Foucault-Barker Masks: Nonconventional Schlieren Technique

1. Introduction

For current applications in optical microscopy and optical testing, researchers have developed techniques for rendering visible transparent structures [1,2,3,4]. Among others, we note Foucault’s remarkable invention of the knife-edge.

This optical technique initiated experimental methods for testing optical elements. visualizing the departures of sphericity of mirrors and lenses [5].

According to Krehl and Engemann [6], independently of Foucault, Toepler was the first person who visualized shock waves, by employing a knife-edge [7].

Renitz also noted that there is an early version of the knife-edge [8]. Furthermore, Renitz has also recognized previous experimental results from Wiesel and Marat [9].

Zernike presented the phase contrast technique for microscopy as an improved version of the knife-edge technique [10]. Independently of Zernike, Lyot proposed a phase contrast technique, used for visualizing polishing inhomogeneities in a coronagraph [11].

Due to the contributions of Schardin [12] and Burton [13], the use of the knife-edge is now recognized as a member of the Schlieren techniques [14,15,16,17].

Here, for implementing nonconventional Schlieren techniques, we unveil the use of 1-D optical masks, coded either with the Barker sequences [18], or with the pseudorandom sequences [19].

For developing our current discussion, in Section 2, we revisit the basics of an effective transfer function, which was initially proposed by Menzel [20,21]. This concept was further developed by one of us [22,23,24,25]. This discussion considers, as input pattern, a transparent grating that has weak phase variations.

For emphasizing the usefulness of the transfer function, in Section 3, we present a statistical model, which describes the presence of a randomly located prism.

In Section 4, we discuss the usage of a coded source. We note that there is a link between the Schlieren techniques and the mathematical expression for the ambiguity functions, as employed in RADAR engineering [26,27,28,29,30]. We disclose the use of optical autocorrelations for implementing Schlieren techniques.

In Section 5, we illustrate the relevance of coding the source, by numerically evaluating the autocorrelation of pseudorandom sequences and of the Barker sequences. And in Section 6, we summarize our contribution.

2. Effective Transfer Function

In Figure 1, we depict the schematics of an optical processor, which is used for describing our proposal.

In Figure 1, the input pattern is placed in the axis denoted as x’. The input pattern is a sinusoidal phase grating, with period d. Its complex amplitude transmittance is

$$t\left(x^{\text{'}},y^{\text{'}}\right)=\exp \left[i2\pi \left({\displaystyle \frac{a}{\lambda }}\right)\sin \left(2\pi {\displaystyle \frac{x^{\text{'}}}{d}}\right)\right].$$

(1)

In Equation (1) the amplitude of the phase delays is denoted with the lower-case Latin Letter a. And the upper-case Greek letter λ denotes the wavelength. Next, we consider that the above grating has phase variations that are weak, in the sense that ${\left({\displaystyle \frac{a}{\lambda }}\right)}^2\ll \left({\displaystyle \frac{a}{\lambda }}\right).$

Under this assumption, the complex amplitude transmittance can be approximated by the initial two terms of its Taylor expansion. Hence, we consider that the input patterns have the following complex amplitude transmittance

$$u_0\left(x^{\text{'}},y^{\text{'}}\right)=1+i2\pi \left({\displaystyle \frac{a}{\lambda }}\right)\sin \left(2\pi {\displaystyle \frac{x^{\text{'}}}{d}}\right).$$

(2)

The Fourier spectrum of Equation (2) is

$$U_0\left(\mu ,\nu \right)=\delta \left(\mu \right)\delta \left(\nu \right)+\pi \left({\displaystyle \frac{a}{\lambda }}\right)\left[\delta \left(\mu -{\displaystyle \frac{1}{d}}\right)-\delta \left(\mu +{\displaystyle \frac{1}{d}}\right)\right]\delta \left(\nu \right).$$

(3)

In Equation (3), the Greek letters, μ and ν denote the spatial frequencies at Fourier domain. We note that the mathematical expression in Equation (3) represents the complex amplitude distribution impinging on the optical mask. If we denote as $\mathcal{Q}\left(\mu ,\nu \right)$ the amplitude transmittance of the optical mask, then after the mask the amplitude transmittance reads

$$U\left(\mu ,\nu \right)=Q\left(\mu ,\nu \right)U_0\left(\mu ,\nu \right).$$

(4)

