Standing Wave Photon Structures in Constraint Spaces

Abstract

Based on the single-photon structure model, the standing wave electric 4-photon (SWE4-P) composite, the standing wave magnetic 4-photon (SWM4-P) composite in one-dimensional longitudinal constraint space, and the standing wave 8-photon (SW8-P) composite structure in a laser microcavity are derived. The electromagnetic field of the TM₀₁₀ mode in a microwave cylindrical resonant cavity is studied and analyzed, and the photon structure basic unit of this mode is identified as the standing wave cylindrical 8-photon composite structure. The cylindrical photon is of the same size as the cavity volume, the photon volume being V = $\mathrm{\pi }{R}^{2}L$. The standing wave 8-photon composite structure contains an SWE4-P composite and an SWM4-P composite, with a phase difference of 90°. Therefor, the energy unit of the TM₀₁₀ mode in the cavity is 8$\hslash \omega$.

1. Introduction

After the canonical quantization of the electromagnetic (EM) field [1,2,3], the concepts of two transverse photons, i.e., the right-spin photon and the left-spin photon, were obtained. Pan et al. wrote in [4] that photons, massless spin 1 particles, have only two eigenvalues of their spin along the direction of their propagation (wave vector k), ±ℏ. These two spin values correspond to right- and left-handed circular polarizations, respectively [4]. However, this single photon does not correspond to the concept of the circularly polarized EM field in classical electrodynamics [5]. Jackson wrote the following in his book: “The circularly polarized states are composite, which is an easy way of allowing for a phase difference between the two linearly polarized (LP) states out of which the circular states are constructed”.

A single-photon state is generally considered to contain only one photon, and it has multiple polarization quantum states [6,7,8,9,10,11,12,13,14,15,16], such as horizontal linear polarization (HLP), vertical linear polarization (VLP), diagonal (45°) linear polarization (DLP), antidiagonal (−45°) linear polarization (ALP), right-handed circular polarization (RCP), left-handed circular polarization (LCP), etc. In fact, these polarization states are not true single-photon states. The results of the single-photon structure model [17] show that these quantum states are all 4-photon composite monomers. Here, to distinguish the circular polarization states, the photons with spin angular momentum of ±ℏ are defined as right-/left-spin photons, while the 4-photon composite monomers with spin angular momentum of ±4ℏ are called right-/left-handed circular polarization 4-photon monomers.

The photon is deemed by some people to be a derived concept: it arises as a ripple of the EM field. Or, instead, they view the EM field as fundamental, with the photon appearing only when we correctly treat the field in a manner consistent with quantum theory. Others view the photon quantum as fundamental, with the EM field arising only in some classical limit from a collection of quantum photons. This study considers that photons and EM fields objectively exist and should not be changed by mathematical processing. It is very necessary to reveal their internal unity and consistency.

Particle physics says that “the excited state of the EM field is manifested by the appearance of photons”. “The excited state of the electron field is manifested by the appearance of electrons”. By analogy, a particle is the manifestation of its field in an excited state. The atomic excited state is defined as the shell electrons are in a high energy level and a photon equal to the energy level difference must be radiated when the electrons transition to a lower energy level. However, the “EM field excited state” is not defined, and the results of this study will clarify this fuzzy concept.

The single-photon structure model [17] constructs the photon quantum with the EM-field vector matter. Conversely, can the classical continuous EM field in constraint spaces be constructed from photons? The answer is yes. For example, the alternating electric fields in parallel-plate capacitors, the pure alternating magnetic fields in current-carrying solenoid coils, the EM fields of single longitudinal mode TEM₀₀ in a laser [18,19] microcavity, and the EM fields of fundamental mode TM₀₁₀ in a microwave cylindrical resonant (MCR) cavity can all be constructed with a multi-photon composite structure.

The constraint spaces can be divided into two categories: longitudinal constraints and transverse constraints. The former will form standing waves and the latter will still be traveling waves. Cavity types are standing wave systems, and waveguide types are traveling wave systems. MCR cavities are three-dimensional constraint spaces; guide-wave systems are laterally two-dimensional constraint spaces. This paper is devoted to a discussion of the photon structure of standing-wave EM fields, and does not enter into a discussion of the traveling wave EM fields of guide-wave systems, in the interests of brevity.

2. LP Standing Wave 4-Photon Composite

When a pair of LP photons are reflected back and forth between two parallel conducting planes, the reflected photon pair at the incident point has a half-wave loss relative to the incident photon pair. There are two cases: One is that the magnetic field component is reversed, and the electric field component remains unchanged. The other is that the electric field component is reversed, and the magnetic field component remains unchanged. Thus, between two parallel conducting planes, when the reflected LP photon pairs meet and superimpose with the incident LP photon pairs to form a standing wave (since their velocity cancels out to zero) 4-photon (SW4-P) composite, there are two types: One is the magnetic field component canceling out to zero, leaving only the electric field component to overlap and form a LP standing wave electric 4-photon (SWE4-P) composite. The second is that the electric field components cancel each other out to zero, leaving only the magnetic field components to overlap and form a standing wave magnetic 4-photon (SWM4-P) composite. Below we discuss them respectively.

