You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


Tom G. Mackay[^1]\
[*School of Mathematics and Maxwell Institute for Mathematical Sciences\
University of Edinburgh, Edinburgh EH9 3FD, UK*]{}\
and\
[*NanoMM — Nanoengineered Metamaterials Group\
Department of Engineering Science and Mechanics\
Pennsylvania State University, University Park, PA 16802–6812, USA*]{}\
Akhlesh Lakhtakia[^2]\
[*NanoMM — Nanoengineered Metamaterials Group\
Department of Engineering Science and Mechanics\
Pennsylvania State University, University Park, PA 16802–6812, USA*]{}

Two active dielectric materials may be blended together to realize a homogenized composite material (HCM) which exhibits more gain than either component material. Likewise, two dissipative dielectric materials may be blended together to realize an HCM which exhibits more loss than either component material. Sufficient conditions for such gain/loss enhancement were established using the Bruggeman homogenization formalism. Gain/loss enhancement arises when (i) the imaginary parts of the relative permittivities of both component materials are similar in magnitude and (ii) the real parts of the relative permittivities of both component materials are dissimilar in magnitude.

[**Keywords:**]{} Bruggeman homogenization formalism; active materials; dissipative materials; gain enhancement; loss enhancement

Introduction
============

Two (or more) particulate materials may be mixed together to realize a homogenized composite material (HCM), provided that the particles making up the component materials are much smaller than the wavelengths involved L96. To be of practical value, an HCM is generally required to exhibit a desirable blend of certain properties of its component materials. Metamaterials are HCMs whose performances exceed those of their component materials Walser,Meta. Within the electromagnetic realm, many instances of such HCMs can be found. For examples: through the process of homogenization, the phenomenon of weak nonlinearity may be enhanced Boyd96,Liao,L01, and the group speed may be enhanced beyond the maximum group speed in the component materials Şolna2,ML\_group\_vel\_1 or weakened below the minimum group speed in the component materials Gao\_06.

In this short article, the prospect of enhancing gain by means of homogenization is explored for HCMs arising from active component materials. The dual process of loss enhancement in HCMs arising from dissipative component materials is also considered. The well–established Bruggeman homogenization formalism Br,Aspnes,MAEH is employed, all component materials being thereby treated on the same footing. Accordingly, this formalism is applicable for all values of volume fraction of the component materials.

Homogenization via the Bruggeman formalism
==========================================

Let us consider a composite material comprising two distinct materials labelled ‘a’ and ‘b’ that are distributed randomly as electrically small spheres. Both component materials are isotropic dielectric materials with relative permittivities $\eps_a = \eps^r_a +i \eps^i_a $ and $\eps_b = \eps^r_b +i \eps^i_b$, respectively, wherein $\eps^{r,i}_{a,b} \in \mathbb{R} $ and $\eps_a \neq \eps_b$. Physical plausibility requires the imposition of the restriction $\eps^r_a \eps^r_b >0 $ on the Bruggeman formalism ML\_Br\_limitation.

The Bruggeman estimate $\eps_{Br} = \eps^r_{Br} +i \eps^i_{Br}$ of the HCM relative permittivity is provided implicitly by the quadratic equation Br $$\l{Br_eqn}
 2 \eps^2_{Br} + \eps_{Br} \les \eps_a \le 1-3f_a \ri + \eps_b \le 3f_a - 2 \ri \ris - \eps_a \eps_b =0\,,$$ with $f_a$ being the volume fraction of component material ‘a’. The limiting conditions $\eps_{Br} \to \eps_b$ as $f_a \to 0$, and $\eps_{Br} \to \eps_a$ as $f_a \to 1$ allow the correct root to be extracted from Eq. .

When both component materials are active (i.e., $\eps^i_{a,b} < 0$), the phenomenon of gain enhancement is signified by $ \eps^i_{Br} < \min\left\{\eps^i_{a},\eps^i_{b}\right\}$. When both component materials are dissipative (i.e., $\eps^i_{a,b} > 0$), the phenomenon of loss enhancement is signified by $ \eps^i_{Br} > \max\left\{\eps^i_{a},\eps^i_{b}\right\}$.

