1. Introduction

We propose an improved ray tracing method using a stochastic process with the phase tracking of a monochromatic laser beam. We focus on only light propagation with Fresnel diffraction effects in the vacuum, showing the duality between wave and ray representations. As an example of diffraction effects of a free propagating laser beam, we will discuss the case of a Gaussian beam after the general construction of “stochastic ray tracing.”

We have employed the ray tracing method [1–4] to design optical elements and systems, and it is based on ray optics (geometrical optics). An optical system includes diffraction and focusing features. Furthermore, a free propagating light beam has diffraction effects generally. Tricks are required to express them in the ray tracing method. For instance, we have described an optical ray by a line in each usual propagation segment and polygonal lines around the Rayleigh range in simple ray tracing of a Gaussian beam. Previous major works for ray tracing of a Gaussian beam were summarized in Ref. [5]. Several remarkable approaches exists for diffraction and focusing optics in ray tracing. The first was with the complex ray (the skew line ray) [6–10] to update a ray’s instantaneous position and direction. We can couple it with an optical system expressed by its ABCD matrix via $q(z)$, the complex beam parameter of a Gaussian beam. The second was with the invariance of the function structure of a Gaussian beam in ray tracing [11,12], similar to the above method. The third was Monte Carlo ray tracing, where random events realized diffraction effects [13,14]. An ensemble of rays illustrated a transverse beam pattern after its propagation. However, their introduction was somehow artificial from the viewpoint of theoretical physics. There should be a ray representation mathematically equivalent to the wave one, the ray tracing method with the highest calculation precision. Then, the users of the previous models can evaluate how accurate those calculations are. We now theoretically consider the diffraction and focusing effects in the ray tracing method from the Maxwell equations via the wave–ray duality.

As one of its solutions, we propose the stochastic ray tracing method. We transform from the Maxwell equations to the Schrödinger equation via the slowly varying envelope approximation and discover the usual ray features by the saddle-point method for Fresnel diffraction by Feynman’s path integral in Sect. 2. We will suppose only the wave equation with this physical approximation and obtain the outcomes by purely mathematical procedures. We will confirm that a ray in diffraction must fluctuate from its path in conventional ray tracing. The Schrödinger equation is a diffusion-type equation that relates to the Gaussian distribution with Wiener processes mathematically. Nelson’s stochastic mechanics employs this idea, providing a stochastic path of a particle. We consider this path to include the fluctuation of a ray in Fresnel diffraction, and assume that the wave nature is equivalent to an ensemble of stochastic rays. We explain Nelson’s stochastic mechanics briefly in Sect. 3. Then, we discuss the structure of stochastic ray tracing as an application of Nelson’s stochastic mechanics in Sect. 4.1. We also confirm the uncertainty relation between the position and divergence of a ray in Sect. 4.2. At this stage, no relationship exists between the propagation distance and time. Hence, we argue how to include this relation by phase tracking in Sect. 4.3. A Gaussian beam by stochastic ray tracing is formulated in Sect. 5, its numerical results are shown to discuss this model in Sect. 6. Realizing the Gaussian beam waist is a remarkable feature in this ray tracing calculation. Finally, we conclude this work in Sect. 7.

2. Issues for rays in diffraction

Let us consider the propagation of a laser beam in the vacuum, expressed by the Maxwell equations:

We separate the domain $[0,z]$ into $N$ segments. With $z^{(0)}=0\le z^{(1)}\le \cdots \le z^{(N)}=z$ and $\boldsymbol {x}^{(n)}=(\boldsymbol {x}_{\perp }^{(n)},z^{(n)})$ for $n=0,1,\ldots,N$, the repeating use of Eq. (6) allows us the following expression,