2.1. SWE4-P Composite

Between two parallel conducting planes, suppose that E₁H₁ and E₂H₂ are the incident LP photon pairs, E₃H₃ and E₄H₄ are the reflected LP photon pairs, and both are defined with respect to one of the reflecting planes. When they meet and co-volume superpose, the electric field components of the photon pairs E₁H₁ and E₂H₂ polarize in the y axis and are in phase with the electric field components of the photon pairs E₃H₃ and E₄H₄. When their magnetic field components are at the 180° phase, as shown in Figure 1a, they form a SWE4-P composite. Figure 1a–d show the orientation of the electric-field and magnetic-field substance vectors in each of the four photons at phase points $0^{°}, {90}^{°}, {180}^{°}$, and ${270}^{°}$, respectively.

When the photon E₁H₁ ($(E_{\mathrm{p}} + \mathrm{i}{H}_{\mathrm{p}})V_{\mathrm{P}}{\mathrm{e}}^{\mathrm{i}(kz-\omega t)}$) and the photon E₃H₃ ($(E_{\mathrm{p}} -\mathrm{i}{H}_{\mathrm{p}})V_{\mathrm{p}}{\mathrm{e}}^{-\mathrm{i}(kz-\omega t)}$) are the right-spin photon along the k direction and left-spin photon along the −k direction, respectively, and their resultant wave vector is zero and the resultant velocity is also zero, becoming a standing wave photon pair, their magnetic field components H₁ and H₃ are always in opposite directions and become hidden components that are invisible, and their electric field components E₁ and E₃ are always in phase and doubled, becoming “manifest components”. Here, E_p and H_p are the electric and magnetic field densities of a single photon, respectively, and V_p is the single-photon volume.

Likewise, when the photons E₂H₂ ($(E_{\mathrm{p}} - \mathrm{i}{H}_{\mathrm{p}})V_{\mathrm{P}}{\mathrm{e}}^{\mathrm{i}(kz-\omega t)}$) and E₄H₄ ($(E_{\mathrm{p}} + \mathrm{i}{H}_{\mathrm{p}})V_{\mathrm{P}}{\mathrm{e}}^{-\mathrm{i}(kz-\omega t)}$) are the left-spin photon along the k direction and the right-spin photon along the −k direction, respectively, and their resultant wave vector is equal to zero, and the resultant velocity is also equal to zero, they also become a standing wave photon pair; the synthetic magnetic field component is also a “hidden component”, and the electric field component is also a “manifest component”.

The total spin angular momentum of the same-body -photon synthesis is equal to zero, the total wave vector is also equal to zero, the total magnetic field component is always equal to zero, and there is only the total electric field component E_y, which is shown a pure electric field oscillating harmonically along the y axis. Looking at the point z = 0, it is easy to see that the total electric field E_y of the standing wave 4-photons E₁H₁, E₃H₃, E₂H₂ and E₄H₄ can be calculated by the following formula:

$$E_{\mathrm{y}}(t)=(E_{\mathrm{p}} + \mathrm{i}{H}_{\mathrm{p}}){\mathrm{e}}^{-\mathrm{i}\omega t} + (E_{\mathrm{p}} - \mathrm{i}{H}_{\mathrm{p}}){\mathrm{e}}^{\mathrm{i}\omega t} + (E_{\mathrm{p}} - \mathrm{i}{H}_{\mathrm{p}}){\mathrm{e}}^{-\mathrm{i}\omega t} + (E_{\mathrm{p}} + \mathrm{i}{H}_{\mathrm{p}}){\mathrm{e}}^{\mathrm{i}\omega t} =2E_{\mathrm{p}}{\mathrm{e}}^{\mathrm{i}\omega t}+2E_{\mathrm{p}}{\mathrm{e}}^{-\mathrm{i}\omega t}=4E_{\mathrm{p}} \cos \omega t$$

(1)

Its energy

$$u_{4\mathrm{p}}={\displaystyle \frac{V_{\mathrm{p}}}{2\mathrm{\pi }}}{\displaystyle {\int }_0^{\mathrm{\pi }}{\mathrm{\epsilon }}_0[{(E_{\mathrm{y}}(\omega t)]}^2\mathrm{d}(\omega t)}={\displaystyle \frac{V_{\mathrm{p}}}{2\mathrm{\pi }}}{\displaystyle {\int }_0^{\mathrm{\pi }}{\epsilon }_0}{[4E_{\mathrm{p}}\cos \omega t]}^2\mathrm{d}(\omega t)=4\hslash \omega$$

(2)

The SWE4-P composite provides a pure alternating electric field, similar to the pure alternating field inside a capacitor. It can be said that the SWE4-P composite is the basic unit that constitutes a pure alternating electric field.

