Since the advent of optical waveguides, guided-wave optics technology has been the focus of optics communication^[1–5]. It has developed quickly and gives powerful impetus to technology innovation in our modern life and has become the focus of the information industry^[6–8]. Its propagation characteristics, such as the near-field and far-field characteristics, are often depicted by the mode-field half-width and divergence half-angle, respectively, and a problem is how to exactly to determine them.

As one of the most important parameters for slab waveguides, mode-field half-width relates to various transmission characteristics and attracts more attention^[9,10]. According to ISO 11146-1: 2005^[11], a combination of mode-field half-width, divergence half-angle, and beam propagation factor produces a general method to evaluate practical beam quality. Therefore, the divergence half-angle of the far-field should be of concern as well.

There are two kinds of definitions of mode-field half-width for slab waveguides based on the second-moment and differential operator^[12,13], and then there are the second-moment and differential operator divergence half-angle according to the two kinds of definitions. Researchers have also presented a definition for the two parameters^[14]. However, they are different from each other.

In this work, definitions for mode-field half-width and divergence half-angle referring to different kinds of methods were researched. Then, numerical results of these mode-field half-width and divergence half-angle of the ${\mathrm{TE}}_0$ mode in step-index slab waveguides are given and numerical calculation proves that some of them are more suited than others to describe the near-field and far-field based on their respective power in the bucket. These conclusions may provide a theoretical foundation for analyzing the propagation characteristics of optical waveguides.

In the space domain, there are many kinds of traditional definitions of laser beam spot size for illuminating the propagation characteristics of laser beams; thus, we try to define mode-field half-width of a slab waveguide in view of these definitions and discuss their properties. The entropy-based mode-field half-width ${\omega }_E$ is presented according to the Shannon’s information-entropy formula^[15]$${\omega }_E=\frac{1}{2}\exp [-{\int }_{-\infty }^{\infty }\frac{{|E(x)|}^2}{{\int }_{-\infty }^{\infty }{|E(x)|}^2\mathrm{d}x}\ln \frac{{|E(x)|}^2}{{\int }_{-\infty }^{\infty }{|E(x)|}^2\mathrm{d}x}\mathrm{d}x],$$(1)where $E(x)$ is the near-field distribution.

The mode-field half-width ${\omega }_{\mathrm{PIB}}$ is defined in terms of the light of the power in the bucket $$\eta ({\omega }_{\mathrm{PIB}})=\frac{{\int }_{-{\omega }_{\mathrm{PIB}}}^{{\omega }_{\mathrm{PIB}}}{|E(x)|}^2\mathrm{d}x}{{\int }_{-\infty }^{\infty }{|E(x)|}^2\mathrm{d}x},$$(2)where $\eta =86.5\%$, and ${\omega }_N$ is given as $$I({\omega }_N)=\frac{I_{\max }}{\mathrm{n}}=\frac{{|E(x)|}_{\max }^2}{\mathrm{n}},$$(3)where $\mathrm{n}={\mathrm{e}}^2\approx {(2.7183)}^2$ and the subscript max refers to the maximum.

For slab waveguides, Liang and Hayata presented two definitions based on the second-order-moment ${\omega }_{\mathrm{SM}}$ and differential operator ${\omega }_{\mathrm{DO}}$, respectively^[12,13]$${\omega }_{\mathrm{SM}}={\left[\frac{4{\int }_{-\infty }^{\infty }{|E(x)|}^2{x}^2\mathrm{d}x}{{\int }_{-\infty }^{\infty }{|E(x)|}^2\mathrm{d}x}\right]}^{\frac{1}{2}},$$(4)$${\omega }_{\mathrm{DO}}={\left[\frac{{\int }_{-\infty }^{\infty }{|E(x)|}^2\mathrm{d}x}{{\int }_{-\infty }^{\infty }{\left|\frac{dE(x)}{dx}\right|}^2\mathrm{d}x}\right]}^{\frac{1}{2}}.$$(5)

Researchers have also introduced a definition for mode-field half-width ${\omega }_{\mathrm{SD}}$^[14]$${\omega }_{\mathrm{SD}}={\left[\frac{4{\int }_{-\infty }^{\infty }{|E(x)|}^2{x}^2\mathrm{d}x}{{\int }_{-\infty }^{\infty }{\left|\frac{dE(x)}{dx}\right|}^2\mathrm{d}x}\right]}^{\frac{1}{4}},$$(6)where the subscript denotes the relationship to the definitions of mode-field half-width.

