1. Introduction

In the last four centuries, scientists have presented fundamentally different theories for light. Isaac Newton (1643-1727) described light behaviors by corpuscular theory [1]. Christiaan Huygens (1629-1695) suggested the wave theory for the treatment of light [2]. Thomas Young (1773-1829) reported the results of experiments that attracted many researchers to the wave optics [3] . The works by Augustin-Jean Fresnel (1787-1827), Gustav Kirchhoff (1824-1887), James Clerk Maxwell (1834-1879), Albert Abraham Michelson (1852-1931), and others expanded the wave optics remarkably [4]. However, in 1900, Max Planck (1858-1947) in his study of the black body radiation used quantized radiation energy [5]. In 1905, Albert Einstein (1895-1955) applied the quantized light energy to describe the photoelectric effect [6]. After 1926, the quantized light energy called photon [7].

Since the beginning of 20th century, the behavior of the photon has somehow confused scientists. In a large number of experiments, it behaves like a wave, in many experiments it appears as a particle, and there are experiments that it behaves like neither of them [8–10]. Here, we briefly review some of the well-known confusing behaviors. For example, by illuminating an aperture with a parallel beam of light we get an intensity pattern that is called diffraction pattern in the wave optics. By reducing the light intensity, the structure of the pattern does not change down to very low intensities. However, for extremely low intensities, a practically random distribution of bright spots appears which may be attributed to the particle behavior. Nonetheless, using the coordinates of a bright spot and the light source, one cannot specify the exact striking point of the particle on the aperture, which accounts for a non-classical physics behavior. Consider that the intensity of the illuminating light is decreased so that only a single photon at the time strikes the aperture and a single spot is recoded on the observation screen, accordingly. Then, for a very large number of such photons the recorded spots will form the normal diffraction pattern. Thus, distribution of the bright spots is not random anymore and they follow a law that cannot be described by classical physics.

Photon is a non-classical physical object and its behavior can be described by applying the “momentum-location” and “time-energy” uncertainty principles. These principles require associating photon probability amplitude at each point of a volume. Superposition of the probability amplitudes at the observation point determines the probability that a photon strikes the point. For the incident light containing a very large number of photons, the normalized density of the photons on the observation plane, i.e. the number of photons per area unit at a point, is practically equivalent to the intensity distribution of the diffraction pattern in the wave theory. In this paper, using “photon approach”, we calculate the probability distribution of a photon that passes through a single slit, a double slit, and a transmission step. The formulation, discussion, and the results are presented in Section 2. In addition, in Section 3, we use the approach to determine the probability distribution of a photon on an observation plane for a bi-prism and a Michelson’s interferometer. Besides, using the uncertainty principles, in Section 4, we describe the coherence behavior of light, and, finally, in Section 5, we derive the laws of the ray optics and the equation of image formation by photon approach. Section 6 concludes the paper and remarks the main points.

2. Photon approach to diffraction

2.1 Diffraction from a single slit

A small light source located at far distance from the experimental setup is regarded as a point source. The emerging photons strike evenly the surface of a large sphere with the source at its center. The exact incident point of each photon on the sphere is not predictable, however, the probability to strike each point is equal. In fact, there is an indeterminable uncertainty in specifying the incident point of a photon. In classical physics, by specifying the position and the velocity of a particle at a point on its trajectory, one can determine, in principle, the exact incident point of the particle. If we install a narrow slit tangential to the sphere, the photons strike the slit surface evenly and perpendicularly. However, since the slit confines the passage of the photons, the uncertainty principle requires photons to leave the slit’s width in different directions, which leads to the diffraction of light. The component of the photon momentum along the slit’s width varies between zero and the quantity that is determined by the uncertainty principle [10]:

In Fig. 1(a) a parallel beam of light illuminates a rather long slit of width $2b$ to realize diffraction from a 1D slit. The impact point of a photon on the slit is indeterminable. Therefore, it is probable for a photon that emerges from any point of the slit’s width to hit a point $P(x_0)$ on the observation screen (Sc.). In order to find the probability that a photon hits the point $P(x_0)$ we associate the probability amplitude $A\:e^{ikr}$ to a photon at each point of the slit, where $r$ stands for the distance between the point and the observation point. The probability amplitude has no analogy in classical physics. However, in the wave approach to diffraction, it resembles the wave amplitude. The probability amplitude that a photon passes through the slit and hits the point $P(x_0)$ is the superposition of the probability amplitudes of a photon along the slit width at the observation point. The resultant amplitude is expressed as follows [11]:

