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Nonlinearity-mediated collimation of optical beams

Guo Liang, Fangjie Shu, Wenjing Cheng, and Longbo Jiao

Open Access Open Access

Abstract

We investigated the evolutions of optical beams in an optical system composed of free spaces and nonlocal nonlinear media layers in a cascaded manner. From an application point of view, two kinds of evolution processes for Gaussian beams, nonlinearity-mediated collimation and switching from breathers to solitons, were discussed in details. By adjusting the input optical power, the collimating, the compressing and the expanding of optical beams are convenient to be controlled.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The Kerr effect, which refers to a light-intensity dependence of the refractive index perturbation, is a third-order nonlinear phenomenon in optics [1,2]. Kerr effect can be generalized to the nonlocal case when the response of the medium at a particular point depends on the intensity in its vicinity on entire beam transverse profile. Nonlocal nonlinearity has attracted considerable interest in various fields of optical spatial solitons [3–5] and optical spatial shock waves [6]. This occurs, e.g., in nematic liquid crystal [7], lead glass [8], thermal nonlinear liquid [9], and nonlinear ion gas [10]. Recently, a giant nonlocal nonlinearity is discovered in the m-cresol/nylon mixed solution [11]. The propagation of the optical beams in the nonlocal nonlinear media is modeled by the nonlocal nonlinear Schrödinger equation (NNLSE), in which the nonlinear refractive index is mathematically expressed by a convolution between optical intensity and the response function [12]. According to the relative width of the response function and the optical beam, there are four categories of the nonlocality [13]: local, weakly nonlocal, generally nonlocal, and strongly nonlocal. Under the strongly nonlocal case, the NNLSE can be linearized to the well-known Snyder-Mitchell model (SMM) [3], which has various analytical solutions, including the Hermite-Gaussian breathers and solitons [3], the Laguerre-Gaussian breathers and solitons [14,15], the Ince-Gaussian breathers and solitons [16], and the spiraling elliptic breathers and solitons [17,18], just to name a few.

Apart from the nonlocal solitons mentioned above, the beam steering in the nonlocal nonlinear media is the other topic. The nonlocal nonlinearity can arrest the catastrophic collapse [19], and can overcome repulsion between out-of-phase bright [20] or in-phase dark solitons [21]. At the interface between two regions of a nematic liquid crystal, it is possible to steer the beams by refraction and total internal reflection by as much as $-18$ and $+22$ degrees [22]. The fractional Fourier transform is found to naturally exist in strongly nonlocal nonlinear media (SNNM), and the propagation of optical beams in SNNM can be simply regarded as a self-induced fractional Fourier transform [23]. The beam steering is also achievable for the beam arrays [24,25]. In the paper, we will explore the nonlinearity-mediated collimation and the switching from breathers to solitons in a cascaded optical system composed of free spaces and nonlocal nonlinear media, and provide a flexible method to control the processes of collimating of beams.

2. Theoretical model and results

The propagation of optical beams in nonlocal nonlinear media can be modeled by the NNLSE in the following dimensionless form [7]

$$i\frac{\partial\psi}{\partial z}+\frac{1}{2}\frac{\partial^2\psi}{\partial x^2}+\Delta n\psi=0,$$
where $\psi (x,z)$ is the complex amplitude envelope, $\Delta n=\int \!\!\!\int R(x-x')|\psi (x',z)|^2dx'$ is the light-induced nonlinear refractive index, $z$ is the longitudinal coordinate, $x$ is the transverse coordinate, and $R$ is normalized symmetrical real spatial response of the media such that $\int \!\!\!\int R(x)dx=1$. We suppose the material response to be the Gaussian function [26]
$$R(x)=\frac{1}{\sqrt{2\pi}w_m}\exp\left(-\frac{x^2}{2w_m^2}\right),$$
where $w_m$ is the width of the response function, and is referred to as the nonlinear characteristic length. The ratio of the width of the response function to the scale in the transverse dimension occupied by the optical beam determines the degree of nonlocality, which reads $\delta \equiv \frac {w_m}{w}$. The larger is $\delta$, the stronger is the degree of nonlocality. For the strongly nonlocal case, we need only keep the first two terms of the expansion of $\Delta n$, and the NNLSE in the strongly nonlocal nonlinear media (SNNM) is simplified to the Snyder-Mitchell model
$$i\frac{\partial \psi}{\partial z}+\frac{1}{2}\frac{\partial^2\psi}{\partial x^2}-\frac{1}{2}\gamma^2 P_0x^2\psi=0,$$
where $\gamma ^2=-\partial _x^2R(x)|_{x=0}=\frac {1}{\sqrt {2 \pi }w_{m}^3}$ and $P_0=\int \!\!\!\int |\psi (x',0)|^2dx'$ is the optical power. Therefore, the larger $\delta$ is, the solution of the Snyder-Mitchell model (3) will approach more closely to the exact one of the NNLSE (1). Besides, the Gaussian response (2) is phenomenological, and in actual systems such as the lead glass and the nematic liquid crystal, the response to light can not be described by Gaussian function. However, the physical properties do not depend closely on the specific shapes of the response functions for the SNNM [27].