To illustrate the phenomenon of gain enhancement, let us consider a specific example. Suppose that the component materials are active ones, specified by $\eps_a = 2 - 0.05 i$ and $\eps_b = 5 - 0.04 i$. The real and imaginary parts of the Bruggeman estimate of the HCM relative permittivity are plotted against volume fraction in Fig. \[fig1\]. Also plotted in this figure are two well–established bounds on the HCM relative permittivity, namely the Wiener bounds Wiener $$\left.
\begin{array}{l}
\mbox{W}_\alpha = f_a \eps_a + f_b \eps_b
\vspace{8pt} \\
\mbox{W}_\beta = \displaystyle{ \le \frac{f_a}{\eps_a} + \frac{f_b}{\eps_b} \ri^{-1}}
\end{array}
\right\}\,\l{Wiener}$$ and the Hashin–Shtrikman bounds ḨS $$\left.
\begin{array}{l}
\displaystyle{\mbox{HS}_\alpha = \eps_b + \frac{3 f_a \eps_b \le \eps_a - \eps_b
\ri}
{\eps_a + 2 \eps_b - f_a \le \eps_a - \eps_b \ri}}  \vspace{8pt} \\
\displaystyle{\mbox{HS}_\beta = \eps_a + \frac{3 f_b \eps_a \le \eps_b - \eps_a
\ri} {\eps_b + 2 \eps_a - f_b \le \eps_b - \eps_a \ri}}
\end{array}
\right\}\,. \l{Hashin}$$ Herein, $f_b = 1- f_a$ is the volume fraction of component material ‘b’. Originally, the Wiener bounds and the Hashin–Shtrikman bounds were derived for HCMs characterized by wholly real–valued constitutive parameters, but generalizations to complex–valued constitutive parameters later emerged Milton.

The Hashin–Shtrikman bound $\mbox{HS}_\alpha$ is equivalent to the Maxwell Garnett estimate of the HCM relative permittivity, based on the homogenization of a random dispersal of spheres of component material ‘a’ embedded in the host component material ‘b’, valid for $f_a  \lesssim 0.3$ MG. Similarly, $\mbox{HS}_\beta$ is equivalent to the Maxwell Garnett estimate of the HCM relative permittivity, based on the homogenization of a random dispersal of spheres of component material ‘b’ embedded in the host component material ‘a’, valid for $f_b \lesssim 0.3$.

The real part of $\eps_{Br}$ is seen in Fig. \[fig1\] to decrease uniformly from $\eps^r_b$ to $\eps^r_a$ as $f_a$ increases from $0$ to $1$. Furthermore, $\eps^r_{Br}$ is tightly bounded by $\mbox{HS}_\alpha$ and $\mbox{HS}_\beta$, and less tightly bounded by $W_\alpha$ and $W_\beta$. The imaginary part of $\eps_{Br}$ follows a more interesting trajectory as $f_a$ increases: $\eps^i_{Br}$ decreases from $\eps^i_b$ at $f_a =0$, reaches a minimum value at $f_a \approx 0.8$, and then increases to reach $\eps^i_a$ at $f_a =1$. Thus, according to the Bruggeman formalism, gain enhancement arises in the vicinity of $f_a \approx 0.8$, with the minimum value of $\eps^i_{Br}$ ($\approx -0.0515$) being approximately $3\%$ smaller than $\min\left\{\eps^i_{a},\eps^i_{b}\right\}$. Furthermore, ${\rm Im}\left(\mbox{HS}_\beta\right) <\min\left\{\eps^i_{a},\eps^i_{b}\right\}$ when $0.7 \lesssim f_a < 1$. Thus, gain enhancement is also predicted by the Maxwell Garnett formalism.