In Eqs. (5), $l^\prime$ is the distance between the slit and the observation plane and $\lambda =2\pi /k$. For the observation point $P{\:^\prime }(x_0)$ in the ray optics shadow of the slit, the probability amplitude is [11]:

The minus sign in the first term indicates that the corresponding distance between the ray optics border and the observation point lies in the shadow of the slit. Multiplying the resultant probability amplitude by its complex conjugate results in the probability that a photon hits the observation point:

If we illuminate the slit by a parallel beam of light containing a very large number of similar photons, the diffraction pattern from a single slit will be formed that is formulated with similar expressions. Equations (7) and (8), indeed, represent the density of photons on the corresponding diffraction pattern. Thus, the normalized probability distribution of a photon, the normalized density of photons, and the normalized intensity distribution on the diffraction pattern are equivalent. The normalized intensity at a point on the observation screen is obtained as the ratio of the intensity at that point to the intensity at a point on the illuminating parallel beam. Figure 1(b) shows the normalized density of photons for the light diffracting from a slit along the z-axis, which is the plot of Eq. (7) versus Fresnel variable $V$, for $\lambda =600\:\mathrm {nm}$, $b=0.2\:\mathrm {mm}$, and $A^2=1$. As can be observed in Fig. 1(b), the density of photons approaches zero at large distances from the slit, it oscillates rather strongly at short distances, and approaches one near the slit. Figure 1(c) shows the normalized diffracted density of photons, i.e. Equations (7) and (8), on the plane parallel to the slit at the distance which corresponds to $V\approx 1.21$. According to Fig. 1(b) the normalized diffracted density of photons at this $V$ is maximum.

 

Fig. 1. (a) The scheme of the setup used to formulate the diffraction of photon from a slit, and (b) the associated normalized density of photons along the axis perpendicular to the slit versus Fresnel variable. (c) The normalized density of photons on the plane located at the distance corresponding to Fresnel variable $V\approx 1.21$ from the slit, where the normalized density of photon is maximum.

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2.2 Diffraction from a double slit

In Fig. 2(a), a parallel beam of light illuminates perpendicularly a double slit of slit width $2b$ and slits’ distance $2d$. A photon with equal probability hits each point of the double slit. The uncertainty principle requires photons to emerge from the slits in different directions in x-z plane. Therefore, for each point in Sc., such as $P_1$, $P_2$, and $P_3$, the probability to receive a photon from each point of the double slit may be calculated by the same approach of the single slit case. The probability amplitudes are obtained as [11]:

Multiplying the above probability amplitudes by their complex conjugates results in the probabilities that a photon strikes the points $P_1$, $P_2$, and $P_3$:

On the y-z plane, $c_1=c_1^\prime$, $c_2=c_2^\prime$, $s_1=s_1^\prime$, and $s_2=s_2^\prime$, which reduces Eq. (13) to:

Figure 2(b) shows the normalized density of photons (Eq. (16)) as a function of Fresnel variable for $b =0.1\:\mathrm {mm}$, $d=0.8\:\mathrm {mm}$, $\lambda =600\:\mathrm {nm}$, and $A^2=1$. As the plot shows the density of photons fluctuates significantly at the distances corresponding to Fresnel variable in the interval 0 to 5. In Figs. 2(c) and 2(d), the normalized densities of photons on the observation planes parallel to double slit at distances $l^\prime =50\:\mathrm {mm}$ and $l^\prime =719\:\mathrm {mm}$ are plotted. In Fig. 2(d), the photon density near the z-axis is quasi-periodic, which we refer to as interference fringes in wave optics. We emphasize that the well-known question “How a photon passes through both slits” is a question in classical physics. In quantum mechanics, the superposition of the probability amplitudes provides the answer.

 

Fig. 2. (a) The sketch used to formulate the Fresnel diffraction from double slit. (b) The plot of the density of photons versus Fresnel variable, along the axis perpendicular to double slit. (c) The plot of the density of photons on a plane parallel to the double slit at distance $50\:\mathrm {mm}$. $x_0$ shows the distance form $z$ axis on the screen. (d) The plot of the density of photons on the plane parallel to the double slit at the distance corresponding to the Fresnel variable $V\approx 1.51$ at $x_0=0$.