The complex $q$ parameter is considered to be an important parameter for Gaussian beams. $q$-parameter description has the advantage that it extends to higher-order Hermite-Gaussian modes and gives a concise description of their transformation by lenses, mirrors, and other optical components [28]. Furthermore, the $q$-parameter description also can be readily applied to the Gaussian beams in the SNNM, and the transformation of the complex $q$ parameter can be expressed by the following ABCD matrix [29]

$$M_{nl}^{(i)}=\left[ \begin{array}{cc} A_{nl}^{(i)} & B_{nl}^{(i)}\\ C_{nl}^{(i)} & D_{nl}^{(i)}\\ \end{array} \right]=\left[ \begin{array}{cc} \cos\left(\frac{z}{z_{pi}}\right) & -z_{pi}\sin\left(\frac{z}{z_{pi}}\right)\\ \frac{1}{z_{pi}}\sin\left(\frac{z}{z_{pi}}\right) & \cos\left(\frac{z}{z_{pi}}\right)\\ \end{array} \right]$$
with $z_{pi}=1/(\gamma _i\sqrt {P_0})$, where $\gamma _i$ can be made different during the propagation in different nonlinear media. While, the ABCD matrix for the $q$ parameter in free space is describe by [28]
$$M_{l}^{(i)}=\left[ \begin{array}{cc} 1 & z_s\\ 0 & 1\\ \end{array} \right].$$

In this paper, we will explore the nonlinearity-mediated collimation of beams in an optical system composed of free spaces and nonlocal nonlinear media in a cascaded manner, which is realized by the following configuration shown in Fig. 1. The incident Gaussian beam at $z=0$ is supposed to

$$\psi(x,0)=\sqrt{\frac{P_0}{\sqrt{\pi}w_0}}\exp\left(-\frac{ix^2}{2q_0}\right),$$
where $q_0=iw_0^2$. The beam is on-waist and off-waist incident in Fig. 1(a) and Fig. 1(b) for the nonlocal nonlinear medium. The $q$ parameter of the output after a cascade of nonlinear and linear propagations can be expressed by
$$q=\frac{Aq_0+B}{Cq_0+D},$$
of which the overall transformation matrix is obtained as
$$M=\left[ \begin{array}{cc} A & B\\ C & D\\ \end{array} \right]=\left\{\begin{array}{c} M_{nl}^{(2)}M_l^{(1)}M_{nl}^{(1)},~\text{in Fig.1 (a)},\\ M_{l}^{(2)}M_{nl}^{(1)}M_{l}^{(1)},~\text{in Fig.1 (b)}. \end{array} \right.$$

The $q$ parameter relates to the radius of curvature $\rho$ and the beam width $w$ by [28]

$$\frac{1}{q}=\frac{1}{\rho}-i\frac{1}{w^2}.$$

At the beam waist, the phase front is a plane, then $\rho =\infty$.

 figure: Fig. 1.

Fig. 1. Configuration for switching from breathers to solitons by a cascade of two segments of free spaces and one segment of nonlinear medium (b), and configuration for beam collimation by a cascade of two segments of nonlinear media and one segment of free space (a).

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3. Switching breathers to solitons by cascade of nonlinear and linear propagations

We will explore the switching from the breather states to the soliton states by the above cascade of nonlinear and linear propagations. During this switching process, the compression or expansion of the beam waist also can be available.

