1 Introduction

It is the development trend of modern communication that optical communication replaces electrical communication, and all-optical waveguides, which include light-controlled optical waveguide (LCOW) and light-induced optical waveguide (LIOW), are key devices of all-optical network in the future. LCOW is a kind of simultaneous “light-controlling-light” technique, where the controlling light and the signal light act on the material at the same time. Generally LCOWs are nonlinear waveguides, because the controlling light and the signal light always influence each other. LCOW are widely used in so-called discrete solitons [1,2,3,4]. As a successive “light-controlling-light” technique, LIOW [5,6,7,8,9] means that the controlling light first induces a waveguide, and then the signal light propagates in it. LIOWs are mostly linear waveguides, as the inducing light is always not affected by the signal light. All-optical waveguides can also be used in all-optical switches [10,11,12].

We have previously studied LCOW based on photoisomerization of organic materials. Photoisomerization is a fascinating area of investigation because it leads to light-induced surface pattern of amorphous azo-containing polymer film and light-induced deformation of liquid crystal elastomers, some of them are completely unexpected and still unexplained [13]. There are countless review papers on this topic [14,15,16]. However, as a logical component of photoisomerization research, nonlinear effects associated with photoisomerization have not been adequately investigated, which can be seen from all these reviews. Related research area centers on micropic mechanism of second-order nonlinear optical polarizabilities, photo-induced poling, photoswitching, etc. [17,18,19]. Every step in the exploration process of photoisomerization nonlinear effects is helpful to systematization of photoisomerization research.

Due to the difference between contribution of a trans molecule and a cis molecule to the refractive index, the trans–cis photoisomerization leads to extremely strong nonlinear effects [20, 21]. We systematically explore nonlinear effects associated with photoisomerization in three steps. For the case of a single light beam, we [22, 23] revealed polarization effects of nonlinearity associated with photoisomerization for the first time. For the case of two light beams, we systematically studied photoisomerization-related LCOW. We found successively that both intensity information [24, 25] and polarization information [26] of the controlling light with stronger self-defocusing effects could be used to control the propagation of the signal light. Especially, we found that with the help of a controlling light, photoisomerization nonlinear effects could even be changed from a defocusing one to a focusing one, and we could achieve the conversion of dark solitons to bright solitons [24, 25]. As for the situation of LIOW based on photoisomerization, it just will be studied in this paper.

The light wave information mainly includes amplitude (or intensity), phase, and polarization. The polarization state of light was explored in photonic devices [27] and other linear fields [28]. However, as far as we know, the optical nonlinear effect is only related to the light intensity, so the problem for LCOW and LIOW is that the polarization information of light has not been used in all-optical waveguide. The more information that is utilized, the better the device performance. Traditional plane image technology that only uses the amplitude information of light wave evolves into three-dimensional holographic image technology when phase information is utilized. Borrowing the concept “holography”, we will propose a holographic all-optical waveguide based on photoisomerization nonlinear effects in this paper. Taking full advantage of the amplitude information and polarization information of light waves, the proposed holographic LIOW scheme will make the controlling of light beams more accurate and simpler.

2 Theoretical analyses

Suppose both inducing light and signal light are general elliptically polarized (EP) light, which is formed by a linearly polarized (LP) light passing through a quarter-wavelength slide, the angle \(\varphi\) between the polarization direction and the principal axis direction of the slide is called polarization angle and \(a_{j} = \cos^{2} \varphi_{j}\)(\(j = c,\;s\)) is the polarization degree of the inducing light (\(j = c\)) or the signal light (\(j = s\)). The subscript “s” is omitted for simplicity in the context. To avoid the rotation of the polarization direction of the signal light during the propagation process due to the phenomenon of photo-induced birefringence, the direction of the main axis of the polarization ellipse of the signal light and the inducing light must be consistent, that is, the fast axis directions of the glass slides are parallel to each other. First, according to the model of angular hole-burning [21], stationary density distribution of molecules in the trans form is given by [22]