2.2. SWM4-P Composite

Between the two parallel conducting planes, the photons H₁E₁ and H₂E₂ are the reflected x-linearly polarized photon pairs (wave vector −k), and the photons H₃E₃ and H₄E₄ are the incident x-linearly polarized photon pairs (wave vector k). They meet and superpose into a 4-photon composite. Among them, H₁E₁ is a left-spin photon along the −k direction, H₃E₃ is a right-spin photon along the k direction, and they form one stationary pair; H₂E₂ is a right-spin photon along the −k direction, H₄E₄ is a left-spin photon along the k direction, and they form the other stationary pair. At 90° phase, the phase of the magnetic field components of the photon pairs H₁E₁ and H₂E₂ is the same as that of the photon pair H₃E₃ and H₄E₄, their magnetic field pointing to the −x axis as shown in Figure 2b, while the phase of their electric field components is always opposite and cancel each other out to zero. Figure 2a–d show the orientation of the magnetic- and electric-field substance vectors of each photon at phase $0^{°}, {90}^{°}, {180}^{°}$, and ${270}^{°}$, respectively.

The synthetic total spin angular momentum of the four photons of the composite is equal to zero, the total wave vector is also equal to zero, the total electric field component is always equal to zero, and there is only one total magnetic field component H_x, without other components, which is shown as a pure magnetic field oscillating harmonically along the x axis as follows:

$$\begin{gathered} H_{\mathrm{x}}(t)=(-E_1 + \mathrm{i}{H}_1){\mathrm{e}}^{\mathrm{i}\omega t} + (E_2 + \mathrm{i}{H}_2){\mathrm{e}}^{\mathrm{i}\omega t} + (E_3 + \mathrm{i}{H}_3){\mathrm{e}}^{-\mathrm{i}\omega t} + (-E_4+ \mathrm{i}{H}_4){\mathrm{e}}^{-\mathrm{i}\omega t} \\ =2H_{\mathrm{p}}{\mathrm{e}}^{\mathrm{i}(\omega t+\mathrm{\pi }/2)} +2H_{\mathrm{p}}{\mathrm{e}}^{-\mathrm{i}(\omega t+\mathrm{\pi }/2)} = 4H_{\mathrm{p}}\cos (\omega t +\mathrm{\pi } / 2) = -4H_{\mathrm{p}}\sin \omega t \end{gathered}$$

(3)

The total magnetic field is a pure alternating magnetic field, just like the pure alternating magnetic field inside the current-carrying solenoid coil. It is seen that the SWM4-P composite is the basic unit that constitutes the alternating magnetic field.

Just like the SWE4-P composite, there is no net energy flow, with a Poynting vector of s = 0. As for the effective energy of the SWM4-P composite, there is the following equation:

$$u_{4\mathrm{p}} = {\displaystyle \frac{V_{\mathrm{p}}{\mu }_0}{2\mathrm{\pi }}} {\displaystyle {\int }_0^{\mathrm{\pi }}{(-4H_{\mathrm{p}} \sin \omega t)}^2}\mathrm{d}(\omega t) = 4{\mu }_0{H}_{\mathrm{p}}^2{V}_{\mathrm{p}} = 4\hslash \omega$$

(4)

This shows that the energy of the SWM4-P composite is also the energy ($4\hslash \omega$) of four single photons, and the conservation of energy is observed.

3. The Photon Structure of the Single Longitudinal Mode TEM₀₀ in a Laser Microcavity

The simplest laser microcavity consists of an active medium and two reflecting mirrors and is an open cavity without boundaries on the sides. The eigenmode photon runs back and forth between the two reflective surfaces, activating the medium particles to produce stimulated emission, and the positive feedback continuously proliferates, producing optical amplification. The function of the resonator cavity is to preferentially amplify the eigenmode of light with a certain frequency and consistent direction, while suppressing light from other frequencies and directions. A strong beam of light is formed in the cavity with the same direction of propagation, frequency, and phase, i.e., a laser. To extract the laser from the cavity, one of the reflecting mirrors can be made partially transmissive [18,19], with the transmitted part becoming the available laser, and the reflected part in the cavity continuing to proliferate photons. The single longitudinal mode TEM₀₀ in the optical resonator cavity is a standing wave mode, and the basic unit of the output laser is a propagating LP4-photon (LP4-P) monomer [17]. The intracavity photon structure is a standing wave 8-photon (SW8-P) composite structure formed by the superposition of the reflected LP4-P monomer and the incident LP4-P monomer, as shown in Figure 3. The LP4-P monomer (E₁H₁, E₁′H₁′, E₂H₂, E₂′H₂′) can either be transmitted out as an output laser unit or reflected back.