In this section, we consider that the beam is transmitted from the first free space into the first SNNM under the on-waist incident condition, then propagates in the second free sapce, and finally is incident on the second SNNM. We can obtain the ABCD matrix at the input plane of the second nonlinear medium

$$M=M_l^{(1)}M_{nl}^{(1)},$$
then $A=\frac {z_{s1}}{z_{p1}}\sin \left (\frac {z_1}{z_{p1}}\right )+\cos \left (\frac {z_1}{z_{p1}}\right )$, $B=z_{s1}\cos \left (\frac {z_1}{z_{p1}}\right )-z_{p1}\sin \left (\frac {z_1}{z_{p1}}\right )$, $C=\frac {1}{z_{p1}}\sin \left (\frac {z_1}{z_{p1}}\right )$, and $D=\cos \left (\frac {z_1}{z_{p1}}\right )$. Substitution of Eq. (10) into Eq. (7) yields
$$\frac{1}{q}=\frac{(w_0^2/z_{p1}^2)\sin(z_1/z_{p1})-(i/z_{p1})\cos(z_1/z_{p1})}{(w_0^2-iz_{s1})\cos(z_1/z_{p1})/z_{p1}+(i+w_0^2z_{s1}/z_{p1}^2)\sin(z_1/z_{p1})},$$
from which the radius of curvature $\rho$ and the beam width $w$ at $z=z_1+z_{s1}$ (input plane of the second nonlinear medium) can be calculated by taking the real and the imaginary parts, respectively. On the other hand, the evolution of beam widths in the SNNM is described by the following differential equation [30]
$$\frac{d^2w}{dz^2}-\frac{1}{w^3}+\gamma_i^2P_0w=0.$$

Therefore, the critical power for solitons can be determined by setting $d^2w/dz^2=0$, that is, $P_c=\frac {1}{\gamma _i^2w^4}$. Meanwhile, to realize the soliton states in the nonlinear media the optical beam must be on-waist incident. Therefore, in order to guarantee the soliton states in the second nonlinear medium, the two conditions that

$$P_0=P_c=\frac{1}{\gamma_2^2w_{wai}^4},$$
where $w_{wai}$ will be given in the following, and $\rho =\infty$ (on-waist incidence) should be met simultaneously. By making the real part of Eq. (11) to be zero, the distance between two segments of nonlinear media should satisfy
$$z_{s1}=\frac{(w_0^4/z_{p1}^2-1)\sin(2z_1/z_{p1})}{(2/z_{p1})\left[\cos^2(z_1/z_{p1})+w_0^4\sin^2(z_1/z_{p1})/z_{p1}^2\right]}.$$

By taking the imaginary part of Eq. (11) and using the above condition (14), we have the beam waist

$$w_{wai}=w_0\sqrt{\frac{1}{1+(w_0^4/z_{p1}^2-1)\sin(z_1/z_{1p})^2}}.$$

Equation (15) indicates that the beam waist at the input plane of the second nonlinear medium depends on the nonlinear characteristic lengths of two nonlinear media, the input power, and the length of the nonlinear media. It can be found that the beam waist $w_{wai}$ oscillates between $w_0$ and $z_{p1}/w_0$. Therefore, different critical power (is also the input power incident on the first nonlinear medium) is needed for different beam waists. Figure 2 shows the cascaded evolutions of beam width under different input powers. While, the corresponding numerical propagations are given in Fig. 3. The input power in Fig. 2 and Fig. 3 is indeed the critical one of solitons in the second medium, is however smaller than the critical one required by the first nonlinear medium. Therefore, the beam waist is expanded ultimately after the cascaded propagations. If the input power is taken to be larger than the critical one required by the first nonlinear medium, as shown in Fig. 4, the beam waist can be compressed.

 figure: Fig. 2.

Fig. 2. Cascaded evolutions of beam width under different input powers, that is $P_0=3177.9$ in the Left figure and $P_0=476.5$ in the Right figure. The other parameters are taken as $z_1=8.0,z_{s1}=0.476$ for the Left figure, $z_1=8.0,z_{s1}=7.86$ for the Right figure, and $w_{m1}=15, w_{m2}=20$ for the both. Red and yellow shadow areas indicate the linear and nonlinear propagations, respectively. The input power $P_0$ is given by Eq. (13), Eq. (15) and $w_0=1$.

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 figure: Fig. 3.

Fig. 3. Cascaded propagations of beams, the region between two red lines is for the linear propagation. (a) and (b) correspond to the left and right ones in Fig. 2

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 figure: Fig. 4.

Fig. 4. Same as Fig. 2 but plots are $P_0=33965, z_1=8.0,z_{s1}=0.347$ for the Left figure, and $P_0=65997, z_1=8.0,z_{s1}=0.436$ for the Right figure.