Loss enhancement mirrors gain enhancement. To support this assertion, let us consider the dissipative counterpart of the active HCM considered in Fig. \[fig1\]. In Fig. \[fig2\], plots are presented which are equivalent to those presented in Fig. \[fig1\] but now the component materials are dissipative ones, specified by $\eps_a = 2 + 0.05 i$ and $\eps_b = 5 + 0.04 i$. As in Fig. \[fig1\], $\eps_{Br}^r$ in Fig. \[fig2\] decreases uniformly from $\eps^r_b$ to $\eps^r_a$ as $f_a$ increases from $0$ to $1$; moreover, $\eps^r_{Br}$ is tightly bounded by $\mbox{HS}_\alpha$ and $\mbox{HS}_\beta$, and less tightly bounded by $W_\alpha$ and $W_\beta$. The plot of $\eps_{Br}^i$ in Fig. \[fig2\] displays loss enhancement with the maximum value of $\eps^i_{Br}$ ($\approx 0.0515$) being approximately $3\%$ larger than $\max\left\{\eps^i_{a},\eps^i_{b}\right\}$. In addition, ${\rm Im}\left(\mbox{HS}_\beta\right) >\max\left\{\eps^i_{a},\eps^i_{b}\right\}$ when $0.7 \lesssim f_a < 1$. Thus, loss enhancement is predicted by both the Bruggeman formalism and the Maxwell Garnett formalism.

Since the active and dissipative scenarios effectively represent two different sides of the same coin, henceforth in this section we focus on gain enhancement. Let us now turn to the gain–enhancement index $$\rho = \frac{\eps^i_{Br}}{\min \lec \eps^i_a, \eps^i_b \ric }$$ estimated using the Bruggeman formalism. Gain enhancement is signified by $\rho > 1$. For $\eps^r_a = 2$, $\eps^r_b = 5$, and $\eps^i_b = -0.04$, $\rho$ is plotted against volume fraction $f_a$ and the ratio $\eps^i_a/\eps^i_b$ in Fig. \[fig3\]. Gain enhancement is evident for mid–ranges value of $f_a$ when $\eps^i_a/\eps^i_b\simeq1$. Specifically for this particular example,

-   $\rho$ is as high as about $1.05$, with its maximum value occurring for $f_a \approx 0.6$ and $
    \eps^i_a/\eps^i_b = 1$; and

-   there is no gain enhancement for $
    \eps^i_a/\eps^i_b  \lesssim  0.95$ and for $
    \eps^i_a/\eps^i_b  \gtrsim  1.07$, regardless of the value of $f_a$.

The dependency of $\rho$ upon $\eps_a^r$ and $\eps_b^r$ is delineated in Fig. \[fig4\], wherein $\rho$ is plotted against $f_a$ and $\eps^r_a/\eps^r_b$ for $\eps^i_a = -0.05$, $\eps^i_b = -0.04$, and $\eps^r_b = 5$. As in Fig. \[fig3\], $\rho$ is high for mid-range values of $f_a$ when the ratio $\eps^r_a/\eps^r_b$ deviates most from unity in Fig. \[fig4\]. Specifically for this particular example,

-   $\rho$ is as high as about $1.4$, with its maximum value occurring for $f_a \approx 0.7$ and $
    \eps^r_a/\eps^r_b = 0.1$;

-   $\rho$ is as high as about $1.2$, with its maximum value occurring for $f_a \approx 0.5$ and $
    \eps^r_a/\eps^r_b  = 10$; and

-   there is no gain enhancement for $ \eps^r_a/\eps^r_b\approx  1$, regardless of the value of $f_a$.

Conditions for gain/loss enhancement
====================================

The foregoing and similar calculations led us to conclude that gain enhancement should be expected when

-   $\eps^i_a < 0$ and $\eps^i_b<0$,

-   the ratio $\eps^i_a/\eps^i_b$ is close to unity, and

-   the ratio $\eps^r_a/\eps^r_b$ is either very small or very large.

Loss enhancement should be expected when $\eps^i_a >0$, $\eps^i_b>0$, and the conditions (ii) and (iii) are satisfied. In order to formally establish this understanding soundly, we used the Bruggeman equation to obtain the gradient $$\l{dBr}
 \frac{d \eps_{Br}}{d f_a} = \frac{3 \eps_{Br} \le \eps_a - \eps_b \ri}{4 \eps_{Br} + \eps_a \le 1-3f_a \ri + \eps_b \le 3f_a - 2 \ri}\,.$$ This expression underlies further analysis.