When the laser output mirror is taken as the reflection reference plane, and the tangential component of the electric field on the reflection surface is required to be equal to zero (boundary conditions), requiring that the electric field component of the reflected photon is reversed (half-wave loss), while the magnetic field component remains unchanged, the reflected monomer (E₃H₃, E₃′H₃′, E₄H₄, E₄′H₄′) and the incident monomer (E₁H₁, E₁′H₁′, E₂H₂, E₂′H₂′) meet and superpose into an 8-photon composite structure along the way. Since the wave vectors cancel out in opposite directions, the velocities also cancel out to zero, resulting in a standing wave 8-photon (SW8-P) composite structure. This composite structure is just a specific structure in the constraint space, and is not a stable structure.

Figure 4a shows that the electric field E₁′ of photon 1′ and the electric field E₃ of photon 3 are always opposite and cancel each other out to zero, and the electric field E₁ of photon 1 and the electric field E₃′ of photon 3′ are also always opposite and cancel each other out to zero. Only E₂, E₂′, E₄, and E₄’ contribute to the x-directed electric field as follows:

$$E_x = 4E_{\mathrm{p}} \cos \omega t$$

(5)

The four photons E₂H₂, E₂’H₂’, E₄H₄, and E₄’H₄’ form a standing wave electric 4-photon (SWE4-P) composite, which produces a pure alternating electric field.

Figure 4b shows that the magnetic field components H₂ of photon 2 and H₄′ of photon 4′ are always in opposite directions, canceling each other out to zero. The magnetic field components H₂′ of photon 2′ and H₄ of photon 4 are always opposite, canceling each other out to zero. Only H₁, H₁’, H₃, and H₃’ contribute to the magnetic field in the y direction as follows:

$$H_{\mathrm{y}}=4H_{\mathrm{p}}\sin \omega t$$

(6)

The four photons H₁E₁, H₁’E₁’, H₃E₃, and H₃’E₃’ form a standing wave magnetic 4-photon (SWM4-P) composite, which produces a pure alternating magnetic field.

It is apparent that in this standing wave 8-photon composite structure, four of the photons have their magnetic fields canceled to zero because of their opposite directions; the electric fields of these four photons are doubled and strengthened because of their same directions, thus only the electric field is shown. In this standing wave 8-photon structure, the other four photons have just canceled their electric fields and only display the magnetic field. Moreover, the electric field and magnetic field are orthogonal, and their time phases differ by 90°, i.e., sine and cosine functions. This is determined by the fact that the electromagnetic field matter in the single photon structure model is an orthogonal vector, which is a universal law. The total energy of the SW8-P composite structure is calculated as follows:

$$u_{8\mathrm{p}}= {\displaystyle \frac{1}{2}}{\displaystyle \frac{V_{\mathrm{p}}}{\mathrm{\pi }}}{\displaystyle {\int }_0^{\mathrm{\pi }}[{\mu }_0{(4H_{\mathrm{p}}\cos \omega t)}^2 + {\epsilon }_0{(4E_{\mathrm{p}} \sin \omega t)}^2](\mathrm{d}\omega t)}=4V_{\mathrm{p}}({\epsilon }_0({E_{\mathrm{p}}}^2+{\mu }_0{H_{\mathrm{p}}}^2)=8\hslash \omega .$$

(7)

4. The Photon Structure of the TM₀₁₀ Mode in an MCR Cavity

The TM₀₁₀ mode of the MCR cavity has the lowest resonant frequency (${\omega }_{010} =J_1(2.405)\mathrm{c}/R$) and the longest resonant wavelength (${\lambda }_{010}= 2\mathrm{\pi }\mathrm{R}/2.405 = 2.62R$). A TM₀₁₀ single-mode operation is often used to accelerate charged particles, with the cavity called an accelerating cavity [20] and the mode called an accelerating mode. Now, we will begin to discuss the microwave photon structures of the TM₀₁₀ mode in the MCR cavity, and compare it with the mathematical description of the classical EM field.

4.1. The EM Field of the TM₀₁₀ Mode in an MCR Cavity

In the MCR cavity with inner radius R and length L, the exact expressions of the EM field distribution of all resonant modes in the cavity are given in classical electrodynamics [21,22]. The EM field distribution of the TM₀₁₀ mode is the simplest, as follows:

$$\left\{\begin{array}{l}{E}_z = E_0{\mathrm{J}}_0 \left({\mathrm{j}}_{01}\rho /R\right)\cos \omega t\hfill \\ \begin{array}{l}{H}_{\phi } = (\omega {\mathrm{\epsilon }}_{0}R/{\mathrm{j}}_{01}){{\mathrm{J}}^{\prime }}_0\left({\mathrm{j}}_{01}\rho /R\right)E_0\sin \omega t\\ E_{\rho } = E_{\phi } = H_{\rho } = H_{\mathrm{z}} = 0\end{array}\hfill \end{array}\right.$$

(8)