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In the nematic liquid crystals, the nonlinear characteristic length $w_m$ can be modulated by changing the bias voltage [31]. Therefore, it is handy in experiment to make the two segments of nonlinear media exhibit different $w_m$ if the nematic liquid crystals serve as the nonlinear media. What we need do is to only apply different bias voltages on the two segments of nematic liquid crystals. In fact, the predicated behavior shown in Fig. 2 and Fig. 3 can be also achieved if two segments of nonlinear media share the same parameters, such as the same length and the same nonlinear characteristic length. Both the compression or expansion of the beam waist in the switching from breathers to solitons are shown in Fig. 5, where the nonlinear media exhibit the same $w_m$.

 figure: Fig. 5.

Fig. 5. Same as Fig. 2 but plots are $P_c=23886, z_1=6.0,z_{s1}=0.312$ for the Left figure, $P_c=1042, z_1=6.0,z_{s1}=3.121$ for the Right figure, and $w_{m1}=w_{m2}=15$ for the both.

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4. Nonlinearity-mediated collimation of beams

The collimating and expanding of Gaussian beams are always realized by the telescope optic system composed of two convex lenses with different focal lengths [32]. Meanwhile, for a given lens its focal length is fixed. In this section, we will explore a kind of nonlinearity-mediated collimation of beams. It will be shown that it is very advantageous to control the processes of collimating and expanding by adjusting the input optical power.

We assume the beam waist locates at $z=z_s$, and consider that the beam propagates from free space into SNNM under the off-waist incident condition. The ABCD matrix after the SNNM is expressed by

$$M=\left[ \begin{array}{cc} 1 & z\\ 0 & 1\\ \end{array} \right]\left[ \begin{array}{cc} \cos\left(\frac{z}{z_{p1}}\right) & -z_{p1}\sin\left(\frac{z}{z_{p1}}\right)\\ \frac{1}{z_{p1}}\sin\left(\frac{z}{z_{p1}}\right) & \cos\left(\frac{z}{z_{p1}}\right)\\ \end{array} \right]\left[ \begin{array}{cc} 1 & z_s\\ 0 & 1\\ \end{array} \right],$$
substitution of which into Eq. (7) with $q_0=iw_0^2$ yields
$$\frac{1}{q}=\frac{-[i\cos(z_1/z_{p1})+(1/z_{p1})(w_0^2+iz_s)\sin(z_1/z_{p1})]/z_{p1}}{[w_0^2-i(z-z_s)]\cos(z_1/z_{p1})/z_{p1}+[i+z/z_{p1}^2(w_0^2+iz_s)]\sin(z_1/z_{p1})}.$$

The location of the beam waist after nonlinear propagation is obtained by making the real part of Eq. (17) to be zero

$$z_{wai}=\frac{2z_s\cos(2z_1/z_{p1})/z_{p1}-[(w_0^4+z_s^2)/z_{p1}^2-1]\sin(2z_1/z_{p1})}{2[\cos^2(z_1/z_{p1})+(w_0^4+z_s^2)\sin^2(z_1/z_{p1})/z_{p1}^2-z_s\sin(2z_1/z_{p1})/z_{p1}]/z_{p1}}.$$

While, the beam waist can be determined by taking the inverse of imaginary part of Eq. (17)

$$w_{wai}=\sqrt{\frac{2w_0^2z_{p1}^2}{w_0^4+z_s^2+z_{p1}^2-(w_0^4+z_s^2-z_{p1}^2)\cos(2z_1/z_{p1})-2z_sz_{p1}\sin(2z_1/z_{p1})}},$$
from which we get the maximal waist
$$w^{max}_{wai}=\sqrt{\frac{w_0^4+z_s^2+z_{p1}^2+\sqrt{(w_0^4+z_s^2-z_{p1}^2)^2+4z_s^2z_{p1}^2}}{2w_0^2}}$$
under the following condition
$$\frac{2z_1}{z_{p1}}-\arctan\left(\frac{w_0^4+z_s^2-z_{p1}^2}{2z_sz_{p1}}\right)-\left(2n+\frac{1}{2}\right)\pi=0.$$

When the input power changes, the beam waist has different maxima as shown in Fig. 6. Besides, the maximal beam waists always locate near the output plane, which is shown the evolutions of beam width in Fig. 7 (Left) and the corresponding numerical simulation in Fig. 8(a). We also can intentionally adjust the input power to make the beam waist appear at our desired positions, which is in Fig. 7 (Right) and Fig. 8(b).

 figure: Fig. 6.

