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Abstract
This paper shows analytically and numerically that a vortex plate coupled to a neutral density filter can deliver a true optical spatial derivative when placed at the focal plane of a 2f lens pair. This technique turns any intensity or phase variations of coherent light into an intensity that is proportional to the square of the norm of the initial variation gradient. Since the optical derivative removes the uniform background, it is possible to measure the mode numbers of spatial phase gradients or fluctuations optically, without using any interferometer.
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1. Introduction
Taking the differentiation of a measured signal has always been a task difficult to accomplish in common scientific applications and, while the operation itself does not amplify the noise levels, the elimination of the signal average (also called “DC" background) decreases the signal-to-noise ratio. Yet, taking the spatial derivative of a signal can be useful. For instance, the derivative can enhance the contrast of an image. When light is used as a probe, the analog derivative operation becomes relatively straightforward, using optical systems to perform the derivative operation before the signal is recorded [1], leading to practical measurements such as mechanical shear [2]. In fact, almost any optical method has its own optical derivation technique, from schlieren imaging to [3] to shearing interferometry [4]. Methods have been perfected over the years to yield high sensitivity [5–7]. Another advantage of using optics is the simplicity of doing a Fourier transform, which naturally appears on the focal plane of a converging lens or mirror. Any optical operation done on this plane is automatically done in Fourier space, and transferred back to the image when it is reconstructed on the other side of the lens. Using different filtering techniques one can perform first order optical phase derivatives [8,9], fractional derivatives [10,11] and even two-dimensional first derivatives [12].
As manufacturing methods for optical devices reach sub-wavelength precision, new possibilities have come to light to form the derivative of an optical signal: vortex plates [13,14]. By carving a spiral (stair-case like) cavity on the order of the light wavelength, one can generate vectorial beams [15–18]. When they are inserted on the focal plane of a thin lens, they can greatly improve the quality of the image from phase contrast imaging [19,20]. However, a vortex plate coupled to a linearly varying neutral density filter placed on the focal plane of the $2f$ lens pair of Fig. 1 can yield a spatial derivative of extremely high quality. In this paper, we show how to construct an optical derivative using the mathematical foundations behind the optical Fourier transforms. We then use numerical simulations to show how this setup can measure phase gradients or fluctuations.
2. True optical derivatives
2.1 Construction of a true optical derivative
Lenses can decompose the image of an object into its spatial frequency components on the lens’ focal plane
2.1.1 Normalized spatial derivative using a vortex plate
Following Ref. [15], using monochromatic coherent light, any image in the object plane $u(x,y)$ can be decomposed using the Fourier theorem as a discrete sum of sine and cosine functions
Back on the image plane we now have
At this point, we can compute the intensity $I_\textbf k$ of the mode wave number $\textbf k$ as
We see that the square root of the intensity on the image plane is which can be written as a normalized derivativeA more general derivation of the Fourier transform of a spiral phase plate can be found in Ref. [22].
2.1.2 True optical derivative using a vortex plate and a neutral density filter
Equation (4) shows that it is relatively straightforward to recover a true derivative by adding next to the vortex plate a neutral density filter, which transmission $T(f_x,f_y)$ varies linearly with the radius $\rho$ defined as
In practice, the transmission $T$ cannot be larger than 1. However, we treat here the purely mathematical case. We now get on the Fourier plane
Note that the combined vortex plate and neutral density filter with spatial variations given by Eq. (5) and Eq. (10) form a transmission plane that is simply the portion of the complex plane $\mathbb {C}$
limited to the values where $\rho <1$. Back to the image plane, Eq. (11) yieldsThe square root of the intensity on the image plane is now
Using the same reasoning Eq. (14) turns into
2.2 Numerical validation
In the rest of the paper, we use ray tracing [23] with Rayleigh-Sommerfeld (RS) diffraction [24] to compute the effect of each optical element on the intensity and phase of a Gaussian laser beam. For practical reasons, we choose to work using a wavelength $\lambda =$532 nm. The beam diameter $d$ is 20 mm, the focal length of each lens is $f=10\textit {cm}$ and the total number of rays is 1024. The input is a collimated beam, which starts at $-f$ in the simulation. All results are shown at the location $3f$. We use numerical simulations here to highlight the difference between the normalized optical derivative and the true optical derivative. We also computed a schlieren image for comparison. The image of the backlit mask is shown in Fig. 2-a. A well known method to enhance contrast uses schlieren imaging [25], shown in Fig. 2-b. We can greatly improve the contrast using vortex plate. As Fig. 2-c shows, the mask edges appear more clearly. The normalized derivation causes the "halo" surrounding the mask edges as large gradients have the same intensities as smaller ones. In other words, the derivative of large $k$ modes is similar to the derivative for low $k$ modes, leading to a spread of the derivative, regardless of its $k$ value. This result matches qualitatively Fig. 3 of Ref. [20] using a spiral phase plate. Finally, if we add the neutral density filter with profile given by Eq. (10) we get the true optical derivative, shown in Fig. 2-d. Based on Eqs. (14) and (15) we expect the intensity to be proportional to the characteristic wave number $k$ of the edge transition.