$$N^{\prime}_{T} = N\frac{{I_{c} \sigma^{\prime}_{c} q^{\prime}_{c} + \gamma }}{{I_{c} \sigma_{c} q_{c} [a_{c} \cos^{2} \theta + (1 - a_{c} )\sin^{2} \theta \cos^{2} \psi ] + I_{c} \sigma^{\prime}_{c} q^{\prime}_{c} { + }\gamma }},$$
(1)

where N is the total density of material molecules, \(I_{c}\) is intensity of the inducing light, \(\sigma_{c}\) and \(q_{c}\) are the absorption cross-section of inducing light and quantum yield of trans molecules, respectively, \(\sigma^{\prime}_{c}\) and \(q^{\prime}_{c}\) are the absorption cross-section and quantum yield of cis molecules, respectively, \(\gamma\) is the thermal relaxation rate from the cis to trans. \(\theta\) and \(\psi\) are polar angle and azimuthal angle of orientation of trans molecule, respectively, in the spherical coordinate system, whose z axis is the main axis of the polarization ellipse.

Second, the density distribution function \(N^{\prime}_{T}\) becomes the initial condition of optical chemistry process of the signal light, and the final density distribution of trans molecules under action of the signal light is

$$N_{T} = N^{\prime}_{T} \frac{{I\sigma^{\prime}q^{\prime} + \gamma }}{{I\sigma q[a\cos^{2} \theta + (1 - a)\sin^{2} \theta \cos^{2} \psi ] + I\sigma^{\prime}q^{\prime} + \gamma }},$$
(2)

where \(I\) is intensity of the signal light, \(\sigma\) and \(q\) are the absorption cross-section of signal light and quantum yield of trans molecules, respectively, \(\sigma^{\prime}\) and \(q^{\prime}\) are the absorption cross-section and quantum yield of cis molecules, respectively.

Finally, refractive index perturbation arises from photoisomerization of the signal light [20].

$$\Delta n = \int_{\psi = 0}^{2\pi } {\int_{\theta = 0}^{\pi } {a\alpha (N_{T} - N^{\prime}_{T} )} } \cos^{2} \theta \sin \theta d\theta d\psi + (1 - a)\int_{\psi = 0}^{2\pi } {\int_{\theta = 0}^{\pi } {\alpha (N_{T} - N^{\prime}_{T} )} } \cos^{2} \psi \sin^{3} \theta d\theta d\psi .$$
(3)

It must be pointed out that the inducing light generates a nonlinear waveguide, which is related to the signal light.

Photoisomerization nonlinearity is wavelength related. We should select light with stronger isomerization activity as inducing light, which is always blue or green light, and select light with weaker activity as signal light, which is generally red light. Thus, we set \(\sigma_{c} q_{c} = 10\), \(\sigma^{\prime}_{c} q^{\prime}_{c} = 5\), \(\sigma q = 1\), \(\sigma^{\prime}q^{\prime} = 1\), \(\gamma { = }1\), \(\alpha { = }1\), \(N = 1\) when numerically calculating with formula (3) in the following, without loss of generality.

Exploring the effect of light intensity, we first consider the situation when both the inducing light and the signal light are LP light, which are of the same polarization direction. In this situation, \(a = a_{c} = 0\). The relation between nonlinear refractive index perturbation \(\Delta n\) and intensities of signal light \(I\) and of inducing light \(I_{c}\) is shown in Fig. 1a.

Fig. 1
figure 1

Intensity dependence of refractive index perturbation \(\Delta n(I,\;I_{c} )\) when a \(a = a_{c} = 0\), b \(a = 0\ominus a_{c} = 0.5\), and c \(a = 0.5\ominus a_{c} = 0\)

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It can be seen from Fig. 1a that the perturbation of refractive index is forever negative for LIOW, regardless of the intensity of the inducing light and the signal light. That is to say, the inducing light affects the nonlinear effects of the signal light; however, it never changes the defocusing nature associated with photoisomerization. It is worthy of notice that as for the case of LCOW, the controlling light could turn the defocusing nonlinear effect to a focusing one under certain conditions. Moreover, intensities undergo similar influence on the nonlinear effects regardless of the polarization states of both the inducing light and the signal light; see Fig. 1(b, c), where the polarization degrees of signal light and inducing light are \(a = 0\ominus a_{c} = 0.5\) and \(a = 0.5\ominus a_{c} = 0\), respectively.