Fig. 6. Dependence of beam waist (solid blue curve) and its positions (dashed red curve) on the input power (in units of $1/\gamma _1^2$). The parameters are taken as $w_0=1,z_1=5$ and $z_s=1$.

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 figure: Fig. 7.

Fig. 7. Cascaded evolution of beam width under input powers, that is $P_0=0.0676/\gamma _1^2, 0.1/\gamma _1^2$ in the Left and the Right figures, and $z_1=5.0,z_{s1}=1,w_{m1}=15$ for the both. Red and yellow shadow areas indicate the linear and nonlinear propagations, respectively.

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 figure: Fig. 8.

Fig. 8. Cascaded propagations of beams. (a) and (b) correspond to the left and right ones in Fig. 7. Red lines denote the positions of beam waists.

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The configuration above can be applied not only in nonlinearity-mediated collimation of beams, but also in the mode matching [32]. In most practical situations, a Gaussian mode excited by a resonator always mismatches with the mode of the other optical system, because every optical system owns its respective eigenmodes. One key parameter characterised by Gaussian modes is the beam waist, and the waist matching can be readily realized in our configuration by adjusting the optical power. Furthermore, the collimating, the compressing and the expanding of optical beams are attributed to the nonlinearity induced phase correction. Therefore, the results obtained in the paper can be extended to the NNLSE under weak degree of nonlocality, and also can be realized in the actual systems such as the lead glass and the nematic liquid crystal.

5. Conclusion

Based on the ABCD matrix and using the $q$ parameter description, we investigated the cascaded evolutions of optical beams in the free spaces and the nonlocal nonlinear media. Two kinds of evolution processes for Gaussian beams, nonlinearity-mediated collimation and switching from breathers to solitons, were discussed in details. By adjusting the input optical power, the collimating, compressing and expanding of beams are convenient to be controlled. Besides, the configuration considered in the paper can also be applied in the mode matching during different optical systems in practical situations.

Funding

National Natural Science Foundation of China (12004238, 12174056); Open Subject of the Key Laboratory of Weak Light Nonlinear Photonics of Nankai University (OS 21-3); Natural Science Foundation of Henan Province (222102230068).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)
Fig. 1.
Fig. 1. Configuration for switching from breathers to solitons by a cascade of two segments of free spaces and one segment of nonlinear medium (b), and configuration for beam collimation by a cascade of two segments of nonlinear media and one segment of free space (a).

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Fig. 2.
Fig. 2. Cascaded evolutions of beam width under different input powers, that is $P_0=3177.9$ in the Left figure and $P_0=476.5$ in the Right figure. The other parameters are taken as $z_1=8.0,z_{s1}=0.476$ for the Left figure, $z_1=8.0,z_{s1}=7.86$ for the Right figure, and $w_{m1}=15, w_{m2}=20$ for the both. Red and yellow shadow areas indicate the linear and nonlinear propagations, respectively. The input power $P_0$ is given by Eq. (13), Eq. (15) and $w_0=1$.

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Fig. 3.
Fig. 3. Cascaded propagations of beams, the region between two red lines is for the linear propagation. (a) and (b) correspond to the left and right ones in Fig. 2

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Fig. 4.
Fig. 4. Same as Fig. 2 but plots are $P_0=33965, z_1=8.0,z_{s1}=0.347$ for the Left figure, and $P_0=65997, z_1=8.0,z_{s1}=0.436$ for the Right figure.

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Fig. 5.
Fig. 5. Same as Fig. 2 but plots are $P_c=23886, z_1=6.0,z_{s1}=0.312$ for the Left figure, $P_c=1042, z_1=6.0,z_{s1}=3.121$ for the Right figure, and $w_{m1}=w_{m2}=15$ for the both.

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Fig. 6.
Fig. 6. Dependence of beam waist (solid blue curve) and its positions (dashed red curve) on the input power (in units of $1/\gamma _1^2$). The parameters are taken as $w_0=1,z_1=5$ and $z_s=1$.

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Fig. 7.
Fig. 7. Cascaded evolution of beam width under input powers, that is $P_0=0.0676/\gamma _1^2, 0.1/\gamma _1^2$ in the Left and the Right figures, and $z_1=5.0,z_{s1}=1,w_{m1}=15$ for the both. Red and yellow shadow areas indicate the linear and nonlinear propagations, respectively.

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Fig. 8.
Fig. 8. Cascaded propagations of beams. (a) and (b) correspond to the left and right ones in Fig. 7. Red lines denote the positions of beam waists.

