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Abstract
For the beam spreading case, the propagation formulae of Gaussian–Schell model (GSM) beams through general optical systems in Kerr media are derived, and the propagation characteristics of GSM beams through an ideal thin lens in Kerr media are studied in detail. It is shown that the size and position of the beam waist can be controlled by the Kerr effect. Furthermore, the formula of the focal shift of the GSM beams focused by an ideal thin lens in Kerr media is also derived. It is found that in self-focusing media the focal shift decreases as the beam power or the beam coherence degree increases. In addition, there exists a maximum of the focal shift, and the formula of the focal shift maximum is derived. On the other hand, for the beam self-focusing case, the focusing characteristics of GSM beams focused by an ideal thin lens in Kerr media are also investigated.
© 2020 Optical Society of America
1. INTRODUCTION
In 1965, Kogelnik introduced the ABCD law to study the propagation of fully coherent Gaussian (GS) beams through optical systems in a vacuum by defining the complex beam parameter [1]. It is known that partially coherent beams (PCBs) may have advantages compared with fully coherent beams. Gbur and Wolf clarified under what circumstances a PCB is less sensitive to the turbulence effect than a fully coherent one [2]. Furthermore, our group found that a PCB has higher threshold critical power than a fully coherent beam in the atmosphere [3]. The Gaussian–Schell model (GSM) beam is a typical example of PCBs. In 1986, Turunen and Friberg defined the complex parameters of GSM beams propagating in a vacuum and generalized the ABCD law for GSM beams [4]. The ABCD law was applied widely to studied the fully and partially coherent laser beams’ propagation through optical systems [5–9].
It is noted that these studies mentioned above were carried out concerning the propagation of fully and partially coherent laser beams in linear media [5–9]. In 1983, Bélanger and Pare defined the complex parameter of GS beams propagating in nonlinear media [10]. After that, the ABCD law was applied to study the nonlinear propagation and transformation of GS beams [11–14]. In 2019, our group proved that the ABCD law is also valid for GSM beams in nonlinear media if a new complex beam parameter is adopted [15]. However, until now, the nonlinear propagation of GSM beams has not been investigated by using the ABCD law.
On the other hand, the focal shift of laser beams is of theoretical and practical interest. In 1982, Li and Wolf found the on-axis actual focal plane for a GS beam is closer to the focusing lens than the geometrical focal plane, and they called the phenomenon the focal shift [16]. Then Carter et al. investigated the focal shift of an unapertured GS beam propagating an ideal thin lens in a vacuum, found the focal shift related to the effective Fresnel number, and successfully observed the focal shift in experiments [17,18]. In 1992, Li presented the focal shift formula of apertured GS beams propagating an ideal thin lens in a vacuum [19]. In 1995, Lü et al. gave the focal shift formulae of GSM beams focused by both an unapertured and apertured lens, and they found that the focal shift decreases as the coherent parameter and the truncation parameter increase [20]. In 2009, Wang et al. confirmed the focal shift of GSM beams focused by an apertured thin lens in a vacuum by experiments [21]. However, until now, the focal shift of GSM beams focused by a thin lens in Kerr media has not been examined.
In this paper, based on the ABCD law, the propagation characteristics of GSM beams through optical systems in Kerr media are studied, where both the beam spreading and the beam self-focusing cases are considered. For the beam spreading case, the propagation formulae of GSM beams through general optical systems in Kerr media are derived, the propagation characteristics of GSM beams through an ideal thin lens in Kerr media are studied, and the focal shift of GSM beams focused by an ideal thin lens in Kerr media is also investigated in detail. On the other hand, for the beam self-focusing case, the focusing characteristics of GSM beams focused by an ideal thin lens in Kerr media is also studied.
2. PROPAGATION OF GSM BEAMS THROUGH OPTICAL SYSTEMS IN KERR MEDIA FOR THE BEAM SPREADING CASE
When a powerful laser beam propagates in a Kerr medium, the refraction index of the medium will change with intensity as $n = {n_0} + {n_2}I$, where $ n_{0} $ is the linear refractive index, $ n_{2} $ is the nonlinear refractive index, and $ I $ is the beam intensity. In addition, the ${n_2} \gt 0$ and ${n_2} \lt 0$ correspond to self-focusing and self-defocusing media, respectively. ${P_{{\rm cr}}} = {\varepsilon _0}cn^2_0\pi ({1 + {\alpha ^{- 2}}})/({{k^2}{n_2}})$ is the self-focusing critical power of GSM beams in Kerr media [15], where ${\varepsilon _0}$ is the vacuum permittivity, $ c $ is the speed of light in a vacuum, $ k $ is the wavenumber in the linear media, and $\alpha$ is the degree of global coherence of the GSM beams and it is independent of the propagation distance $ z $ [15]. A GSM beam will spread and its intensity keep in a Gaussian profile when the GSM beam propagates in self-defocusing or in self-focusing media (but the beam power $ P_{0} $ should satisfy ${P_0} \lt {P_{{\rm cr}}}$ in self-focusing media), while a GSM beam will focus when ${P_0} \gt {P_{{\rm cr}}}$ in self-focusing media. It is noted that the beam spreading case is considered in Section 2.