The near-field characteristic of the ${\mathrm{TE}}_0$ mode in a step-index slab waveguide is presented as an example to find whether these definitions are suitable for slab waveguides. From the analytical field solution of the ${\mathrm{TE}}_0$ mode $$E(x)=\cos \left(U\frac{x}{a}\right)|x|\le a,$$(7)$$E(x)=\cos (U)\exp (W)\exp (-W\left|\frac{x}{a}\right|)|x|>a,$$(8)where $U=a{[{(k_0{\mathrm{n}}_1)}^2-{\mathrm{\beta }}^2]}^{1/2}$ and $W=a{[{\mathrm{\beta }}^2-{(k_0{\mathrm{n}}_2)}^2]}^{1/2}$ are the normalized standing wave parameter and the normalized evanescent wave parameter, respectively; the normalized frequency $V^2=U^2+W^2$; $a$ is the half thickness of the core layer; $k_0=2\pi /\lambda$ is the wave number in vacuum; β is the propagation constant; $\lambda$ is the wavelength of the electromagnetic wave in vacuum; ${\mathrm{n}}_1$ and ${\mathrm{n}}_2$ are the refractive indices of the core layer and cladding layer, respectively. We can obtain the mode-field half-width and make various comparisons.

Numerical results of mode-field half-width based on different definitions are displayed in Fig. 1. Figure 1 shows that with the increase of normalized frequency, all the mode-field half-width decreases and the difference among them is large when the normalized frequency is small. We can also see that ${\omega }_{\mathrm{PIB}}$ is smaller than the other kinds of mode-field half-width.

Furthermore, the power in the bucket as function of mode-field half-width versus normalized frequency is shown in Fig. 2. It is obvious that all kinds of mode-field half-widths can exactly reflect the energy concentration except for power in the bucket-based mode-field half-width ${\omega }_{\mathrm{PIB}}$, and most of beam energy is occupied in the range of mode-field half-width. If $V=1.00$ as an example, the power in the bucket is $\eta ({\omega }_E)=94.65\%$, $\eta ({\omega }_{\mathrm{PIB}})=86.5\%$, $\eta ({\omega }_N)=91.92\%$, $\eta ({\omega }_{\mathrm{SM}})=94.59\%$, $\eta ({\omega }_{\mathrm{DO}})=92.90\%$, and $\eta ({\omega }_{\mathrm{SD}})=93.78\%$. Thus, we can draw a conclusion that similarly to the mode-field half-widths of a slab waveguide ${\omega }_{\mathrm{SM}}$, ${\omega }_{\mathrm{DO}}$ and ${\omega }_{\mathrm{SD}}$, ${\omega }_E$ and ${\omega }_N$ can be a good choice for description of the near-field characteristic.