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Equations (21)

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i ∂ ψ ∂ z + 1 2 ∂ 2 ψ ∂ x 2 + Δ n ψ = 0 ,
R ( x ) = 1 2 π w m exp ⁡ ( − x 2 2 w m 2 ) ,
i ∂ ψ ∂ z + 1 2 ∂ 2 ψ ∂ x 2 − 1 2 γ 2 P 0 x 2 ψ = 0 ,
M n l ( i ) = [ A n l ( i ) B n l ( i ) C n l ( i ) D n l ( i ) ] = [ cos ⁡ ( z z p i ) − z p i sin ⁡ ( z z p i ) 1 z p i sin ⁡ ( z z p i ) cos ⁡ ( z z p i ) ]
M l ( i ) = [ 1 z s 0 1 ] .
ψ ( x , 0 ) = P 0 π w 0 exp ⁡ ( − i x 2 2 q 0 ) ,
q = A q 0 + B C q 0 + D ,
M = [ A B C D ] = { M n l ( 2 ) M l ( 1 ) M n l ( 1 ) ,   in Fig.1 (a) , M l ( 2 ) M n l ( 1 ) M l ( 1 ) ,   in Fig.1 (b) .
1 q = 1 ρ − i 1 w 2 .
M = M l ( 1 ) M n l ( 1 ) ,
1 q = ( w 0 2 / z p 1 2 ) sin ⁡ ( z 1 / z p 1 ) − ( i / z p 1 ) cos ⁡ ( z 1 / z p 1 ) ( w 0 2 − i z s 1 ) cos ⁡ ( z 1 / z p 1 ) / z p 1 + ( i + w 0 2 z s 1 / z p 1 2 ) sin ⁡ ( z 1 / z p 1 ) ,
d 2 w d z 2 − 1 w 3 + γ i 2 P 0 w = 0.
P 0 = P c = 1 γ 2 2 w w a i 4 ,
z s 1 = ( w 0 4 / z p 1 2 − 1 ) sin ⁡ ( 2 z 1 / z p 1 ) ( 2 / z p 1 ) [ cos 2 ⁡ ( z 1 / z p 1 ) + w 0 4 sin 2 ⁡ ( z 1 / z p 1 ) / z p 1 2 ] .
w w a i = w 0 1 1 + ( w 0 4 / z p 1 2 − 1 ) sin ⁡ ( z 1 / z 1 p ) 2 .
M = [ 1 z 0 1 ] [ cos ⁡ ( z z p 1 ) − z p 1 sin ⁡ ( z z p 1 ) 1 z p 1 sin ⁡ ( z z p 1 ) cos ⁡ ( z z p 1 ) ] [ 1 z s 0 1 ] ,
1 q = − [ i cos ⁡ ( z 1 / z p 1 ) + ( 1 / z p 1 ) ( w 0 2 + i z s ) sin ⁡ ( z 1 / z p 1 ) ] / z p 1 [ w 0 2 − i ( z − z s ) ] cos ⁡ ( z 1 / z p 1 ) / z p 1 + [ i + z / z p 1 2 ( w 0 2 + i z s ) ] sin ⁡ ( z 1 / z p 1 ) .
z w a i = 2 z s cos ⁡ ( 2 z 1 / z p 1 ) / z p 1 − [ ( w 0 4 + z s 2 ) / z p 1 2 − 1 ] sin ⁡ ( 2 z 1 / z p 1 ) 2 [ cos 2 ⁡ ( z 1 / z p 1 ) + ( w 0 4 + z s 2 ) sin 2 ⁡ ( z 1 / z p 1 ) / z p 1 2 − z s sin ⁡ ( 2 z 1 / z p 1 ) / z p 1 ] / z p 1 .
w w a i = 2 w 0 2 z p 1 2 w 0 4 + z s 2 + z p 1 2 − ( w 0 4 + z s 2 − z p 1 2 ) cos ⁡ ( 2 z 1 / z p 1 ) − 2 z s z p 1 sin ⁡ ( 2 z 1 / z p 1 ) ,
w w a i m a x = w 0 4 + z s 2 + z p 1 2 + ( w 0 4 + z s 2 − z p 1 2 ) 2 + 4 z s 2 z p 1 2 2 w 0 2
2 z 1 z p 1 − arctan ⁡ ( w 0 4 + z s 2 − z p 1 2 2 z s z p 1 ) − ( 2 n + 1 2 ) π = 0.
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