Figure 2a reveals the influence of inducing light intensity on nonlinear effects of the signal light when intensities of uniform inducing lights are \(I_{c} { = }0,\;\;0.1,\;\;1,\;\;10,\)respectively. The bigger the intensity of inducing light (\(I_{c}\)), the smaller the absolute value of refractive index perturbation (\(\left| {\Delta n} \right|\)). So the inducing light weakens the self-defocusing effects. When \(I_{c}\) increases, defocusing nonlinear effects become weaker. As the initial state of photoisomerization of signal light, the distribution of cis molecules arises from photoisomerization of the inducing light weakens the self-defocusing effects of the signal light. Figure 2b reveals the influence of distribution of inducing light intensity by illustrating dependence of refractive index change on inducing light intensity when the signal light intensity is 0.1, 1, and 10, respectively. Here, two LP light beams (\(a = a_{c} = 0\)) are considered. Apparently, \(\Delta n\) increases as \(I_{c}\) increases. Assume that a light beam with Gaussian distribution \(I_{c} = e^{{ - r^{2} /\omega^{2} }}\)(\(\omega { = 1}\)) excites the material and then the signal light with uniform distribution propagates in the material. The LIOW shown as the relation of index change and the signal light intensity is illustrated by Fig. 3a. The nonlinear effect felt by signal light looks like a focusing one, although it essentially is not a nonlinear effect because refractive index change is not the function of the signal light. Nonlinear effect induced by the inducing light is some of a focusing one. A bright beam experiences self-defocusing effects itself; however, it induces a focusing waveguide. This is actually an interesting phenomenon. Considering that the signal light experiences self-defocusing effects, a bright inducing beam weakens the self-defocusing effects of signal light. Figure 3b is the waveguide induced by a light with periodically distributed intensity, which is generated by an amplitude mask and is described as \(I_{c} = I_{0} \cos x^{2} \cos^{2} y\). So photonic crystals can be generated with periodic bright beams. When a light with uniform distribution passes through the photonic crystal, the focusing effect associated with the photonic crystal, the self-defocusing effect, and natural diffraction of the signal light will achieve balance and discrete solitons may be formed.

Fig. 2.
figure 2

The relation between nonlinear refractive index perturbation and a signal light intensity under different uniform inducing light intensities; b inducing light intensity under different signal light intensities

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Fig. 3
figure 3

a Waveguide induced by a Gaussian beam; b photonic crystal generated by a uniform broad beam passed through an amplitude mask

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Figure 4 shows the polarization dependence of nonlinear effects, where intensities are set as \(I_{c} = 0.3\) and \(I = 1\). It is seen that from Fig. 4a that the dependence of polarization is not always monotonous, and there is a twist in the \(\Delta n(a,\;\;a_{c} )\) surface. Complicated polarization dependence of nonlinear effects is elaborated as follows.

Fig. 4
figure 4

a The relation between nonlinear refractive index perturbation and polarization degree of signal light and inducing light; influence of polarization of inducing light b and signal light (self-modulation) c

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Figure 4b illustrates the influence of polarization of the inducing light on nonlinear effects. As for the circularly polarized (CP) signal light (\(a = 0.5\)), when the polarization degree of the inducing light turns smaller (\(a_{c} = 0 \to 0.5\), from linear polarization to circular polarization), \(\left| {\Delta n} \right|\) decreases, which denotes that the defocusing effect becomes weaker. As for a LP signal light (\(a = 0\)), when the degree of polarization of the inducing light turns smaller, \(\left| {\Delta n} \right|\) increases, and the defocusing effect becomes stronger. Figure 4c illustrates the influence of polarization of the signal light on nonlinear effects. As for the CP inducing light (\(a_{c} = 0.5\)), when the polarization degree of the signal light turns smaller, \(\left| {\Delta n} \right|\) decreases monotonically, which denotes that the effect becomes weaker gradually. As for the LP inducing light (\(a_{{\text{c}}} = 0\)), when the polarization degree of the signal light turns smaller, \(\left| {\Delta n} \right|\) first decreases until \(a \approx 3\), and then increases, which denotes that the defocusing effect first becomes weaker and weaker, and then become stronger and stronger.