A. Propagation Formulae of GSM Beams through General Optical Systems in Kerr Media
As is shown in Fig. 1, a GSM beam propagates through a general optical system in Kerr media. The $ q_{1} $, $ w_{1} $, and $ R_{1} $ are the complex beam parameter, the beam width, and the curvature radius of the GSM beam at the $ s_{1} $ plane in the object space, respectively. The transfer matrix of the optical system is expressed as $\left[{\begin{array}{cc}a&b\\c&d\end{array}}\right]$. The $ q_{2} $, $ w_{2} $, and $ R_{2} $ are the complex beam parameter, the beam width, and the curvature radius of the GSM beam at the $ s_{2} $ plane in the image space, respectively.
The complex beam parameter of GSM beams propagating in Kerr media is defined as [15]
where ${M^{2}} = [({1 + {\alpha ^{- 2}}}){({1 - \eta})]^{1/2}}$ is the $ M^{2} $ factor of GSM beams propagating in Kerr media and $\eta = {P_0}/{P_{{\rm cr}}}$ is the ratio of the effect of the photorefraction to the diffraction of GSM beams. It is shown that the complex beam parameters $ q_{1} $ and $ q_{2} $ of the GSM beams propagating through a general optical system before and after should satisfy the ABCD law [15], i.e.,The transfer matrix from the $ s_{1} $ plane to the $ s_{2} $ plane is expressed as
Substituting Eq. (1) into Eq. (2) and separating the real part and the imaginary part, we obtain
As ${R_1} \to \infty$ and ${R_2} \to \infty$, Eqs. (4) and (5) can be simplified as
It is noted that Eqs. (6) and (7) show the transformation of GSM beams propagating through optical systems in Kerr media between the two beam waists.
Substituting Eq. (3) into Eqs. (6) and (7), we can obtain the position and the width of the beam waist in the image space, i.e.,
When $\alpha \to \infty$, Eqs. (8) and (9) reduce to those of fully coherent GS beams propagating through general optical systems in Kerr media, and they reduce to those of GSM beams propagating through general optical systems in vacuum when ${n_2} = 0$, which are in agreement with those in Ref. [14] and Ref. [8], respectively.
B. Propagation Characteristics of GSM Beams through an Ideal Thin Lens in Kerr Media
For an ideal thin lens with focal length $ f $, the transfer matrix is
Substituting Eq. (10) into Eqs. (8) and (9), we obtain the expressions of the position $ s_{2} $ and the width $ w_{2} $ of the beam waist in the image space, i.e.,
Letting $\partial {w_2}/\partial {s_1} = 0$ in Eq. (12), we obtain the expression ${w_{2{\max }}}$ of the waist size maximum in the image space when ${s_1} = f$, i.e.,
Equation (13) indicates that the ${w_{2{\max }}}$ decreases as $ \alpha $ or $ P_{0} $ increases because ${Z_{01}}$ increases (i.e., the beam collimation in the object space is improved).
It is interesting that the size and position of the beam waist can be controlled by the Kerr effect. For example, to reach two GSM beams having the same size and position of the beam waist, the following equation should be satisfied:
To study the characteristics of GSM beams through a thin lens in Kerr media, several numerical calculation examples are given. Unless otherwise stated, in this paper the numerical calculation parameters are taken as $f = 3\;{\rm m} $, ${w_1} = 0.8\;{\rm mm} $, $\lambda = 0.8\; \unicode{x00B5}{\rm m}$, and only the self-focusing media are considered. The Kerr effect is also called the self-focusing effect in self-focusing media. It is mentioned that the numerical calculation examples in self-defocusing media are omitted here because the propagation characteristics are straightforward.
Changes of the relative beam width ${w_2}/{w_1}$ versus the relative position $s_{1}/\!{f}$ of GSM beams propagating through a thin lens in Kerr media are shown in Fig. 2. It shows that there exists a maximum of the image waist size when ${s_1}/f = 1$, which is consistent with the result in Eq. (13). In addition, $ w_{2} $ decreases as $ \alpha $ or $ P_{0} $ increases because the self-focusing effect becomes stronger. It is noted that the value of ${{s}_1}/{f}$ is minus that of the position of the beam waist is located on the right side of the lens.