Fig. 2. A mask of the spirit logo of the authors’ university is placed at the object location shown in Fig. 1 leading to a) the bright-field image of this logo. b) When a blade is inserted so its edge rests on the optical axis, we obtain a schlieren image. c) The image edges are sharper when a vortex plate is placed on the focal plane, yielding the normalized optical derivative. d) The image can be made much sharper by adding a neutral density filter, which produces the true optical derivative. All intensities are absolute. The axis units are in $\mu$m.
Fig. 3. The square root of the intensity for the vortex plate alone (VP) and the vortex plate coupled to a neutral density filter (VP+ND) for a backlit cylinder with a transmission profile given by $P(x)=1-\exp (-(x/\sigma )^2)$. The analytic derivative of the profile $P'(r)=\frac {2x}{\sigma ^2}\exp (-(x/\sigma )^2)$ is also given for reference (Derivative). The mask is also shown for reference. All line-outs have been normalized.
We now use a mask that changes smoothly as
to track numerically the value of the derivative and shown on Fig. 3. This approach allows to check numerically the spatial dependence of the true optical derivative. Figure 3 shows that the optical derivative matches the analytical derivative up to the simulation precision limited by the discretization inherent to the Rayleigh-Sommerfeld propagation scheme [24]. The separation between rays lead to some oscillations clearly visible on the line-out. These oscillations are due to the reduction in intensity, which decreases the signal to noise ratio of the computation. However, this is the maximum intensity for each ray that counts, and the intensity there always matches the analytical derivative. We can see that the normalized derivative using the vortex plate alone does not match as well the analytical derivative. Note that the RS propagation error is less pronounced for the vortex plate line-out (though still visible), since the actual intensity is an order of magnitude larger compared to the vortex plate coupled with the neutral density filter.3. Direct phase measurements
Overall, we can find multiple optical systems measuring intensity variations, small or large, and the proposed arrangement could simply be seen as a minor improvement over the work done in Refs. [19,22]. However, there is an inherent symmetry between the intensity both in the filter of Eq. (12) and the signal decomposition of Eq. (2). As a result, we see that the spatial derivative works on intensity and phase in the exact same way. So, in media where the change in intensity is negligible compared to the change of phase, the proposed setup turns a phase change into intensity change. We look at two cases here. The first case looks at large scale gradients, where the phase changes smoothly across the beam. The second case looks at small scale fluctuations. While both cases are related, the optical derivative brings a key component to an-all-optical Fourier transform: the absence of a DC component, allowing to measure the wavenumber of the phase fluctuations close to the origin.
3.1 Phase gradients
We looked at two cases and used one to calibrate the simulation, as we would do experimentally, though it is technically possible to compute it absolutely here: the case where $P_1(x,y)=5\pi P(x,y)$ and the case $P_2(x,y)=15\pi P(x,y)$. The laser beam is processed by the setup shown in Fig. 1, leading to the results presented in Fig. 4. After computing the intensity for the profile corresponding to the distribution $P_1$, we scaled $\sqrt {I}$ to match the analytical derivative of $P_1$, in order to calibrate the simulation easily. Then we computed the intensity variation caused by the profile $P_2$ keeping the scaling unchanged. Figure 4-a shows that the analytical derivatives are following exactly the derivatives obtained optically for both profiles. We see that the optical and analytical derivatives virtually fall on top of each other for both $P_1$ and $P_2$. Note that when using a neutral density filter alone, we get an approximate optical derivative that does not match as well the analytical derivative, especially on axis.
Fig. 4. a) The square root of the intensity for two phase profiles $P_1$ and $P_2$. b) The integration of the square root of the intensity obtained for the phase profile $P_2$ (integrated) matches exactly the analytical form of the phase profile $P_2$ (true). Without the optical derivative, only the wrapped phase can be measured (measured). c) the normalized error between the phase profile $P_2$ and the square root of the intensity.