Gain enhancement
----------------

ł[Sec\_gain\_enhancement]{}

Suppose that both component materials are active, i.e., $\eps^i_a <0 $ and $ \eps^i_b<0$. If $\eps^i_a \geq  \eps^i_b$, then a *sufficient* condition for gain enhancement is that the gradient $$\l{c1}
\lim_{f_a \to 0}   \frac{d \eps^i_{Br} }{d f_a}   < 0\,.$$ Given that $$\lim_{f_a \to 0}    \eps_{Br} =\eps_b\,,$$ Eq.  yields $$\lim_{f_a \to 0}   \frac{d \eps^i_{Br} }{d f_a}  = \frac{3 \eps_b \le \eps_a - \eps_b \ri}{2 \eps_b + \eps_a  }\,,$$ and hence $$\lim_{f_a \to 0}   \frac{d \eps^i_{Br} }{d f_a}
 = \frac{ 3 \lec \les \eps^i_b \le  \eps^r_a - \eps^r_b \ri + \eps^r_b \le \eps^i_a - \eps^i_b \ri \ris
  \le 2 \eps^r_b + \eps_a^r \ri
  -  \les  \eps^r_b \le  \eps^r_a - \eps^r_b \ri - \eps^i_b\le \eps^i_a - \eps^i_b \ri  \ris \le 2 \eps^i_b + \eps^i_a \ri \ric}{\le 2 \eps^r_b + \eps_a^r \ri^2 + \le 2 \eps^i_b + \eps^i_a \ri^2 }\,.$$ The sufficient condition for gain enhancement is therefore logically equivalent to $$\begin{aligned}
 \l{cond1}
&& \les \eps^i_b \le  \eps^r_a - \eps^r_b \ri + \eps^r_b \le \eps^i_a - \eps^i_b \ri \ris
  \le 2 \eps^r_b + \eps_a^r \ri
  <  \les  \eps^r_b \le  \eps^r_a - \eps^r_b \ri - \eps^i_b\le \eps^i_a - \eps^i_b \ri  \ris \le 2 \eps^i_b + \eps^i_a \ri \,.\end{aligned}$$

If $\eps^i_b \geq  \eps^i_a$ then a *sufficient* condition for gain enhancement is that the gradient $$\l{c2}
\lim_{f_a \to 1}   \frac{d \eps^i_{Br} }{d f_a}  > 0\,.$$ Following the same argument as used to derive condition , we found that the sufficient condition for gain enhancement is logically equivalent to $$\begin{aligned}
 \l{xcond1}
&& \les \eps^i_a \le  \eps^r_a - \eps^r_b \ri + \eps^r_a \le \eps^i_a - \eps^i_b \ri \ris
  \le 2 \eps^r_a + \eps_b^r \ri
  >  \les  \eps^r_a \le  \eps^r_a - \eps^r_b \ri - \eps^i_a\le \eps^i_a - \eps^i_b \ri  \ris \le 2 \eps^i_a + \eps^i_b \ri .\end{aligned}$$

The special case $ \eps^i_a = \eps^i_b$ is noteworthy. Both the conditions and then reduce to $$\l{cond3}
 \le  \eps^r_a - \eps^r_b \ri^2 >0.$$ Since condition is always satisfied because $\eps_a\ne\eps_b$, gain enhancement is guaranteed for all values of $ \eps^r_a$ and $ \eps^r_b$, provided that $ \eps^i_a = \eps^i_b$.