Suppose the path of the above $\omega :\boldsymbol {x}^{(0)}\rightarrow \boldsymbol {x}^{(1)}\rightarrow \cdots \rightarrow \boldsymbol {x}^{(\infty )}$ represents a stochastic process that happened with a probability $\mathcal {P}(\omega )$. Moreover, let $\mathcal {P}(\omega )$ be given by Eq. (5) a priori. We will denote $\boldsymbol {x}_{\perp }^{(i)}$ at $z_{i}$ for $\omega$ by $\hat {\boldsymbol {X}}(z_{i},\omega )$ in Sect. 4. We employ the hat sign for random variables. The set $\{(\hat {\boldsymbol {X}}(z,\omega ),z)\}_{z\in \mathbb {R}}$ illustrates a continuous and stochastic curve in $\mathbb {R}^{3}$, which we regard as the path of $\omega$ in optical ray tracing. We use $\hat {\boldsymbol {X}}(\omega )=\{\hat {\boldsymbol {X}}(z,\omega )\}_{z\in \mathbb {R}}$ to be the symbolic expression of the path for $\omega$.

3. Method: stochastic mechanics

In Eqs. (5, 9), we can regard that the triple $(\hbar /m,t,\psi )$ in quantum mechanics is replaced by $(k^{-1},z,a)$ in ray optics. Recall that the Schrödinger equation (5) is categorized into diffusion-type equations, and can be equivalent to a diffusion process (a stochastic process). “Nelson’s stochastic mechanics [18–20]” is one of the formalisms of quantum mechanics that employ a stochastic process representing each particle’s path. We aim to employ this for optical ray tracing. Here, we briefly confirm the result of stochastic mechanics for a later discussion. We consider the quantization of the classical system, such as $d\boldsymbol {x}=\boldsymbol {v}dt$ and $d\boldsymbol {v}/dt=-(Q/M)\nabla \phi$, for a particle of its mass $M$ and charge $Q$ in the external field by an electromagnetic scalar potential $\phi$. Let $\hat {\boldsymbol {x}}(\omega )$ be a sample path of a stochastic process labeled by the identifier $\omega$, a different $\omega$ generates a different path due to randomness. When we record positions, $\hat {\boldsymbol {x}}(t,\omega )$ for a given $\omega$ at each time $t$ in the recording period $\mathbb {T}$, the set $\hat {\boldsymbol {x}}(\omega )=\{\hat {\boldsymbol {x}}(t,\omega )\}_{t\in \mathbb {T}}$ becomes a continuous path in the space. The above is the general idea of a stochastic process [17]. Then, we must restrict $\hat {\boldsymbol {x}}(\omega )$ equivalent to the Schrödinger equation in stochastic mechanics. The following is the mathematical fact for a Schrödinger particle [18–20]. Suppose $\hat {\boldsymbol {x}}(t,\omega )\in \mathbb {R}^{n}$ the position of a particle for $\omega$ at $t$, stochastic kinematics was defined by

We define $\boldsymbol {v}_{\pm }$ in Eq. (10) by proposing its dynamics [18–20]

Our strategy with stochastic mechanics in ray tracing is below: (A) we compare Eq. (5) to Eqs. (11), (14) with Eqs. (12), (13) for $\boldsymbol {v}$, then, (B) we consider the stochastic kinematics (10) with $\boldsymbol {v}_{\pm }=\mathrm {Re}\{\boldsymbol {v}\}\mp \mathrm {Im}\{\boldsymbol {v}\}$ as stochastic ray tracing. For $\boldsymbol {v}$, we must prepare the solution $\psi$.

4. Stochastic ray tracing

4.1 Basic structure

Let us apply Eqs. (10–14) of stochastic mechanics to Eq. (5). We pick out a stochastic path from a laser beam by Eqs. (5), (9) and name it a “photon” for convenience in our discussion. In this article, we emphasize that the word “photons” does not mean the quantization of an electromagnetic field. A photon is an elementary particle in quantum electrodynamics [22–25]. We can distinguish each type of elementary particle by Wigner’s classification based on the representation of the Poincaré group [26,27]. This classification includes the idea of the relativistic invariance. In addition to Wigner’s classification, the quantum field of a photon must satisfy Eq. (3), treated as Heisenberg’s equation of motion in quantum field theory, holding the relativistic invariance [22–25]. On the other hand, Eq. (5) is not a relativistic form, a quantum field by Eq. (5) does not express any quantum field for elementary particles. Namely, we abandoned the relativistic invariance for the right as an elementary particle, by the slowly varying envelope approximation. The present model with Eqs. (5), (6) and (9) does not satisfy the rules of elementary particles and considers a “photon” a ray for a classical electromagnetic field in Fresnel diffraction.