The TM₀₁₀ mode is an accelerating mode with only two field components. The electric field component E_z is in the z axis direction, while the magnetic field component H_φ is in the angular φ direction, and the phase difference between the electric field component and the magnetic field component is ${90}^{°}$, where J₀(x) is the zero-order Bessel function, J₀’(x) is the derivative of zero-order Bessel function, and ${{\mathrm{J}}^{\prime }}_0(x) =-{\mathrm{J}}_1(x)$, ${\mathrm{J}}_1(x)$ is the first-order Bessel function. The ${\mathrm{j}}_{01}$ is the first zero point of the zero-order Bessel function, and its value ${\mathrm{j}}_{01}$ = 2.405. The Bessel function curves are shown in Figure 5a. On the $\rho$ = 0 axis of the cavity, J₀(0) = 1. This means that the electric field has the maximum value, E_z (0, z) = E₀. At $\rho$ = R, the side wall of the cavity, because J₀(2.405) = 0, the electric field is zero, i.e., E_z (R, z) = 0. At the magnetic field at the $\rho$ = 0 axis, because ${\mathrm{J}}_1(0)=0$, the magnetic field is zero, i.e., H_φ(0, z) = 0. However, near the side wall of the cavity, where ρ = R, J₁(2.405) = 0.5019, and the magnetic field H_φ(R, z) has a bigger value. Using the relation ${J^{\prime }}_0(x) =-J_1(x)$, the H_φ in Equation (8) can be calculated as follows:

$$H_{\phi } = -\omega {\mathrm{\epsilon }}_0\left(R/{\mathrm{j}}_{01}\right) E_0{\mathrm{J}}_1({\mathrm{j}}_{01}\rho /R)\sin \omega t$$

(9)

The amplitude of the angular magnetic field at $\rho$ = R is proportional to J₁(2.405) (Figure 5a). The spatial distribution of the classical electric and magnetic fields in most spaces between the cavity axis and the cavity side wall is shown in Figure 5b (note there is a 90° phase difference between the electric field and the magnetic field). The field in the cavity is axisymmetric; there is no periodicity in both the e_φ direction and the e_z direction. The electron beam bunch is injected along the z axis in a suitable half-period, which can be accelerated by the axial electric field E_z. When the electric field peaks to accelerate the charged particles, the transverse circular magnetic field is nearly zero, since there is a 90° phase difference between the electric field E_z and the magnetic field H_φ.

4.2. Consideration to Construct the EM Field of the TM₀₁₀ Mode with Photons

The intracavity field is a standing wave field. A single photon cannot be localized; a single LP photon pair cannot be localized either. Only co-volume identical LP photon pairs with opposite wave vectors k can be stationary since the wave vectors cancel out to zero, causing their velocities to also cancel out to zero (additionally, the photons in an MCR cavity are zero velocity photons). That is to say, the photon structure allowed in the resonant cavity can only be the standing wave 4-photon composite. Although they have no translational velocity, they do keep a spin velocity, i.e., a spin angular velocity.

Next, we will look at the resonant wavelength of the TM₀₁₀ mode, ${\lambda }_{010}=2.62R$, and the photon diameter in free space, d = 0.5032λ. Now, let $d = 0.5032{\lambda }_{010} \approx 1.32R$ < 2R. Compared with the cavity diameter 2R and cavity length L < 2R, the photon size is not small and is comparable to the cavity size, so the cavity volume can only accommodate the volume of one photon.

Furthermore, the photons in the cavity must be deformed into a cylinder to satisfy the Helmholtz equation and the boundary conditions of the MCR cavity. The cylindrical photons of length L and radius R have the cavity axis as their symmetry axis. The electric-field substance vector, the magnetic-field substance vector, the wave vector, and the spin angular momentum vector of the photon in free space are all Cartesian vectors [17]. After entering the constraint cavity space, due to the interaction with the cavity wall, these vectors of each cylindrical photon turn into the orthogonal curve vectors (Ee_z, He_φ, ke_ρ) of the cylindrical coordinate system (ρ, φ, z); e_ρ, e_φ, and e_z are unit vectors on the ρ, φ, and z axis, respectively.

On the ρ = 0 axis, J₀(0) = 1 and J₁(0) = 0, so E_z(0, z) = E₀cosωt and H_φ(0, z) = 0. Without a magnetic field, there is a pure alternating electric field. And, at the side wall of the cavity where ρ = R, J₀(2.405) = 0 and J₁(2.405) $\approx$ 0.519, E_z (R, φ, z) = 0 and H_φ (R, φ, z) = −ωε₀(R/2.405)J₁(2.405)E₀sinωt. Without an electric field, there is a pure alternating magnetic field. The electric-field and magnetic-field components are mutually phase-shifted by 90° and superimposed orthogonally elsewhere.

For the amplitude of the electric field E_z given by Equation (8) and the amplitude of the magnetic field H_φ given by Equation (9), it is observed in Figure 5a that, at x ≈ 1.44, the zero-order Bessel function J₀(x) and the first-order Bessel function J₁(x) have a point of intersection, i.e., J₀(1.44) $\approx$ J₁(1.44). Using this value, for any microwave frequency, the amplitude ratio can be calculated E_z/H_φ= 1/(ωε₀R/2.405) $\approx$ 377Ω, which is consistent with the wave impedance of the photon [17].