By taking an inverse Fourier transformation of Equation (4), we obtain the complex amplitude distribution at the image plane. That is,

$$\begin{array}{c}u\left(x,y\right)=Q\left(0,0\right)+\\ \pi \left({\displaystyle \frac{a}{\lambda }}\right)\left[Q\left({\displaystyle \frac{1}{d}},0\right)\exp \left(i2\pi {\displaystyle \frac{x}{d}}\right)-Q\left(-{\displaystyle \frac{1}{d}},0\right)\exp \left(-i2\pi {\displaystyle \frac{x}{d}}\right)\right].\end{array}$$

(5)

Here, it is convenient to recognize that $\mathcal{Q}\left(\mu ,0\right)$ can be expressed in a unique manner as the sum of two functions. The function $E\left(\mu \right)=E\left(-\mu \right)$ is an even function. And the function $\mathrm{\Theta }\left(\mu \right)=-\mathrm{\Theta }\left(-\mu \right)$ is an odd function. For further details, please see Appendix A.

$$\mathcal{Q}\left(\mu ,0\right)=E\left(\mu \right)+\mathrm{\Theta }\left(\mu \right).$$

(6)

In Figure 2, we depict the even component, $E\left(\mu \right)$, and the odd component, $\mathrm{\Theta }\left(\mu \right)$, of the amplitude transmittance of the function $\mathcal{Q}\left(\mu ,0\right).$

If we substitute Equation (6) in Equation (5) we have that

$$\begin{array}{c}u\left(x,y\right)=Q\left(0,0\right)+\\ +\left(2\pi {\displaystyle \frac{a}{\lambda }}\right)\mathrm{\Theta }\left({\displaystyle \frac{1}{d}}\right)\cos \left(2\pi {\displaystyle \frac{x}{d}}\right)+i\left(2\pi {\displaystyle \frac{a}{\lambda }}\right)E\left({\displaystyle \frac{1}{d}}\right)\sin \left(i2\pi {\displaystyle \frac{x}{d}}\right).\end{array}$$

(7)

The definition of the normalized image irradiance distribution is

$$I\left(x,y\right)\stackrel{\Delta }{=}{\left|{\displaystyle \frac{u\left(x,y\right)}{Q\left(0,0\right)}}\right|}^2.$$

(8)

From Equations (7) and (8) we obtain the normalized image irradiance distribution

$$I\left(x,y\right)=1+\left({\displaystyle \frac{2\pi a}{\lambda }}\right)\left({\displaystyle \frac{2}{E\left(0\right)}}\mathrm{\Theta }\left({\displaystyle \frac{1}{d}}\right)\right)cos\left(2\pi {\displaystyle \frac{x}{d}}\right).$$

(9)

Now, it is relevant to make the following comparison between Equations (2) and (9). The complex amplitude distribution at the input plane, in Equation (2), is mapped as an image irradiance distribution, in Equation (9). The initial sinusoidal variation is transferred as a cosinusoidal variation, with modulation that is equal to

$$H\left({\displaystyle \frac{1}{d}}\right)\stackrel{\Delta }{=}{\displaystyle \frac{2}{E\left(0\right)}}\mathrm{\Theta }\left({\displaystyle \frac{1}{d}}\right).$$

(10)

In other words, there is a spatial frequency mapping, which can be associated with an effective transfer function, which is sampled at the spatial frequency 1/d. This remarkable simple result is conveniently depicted in Figure 3.

