The linear coupler occurs between two waveguides placed close to each other, with a distance between the cores of the waveguides of several micrometres, and this distance has a strong effect on the coupling between waveguides Figure 1.

In general, coupled mode theory may explain the coupling between waveguides when two waveguides are positioned infinitely far apart, and their amplitudes are a 1 and a 2 with propagation constants β 1 and β 2 for propagation along the z-direction unperturbed as the Equation 1-3. d dz a 1 ( z ) = i β 1 a 1 ( z ) d dz a 2 ( z ) = i β 2 a 2 ( z ) (1)

When there is a weak coupling between two waveguides in an evanescent field, the mode amplitude of one waveguide will affect the second waveguide, and the coupled equations will look as follows ^{8, 23– 27} d dz a 1 ( z ) = i β 1 a 1 ( z ) + k 12 a 2 ( z ) d dz a 2 ( z ) = i β 2 a 2 ( z ) + k 21 a 1 ( z ) (2) where κ _ij is the coupling coefficient that is defined as the modal overlap of the two waveguides and can be expressed as below ^{1, 2, 28} k 12 = k ° 2 2 β ∬ − ∞ ∞ ( n 1 2 − n ° 2 ) F 1 ∗ ( x , y ) F 2 ( x , y ) dxdy (3) where F i ( x , y ) indicates the mode’s radial distribution of the mode ( i = 1, 2) and β is the average of propagation constants β 1 and β 2 as β = 1 2 ( β 1 + β 2 ) , n 1 is the waveguide core’s refractive index, and n o is the surrounding refractive index.

Also, the coupling can be expressed according to supermode theory as two individual modes, either a symmetric (identical) supermode, where the phase between these modes is synchronous or an anti-symmetric (non-identical) supermode that has out-of-phase, depending on the structure parameter of waveguides. ^{1– 3} The value of the effective index of the mode splitting is determined by the coupling strength κ, which becomes a perturbation of the individual modes. The effective index of even and odd supermodes and their corresponding propagation constants can be calculated to determine the coupling strength and coupling length using COMSOL Multiphysics, which relies on FEM to solve the modes easily. The coupling strength can then be obtained from; k = 1 2 ( β even − β odd ) (4) L c = π 2 k = π ( β even − β odd ) = λ 2 ( n even − n odd ) (5)

The analytical solution of these equations can find the power distribution in each waveguide when these waveguides are identical, i.e. β 1 = β 2 = β and k 12 = k 21 = k .

The power in one waveguide is P 1 ( z ) = | a 1 ( z ) | 2 and the other is P 2 ( z ) = | a 2 ( z ) | 2 , at z = 0, the light is only the input of one waveguide P 1 ( z = 0 ) = P ° and P 2 ( z = 0 ) = 0 . P 1 ( z ) = P ° cos 2 ( kz ) P 2 ( z ) = P ° sin 2 ( kz ) (6)

The light is initially confined to one waveguide and completely transferred to another waveguide within a distance L c = π 2 k , where the power in each waveguide oscillates sinusoidally back and forth due to the light propagation, and the coupling length is a measure of the coupling between the two cores along the propagation distance L c . ^{1– 2, 8, 29} Consequently, the minimal distance at which a maximum power transfer between the waveguides occurs is known as the coupling length, and then 100% of the optical power can be transferred when two waveguides are identical. ^{2, 8} The linear coupling can be used as a fiber beam-splitter, and the splitting ratio of power depends on both the coupling strength κ and the length of the coupler L, when z = L c / 2 , show 50% of power is transferred, and the coupler in this length is defined as -3dB coupler because the loss is (10 log 0.5 = -3dB), then the coupler is a 1:1 beam splitter. ^{2, 11, 29}