Considering that the field in the cavity is axisymmetric, there is no periodicity in either the angular e_φ direction or the longitudinal e_z direction. For cylindrical photons, the electric and magnetic field matter vectors are located on the cylindrical surface (z, φ); the wave vector and angular momentum are along the radial e_ρ or −e_ρ.

In order to satisfy the conditions of pure electric field E_z (near axis) and pure magnetic field (near side wall), and to satisfy the condition of a phase difference of 90° between them, we need two kinds of SW4-photon composite. The SWE4-P composite can provide a pure electric field along the z direction. The field substance distribution of each of its photons along the e_ρ direction obeys the ${\mathrm{J}}_0\left(2.405\rho /R\right)$ function, and is uniform along both the e_φ and e_z directions. The SWM4-P composite can provide a pure magnetic field H_φ along the e_φ direction. The field substance distribution of each of its photons along the e_ρ direction obeys the ${\mathrm{J}}_1\left(2.405\rho /R\right)$ function, and is uniform along both the e_φ and e_z directions.

The superposition of these two kinds of SW4-P composite provides both the harmonic oscillating electric field along the z direction and the harmonic oscillating magnetic field along the e_φ direction in the MCR cavity. The SWE4-P composite and the SWM4-P composite are configured in a 1:1 ratio, satisfying the principle of equal partition for EM energy in the cavity.

In regard to the changes over time, it is interesting that the SWE4-P composite obeys the cosωt function and the SWM4-P composite obeys the sinωt function. Looking at the structure of the EM field given by Equations (8) and (9) and comparing it with the electric and magnetic field expressions for the SW8-P composite structure given in Section 3, one easily realizes that this TM₀₁₀ mode’s photon structure is a typical SW8-P composite structure. There is an inherent correlation between the SWE4-P composite and the SWM4-P composite. That is, the spatial orthogonal, the time phase difference in the 90° relation, and the volume integral of its electric field energy are equal to that of the magnetic field energy (i.e., each half of the EM energy).

This SW8-P composite structure is the basic unit of the TM₀₁₀ mode photon structure. Below, we will discuss each of these in detail.

4.2.1. The SWE4-P Composite of the TM₀₁₀ Mode

The EM-field matter distribution of each photon in the cylindrical-shaped SWE4-P composite follows the zero-order Bessel ${\mathrm{J}}_0({\mathrm{j}}_{01}\rho /R)$ function along the radial ρ and is uniformly distributed along both the angular φ and the z axis. For a cylindrical photon, the electric and magnetic field matter vectors (E, H) of each photon are located on the cylindrical surface (φ, z), while the wave vector k and angular momentum j are in the radial ($\pm {\mathrm{e}}_{\rho }$) direction. Right-/left-spin photons in free space obey the $\mathit{E}\times (\pm \mathit{H})\to \mathit{k}$ spin rule. How do we implement the spin rule for right-/left-spin cylindrical photons? We divide each cylindrical photon into an infinite number of differential volumes (dV) by using the three orthogonal curve surfaces ($(\rho ,\phi ),(\phi ,z),(z,\rho )$). In each dV the EM-field matter vectors (E, H) of each photon spin/rotate about the local wave vector k at ω according to the $\mathit{E}\times (\pm \mathit{H})\to \mathit{k}$ spin rules, as in the Cartesian coordinate system shown in Figure 6.

Intracavity four cylindrical photons E₁H₁, E₂H₂, E₃H₃, and E₄H₄, at ωt = 0°, are shown in Figure 6a–d, respectively. The electric field substance vectors are all along the z axis and become manifest components, while the magnetic field substance vectors cancel out to zero and become hidden components. The direction of the electric field is along the z axis, and the magnetic field direction is along the angular φ and is connected to the head and the tail. In order to see this clearly, take the same dV as shown in a’–d’ of Figure 6. The H₁ of photon E₁H₁ is along the −e_φ direction, with an E₁H₁ right-spin. Both its wave vector k and its angular momentum j are along the e_ρ direction. The H₂ of photon E₂H₂ is also along the −e_φ direction, with an E₂H₂ left-spin. Its wave vector k is along the e_ρ direction, but its angular momentum j is along the −e_ρ direction. The H₃ of photon E₃H₃ is along the e_φ direction, with an E₃H₃ left-spin. Its wave vector k is along the −e_ρ direction, but its angular momentum j is along the e_ρ direction. The H₄ of photon E₄H₄ is along the e_φ direction, with an E₄H₄ right-spin. Both its wave vector k and angular momentum j are along the −e_ρ direction. In a word, the field substance superposition of the same dV belonging to these four photons results in a zero magnetic-field vector synthesis, zero total wave vector synthesis, zero total angular momentum synthesis, and four times electric-field vector synthesis of a single photon. This result is applicable for every SWE4-P composite.