Loss enhancement
----------------

Suppose both component materials are dissipative, i.e., $\eps^i_a > 0 $ and $ \eps^i_b>0$. If $\eps^i_a \leq  \eps^i_b$, then a sufficient condition for loss enhancement is that the gradient $$\lim_{f_a \to 0}   \frac{d \eps^i }{d f_a}   > 0\,,$$ which, in the manner described in §\[Sec\_gain\_enhancement\], is logically equivalent to the condition $$\begin{aligned}
 \l{cond4}
&& \les \eps^i_b \le  \eps^r_a - \eps^r_b \ri + \eps^r_b \le \eps^i_a - \eps^i_b \ri \ris
  \le 2 \eps^r_b + \eps_a^r \ri
  >  \les  \eps^r_b \le  \eps^r_a - \eps^r_b \ri - \eps^i_b\le \eps^i_a - \eps^i_b \ri  \ris \le 2 \eps^i_b + \eps^i_a \ri \,.\end{aligned}$$ If $\eps^i_b \leq  \eps^i_a$, then a sufficient condition for loss enhancement is that the gradient $$\lim_{f_a \to 1}   \frac{d \eps^i }{d f_a}    < 0\,,$$ which is logically equivalent to the condition $$\begin{aligned}
 \l{cond5}
&& \les \eps^i_a \le  \eps^r_a - \eps^r_b \ri + \eps^r_a \le \eps^i_a - \eps^i_b \ri \ris
  \le 2 \eps^r_a + \eps_b^r \ri
  <  \les  \eps^r_a \le  \eps^r_a - \eps^r_b \ri - \eps^i_a\le \eps^i_a - \eps^i_b \ri  \ris \le 2 \eps^i_a + \eps^i_b \ri \,.\end{aligned}$$

As in §\[Sec\_gain\_enhancement\], both conditions and reduce to condition for the special case $ \eps^i_a = \eps^i_b$. Therefore, loss enhancement is guaranteed for all values of $ \eps^r_a$ and $ \eps^r_b$ when $ \eps^i_a = \eps^i_b$.

Numerical illustration
----------------------

The conditions and provide a convenient means of exploring the parameter space of the relative permittivities of the component materials that support gain enhancement, and conditions and play the same role for loss enhancement. Let us illustrate this assertion with a numerical example.

In Fig. \[fig5\], the parameter spaces that support gain enhancement are mapped for: (i) $\le -\eps^i_a, -\eps^i_b \ri \in \le 0, 1 \ri \times \le 0, 1 \ri $ with $\eps^r_a = 2$ and $\eps^r_b = 5$; and (ii) $\le \eps^r_a, \eps^r_b \ri \in \le 0.5, 10 \ri \times \le 0.5, 10 \ri $ with $\eps^i_a = - 0.05$ and $\eps^r_b = -0.04$. For $\eps^r_a = 2$ and $\eps^r_b = 5$, the gain-enhancement subspace in the $\le -\eps^i_a, -\eps^i_b \ri$ space is a window that contains $\eps^i_a = \eps^i_b$ and becomes narrower as the magnitudes of $\eps^i_a$ and $\eps^i_b$ are decreased. For $\eps^i_a = -0.05$ and $\eps^i_b = -0.04$, two gain-enhancement subspaces in the $\le \eps^r_a, \eps^R_b \ri$ space exist where $\eps^r_a$ and $\eps^r_b$ are dissimilar in magnitude with greater scope for gain enhancement arising when the magnitudes of $\eps^r_a$ and $\eps^r_b$ are increased. These trends gleaned from Fig. \[fig5\] are wholly consistent with those evident in Figs. \[fig3\] and \[fig4\].

Non-dissipative and non-active component materials
--------------------------------------------------

In passing, let us remark on the special case when both component materials are neither dissipative nor active, i.e., $\eps^i_a = \eps^i_b = 0$. Provided that the possibility $\eps_{Br} = 0$ is excluded from consideration (which is not physically plausible for the situation $\eps_a \eps_b >0 $ considered here), we infer from Eq.  that $ \displaystyle{ {d \eps_{Br}}/{d f_a} \neq 0}$. Therefore, $\eps_{Br}$ is either a uniformly increasing or a uniformly decreasing function of $f_a$. Hence, $\eps_{Br}$ must lie between $\eps_a$ and $\eps_b$ for all values of $f_a\in[0,1]$.

Closing remarks
===============

Using the Bruggeman formalism, we have established in the foregoing sections that an HCM comprising two active (resp. dissipative) component materials may exhibit more gain (resp. loss) than either of its component materials. For the range of $\eps_a$ and $\eps_b$ values explored in numerical examples here, gain enhancements of up to 40% were found. Furthermore, sufficient conditions for such gain enhancement and loss enhancement have been established in conditions and , and and , respectively. These enhancements arise when (i) the imaginary parts of the relative permittivities of both component materials are similar in magnitude and (ii) the real parts of the relative permittivities of both component materials are dissimilar in magnitude. Similar gain/loss enhancements also emerge from the Maxwell Garnett formalism for dilute composite materials.