Theoretically comparing the influence of intensity and polarization on index change, we find that effects of polarization are generally weaker than that of intensity; see Fig. 5. Thus, when inducing waveguide, we can coarsely control the waveguide by adjusting intensity and distribution of the inducing light, and then experience fine control by adjusting the polarization of the inducing light; thereupon, a holographic LIOW scheme is proposed. Holographic all-optical waveguide makes the guide of light beam more precise and more convenient. Moreover, influence of intensity of signal light is obviously greater than that of inducing light, from (a) and (b) of Fig. 5

Fig. 5
figure 5

Comparison of the influence of intensity and that of polarization on index change

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3 Experimental confirmation

Now we give some of our qualitative experimental confirmation. The Z-scan experimental12 setup is shown in Fig. 6. The part enclosed by the dotted line is the inducing light path. The intensity of uniform inducing light with a wavelength of \(\lambda_{2} { = }514.5nm\) emitted from the Ar + laser is controlled by the combination of two polarizers, P3 and P4. The LP light becomes EP light through the \(\lambda_{1} {/}4\) (\(\lambda_{2} {/}4\)) slide (quarter-wave plate). The concave lens X and the convex lens L2 are combined to expand the beam. Similarly, the intensity of the signal light with a wavelength of \(\lambda_{1} { = }632.8nm\) is determined by polarizers P1 and P2. To avoid the change of the polarization state of the light in the sample due to the photo-induced birefringence (including the rotation of the polarization direction), the fast axis direction of the slide \(\lambda_{1} {/}4\) and \(\lambda_{2} {/}4\) is kept parallel. P1 and P2 are fixed on a platform and by rotating the platform, the polarization state of the signal light can be changed without changing the intensity. A lens of focal length \(f = 111mm\) provides a tight focus of the laser beam, whose spatial mode of the laser is close to Gaussian TEM00, and the beam-waist radius is \(\omega_{0} \approx 62\mu m\) at the focal point. An aperture with a linear transmittance of 0.15 is fixed at the position about 70 cm from the focal plane. A photodetector (Field Master, Coherent Inc.) D is used to detect the light power behind the aperture. The sample is poly(methyl methacrylate) (PMMA) film doped with disperse red 13 (DR13) (1.6% by weight), which was prepared from a solution containing an appropriate amount of dye, polymer, and chloroform. The thickness of the film is measured to be \(l \approx 52\mu m\). The absorption spectrum is presented in reference [22].

Fig. 6.
figure 6

Setup of Z-scan experiment

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Normalized transmittance of the signal light passing through the aperture is measured as the sample is moved along the propagation path z of the focused Gaussian beam. Peak-valley difference \(\Delta T_{p - v}\) of Z-scan curves denotes the magnitude of the refractive index perturbation.

Experimental results are shown in Fig. 7 when central intensity of the signal light at the focal point is \(I_{0} = 0.80W/cm^{2}\). Py3 is the Z-scan curve of signal CP light (\(a = 0.5\)) before the sample is induced (\(I_{c} = 0\)). py1, py2, and py4 are the Z-scan curves of LP light (\(a = 0\)), EP light (\(a = 0.25\)), and CP light, respectively, after the sample is induced by an uniform light with intensity of \(I_{c} = 0.42W/cm^{2}\). \(\Delta T_{p - v}\) of py1–py4 are 0.39, 0.34, 0.23, 0.17, respectively.