Figure 6 shows only the vector direction at the initial phase (ωt = 0°) of photon E₁H₁, E₂H₂, E₃H₃, and E₄H₄ within the SWE4-P composite. Figure 7a–d show the direction of the electric field substance vector and the magnetic field substance vector in the same differential volume belonging to four different photons at 0°, 90°, 180°, and 270° phase points, respectively. Note that H₁ and H₃ are always in opposite directions and cancel each other out to zero, and H₂ and H₄ are always in opposite directions and cancel each other out to zero. The dV, a′–d′ of each photon in Figure 6 only corresponds to Figure 7a.

The result of the superposition of four photons is that the total magnetic field is always zero and invisible, and the total electric field E is a manifest component, which is four times the intensity of the single photon. There is no net wave vector and no net angular momentum in any dV, but the substances in each dV which belong to each photon keep an independent spin. The z-direction electric field of the TM₀₁₀ mode is contributed by the SWE4-P composite alone. Obviously, this is the basic unit of the z-direction electric field of the TM₀₁₀ mode in the cavity. As a result, the pure electric field oscillates along the z direction with angular frequency ω. The electric field generated by the SWE4-P composite can be expressed as follows:

$${(E_z)}_{4\mathrm{p}} = 4E_{\mathrm{p}}{\mathrm{J}}_0({\mathrm{j}}_{01}\rho /R)\cos \omega t$$

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4.2.2. The SWM4-P Composite of the TM₀₁₀ Mode

Each photon in the SWM4-P composite is also deformed into a cylinder of length L and radius R. The EM-field matter distribution of each photon in the SWM4-P composite follows the first-order Bessel ${\mathrm{J}}_1({\mathrm{j}}_{01}\rho / R)$ function along the radial ρ direction and is uniformly distributed along both the angular φ and the z-axis direction. The electric field E and magnetic field H of each photon are located on the cylindrical surface (φ, z), while the wave vector k and angular momentum j are along the $\pm$e_ρ direction, as shown in Figure 8. A significant difference from the SWE4-P composite is that the superposition of four photons results in the total electric field component being zero. Instead, the total magnetic field component along the e_φ direction is oscillating harmonically at the angular frequency ω.

We will use a dV to illustrate the physical mechanism. As shown in Figure 8, at ωt = 90° phase, the magnetic field H_i (i = 1, 2, 3, 4) belonging to the four photons of the SWM4-P composite are all along the −e_φ direction, while their electric field components are pairwise opposite to each other and cancel out to zero. Specifically, in the dV, the magnetic field H₁ belonging to photon 1 is along the −e_φ direction, the photon H₁E₁ is left-spin, its wave vector k₁ is along the −e_ρ direction, and its angular momentum j₁ is along the e_ρ direction as shown in Figure 8a’. The magnetic field H₂ belonging to photon 2 is also along the −e_φ direction, the photon H₂E₂ is right-spin, and both its wave vector k₂ and angular momentum j₂ are along the −e_ρ direction as shown in Figure 8b. The magnetic field H₃ belonging to photon 3 is along the −e_φ direction, the photon H₃E₃ is right-spin, and both its wave vector k₃ and angular momentum j₃ are along the e_ρ direction as shown in Figure 8c’. The magnetic field H₄ belonging to photon 4 is along the −e_φ direction, the photon H₄E₄ is left-spin, its wave vector k₄ is along the e_ρ direction, and its angular momentum j₄ is along the −e_ρ direction as shown in Figure 8d’. In a word, in the same dV belonging to the four photons, the superposition result is that the total electric-field vector synthesis is zero, the total wave vector synthesis is zero, the total angular momentum synthesis is zero, and the total magnetic-field vector synthesis is four times that of the single photon. This result is applicable for every SWM4-P composite.

Figure 9a–d show the direction of the electric field substance vector and the magnetic field substance vector in the dV shared by four cylindrical photons at the four phase points of 0°, 90°, 180°, and 270°, respectively. Note the E₁ and E₃ always cancel each other out to zero, and E₂ and E₄ also always cancel each other out to zero. The field synthesized by the four cylinder photons is a pure magnetic field. The magnetic field produced by the SWM4-P composite harmonically oscillates at an angular frequency ω along the e_φ direction.

$${(H_{\phi })}_{4\mathrm{p}}= -4H_{\mathrm{p}}{\mathrm{J}}_1(2.405\rho /R)\sin \omega t$$

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The SWM4-P composite is the basic unit of the angular magnetic field of the TM₀₁₀ mode in the MCR cavity.

4.3. The SW8-P Composite Structure of the TM₀₁₀ Mode

The superposition of an SWE4-P composite and an SWM4-P composite with a total of 8 photons as a basic unit constitutes the EM field of the TM₀₁₀ mode. The pure electric field E_z near the axis is contributed by the SWE4-P composite alone, the pure magnetic field H_φ near the side wall is contributed by the SWM4-P composite alone, and the mixed electric and magnetic fields with a phase difference of 90° in other regions are jointly contributed by both the SWE4-P composite and the SWM4-P composite. The co-volume superposition of the SWE4-P and SWM4-P composites (total of eight photons) as the basic unit provides the energy storage for the TM₀₁₀ of the MCR cavity. That is to say, the energy storage in the cavity increases step by step with the unit ($8\hslash \omega$) of eight photons. This is a result deduced from a single-photon structure model [17].