Furthermore, it is apparent from Equation (10) that the action of effective transfer function can be enhanced if the value at the origin of the even component, E(0), is less than unity. This observation is useful for proposing improvements to two previous Schlieren techniques, which are depicted in Figure 2. For Wolter’s phase edge, the coded mask has the following amplitude transmittance

$$\mathcal{Q}\left(\mu ,\nu \right)=\left\{\begin{array}{ccc}-1& if& -\mathrm{\Omega }<\mu <0\\ 1& if& 0<\mu <\mathrm{\Omega }\end{array}\right\}.$$

(11)

In Equation (11), the Greek letter Ω denotes the cut-off spatial frequency. In Figure 4a we plot Equation (11). Now, for the modulation contrast technique [31], the amplitude transmittance is

$$Q\left(\mu ,\nu \right)=\left\{\begin{array}{ccc}0& if& -\mathrm{\Omega }<\mu <-\epsilon \mathrm{\Omega }\\ \alpha & if& -\epsilon \mathrm{\Omega }<\mu <\epsilon \mathrm{\Omega }\\ 1& if& \epsilon \mathrm{\Omega }<\mu <\mathrm{\Omega }\end{array}\right\}.$$

(12)

In Equation (12), the Greek letters α and ε denote two real numbers with values lower than unity. The letter α represents an attenuation coefficient at the center of the mask. And the letter ε indicates the length covered by the central section, as compared with the length covered by the mask.

In Figure 4 , we plot the amplitude for an improved version, which incorporates a transmission equals minus unity; as in Wolter’s phase edge. And it also incorporates a central attenuation coefficient, as in the modulation contrast technique. Its amplitude distribution reads

$$Q\left(\mu ,\nu \right)=\left\{\begin{array}{ccc}-1& if& -\mathrm{\Omega }<\mu <-\epsilon \mathrm{\Omega }\\ \alpha & if& -\epsilon \mathrm{\Omega }<\mu <\epsilon \mathrm{\Omega }\\ 1& if& \epsilon \mathrm{\Omega }<\mu <\mathrm{\Omega }\end{array}\right\}.$$

(13)

Next, we note that some authors use, in a lax manner, a heuristic approach for describing phase gradients. This description considers that a phase gradient is a randomly located prism. To our knowledge, this heuristic approach has not been described in terms of scalar diffraction. For gaining physical insight, on the heuristic approach, we present next a diffraction-based description.

3. Randomly Located Prism

In Equation (14) we denote the amplitude transmittance of a prism that has a random location. That is,

$$t\left(x^{\text{'}},y^{\text{'}};X\right)=1+\left[\exp \right(i2\pi \left({\displaystyle \frac{a}{\lambda }}\sin \left(\theta \right)\right)\left)-1\right]\delta \left(x^{\text{'}}-X\right).$$

(14)

In Equation (14) the random variable X takes values that obey to a probability density function p(X). Its Fourier transform pair is the characteristic function

$${\mathrm{\Phi }}_X\left(\mu \right)=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}p\left(X\right)\exp \left(-i2\pi \mu X\right)dX.$$

(15)

Under the assumption of weak phase delays, we can approximate Equation (14) as follows

$$u_0\left(x^{\text{'}},y^{\text{'}};X\right)=1+i2\pi \left({\displaystyle \frac{a}{\lambda }}\sin \left(\theta \right)\right)\delta \left(x^{\text{'}}-X\right).$$

(16)

Its Fourier spectrum reads

$$U_0\left(\mu ,\nu ;X\right)=\delta \left(\mu \right)\delta \left(\nu \right)+i\left(2\pi {\displaystyle \frac{a}{\lambda }}\sin \left(\theta \right)\right)\exp \left(-i2\pi \mu X\right)\delta \left(\nu \right).$$

(17)

After passing through the coding mask, Q(μ, ν), the Fourier spectrum reads

$$U\left(\mu ,\nu ;X\right)=\varrho \left(\mu ,\nu \right)U_0\left(\mu ,\nu ;X\right).$$

(18)

By substituting Equation (17) in Equation (18) we obtain the explicit expression

$$\begin{array}{c}U\left(\mu ,\nu ;X\right)=Q\left(0,0\right)\delta \left(\mu \right)\delta \left(\nu \right)+\\ +i\left(2\pi {\displaystyle \frac{a}{\lambda }}\sin \left(\theta \right)\right)Q\left(\mu ,0\right)\exp \left(-i2\pi \mu X\right)\delta \left(\nu \right).\end{array}$$

(19)