The far-field characteristic of a slab waveguide is described by the divergence half-angle and we can introduce their definitions according to the mode-field half-width definitions. The entropic divergence half-angle ${\theta }_E$ is defined as $${\theta }_E=\frac{1}{2}\exp [-{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{{|s(\gamma )|}^2}{{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{|s(\gamma )|}^2\mathrm{d}\gamma }\ln \frac{{|s(\gamma )|}^2}{{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{|s(\gamma )|}^2\mathrm{d}\gamma }\mathrm{d}\gamma ],$$(9)where $S(\gamma )={\int }_{-\infty }^{\infty }E(x)\exp (-ikx\sin \gamma )\mathrm{d}x$ is the spatial frequency spectrum, $\gamma$ is the inclination angle, and ${\theta }_{\mathrm{PIB}}$ and ${\theta }_N$ are given in the following $$\eta =\frac{{\int }_{-{\theta }_{\mathrm{PIB}}}^{{\theta }_{\mathrm{PIB}}}{|S(\gamma )|}^2\mathrm{d}\gamma }{{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{|S(\gamma )|}^2\mathrm{d}\gamma },$$(10)$$I({\theta }_N)=\frac{I_{\text{max}}}{\mathrm{n}}=\frac{{|S(\gamma )|}_{\max }^2}{\mathrm{n}}.$$(11)Accordingly, the second-order moment ${\theta }_{\mathrm{SM}}$ as well as differential operator divergence half-angle ${\theta }_{\mathrm{DO}}$ are introduced $${\theta }_{\mathrm{SM}}={\left[\frac{4{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{|S(\gamma )|}^2{\gamma }^2\mathrm{d}\gamma }{{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{|S(\gamma )|}^2\mathrm{d}\gamma }\right]}^{\frac{1}{2}},$$(12)$${\theta }_{\mathrm{DO}}={\left[\frac{{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{|S(\gamma )|}^2\mathrm{d}\gamma }{{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\left|\frac{dS(\gamma )}{d\gamma }\right|}^2\mathrm{d}\gamma }\right]}^{\frac{1}{2}},$$(13)and the new definition ${\theta }_{\mathrm{SD}}$ is $${\theta }_{\mathrm{SD}}={\left[\frac{4{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{|S(\gamma )|}^2{\gamma }^2\mathrm{d}\gamma }{{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\left|\frac{dS(\gamma )}{d\gamma }\right|}^2\mathrm{d}\gamma }\right]}^{\frac{1}{4}}.$$(14)

For the ${\mathrm{TE}}_0$ mode, the spatial frequency spectrum of the far-field is given as^[16]$$S(\gamma )=\frac{a{V}^2}{F^2+W^2}[\mathrm{sinc}\left(\frac{F-U}{\pi }\right)+\mathrm{sinc}\left(\frac{F+U}{\pi }\right)],$$(15)where $F=ka\sin \gamma$ is the spatial frequency.

Combining Eqs. (9)–(14) with Eq. (15), relational curve between divergence half-angle and normalized frequency is shown in Fig. 3. It is obvious that the divergence half-angle increases with increasing normalized frequency and deviation also becomes large. Similarly to the mode-field half-width, ${\theta }_{\mathrm{PIB}}$ is the smallest compared to other kinds of divergence half-angles.

More specifically, the power in the bucket based on all other kinds of divergence half-angles is far greater than $\eta ({\theta }_{\mathrm{PIB}})=86.50\%$ displayed in Fig. 4. For example, $\eta ({\theta }_E)=94.65\%$, $\eta ({\theta }_{\mathrm{PIB}})=86.50\%$, $\eta ({\theta }_N)=91.71\%$, $\eta ({\theta }_{\mathrm{SM}})=94.56\%$, $\eta ({\theta }_{\mathrm{DO}})=92.86\%$, and $\eta ({\theta }_{\mathrm{SD}})=93.75\%$ when $V=1$ according to the aforementioned parameters. Consequently, ${\theta }_E$ and ${\theta }_N$ can be used for describing the far-field characteristic compared to the divergence half-angle of slab waveguides ${\theta }_{\mathrm{SM}}$, ${\theta }_{\mathrm{DO}}$, and ${\theta }_{\mathrm{SD}}$.

In conclusion, mode-field half-widths and divergence half-angles based on different kinds of definitions are investigated. Numerical calculation proves that ${\omega }_E$, ${\omega }_N$, ${\omega }_{\mathrm{SM}}$, ${\omega }_{\mathrm{DO}}$, as well as ${\omega }_{\mathrm{SD}}$ are fit to depict the near-field of slab waveguides, and ${\theta }_E$, ${\theta }_N$, ${\theta }_{\mathrm{SM}}$, ${\theta }_{\mathrm{DO}}$, as well as ${\theta }_{\mathrm{SD}}$ are suitable for illuminating the characteristic of the far-field. These conclusions may provide theoretical groundwork for further researching beam propagation characteristics of slab waveguide.