The physical process that occurs in the cavity can be understood through the classical EM theory. The electric and magnetic fields of eigenmodes that satisfy the Helmholtz equation and boundary conditions of the MCR cavity oscillate at an angular frequency ω. The electric energy is converted into magnetic energy, and the magnetic energy is converted into electric energy, and so on. The electric and magnetic energies are converted into each other.

From the photon’s point of view, the physical process occurring in the cavity is the result of the EM-field vector matter spinning/rotating independently about their respective wave vectors ($\pm e_{\rho }$) at the angular velocity ω in each dV of the eight cylindrical photons. This physical mechanism is more intuitive, concise, and clear than the abstract image of “electric and magnetic energy can be converted into each other”. Of course, the classical theory of the EM field is still very accurate in its mathematical description, which is unrivaled from the photon′s point of view.

5. Discussion

The standing wave 8-photon composite structure is the inevitable result of the superposition of two running wave 4-photon monomers moving in opposite directions in a longitudinal-constraint space. The standing wave electric 4-photon composite and the standing wave magnetic 4-photon composite, which being out of phase by 90°, are specific states in which the running wave 4-photon monomers exist in a longitudinal-constraint space. Once it leaves this constraint space, the standing wave 4-photon composite ceases to exist. You absolutely cannot extract the standing wave electric 4-photon composite or the standing wave magnetic 4-photon composite from the cavity space. What can be extracted from the cavity space is only the running wave 4-photon monomer, which is commonly referred to as the “single photon” in the current literatures. The 2-photon generated by type II spontaneous parametric down-conversion (type II SPDC in BBO) [23] is actually two 4-photon monomers with cross-polarization. And, the so-called “non-Gaussian triphotons entangled state” [24,25] is actually an entangled state of three “running wave 4-photon monomers”. According to the single-photon structure model [17], the normal unit of the pump laser is a running wave 4-photon monomer, and whether it is divided into two monomers or three monomers in the nonlinear crystal, each of its monomers is still the running wave 4-photon monomer (although people still refer to it as a “single photon”).

6. Summary and Conclusions

Based on the single-photon structure model [17], the SWE4-P composite and the SWM4-P composite are deduced in one-dimensional constraint space. They are the basic units of a pure alternating electric field and a pure alternating magnetic field, respectively. The SW8-P composite structure is deduced from the laser microcavity. It can simulate the resonant EM field of the TM₀₁₀ mode in the MCR cavity, but the photon has already been deformed into a cylinder, its wave vector is radial, and the EM-field matter vectors are located inside the cylindrical surface (φ, z).

On the one hand, the TM₀₁₀ mode has the lowest resonance frequency and the largest resonant-photon volume compared to photons of other modes. The volume of the entire cavity is one photon’s volume, and its EM field, though it is at least 8-photons superimposed, is always confined inside of a single photon’s volume, so it only shows a classical continuity and cannot possibly show particle nature. This is the root cause of why the EM fields and photons look so different. This way, photons and classical EM fields no longer appear completely different and unrelated, but rather have intrinsic unity and consistency.

In a popular sense, the standing wave EM field and the corresponding standing wave photon structure are the same thing. There is no so-called “excited state” and “ground state” in an EM field. Photons and EM fields are just two different ways of description. The lower the frequency, the more convenient it is to describe it as a field (within the photon volume), and, the higher the frequency, the more convenient it is to describe it as a photon (outside the photon volume). This is also significant for quantum optics.

On the other hand, for a cylindrical photon, not only does its exterior shape vary extremely, but its content changes even more significantly. First of all, its wave vector becomes a “variable vector (e_ρ)”; secondly, one of its field substance vectors (e_φ, e_z) also becomes a variable vector. In this way, the spin of each photon becomes extremely complicated. How do we perform the spin rule $\mathit{E}\times (\pm \mathit{H})\to \mathit{k}$? We perform it only by cutting the photon into infinite differential volumes with three orthogonal curved surfaces, allowing the field substance in each dV to spin around the local wave vector with angular velocity ω. The execution of the spin of each photon is so precise that the four photons (SWE4-P) cancel their magnetic field components and double the strength of their electric field components, and the other four photons (SWM4-P) cancel their electric field components and double the strength of their magnetic field components. What is it that is playing the role of manipulator so precisely? It is the mission of future studies to reveal this unknown substance. This is where the significance of this paper lies.

It can be said that the photon is a quantized EM-field particle, and that the electric and magnetic fields are the collection of the SWE4-P composite and SWM4-P composite, respectively, or said that the EM fields are the collection of the SW8-P composite structures.