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2.3 Diffraction from a transmission phase step

A phase step is an object that imposes step discontinuity in the phase or in the phase gradient of a quasi-monochromatic beam of light that emerges from a point source. For example, when a parallel beam of light reflects from a physical step or transmits through a boundary region of two transparent media with different refractive indices, it experiences discontinuous change in phase. However, when it passes through a bi-prism or a plate with wedge part, it experiences discontinuous change in phase gradient. In recent decades, the subject has developed considerably and has found numerous interesting metrological and technological applications [12–20]. In Fig. 3(a), a transmission phase step, located at far distance from a 1D light source with the step edge parallel to the source, is shown. An incident photon on the phase step strikes each point of the back surface perpendicularly and with equal probability. The step edge divides the phase step into two parts. This imposes different phases on the probability amplitudes of a photon that diffracts from the phase step to the observation point. Thus, the resultant probability amplitudes at the points $P(x_0)$ and $P^\prime (x_0)$ on the observation screen Sc., located at distance $l^\prime$ from the step are [11]:

In Eqs. (19), $\varphi$, $n$ and $h$ stand for the phase difference between the two sides of the step edge, the refractive index of the step, and the step height, respectively. The ray optics borders of the transmitting beams are y-z plane and the two planes parallel to y-z plane at $\pm \infty$. We determine the largest effective optical path difference (OPD) by considering the uncertainty principle concerning time and energy [10]:

Substituting from the relations, $\Delta E = h\:\Delta \nu$, $\nu =c/\lambda$, and $c\:\Delta T=L$, in Eq. (20) we get:

In Eq. (21), $L$ is the length in the propagation direction of light on which the location of a photon is uncertain. In other words, at each point of the length $L$ at the same time there is a probability to detect a photon that an atom of the source emits. In wave optics, Eq. (21) is derived by assuming a correlation time or length between the fluctuations of two interfering beams. For OPD$\:<L$, by considering the phase difference, we add the probability amplitudes of a photon that diffracts from both sides of the step edge to the observation points (Eqs. (17) and (18)). In fact, the phase difference is effective on the probability amplitude of a photon when its equivalent time is less than the uncertainty time. For OPD$\:>L$, we add the probabilities together. The probability is obtained by multiplying the amplitude by its complex conjugate. In the case of Eqs. (17) or (18) this leads to:

For the incident beam containing a large number of different photons, $A^2$ can be replaced by $I(\lambda )$ in Eq. (22), where $I(\lambda )$ is the density of photons that corresponds to $\lambda$. Integrating the latter equation over $\lambda$ the sum of the densities of the photons or the light intensity at distance $x_0$ from the step edge is obtained. One can evaluate $I(\lambda )$ by fitting a suitable function on the experimental intensity distribution. Multiplying Eqs. (17) and (18) by their complex conjugates, the following probabilities for a photon to hit point $P(x_0)$ or $P^\prime (x_0)$ when OPD$\:<L$ are obtained:

For the incident beam containing a large number of similar photons, Eqs. (23) and (24) represent the normalized densities of the photons on the corresponding parts of the observation plane. For the observation points symmetrical to the step edge, the difference between Eqs. (24) and (23), for normalized densities of the photons leads to the following:

By measuring $\delta I$ for different $(c-s)$, $\varphi$ can be evaluated more accurately. The variation of the photon density on the observation plane appears as optical fringes. In Fig. 3(b) to 3(d), we plot the normalized densities, Eqs. (23) and (24) for $A^2=1$, versus the distance from the step edge, on the plane parallel to the phase step at distance $l^\prime =500\:\mathrm {mm}$, for $\lambda =600\:\mathrm {nm}$ and $\varphi =\pi /2$, $\pi$, $3\pi /2$.

 

Fig. 3. (a) The sketch used to formulate Fresnel diffraction from a transmission phase step. (b)-(d) The plots of the normalized densities of photons versus the distance from the step edge, on the plane located at distance $l^\prime =500\:\mathrm {mm}$ from the step, for three different phases. $x_0$ shows the distance form $z$ axis on the screen.