Fig. 2. Relative beam width ${{w}_2}/{{w}_1}$ versus relative position ${{s}_1}/{f}$. Solid curves: ${P}_0/{{P}_{{\rm crGS}}} = 0.2$; dashed curves: ${P}_0/{{P}_{{\rm crGS}}} = 0.8$.
Figure 3 shows the changes of the relative position ${s_2}/f$ in the image space versus the relative position ${s_1}/f$ in the object space. One can see that the curves are symmetric about ${s_1}/f = 1$, and there exist a minimum and a maximum of $ s_{2} $ versus $ s_{1} $. Letting $\partial {s_2}/\partial {s_1} = 0$ in Eq. (11), we can obtain the minimum and maximum of $ s_{2} $ when ${s_{1{\min }}} = f - {Z_{01}}$ and ${s_{1{\max }}} = f + {Z_{01}}$, respectively, i.e.,
Fig. 3. Relative position ${{s}_2}/{f}$ in the image space versus relative position ${{s}_1}/{f}$ in the object space. Solid curves: ${P}_0/{{P}_{{\rm crGS}}} = 0.2$; dashed curves: ${P}_0/{{P}_{{\rm crGS}}} = 0.8$.
Obviously, the distance between ${s_{1{\min }}}$ and ${s_{1{\max }}}$ is $\Delta s_1 = {s_{1{\max }}}-{s_{1{\min }}}=2Z_{01}$, and the distance between ${s_{2{\min }}}$ and ${s_{2{\max }}}$ is $\Delta s_2 = {s_{2{\max }}}-{s_{2{\min }}}=f^2/Z_{01}$. From Fig. 3 one can see that, as $ \alpha $ or $ P_{0} $ increases, ${s_{1{\max }}}$ increases and ${s_{2{\max }}}$ decreases, while ${s_{1{\min }}}$ decreases and ${s_{2{\min }}}$ increases, and so ${{\Delta s}_1}$ increases and ${{\Delta s}_2}$ decreases. The self-focusing effect becomes stronger when $ \alpha $ or $ P_{0} $ increases, which results in the beam collimation in the object space being improved. Thus, the position of the beam waist in the image space is closer to the ${z} = {f}$ plane. This is the physical reason why ${{s}_{2{\max }}}$ decreases while ${{s}_{2{\min }}}$ increases when $ \alpha $ or $ P_{0} $ increases. In addition, $ s_{2} $ reaches the asymptotic value $ f $ when $| {{s_1}/{f}} | \to \infty$, and this result is independent of both $ \alpha $ and $ P_{0} $.
C. Focal Shift of GSM Beams Focused by an Ideal Thin Lens in Kerr Media
It is known that, when a GS beam propagates through a thin lens, the actual focal plane is shifted from the plane given by geometrical optics [16,17]. This apparent shift in the geometrical focal plane is called the focal shift, which is expressed as [17]
In this paper, this definition of the focal shift is adopted for GSM beams propagating through an ideal thin lens in Kerr media. According to Ref. [17], we have
where ${Z_{02}} = \pi w^2_2/({\lambda {M^2}})$ is the Rayleigh range of GSM beams in the image space. Substituting Eq. (17) into Eq. (16), we obtain the formula of focal shift $\Delta$ asSubstituting Eqs. (11) and (12) into Eq. (18), the focal shift $\Delta$ can be rewritten as
It is noted that Eq. (19) reduces to the focal shift of GSM beams propagating through a thin lens in a vacuum when ${n_2} = 0$, and to the focal shift of fully coherent GS beams propagating through a thin lens in vacuum when ${n_2} = 0$ and $\alpha \to \infty$, which is in agreement with those in Ref. [20] and Ref. [19], respectively.
Changes of the relative focal shift ${\Delta}/f$ versus relative position ${s_1}/f$ are shown in Fig. 4. One can see that the focal shift ${\Delta}$ decreases as $ \alpha $ or $ P_{0} $ increases because the self-focusing effect becomes stronger. It is noted that ${\Delta}$ decreases as $ P_{0} $ decreases or $ \alpha $ increases in self-defocusing media. In addition, the focal shift approaches zero when $| {{s_1}} |/{Z_{01}} \to \infty$, and this result is independent of both $ \alpha $ and $ P_{0} $. There exists a maximum of the focal shift versus $ s_{1} $. By setting $\partial {\Delta}/\partial {s_1} = 0$ in Eq. (19), we obtain the expression ${\Delta _{{\max }}}$ of the focal shift maximum when ${s_{1{\max }}} = Q + 3f/4$, i.e.,
Fig. 4. Relative focal shift ${\Delta}/{f}$ versus relative position ${{s}_1}/{f}$. Solid curves: ${P}_0/{{P}_{{\rm crGS}}} = 0.2$; dashed curves: ${P}_0/{{P}_{{\rm crGS}}} = 0.8$.