Next, as discussed in Appendix B, we take the ensemble average of Equation (19). The average Fourier spectrum is

$$\begin{array}{c}\langle U\left(\mu ,\nu \right)\rangle =Q\left(0,0\right)\delta \left(\mu \right)\delta (\nu )+\\ +i\left(2\pi {\displaystyle \frac{a}{\lambda }}\sin \left(\theta \right)\right)Q\left(\mu ,0\right){\mathrm{\Phi }}_X\left(\mu \right)\delta \left(\nu \right).\end{array}$$

(20)

Then, we recognize the convenience of defining the transfer function

$$H\left(\mu \right)\stackrel{\Delta }{=}\mathcal{Q}\left(\mu ,0\right){\mathrm{\Phi }}_X\left(\mu \right).$$

(21)

The characteristic function is assumed to be an even function. Then, the even component of $H\left(\mu \right)$ has the following expression

$$E_H\left(\mu \right)=E\left(\mu \right){\mathrm{\Phi }}_X\left(\mu \right).$$

(22)

And the odd even component of $H\left(\mu \right)$ has the following form

$${\mathrm{\Theta }}_H\left(\mu \right)=\mathrm{\Theta }\left(\mu \right){\mathrm{\Phi }}_X\left(\mu \right).$$

(23)

Consequently, Equation (20) can be written as

$$\begin{array}{c}\langle U\left(\mu ,\nu \right)\rangle =Q\left(0,0\right)\delta \left(\mu \right)\delta \left(\nu \right)+\\ +i\left(2\pi {\displaystyle \frac{a}{\lambda }}\sin \left(\theta \right)\right)\left[E_H\left(\mu \right)+{\mathrm{\Theta }}_H\left(\mu \right)\right]\delta \left(\nu \right).\end{array}$$

(24)

As discussed in Appendix B, the average complex amplitude distribution is

$$\begin{array}{c}\langle u\left(x,y\right)\rangle =Q\left(0,0\right)-\left(2\pi {\displaystyle \frac{a}{\lambda }}\sin \left(\theta \right)\right){\vartheta }_H\left(x\right)+\\ +i\left(2\pi {\displaystyle \frac{a}{\lambda }}\sin \left(\theta \right)\right)e_H\left(x\right).\end{array}$$

(25)

And as noted in Appendix B, the average irradiance distribution is

$$\langle I\left(x,y\right)\rangle =1-\left(4\pi {\displaystyle \frac{a}{\lambda }}\sin \left(\theta \right)\right){\vartheta }_H\left(x\right).$$

(26)

From Equation (26) and from Figure 5, we observe two relevant features. The presence of a randomly located prism is imaged as a uniform average irradiance distribution, plus a term that contains the phase modulation caused by the prism.

In other words, the phase gradient is rendered visible, as an irradiance variation, due to the presence of the function ${\vartheta }_H\left(x\right).$ Hence, like Equation (10), it is convenient define the following effective transfer function

$$H\left({\displaystyle \frac{1}{d}}\right)\stackrel{\Delta }{=}{\mathrm{\Theta }}_H\left({\displaystyle \frac{1}{d}}\right).$$

(27)

From Equations (23) and (27), we claim that the effective transfer function is the product of a characteristic function times the odd component of coding mask. And consequently, as depicted in Figure 5, the current effective transfer function has now an apodizing effect. To our knowledge, this is a novel formulation.

4. Coded Source: An Imaging Process

We note the existence of heuristic descriptions for considers a transparent structure as a set of randomly located prisms. This heuristic description ignores any discussions on scalar diffraction. For rectifying this limitation, next we develop an imaging process, which considers a coded source. As part of our rationale, we consider first a simple case. As depicted in Figure 6, we consider the source as a slit. At its conjugated image, (μ, ν), we place a second slit that acts as the spatial filter.