A photon $\omega$ of a path $\hat {\boldsymbol {X}}(\omega )$ exists with a probability of $\mathcal {P}(\omega )$. The label $\omega$ is the identifier of a path in sampling. Let the set of all photons be $\varOmega$: $\omega \in \varOmega$. With $\int _{\omega \in \varOmega }d\mathcal {P}(\omega )=1$, a probability space is constructed for our discussion [28]. To compare Eq. (5) and Eq. (14), we can find $\varLambda =\sqrt {1/k}$ and $\varXi =0$ in $\mathbb {R}^{2}$. Then, a photon $\omega$ satisfying Eq. (5) follows the 2D stochastic kinematics

4.2 Probability density and uncertainty relation

On the probability $\mathcal {P}(\omega )$, we translate it to the 2D probability density at $z$ for Eq. (16):

Confirm $\langle \mathrm {Im}\{\boldsymbol {V}(\hat {\boldsymbol {X}}(z),z)\}\rangle =\boldsymbol {0}$ for Eq. (27) as $p(\boldsymbol {x}_{\perp },z)=0$ at $|\boldsymbol {x}_{\perp }|\rightarrow \infty$. By following the discussion in Ref. [31],

4.3 Phase tracking

The above discussion for stochastic ray tracing has not focused on the relation with time $t$; we find this by phase tracking. To consider this, let us rewrite Eq. (4) by

5. Stochastic ray model of a Gaussian beam

We apply stochastic ray tracing to a Gaussian beam. We employ the forward (+) stochastic kinematics by Eq. (16) and phase tracking by Eq. (34). Let the initial distribution be the following 2D Gaussian distribution,

Note that this result does not include the Gouy phase $\phi _{\mathrm {Gouy}}(z)$. The operator $\nabla _{\perp }(\ln \bullet )$ in the definition of $\boldsymbol {V}(\boldsymbol {x}_{\perp },z)$ eliminates it. However, the contribution of $\phi _{\mathrm {Gouy}}(z)$ appears in Eq. (34). Since

We can consider a plane wave in stochastic ray tracing as one of its applications. A plane wave is, of course, the easiest solution for Eq. (3). However, we need its indirect method since stochastic ray tracing is based on the Schrödinger equation (5). We use the Gaussian beam formula (36) with the limit $w_{0}\rightarrow \infty$ to derive the plane wave solution $a(\boldsymbol {x}_{\perp },z)=A/i=\mathrm {constant}$. Equally, when $\Delta \hat {\boldsymbol {X}}(z)\rightarrow \infty$ and $\Delta \mathrm {Im}\{\boldsymbol {V}(\hat {\boldsymbol {X}}(z),z)\}\rightarrow 0$, no beam divergence occurs as the plane wave. Furthermore, $z_{\mathrm {R}}=kw_{0}^{2}/2\rightarrow \infty$ allows us to consider no caustic changing as $dw(z)/dz\approx 0$ at any finite $z$. Thus, stochastic ray tracing of a plane wave is realized by Eq. (43) with the quasi-infinite Rayleigh length, $|z|\ll z_{\mathrm {R}}\rightarrow \infty$.

