendix A

In this appendix we will justify the use of the measurement of the length of a ray in order to calculate the phase that is found at the detector. The argument will be made for a transmission grating, though the result should hold for reflection gratings with only slight modifications to take into account the effect of the reflective materials. The situation is illustrated in Fig. 6, though it should be noted that the field is more complex than is drawn in the region close to the screen. The fields away from the screen will take the form of a collimated bean being incident on it and a collimated beam diffracted at the angle given by the grating equation after it, both these cases are just extended back to the screen in the diagram.

Fig. 6. Two rays, ABC and DEF, passing through two adjacent slits in a diffraction grating. The ray GHD is an arbitrary ray between these two rays.

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The first important point to note is that it is not the true path length that we are interested in, all we are interested in calculating is the phase at a point in space a significant distance away from the grating. In Fig. 6 the red lines represent the maxima in the wavefront both before and after the grating, which in this case takes the form of a periodic set of slits in a screen. The screen is assumed to perfectly adsorb the incident field at the points where is it present, and to leave the incident field unaffected at the points where there are holes in the screen. Thus if we know the field at points A or B in Fig. 6, we can calculate the field at points B and E. Beyond the screen the field is the same as that given by a set of coherent secondary source, and we assume the field at the centre of the slit is unchanged from that incident upon it from the left hand side. We again note that the field in the vicinity of the slits is more complex than illustrated in the figure, however away from the screen the behaviour of the light away from the screen is expected to be that of a flat wavefront propagating as a beam at the diffracted angle. Thus if we consider the length of the path along a ray through the centre of a slit, the phase is the same as that of the incident wave, and is the source of the scattered wave, thus knowing the distance AC, we can relate the phase at these two points. If we now consider ray DEF, which passes through that adjacent slit, we can see that there is an additional distance $m\lambda$ in the path, where in the case illustrated clearly $m=1$. Thus while the path length is different, the phase is the same. Thus if we take any ray through the centre of a slit and calculate the path length, while the distances may be different the phases will not. Thus there is nothing special about the path chosen in Fig. 2.

Looking at a ray that starts on the wavefront between points A and D, and propagates parallel to ABC to a point between points C and F, as illustrated by ray GHI on Fig. 6, it can be seen that the length of the ray will not give the correct phase at point I, as the ray has the same length as ray ABC when it meets the blue dashed line parallel to the screen shown below point C. However, it can also be seen that the error in phase is linear, varying from 0 to $2\pi$ as the starting point moves from point A to point C. If the path to the screen is the same for all of the incident waves, then the error in phase will be the same for all wavelengths. Thus the maximum error when using this ray tracing method will correspond to all of the wavelengths being offset by the same amount of phase in the range $0$ the $2\pi$. If $m$ is greater than one, the error in phase will pass through the range $0$ to $2m\pi$ times as the starting point moves from point A to D. Indeed a similar ray length type argument is made for broadband interferometric systems using pairs of gratings such as in [17] the results obtained do agree with the geometrical picture.

Appendix B

In this appendix we will include details of the approximations that are made in order to obtain Eq. (9) from Eqs. (5) and (6). This very much follows the mathematics laid out very clearly in chapter 7 of [16] and is only included for completeness so that the approximations that have been made can be seen and allow the reader to make a judgment their accuracy and applicability for different systems.

Starting from the fields incident on the detector as given by Eqs. 6 and 5. The intensity recorded by the detector $I = [E_{\textrm {meas}} + E_{\textrm {ref}}][E_{\textrm {meas}} + E_{\textrm {ref}}]^{\ast }$, where the asterisk indicates the complex conjugate. The first approximation that we will apply is that $\left (x-h\right )^2 + y^2 + z_0^2 \approx x^2 + y^2 + z_0^2$ in the denominator of the coefficient of the exponential term in Eq. (5), this is justified as $z_0 >> h,x,y$. This approximation just corresponds to the condition that the variation in the field magnitude due to the different distances propagated is negligible. The intensity is then,

(14)$$\begin{aligned}I\{x\} &= (E_{\textrm{ref}} + E_{\textrm{meas}})(E_{\textrm{ref}} + E_{\textrm{meas}})^{{\ast}} = A\Bigg(\exp\left[i\left(k\sqrt{\left(x-h\right)^2 + y^2 + z_0^2 }+ \phi_{\textrm{ref}}\right)\right] \\ &+ \exp\left[i\left(k\sqrt{x^2 + y^2 + z_0^2 }+ \phi_{\textrm{meas}}\right)\right]\Bigg)\times c.c. \\ &= 2A^2\left[1 + \cos\left(k\sqrt{\left(x-h\right)^2 + y^2 + z_0^2 } - k\sqrt{x^2 + y^2 + z_0^2 } + \phi_{\textrm{ref}}- \phi_{\textrm{meas}}\right)\right] \end{aligned}$$

where $A =E_0\{\lambda \}/(x^2 + y^2 + z_0^2)$, and $c.c$ is the complex conjugate of $(E_{\textrm {ref}} + E_{\textrm {meas}})$. To simplify the next step, the $x$ coordinate is transformed to $\hat {x} = x-h/2$

(15)$$I\{\hat{x}\} = 2A^2\left[1 + \cos\left(k\sqrt{\left(\hat{x}-\frac{h}{2}\right)^2 + y^2 + z_0^2 } - k\sqrt{\left(\hat{x}+\frac{h}{2}\right)^2 + y^2 + z_0^2 } + \phi_{\textrm{ref}}- \phi_{\textrm{meas}}\right)\right]$$

defining

(16)$$r_1 = \sqrt{\left(\hat{x}+\frac{h}{2}\right)^2 + y^2 + z_0^2 }, \hspace{10mm} r_2 = \sqrt{\left(\hat{x}-\frac{h}{2}\right)^2 + y^2 + z_0^2 }$$

then

(17)$$\Delta r = r_2 - r_1 = \frac{r_2^2 + r_1^2}{r_2 + r_1} = \frac{2\hat{x}h}{r_2 + r_1} \approx \frac{\hat{x}h}{z_0}$$

substituting this back into Eq. (16) and reverting to $x$ gives,

(18)$$I\{x\} = 2A^2\left[1 + \cos\left(\frac{(x-h/2)hk}{z_0} + \phi_{\textrm{ref}}- \phi_{\textrm{meas}}\right)\right]$$

Funding

Engineering and Physical Sciences Research Council (EP/P006930/1); Renishaw PLC/Royal Academy of Engineering (RCSRF1516/2/7).

References

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Improved description of the signal formation in grating generated-optical coherence tomography

Abstract

1. Introduction

2. Signal generation with broadband sources

3. Numerical results

4. Experimental verification

5. Discussion and conclusions

Appendix A

Appendix B

Funding

References

Cited By

Figures (6)

Equations (18)

Optics Express