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3. Photon approach to interference

3.1 Bi-prism

In Fig. 4, a parallel beam of light illuminates the base of a bi-prism perpendicularly. In such an illumination, there is an equal probability for an incident photon to hit each point of the base. Two beams emerge from the bi-prism and make an angle of $\pi -2\gamma$ with each other, $\gamma$ being the angle between the incident and the refracted beams. For a bi-prism of rather large dimension along y-axis, the diffraction of light can be regarded as the diffraction from two long slits of widths $OW_1$ and $OW_2$ (Fig. 4). The probability amplitude of a photon at the observation point can be expressed in terms of its distances from the ray optics borders of the beams and from the slits. For the observation point $P(x_0)$ on the intersection plane of the beams, the plane with trace Sc. in Fig. 4, the required distances are: $PH_1$, $PH_2$, $PH^\prime _1$, $PH^\prime _2$, $PN_1$, and $PN_2$. Expressing the latter distances in terms of the given parameters and using the method we applied to the diffraction from a slit, the probability amplitude of a photon that diffracts from the bi-prism to the point $P(x_0)$ is obtained as follows:

For the calculation of $\Psi _{P(x_0)}$ the following equations are required:

For observation points in the ray optics bright area of both beams and away from the borders of the beams, substituting $c$, $c^\prime$, $s$, and $s^\prime$ by $1/2$ and the following equation can be obtained:

Equation (30) describes the interference of two plane waves in the wave optics. For a large number of similar photons that strike the base of the bi-prism, we just multiply Eq. (29) by the number. In fact, we add together the probability distributions of the photons; therefore, it can be referred as the density of the photons. The phase difference $\varphi$ varies as $x_0$ changes and makes the density, Eq. (29) or Eq. (30), to vary periodically. For a light containing photons of different energies, as $x_0$ increases, the maxima and minima of the corresponding densities gradually separate from each other and reduce the visibility of the periodic distribution. For OPD larger than the coherence length, $2x_0 \sin \gamma >L$, the visibility becomes zero and the probability distribution becomes:

Equation (31) represents the sum of the probabilities or density of photons on the common bright area of the beams emerging from the bi-prism. We remark that the photon approach similarly can describe the Loyd’s mirror arrangement.

 

Fig. 4. The sketch used to describe Fresnel diffraction and interference by a bi-prism.

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3.2 Michelson’s Interferometer

In Fig. 5, a parallel beam of light illuminates a Michelson’s interferometer with rather large dimensions along y-axis. This permits us to consider the light diffraction from the two 1D mirrors of the interferometer. We assume that the reflected and transmitted beams from the beam splitter, $r$ and $t$, uniformly illuminate the mirrors. The beam splitter splits the beams reflecting from the mirrors into two pairs of parallel beams, specified by $tr$, $rt$ and $tt$, $rr$, respectively, in Fig. 5. The first pair illuminates the observation screen and the second pair returns back to the source. The ray optics widths of the beams are practically equal to the widths of the mirrors. For parallel illumination, each photon strikes each point of the mirrors with equal probabilities. It is probable that a photon to strike any point of the observation plane from each point of the mirrors. Applying the method we used for a single slit, we express the sum of the probability amplitudes that a photon strikes point $P(z_0)$ on the observation plane by the following equation:

The upper limits of the corresponding Fresnel integrals for the mirrors of width $2b$ are the followings:

In Eq. (33), $l^\prime$ is the distance between the mirror $\mathrm {M_1}$ and the observation plane, and $z_0$ is the distance form $x$ axis on the screen. In order to apply Eq. (33) to both mirrors, we should ignore the small distance between the mirror $\mathrm {M_1}$ and the image of the mirror $\mathrm {M_2}$, $\mathrm {M^\prime _2}$. In Eq. (32), $\varphi = 4\pi h/\lambda$ stands for the phase difference between the beams. Adding the physical phase difference $\pi$, due to twice reflection of the beam $rr$ from the beam splitter, to $\varphi$ in Eq. (32), we can use the equation for the beams that propagate toward the source. To get the corresponding probability, Eq. (32) is multiplied by its complex conjugate:

For $\varphi = 2m\pi$, $m$ being an integer, all the photons illuminating the interferometer strike the observation screen, while for $\varphi =(2m+1)\pi$ all the photons return toward the source. For other values of $\varphi$, the sum of the probabilities for a photon to appear in both directions is equal to one. When a very large number of photons illuminates the interferometer, by the change of $h$, the density of photons on the observation plane, varies periodically. For photons of different energies, the visibility of the variation reduces as $h$ increases. For $2h \ge L$, the visibility reduces to zero and we can evaluate the coherence length of the incident light.