The integral of the optical derivative of $P_2$ is shown in Fig. 4-b. As expected, the integrated phase follows the true phase given by profile $P_2$. If we wanted to measure the true phase, we would get the phase wrapped modulo $2\pi$, also show in 4-b. Note that the proposed setup allows to measured variations in phase or in intensity in the exact same manner. The error between the profile and $\sqrt I$ is shown in Fig. 4-c.
3.2 Phase fluctuations
Analog Fourier transforms using optical systems to study the dominant modes inside an image preceded their digital counterpart by several decades. However, the removal of the zeroth order (i.e. DC) component [26] has always been necessary when one is interested in measuring all the modes [27], especially for low wave numbers. Going back to Eqs. (14) and (15), we clearly see that the optical derivative naturally cancels the constant light background (with wave number $k=0$). With the DC component removed from the Fourier plane, it is now possible to record low-k modes. As a proof of principle, we modelled fluctuations using the products of two sine functions,
We included here two distinct modes $k_1$ and $k_2$. The corresponding phase fluctuations are shown on the left in Fig. 5 and the optical derivatives are on the right of Fig. 5. We took $\gamma _1=\gamma _2=10$. When a vortex plate alone is used (center of Fig. 5), the periodicity of the image is lost. But when the neutral density filter is added, the derivative recovers the expected periodic structure. For our mode analysis, we now look at the Fourier plane where we can find the wave number of the turbulence directly.Fig. 5. Phase fluctuations generated by Eq. (17) to simulate turbulence, with $k_1=3\pi /d$, $k_2=8\pi /d$, and $\gamma _1=\gamma _2=10$ (left). The square root of the intensity for the vortex plate alone (center) and the vortex plate coupled with neutral density filter (right).
Now starting with the vortex plate alone, Fig. 6-a shows the DC component only when the intensity is plotted on the linear scale. Numerically, it is always possible to clip the color scale to regain some information on the dominant modes, even if the mode amplitude cannot be measured accurately (see Fig. 6-b). Practically, the DC component tends to saturate a CCD, swamping the nearby pixels and limiting the measurement of low-k modes. By adding the neutral density filter, Fig. 7 shows the DC component of the signal has disappeared when taking the spatial derivative of the image and all the modes are now visible. Clipping the color only helps in seeing the mode harmonics.
Fig. 6. The intensity on the Fourier (focal) plane, giving directly the spatial spectrum of the turbulence. a) Intensity plotted on the linear scale does not give much information on the turbulence spectrum. b) While the spectrum can be recovered on a saturated image, the DC component tends to wash out the low-k modes in practical implementations.
Fig. 7. The same phase fluctuations as Fig. 6 but with the vortex plate coupled to the neutral density filter. a) Even with the intensity on the linear scale we can see the different modes clearly. Due to the low DC level, even a saturated spectrum is fully usable.
4. Conclusions
In this paper, we have shown that a 2f lens pair can be used to perform a true optical derivative by combining a vortex plate on the Fourier plane with a neutral density filter with linear radial dependence. It is relatively straightforward to obtain the derivative of two-dimensional intensity or phase profiles, though an absolute calibration is required to measure the phase derivative directly, without an interferometer. However some uncertainties remain since we do not have access to the sign of the derivative, but only its absolute value. It is also possible to record directly the wave number of the turbulence by using the analog Fourier transform offered by the lens.
While this method is straightforward to implement numerically, some practical considerations still remain. Since we solely rely on intensity measurements, absolute calibration of intensity is necessary. However, the absolute calibration is not required to measure the turbulence wave number. Ultimately, practical implementation will require a uniform laser beam, as speckling will be amplified by the derivative operation. Ambient light can be another source of noise that should be taken under consideration. So the beam needs to be filtered properly to generate a high quality backlighter.
In conclusion, we were able to demonstrate analytically and numerically that a vortex plate coupled to a neutral density filter can deliver a true optical derivative when placed at the focal plane of a $2f$ lens pair. The system can be used to turn spatial variation in intensity into an intensity, which square root is the spatial derivative of the initial intensity variation. What is more surprising, the system also turns any spatial variation in phase into an intensity which square root is the spatial derivative of the initial phase variation, allowing to measure phase without an interferometer.
Funding
National Science Foundation (PHY-1943939) and the Horton Fellowship of the Laboratory for Laser Energetics.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data is available upon request to the first author.
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