The reported phenomenons of gain enhancement and loss enhancement are likely to be exacerbated by directional effects in anisotropic HCMs, as has been established for nonlinearity enhancement LL01,M03 and group-velocity enhancement ML\_group\_vel\_2.

[**Acknowledgments:**]{} TGM acknowledges the support of EPSRC grant EP/M018075/1. AL thanks the Charles Godfrey Binder Endowment at Penn State for ongoing financial support of his research activities.

  -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
   ![The real and imaginary parts of the HCM relative permittivity $\eps_{Br}$ estimated by the Bruggeman formalism (red, solid curves) plotted against volume fraction $f_a$, when $\eps_a = 2- 0.05 i$ and $\eps_a = 5- 0.04 i$. Also plotted are the Hashin–Shtrikman bounds: $\mbox{HS}_\alpha$ (thin, green, broken dashed curves) and $\mbox{HS}_\beta$ (thick, green, broken dashed curves); and the Wiener bounds: $\mbox{W}_\alpha$ (thin, blue, dashed curves) and $\mbox{W}_\beta$ (thick, blue, dashed curves). []{data-label="fig1"}](Fig1a.eps "fig:"){width="14.0cm"}
   ![The real and imaginary parts of the HCM relative permittivity $\eps_{Br}$ estimated by the Bruggeman formalism (red, solid curves) plotted against volume fraction $f_a$, when $\eps_a = 2- 0.05 i$ and $\eps_a = 5- 0.04 i$. Also plotted are the Hashin–Shtrikman bounds: $\mbox{HS}_\alpha$ (thin, green, broken dashed curves) and $\mbox{HS}_\beta$ (thick, green, broken dashed curves); and the Wiener bounds: $\mbox{W}_\alpha$ (thin, blue, dashed curves) and $\mbox{W}_\beta$ (thick, blue, dashed curves). []{data-label="fig1"}](Fig1b.eps "fig:"){width="14.0cm"}
  -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
   ![As Fig. \[fig1\] except that both component materials are dissipative, having the relative permittivities $\eps_a = 2+ 0.05 i$ and $\eps_a = 5+ 0.04 i$. []{data-label="fig2"}](Fig2a.eps "fig:"){width="14.0cm"}
   ![As Fig. \[fig1\] except that both component materials are dissipative, having the relative permittivities $\eps_a = 2+ 0.05 i$ and $\eps_a = 5+ 0.04 i$. []{data-label="fig2"}](Fig2b.eps "fig:"){width="14.0cm"}
  ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
   ![The gain–enhancement index $\rho$ plotted against $f_a\in[0,1]$ and $\eps^i_a / \eps^i_b\in[0.9,1.1]$, when $\eps^r_a = 2$, $\eps^r_b = 5$, and $\eps^i_b = -0.04$. \[fig3\]](Fig3.eps "fig:"){width="14.0cm"}
  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
   ![The gain–enhancement index $\rho$ plotted against $f_a\in[0,1]$ and $\eps^r_a / \eps^r_b\in[0.9,1]\cup[1,10]$, when $\eps^i_a = -0.05$, $\eps^i_b = -0.04$, and $\eps^r_b = 5$. []{data-label="fig4"}](Fig4a.eps "fig:"){width="14.0cm"}
   ![The gain–enhancement index $\rho$ plotted against $f_a\in[0,1]$ and $\eps^r_a / \eps^r_b\in[0.9,1]\cup[1,10]$, when $\eps^i_a = -0.05$, $\eps^i_b = -0.04$, and $\eps^r_b = 5$. []{data-label="fig4"}](Fig4b.eps "fig:"){width="14.0cm"}
  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
   ![Top: Gain-enhancement subspace in the $\le -\eps^i_a, -\eps^i_b \ri$ space, when $\eps^r_a = 2$ and $\eps^r_b = 5$. Bottom: Gain-enhancement subspaces in the $\le \eps^r_a, \eps^r_b \ri$ space, when with $\eps^i_a = - 0.05$ and $\eps^r_b = -0.04$. []{data-label="fig5"}](Fig5a.eps "fig:"){width="12.0cm"}
   ![Top: Gain-enhancement subspace in the $\le -\eps^i_a, -\eps^i_b \ri$ space, when $\eps^r_a = 2$ and $\eps^r_b = 5$. Bottom: Gain-enhancement subspaces in the $\le \eps^r_a, \eps^r_b \ri$ space, when with $\eps^i_a = - 0.05$ and $\eps^r_b = -0.04$. []{data-label="fig5"}](Fig5b.eps "fig:"){width="12.0cm"}
  --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