 

Fig. 5. The sketch used to describe diffraction and interference in Michelson’s interferometer in photon approach.

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The materials presented so far show that the probability approach to the density of photons, based on the uncertainty principles, adequately describes the diffraction and interference of light.

4. Photon approach to optical coherence

In wave optics, we assume that light is made of frequency-modulated wave trains. The wave trains have correlated fluctuations in limited intervals along the propagation and transverse directions, which are referred as the longitudinal coherence and the transversal coherence, respectively [21]. In photon approach, the transversal coherence depends on the size of the source and its distance from the experimental setup. At very large distance from a quasi-monochromatic point source, light behaves as a parallel beam. The photons of the beam diffract from an aperture with similar probability distributions and lead to a diffraction pattern. The diffraction pattern posses a rather high fringe visibility, which is attributed to the high transversal coherence. If a source of finite size is used, there will be several point sources, which will lead to diffraction patterns with slightly shifted locations of the fringes. Appreciable shift reduces the visibility of the fringes significantly and implies low transversal coherence. Thus, the visibility of the fringes is a measure of transversal coherence for quasi-monochromatic light and, consequently, can be used to determine the size and the distance of a light source [22].

In Section 2.3, we defined the temporal or longitudinal coherence by considering the uncertainties in the time and in the energy of the emitting photons, respectively. The coherence length, $L$, covers the probable locations of a photon along the propagation direction. In other words, each point of $L$ is the probable location of the same photon at the same time. The periodic distribution of photons i.e. the interference, occurs when the OPD of the superimposed probability amplitudes of a photon is less than $L$. In Sections 2 and 3, we discussed the different ways of measuring the temporal coherence of photon or light. For a point source, the probability amplitude of a photon, $A e^{\mathrm {i} k r}$, is non-zero in the volume obtained by multiplying the longitudinal coherence length by the area of the sphere surface of the radius equal to the distance between the source and the observation point. The probability a photon to appear at each point of the volume is inversely proportional to the volume. This volume is small for a photon with a large momentum.

5. Photon approach to image formation

The OPD between the lights at two arbitrary points on a plane perpendicular to a parallel beam of light is zero and it does not change by propagation, reflection, and refraction. Thus, by applying OPD$\:=0$ to the reflection and refraction of two parallel rays of light from the interface of two transparent media we deduce the laws of reflection and refraction. However, in this section, we use the superposition of the probability amplitudes of a photon to find the image of a point object. In Fig. 6, a lens with convex sides and a radius of $R$ is located in air at the distance $l$ from the point object $S$. It is probable for a photon that emerges from the object to strike a point on the lens and pass through to the other side. The probability amplitude of a photon at an observation point on the right side of the lens is the superposition of the probability amplitudes of the photon at each point on the lens at the observation point. The image is, indeed, the observation point with the maximum probability amplitude. In other words, the image is the point with equal optical paths from the object to the image that pass through the lens. For point $S^\prime$ as the image of point $S$, all the optical paths from $S$ to $S^\prime$ that pass through the lens, including the paths specified by ($SCS^\prime$) and($SQS^\prime$), are equal. Representing the diameter and the thickness of the lens by $2b$ and $d$, respectively, we can express the latter optical paths as follows:

Considering Fig. 6, we can show:

Substituting the latter in Eq. (35) and equating the Eqs. (35) and (36) we get the ray optics formulae for the image formation:

 

Fig. 6. The sketch used to derive the equation of the image in photon approach.

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6. Conclusions

In conclusion, we showed that the uncertainty principles concerning “location-momentum” and “time-energy” permit us to evaluate the probability distribution of a photon that passes through apertures. We named the method as “photon approach” and showed that it can describe the diffraction phenomena. The probability distribution depends on the distances of the observation point from the apertures and from ray optics borders of the emerging beams. For beams resulting by division of a parallel beam, the probability distribution of a photon on the observation plane also depends on the OPDs of the beams. Thus, photon approach can also describe the interference in different interferometers, adequately. Moreover, we showed that the photon diffraction from apertures describes the longitudinal and traversal coherences appropriately and provides techniques for their measurements. Photon approach also describes the image formation and provides the ray optics laws. It seems that the presented approach can be considered as an alternative approach to explain the physics behind various optical phenomena and results in providing a fortiori definitions for optical concepts.