In the absences of phase gradients, the two slits coincide. And the light throughput achieves its maximum value. On the other hand, if a phase gradient is present then the image of the source is laterally shifted (say by the amount ζ / λ). This mismatch generates a drop light throughput. That is,

$$T={\left|\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}rect\left({\displaystyle \frac{\mu }{\mathrm{\Omega }}}\right)rect\left({\displaystyle \frac{\mu -\frac{\zeta }{\lambda }}{\mathrm{\Omega }}}\right)d\mu \right|}^2.$$

(28)

In Equation (28), the Greek letter omega denotes the width of the slit used to cover the source. If one evaluates Equation (28), then one obtains

$$T=\left[1-{\displaystyle \frac{\left|\zeta \right|}{2\mathrm{\Omega }\lambda }}\right]rect\left({\displaystyle \frac{\zeta }{4\mathrm{\Omega }\lambda }}\right).$$

(29)

It is apparent from Equation (29) that the light throughput varies slowly with the variable ζ, except for a very narrow slit. However, the light throughput is reduced if one uses narrows slits. Hence, we search next for other options. To conclude, we consider that the source is coded with multiple slits. Then, at the source plane the amplitude transmittance is

$$U_0\left({\mu }^{\text{'}},V^{\text{'}}\right)=Q\left({\mu }^{\text{'}}\right).$$

(30)

In Equation (30) the Greek letters μ’ and ν’ denote spatial frequency coordinates at the source plane. At the input plane, the coordinates we employ the coordinates x’ and y’. At this plane, the amplitude distribution is the Fourier transform of Equation (30). That is,

$$u_0\left(x^{\text{'}},y^{\text{'}}\right)=q\left(x^{\text{'}}\right)\delta \left(y^{\text{'}}\right).$$

(31)

In Equation (31) the function $q\left(x^{\text{'}}\right)$ is the 1-D Fourier spectrum of the coded source. Next, we consider that a prism alters the amplitude distribution in Equation (28). Then, the new amplitude distribution reads

$$u\left(x^{\text{'}},y^{\text{'}}\right)=\exp \left(-i2\pi {\displaystyle \frac{\zeta }{\lambda }}{x}^{\text{'}}\right)u_0\left(x^{\text{'}},y^{\text{'}}\right).$$

(32)

In Equation (32) we consider the presence of a weak phase gradient. Its deflection angle is denoted with the Greek letter ζ. The amplitude distribution, at the pupil aperture, is the inverse Fourier transform of Equation (32). That is,

$$U\left(\mu ,\nu \right)=Q\left(\mu -{\displaystyle \frac{\zeta }{\lambda }}\right).$$

(33)

Next, we consider that the amplitude distribution, in Equation (33), impinges on a second coded mask. For this application, the second mask acts as the spatial filter. Its amplitude transmittance is Q*(μ). Then, just after the mask, the amplitude transmittance is

$$\mathrm{\Psi }\left(\mu ,\nu \right)=Q^{*}\left(\mu \right)U\left(\mu ,\nu \right).$$

(34)

It is straightforward to substitute Equation (33) in Equation (34), for obtaining

$$\mathrm{\Psi }\left(\mu ,\nu \right)=Q^{*}\left(\mu \right)\mathcal{Q}\left(\mu -{\displaystyle \frac{\zeta }{\lambda }}\right).$$

(35)

As expected, at the pupil aperture, the amplitude distribution is a product between a lateral shifted version of the coded source, times the complex conjugate transmittance of the coded mask. And consequently, at the image plane the amplitude distribution is

$$\psi \left(x,y;\zeta \right)=\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}{Q}^{*}\left(\mu \right)Q\left(\mu -{\displaystyle \frac{\zeta }{\lambda }}\right)\exp \left(i2\pi x\mu \right)d\mu .$$

(36)

The mathematical expression in Equation (36) is like the definition of the ambiguity function, as is used in RADAR engineering. This is a previously unknown feature of the Schlieren masks. However, any further discussion on this topic is beyond our current scope. We observe that Equation (36), at the origin (x =0, y = 0), is the light throughput T in Equation (28). That is,

$${\left|\psi \left(0,0;\zeta \right)\right|}^2={\left|\underset{-\infty }{\overset{\infty }{{\displaystyle \int }}}{\mathcal{Q}}^{*}\left(\mu \right)\mathcal{Q}\left(\mu -{\displaystyle \frac{\zeta }{\lambda }}\right)d\mu \right|}^2.$$

(37)

From Equation (37) we state the following working hypothesis. By coding the source as well as the spatial filter with the same sequence, its autocorrelation usefully detects phase gradients. This working hypothesis leads us to the following proposal.