[99]{}

Lakhtakia A (ed.). Selected papers on linear optical composite materials. Bellingham (WA): SPIE Optical Engineering Press; 1996.

Walser RM. Metamaterials: an introduction. In: Introduction to complex mediums for optics and electromagnetics. (W.S. Weiglhofer and A. Lakhtakia, eds). Bellingham (WA): SPIE Press; 2003, pp. 295–316. Cui TJ, Smith D, and Liu R (eds.). Metamaterials: theory, design, and applications. New York (NY): Springer; 2010.

Boyd RW, Gehr RJ, Fischer GL, and Sipe JE. Nonlinear optical properties of nanocomposite materials. Pure Appl. Opt. 1996;5:505–512.

Liao HB, Xiao RF, Wang H, Wong KS, and Wong GKL. Large third–order optical nonlinearity in $ \mbox{Au:TiO}_2$ composite films measured on a femtosecond time scale. Appl. Phys. Lett. 1998;72:1817–1819.

Lakhtakia A. Application of strong permittivity fluctuation theory for isotropic, cubically nonlinear, composite mediums. Opt. Commun. 2001;192:145–151. S[ø]{}lna K and Milton GW. Can mixing materials make electromagnetic signals travel faster? SIAM J. Appl. Math. 2002;62:2064–2091. Mackay TG and Lakhtakia A. Enhanced group velocity in metamaterials. J. Phys. A: Math. Gen. 2004;37:L19–L24. Gao L. Decreased group velocity in compositionally graded films. Phys. Rev. E 2006;73:036602.

Bruggeman DAG. Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen, I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Ann. Phys. Lpz. 1935;24:636–679. (Reproduced in L96). Aspnes DE. Local–field effects and effective–medium theory: a microscopic perspective. Am. J. Phys. 1982;50:704–709. (Reproduced in L96). Mackay TG and Lakhtakia A. Modern analytical electromagnetic homogenization. Bristol (UK): IOP Publishing; 2015.

Mackay TG and Lakhtakia A. A limitation of the Bruggeman formalism for homogenization. Opt. Commun. 2004;234:35–42. Erratum 2009;282:4028.

Wiener O. Die Theorie des Mischkörpers für das Feld der Stationären Strömung. Abh. Math.–Phys. Kl. Sächs. 1912;32:507–604. (Reproduced in L96, which also provides an English synopsis of Wiener by B. Michel).

Hashin Z and Shtrikman S. A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 1962;33:3125–3131.

Milton GW. Bounds on the complex dielectric constant of a composite material. Appl. Phys. Lett. 1980;37:300–302. Maxwell Garnett JC. Colours in metal glasses and in metallic films. Phil. Trans. R. Soc. Lond. A 1904;203:385–420. (Reproduced in L96).

Lakhtakia MN and Lakhtakia A. Anisotropic composite materials with intensity–dependent permittivity tensor: the Bruggeman approach. Electromagnetics 2001;21:129–138. Mackay TG. Geometrically derived anisotropy in cubically nonlinear dielectric composites. J. Phys. D: Appl. Phys. 2003;36:583–591.

Mackay TG and Lakhtakia A. Anisotropic enhancement of group velocity in a homogenized dielectric composite medium. J. Opt. A: Pure Appl. Opt. 2005:7;669–674.

[^1]: E–mail: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>.

[^2]: E–mail: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>