Fig. 7
figure 7

Normalized transmittance data (y1–y4) and Z-scan curves (py1–py4). The suffixes 1, 2, and 4 correspond to situations LP, EP, and CP, respectively, and inducing light intensity is \(I_{c} = 0.42W/cm^{2}\), suffix 3 corresponds to the case of CP before the sample is induced

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The above experimental results preliminarily and qualitatively confirm some of theoretical results. Comparing py3 with py4, \(\Delta T_{p - v}\) of the latter is smaller, which indicates that optical induction weakens the self-defocusing effects. This result is consistent with Figs. 1 and 2a. Comparing py1, py2, and py4, \(\Delta T_{p - v}\) declines in turn. That is to say, as for the case of CP inducing light, nonlinear effects become weaker and weaker when signal light changes gradually from LP light to CP light. This is consistent with Fig. 4c (the case of \(a_{c} = 0.5\)). Moreover, the influence of intensity of inducing light is rather smaller than that of polarization information of the signal light. In this way, the dependence of nonlinear effects on intensity information of inducing light and on polarization information of signal light is qualitatively verified. However, we failed to find obvious influence of polarization of the inducing light on nonlinear effects. Perhaps this dependence is too weak to be observed for the sample we used. Actually this is somehow consistent with Fig. 5a.

Up to now, the role of polarization has been explored in three situations: polarization information of a signal light6,7, polarization information of the controlling light in a light-controlled waveguide10, and polarization information of the inducing light in a light-inducing waveguide. This work deals with LIOW, while Ref.10 dealt with LCOW; they are two parallel works. The theoretical models and results of the two papers differ from each other. For example, for the case of LIOW, inducing light affects the nonlinear effects of the signal light; however, it never changes the defocusing nature associated with photoisomerization, while for the case of LCOW, the controlling light could turn the defocusing nonlinear effects to a focusing one under certain conditions.

The more information a photonic device utilizes, the better its performance. For holographic imaging technology, by additional use of phase information besides amplitude information of light wave, the two-dimensional plane image can be evolved into a three-dimensional image. Similarly, the application of polarization information is joined to the proposed holographic all-optical waveguide scheme. Compared with the traditional all-optical waveguide, the performance of holographic waveguide will be greatly improved, making the control of light more precise and easier. The influence of polarization degree on the beam propagation is much smaller than that of the light intensity, thus the polarization information can be used for fine-tuning of the waveguide. In addition, the operation of changing the degree of polarization is simpler than changing the light intensity distribution. The former only needs to rotate the glass slide, while the latter requires complex operations such as beam shaping. Generally speaking, defocusing effects associated with photoisomerization are too strong to form dark solitons, only if in materials with weaker nonlinear effects, such as DR1/PMMA with low doping concentration and with red light. To form dark solitons in materials with strong defocusing effects such as DR13/PMMA, one should weaken defocusing effects by forming holographic waveguide. To precisely control the beam propagation, such as to form optical spatial solitons, we can adjust the intensity and distribution of the signal light at first, and when the beam is close to the soliton form, we modify the intensity and polarization degree of the inducing light in turn to fine-tune the signal light, so that the optical solitons can be precisely and easily formed.

4 Summary

In summary, the all-optical waveguide theory based on photoisomerization nonlinear effects is systematically and deeply studied, and a nonlinear holographic all-optical waveguide scheme is proposed for the first time. We find that LIOW based on photoisomerization weakens the defocusing effects; however, it never changes the defocusing nature. Besides, a bright inducing light beam generates a focusing waveguide. Polarization states of both inducing light and signal light also affect propagation of signal light. For a CP signal light, when the polarization degree of the inducing light turns smaller, the defocusing effect becomes weaker; as for a LP signal light, when the polarization degree of the inducing light turns smaller, the defocusing effect becomes stronger. For the CP inducing light, when the polarization degree of the signal light turns smaller, the defocusing effect becomes weaker gradually; as for the LP inducing light, when the polarization degree of the signal light turns smaller, the defocusing effect first becomes weaker and weaker, and then becomes stronger and stronger. Part of the theoretical results has been qualitatively confirmed by Z-scan experiments.

The proposed holographic all-optical waveguide scheme will fundamentally improve the performance of the all-optical waveguide, making the control of the beam propagation more accurate and more convenient. The study of double-beam nonlinear effects based on photoisomerization systematizes the study of nonlinear effects of photoisomerization. The new “holography” idea will provide important enlightenment to the research and development of nonlinear optics in the future.