6. Numerical results and discussion

We performed two numerical calculations of Eq. (43) with Eq. (48). First, we calculated a free propagation of $10^{4}$ photons following the initial distribution of Eq. (44) at $z=0$, with its laser wave length $\lambda =1080\,\mathrm {nm}$ and initial spot size $w_{0}=4.0\,\mathrm {\mu m}$. These photons initially stay at the same phase at $z=0$. We suppose their propagation in free space until $t_{f}=2000\times (\lambda /2\pi c)$. It is expected that the ensemble of $\{(\hat {\boldsymbol {X}}(z,\omega ),z)\}$ at $t=t_{f}$ shows us a wavefront after the propagation. Figure 1 shows its visualization. Figure 1(a) shows stochastic paths in the 3D space. Each blue curve represents a stochastic path of a photon propagating from a black point at $t=0$ to another at $t=t_{f}$. The shape of the black points at $t=0$ and $t_{f}$ represent the wavefronts before and after propagation, respectively. Figure 1(b) illustrates its top view. An arbitrary white curve in Fig. 1(b) is a sample path of $\omega$ by Eq. (43), generated with the probability of $\mathcal {P}(\omega )$ relating to the probability density (44). We also depicted the black dashed curves indicating the spot size $w(z)$ by Eq. (37). The figures include the beam waist and divergence features derived from Eq. (43). The first term in the right-hand side (RHS) of Eq. (43) separates the regions where $z<z_{\mathrm {R}}$ and $z>z_{\mathrm {R}}$. The global beam divergence is occurred by $d_+\hat {\boldsymbol {X}}(z,\omega )\approx [\hat {\boldsymbol {X}}(z,\omega )/z]dz$, where $z\gg z_{\mathrm {R}}$ corresponding to the result of the saddle point method in Sect. 2, namely, the usual ray tracing. Equation (43) becomes $d_+\hat {\boldsymbol {X}}(z,\omega )\approx -[\hat {\boldsymbol {X}}(z,\omega )/z_{\mathrm {R}}]dz+\sqrt {1/k}d\hat {\boldsymbol {W}}_+(z,\omega )$ if $z\ll z_{\mathrm {R}}$. The beam waist is realized by the balance between the restricting effects by $-[\hat {\boldsymbol {X}}(z,\omega )/z_{\mathrm {R}}]dz$ and the diffusion effects by $\sqrt {1/k}d\hat {\boldsymbol {W}}_+(z,\omega )$. We regard that as the higher-order corrections for path fluctuation. Equation (43) agrees with the characteristics of a Gaussian beam (36) and the advantage of this method is the natural realization of the beam waist feature in the ray representation.

Figure 2(a) is the 2D histogram of photon counts at $z/\lambda =200$, with $50\times 50$ bins in the $(x/\lambda,y/\lambda )$-space, using the same calculation result of Fig. 1. The standard deviations were $\sigma _{x}=9.04$ and $\sigma _{y}=9.09$ in the $x/\lambda$ and $y/\lambda$ directions, respectively. We illustrated that by the white ellipse in the figure:

 

Fig. 1. Stochastic ray tracing of $10^{4}$ photons for a Gaussian beam. (a) 3D View and (b) the projection on the $x\text{--}z$ plane. The blue curves are the sample paths by stochastic ray tracing, the white one is a sample from the blue ones, and the black dashed curves in (b) are the spot size dependence $w(z)$ by Eq. (37).

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Fig. 2. The 2D distribution of photons at $z/\lambda =200$. (a) The 2D histogram by stochastic ray tracing. We employed the same calculation result of Fig. 1 and the number of bins is $50\times 50$. The white ellipse follows Eq. (50) with the standard deviations $\sigma _{x}=9.04$ and $\sigma _{y}=9.09$. (b) The analytic model by Eq. (51). The white circle represents the domain of its standard deviation: $\sigma (200\lambda )=8.79$.