Linear coupler results from evanescent field coupling, where the overlap between the modes of each core, input power to the waveguide is low, follows a sinusoidal pattern. As a result, only the coupling length and coupling strength, which are based on the refractive index difference between the waveguide’s core and cladding, can determine the output power ratio. Since Equation 3 states that the same waveguide is constant when the refractive indices are taken into account, the cross-phase-induced coupling is typically negligible and can be disregarded. ²⁸ But the nonlinear coupling is due to the cross-phase modulation, and the input power to the waveguide will increase the linear coupler and consequently become power uncoupled, as a result a change in refractive index. Therefore, when calculating nonlinear coupling, it should add the phase to the linear coupler equation as follows ^{8, 30, 31} : d dz a 1 ( z ) = iβ a 1 ( z ) + k a 2 ( z ) + ( γ | a 1 ( z ) | 2 + μ | a 2 ( z ) | 2 ) a 1 ( z ) d dz a 2 ( z ) = iβ a 2 ( z ) + k a 1 ( z ) + ( γ | a 2 ( z ) | 2 + μ | a 1 ( z ) | 2 ) a 2 ( z ) (7)

Here γ = n 2 ϰ o A eff = 2 π λ . represents n 2 A eff represent the nonlinear coefficient, and the term μ has been described. Since the self-generated nonlinear phase γ assumes the nonlinear effect, take into consideration μ = 0 for the phase that is caused by the nonlinear interaction from one mode to the other mode in the adjacent waveguide. ²² When the weak coupling is taken into account by the amplitudes alone, the nonlinear Equation 7 may then be solved analytically using the Jensen approximation, ^{8, 30} where the solution for the linear coupler is obtained when m ≤ 1 . P 1 ( z ) = 1 2 P 1 ( z = 0 ) ( 1 + cn ( ( πz / 2 L c ) / m ) ) ⇛ P 1 ( z ) = 1 2 P 0 ( 1 + cn ( 2 kz / m ) ) (8) where cn represents the Jacobi elliptical function. ⁸

When m =1, the value of the input power P 0 (where m = P 0 2 P c 2 ) is called the critical power P c which is the power that corresponds nonlinear phase shift of about 2 π occurred in the coupling length, where the coupling length reaches infinity at this power and is found to be inversely proportional to the coupling length. ^{5, 32} P c = 4 k γ = λ A e f f n 2 L c                                 (9)

Although the coupling is linear at m = 0, the smallest value of z at which the waveguide’s power drops from an initial value of P 0 to zero has been represented by the coupling length, L c ,

cn ( 2 k L c m ) = − 1 , therefore, the coupling length is defined as. ⁸ cos ( 2 k L c ) = − 1 or L c = π 2 k (10)

But the input power is critical when m >1, the solution is P 1 ( z ) = 1 2 P 0 ( 1 + dn ( 2 kzm m − 1 ) ) (11)

From Equation 11, for a linear coupler with m = 0 and low input power, light couples back and forth sinusoidally between two waveguide cores with a periodicity of 2 L c . The value of m, on the other hand, rises with increasing input power, suggesting that full power transfer is preserved between the two waveguide cores. The oscillation period also rises with increasing power, the coupling length increases, and the solution starts to diverge from the linear coupler solution.

As the input power P ₀ increases to the critical power P _c at m = 1, the coupling duration period gets closer to infinity. This indicates that the light is distributed equally between the coupler’s two cores, see Figure 2-a. When the coupler is given extremely high power at m > 1, the light propagation period of oscillation decreases with a periodicity of L _c. The induced phase that results from the nonlinear effects subsequently stops the two waveguides from connecting where the phase-mismatch rises. ⁸ Light cannot, therefore, be entirely linked from one waveguide to another; rather, it remains in the waveguide into which it was first released, ^{8, 32, 33} as illustrated in Figure 2-b.

Also, the linear coupler cannot be controlled on the output power ratio of a fixed length, but by using nonlinear coupler it possible to control it by amount uncoupled power, therefore, nonlinear coupler works as optical switch in photonic circuits for all optical powers, and by choosing the desired intensity can be obtained on clearly switching between two cores of PCF coupler as the Figure 3. ²⁵