5. Optical Visualization and Automatic Detection

Since a pseudorandom sequence or a Barker sequence have sharp autocorrelations, then they can be useful for automatically detecting phase gradients. For elaborating on this working hypothesis, we note that either the pseudorandom sequences, or the Barker sequences have a finite number of elements, say L. This integer number L is denoted as the sequence length. In electronics, the sequences are commonly employed as cyclic sequences. However, in optics, due to the finite size of the mask the coding sequences are noncyclic. Next, we elaborate on this difference.

The pseudorandom sequences exhibit a high peak over an otherwise uniform value, as depicted in Figure 7. In this pictorial, we show two examples of the pseudorandom sequences. In these sequences the amplitude transmittance has values that are equal to unity, or equal to zero. In Figure 7a, the length is L = 7. And in (b) the length is L = 15. These examples were selected for making comparisons with the results of other authors [32].

In Figure 8, we show the cyclic autocorrelations associated with the sequences in Figure 7. The autocorrelations exhibit peaks that overshoot a uniform value.

In Figure 9, we show that the Barker sequences of length L = 7 and L = 13.Now, the amplitude transmittance has values equal to unity, or equal to minus unity.

In Figure 10, we plot the cyclic autocorrelation associated with the sequences in Figure 9. It is apparent from Figure 10 that the autocorrelations also have peaks that overrun the values of the adjacent sidelobes. The peaks baselines narrow down from the length L = 7 to the length L = 13.

As previously indicated, in optics one is dealing with noncyclic autocorrelations. This is due to the finite size of the masks. The autocorrelations do not repeat themselves. To visualize the impact of this feature, we present the noncyclic autocorrelations of the pseudo random sequences, as well as the noncyclic autocorrelations of the Barker sequences.

From Figure 11 and from Figure 12, we note that the noncyclic autocorrelations do not have replicated peaks. Furthermore, associated with the pseudorandom sequences, the noncyclic autocorrelations exhibit sidelobes variable values.

From Figure 12, we observe that the Barker sequences generate noncyclic autocorrelations, which exhibit a distinctives central peak, with replicated sidelobes. Hence, Barker codes can be a desirable choice implementing an automatic sensor, based on optical autocorrelations. In the absence of phase gradients, the noncyclic autocorrelations have a central distinctives peak. On the other hand, if there are phase gradients, then the noncyclic autocorrelations have reduced values.

For the implementation of an automatic sensor, one has two choices. In Figure 13a, we show an unfolded version of the sensor. optical setup is shown in Figure 13a. And in Figure 13b, we depict folded optical setup.

Hence, Barker codes can be a desirable choice for coding the source and the spatial filter. And in this manner one can have a correlation method for automatically detecting the presence of phase gradients. For this purpose, the optical setup is shown in Figure 13. An unfolded version of the optical setup is shown in Figure 13a. And in Figure 13b, we use a spherical mirror for depicting a folding version of the optical setup.

6. Conclusions

We have described a theoretical framework for designing nonconventional Schlieren techniques. These techniques employ optical masks for rendering visible phase gradients, like the Foucault knife-edge.

We have revisited the use of an effective transfer function. This discussion clarifies the relevance of employing arguments of symmetry, when designing Schlieren techniques.

We have disclosed a statistical model that describes phase gradients as randomly located prisms.

We have unveiled a link between the generation of optical autocorrelations and the use of Schlieren techniques. We employ two masks that at geometrical optics conjugates planes. The first mask is for coding the source. And a second mask is for coding the spatial filter. In this manner, we obtain a Schlieren technique with high light throughput.

We have indicated that the use of two masks is linked to the implementation of an optical correlator. This link is useful for implementing an automatic method that detects phase gradients.

The optical correlator exploits the use of Barker sequences for generating highly peaked, noncyclic irradiance distributions.