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Second, Fig. 3 is the advanced result from Fig. 1 to include the focusing features of a Gaussian beam. We made the new initial values for this calculation such as $t_{i}^{\mathrm {new}}=-t_{f}$, $\hat {\boldsymbol {X}}^{\mathrm {new}}(z_{i}^{\mathrm {new}},\omega ^{\mathrm {new}})=-\hat {\boldsymbol {X}}(z_{f},\omega )$, and $z_{i}^{\mathrm {new}}=-z_{f}$ by using the numerical result $\{(t_{f},\hat {\boldsymbol {X}}(z_{f},\omega ),z_{f})\}$ w.r.t. $z_{f}=z(t_{f})$ of Fig. 1. The evolution in $t\in [-t_{f},t_{f}]$ was recalculated by Eq. (43) with Eq. (48) without any modification of the equations. This result shows us the focusing effects ($z<-z_{\mathrm {R}}$), the Rayleigh range ( $-z_{\mathrm {R}}\le z\le z_{\mathrm {R}}$), and the diffraction effects at ($z>z_{\mathrm {R}}$). This calculation covered the result of Fig. 1 for $z\ge 0$. Again, the balance between the restricting and diffusion effects by the RHS of Eq. (43) contributes to forming the focusing features.

 

Fig. 3. Focusing and defocusing features of $10^{4}$ photons of a Gaussian beam in stochastic ray tracing. The blue and the white curves are sample paths by stochastic ray tracing, and the dashed curves represent $w(z)$ by Eq. (37).

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7. Conclusions

We proposed “stochastic ray tracing” expressed by randomly fluctuating rays for an optical wave in Fresnel diffraction as Eq. (6). Our investigation addressed how to reveal the wave–ray duality in the optics theory. We transformed from the wave equation (3), to the 2D Schrödinger equation (5) by the slowly varying envelope approximation for a monochromatic laser beam. We found the set of equations for stochastic ray tracing, equivalent to the 2D Schrödinger equation via the analogy to Nelson’s stochastic mechanics [18–20]. The 2D Schrödinger equation (5) was transformed into Eq. (17), and coupled with stochastic kinematics by Eqs. (15), (16) illustrating the path of a “photon” in this model. The uncertainty relation (30) was derived directly from the present framework. We introduced the phase tracking for the relation between the propagation distance $z$ and time $t$. With the proposed stochastic ray tracing method, we considered a Gaussian beam and performed its numerical calculations. The balance between the first and second terms in the RHS of Eq. (43) realized the diffraction and focusing features around the Rayleigh range. The numerical calculation results depicted the beam waist and correlated with the well-known characteristics of a Gaussian beam function. We derived its applicable range, such as $kw_{0}>O(1)$, for a Gaussian beam in the phase tracking method, where $k$ and $w_{0}$ are a laser wave number and minimum spot size, respectively. We remark that the mathematical role of an optical wave amplitude or the wave equation was to give a priori probability $\mathcal {P}(\omega )$ and density distribution $p(\boldsymbol {x}_{\perp },z)$ of stochastic rays in expressing the ray feature precisely. To conclude, stochastic ray tracing is the mathematical interpretation of an electromagnetic field to the probability density of an ensemble of rays throughout this article.

We have four issues to extend the present stochastic ray tracing method. The first is to include optical elements for its realistic use. It is necessary to develop theoretical techniques for the reflection effects by a mirror and beam propagation in a non-vacuum medium, i.e., the theoretical model with refractive indices in stochastic ray tracing. The second is the extension to a more general transverse beam profile $a(\boldsymbol {x}_{\perp },0)$ in numerical simulations. We can perform it theoretically with $\mathcal {A}$ and $\mathcal {S}_{0}$ in Eq. (20) derived from Eq. (6). An easy scheme is necessary to obtain $\mathcal {A}$ and $\mathcal {S}_{0}$ as functions of $(\boldsymbol {x}_{\perp },z)$ for numerical simulations. Then the third is describing a laser pulse in stochastic ray tracing. That is the issue of expressing interference of waves in stochastic ray tracing since the superposition of plane waves gives a pulsed wave. The last is developing of theory beyond the slowly varying envelope approximation. In this article, we emphasized that “photons” did not mean the quantization of an electromagnetic field, which was derived from the reason why we could not use Eq. (3) directly. Such a situation broke the relativistic invariance of an electromagnetic field in quantum field theory [22–25]. Overcoming without approximation could allow us an opportunity to discuss a “photon” of an elementary particle in stochastic ray tracing.