Changes of the relative focal shift maximum ${{\Delta}_{{\max }}}/f$ versus the degree of global coherence $ \alpha $ are shown in Fig. 5. One can see that ${{\Delta}_{{\max }}}$ decreases as $ \alpha $ or $ P_{0} $ increases. In addition, ${{\Delta}_{{\max }}}$ approaches a certain value when $\alpha \to \infty$ (i.e., the fully coherent GS beam case).
Fig. 5. Relative focal shift maximum ${{\Delta}_{{\max }}}/{f}$ versus the degree of global coherence $ \alpha $.
3. FOCUSING CHARACTERISTICS OF GSM BEAMS FOCUSED BY AN IDEAL THIN LENS IN SELF-FOCUSING MEDIA FOR THE BEAM SELF-FOCUSING CASE
When a powerful laser beam propagates in a self-focusing medium (${n_2} \gt 0$) and its beam power satisfies ${P_0} \gt {P_{{\rm cr}}}$, beam self-focusing will take place. Ideally, on propagation, the value of the beam width will reach zero due to the self-focusing effect, and the laser beam will extend to the focus point [15]. It is noted that only the beam self-focusing case is considered in Section 3.
A GSM beam is focused by a thin lens in self-focusing media is shown in Fig. 6. Based on the ABCD law and the propagation formula of the beam width of GSM beams in self-focusing media [15], we obtain the expression of the focal length $ z_f $, i.e.,
where $ f_s = kw^2_1/[4(1+\alpha^{-2})(\eta - 1)]^{1/2}$ and $ f_s $ denotes the beam focusing due to the self-focusing effect. The $ f_s $ decreases as $ \alpha $ or $ P_{0} $ increases because of the self-focusing effect becoming stronger.Equation (21) shows that two situations (i.e., one focus and two foci) may appear. Furthermore, one focus or two foci of the GSM beams can be controlled by the self-focusing effect. Letting $ -1/R_1+1/f=1/f_s $, we obtain
There exist two foci when ${{ P }_{{\rm cr}}} \lt {P}_0 \lt {P}_0^\prime$, while there exists only one focus when ${P}_0 \ge {P}_0^\prime$. In addition, there also exists only one focus when ${P_0} = {P_{{\rm cr}}}$ because of ${f_{s}} \to \infty$. In addition, we have ${z_f} = {f_{s}}/2$ when ${P}_0 = {P}_0^\prime $. These results can be shown clearly in Fig. 7.
Fig. 7. Focal length $ z_f $ versus the relative beam power ${P}_0/{{P}_{{\rm crGS}}}$, ${R}_1 \to \infty$.
Furthermore, the range $\delta$ of the beam power for two foci should satisfy
Equation (23) indicates that the range $\delta$ is the same for different values of $\alpha$, which can also be shown in Fig. 7 (i.e., the distances between two red dashed lines and two blue dashed lines are the same).
4. SUMMARY
In this paper, the propagation of GSM beams through optical systems in Kerr media is studied, where two cases (i.e., beam spreading and beam self-focusing) are considered. For the beam spreading case, the propagation formulae of GSM beams through general optical systems in Kerr media are derived, and the propagation characteristics of GSM beams through an ideal thin lens are studied in detail. It is shown that the size and position of the beam waist can be controlled by the Kerr effect. Furthermore, the formula of the focal shift of the GSM beams focused by an ideal thin lens in Kerr media is also derived. It is found that in self-focusing media the focal shift decreases as the beam power $ P_{0} $ or the beam coherence degree $\alpha$ increases, while in self-defocusing media the focal shift decreases as $ P_{0} $ decreases or $\alpha$ increases. In addition, there exists a maximum of the focal shift versus the position of the beam waist in the object space, and the formula of the focal shift maximum is derived.
On the other hand, for the beam self-focusing case, the focusing characteristics of GSM beams focused by an ideal thin lens in Kerr media are also investigated. It is shown that the focal length decreases as $ P_{0} $ or $ \alpha $ increases. It is found that one focus or two foci can be controlled by the self-focusing effect, e.g., there exist two foci when ${P_{{\rm cr}}} \lt {P_0} \lt {P}_0^\prime$, while there exists only one focus when ${P_0} \ge {P}_0^\prime$. Furthermore, the formula of the ${P}_0^\prime$ is derived. It is noted that the formulae obtained in this paper are more general, and they can be reduced to those of fully coherent GS beams propagating through optical systems in Kerr media. The results obtained in this paper are of theoretical and practical interest.
Funding
National Natural Science Foundation of China (61775152).
Disclosures
The authors declare no conflicts of interest.
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