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]


[**Electron Bernstein waves in spherical tokamak plasmas with “magnetic wells”** ]{}\
[Piliya A.D., Popov A.Yu., Tregubova E.N.]{}\
\[6pt\] [*Ioffe Physico-Technical Institute, St.Petersburg, Russia*]{}\
\[6pt\] [e-mail: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>, <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>]{}\

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

Propagation and electron cyclotron resonance (ECR) damping of electron Bernstein waves (EBWs) in spherical tokamaks (ST) is usually analyzed assuming that absolute value of the tokamak magnetic field $|B|$ increases inward the plasma. In this case the perpendicular index of refraction $n_\perp$ grows as the wave approaches the ECR layer, where the wave is fully absorbed regardless of the resonance harmonic number  [@PPT].

However, in addition to traditional regimes with monotonously increasing $|B|$, regimes with “magnetic wells” also occur in STs  [@NSTXfig]. The the magnetic field profile inversion modifies significantly the whole picture of the wave propagation and damping. Since the magnetic wells may become quite common with further improvement of ST performance (fig.1), analysis of such configurations is of interest for assessment of EBW plasma heating an CD perspectives.\

Consider CMA diagram (fig.2) illustrating why this case was not considered in literature yet. In the conventional tokamaks EBWs produced via the linear conversion of the incident electromagnetic waves close to UHR are capable for propagating to the fundamental harmonic along dotted blue arrow. If the inequality $d[ln(n_e)]/dx>2\omega_{ce}^2/\omega_{pe}^2d[ln(|B|)]/dx$ is fulfilled (it’s the case of STs) EBWs can propagate along solid red arrow to the nearest EC harmonic. In this paper we consider basic features of the EBWs propagation and damping for the second cyclotron harmonic, which is now the lowest possible resonance harmonic, in a slab model. This case is illustrated in fig.1 by solid red arrow.

Bernstein waves in electrostatic approximation
==============================================

Assume all plasma parameters depending on the single dimensionless co-ordinate $x$ scaled in units of $c/\omega$ with $\omega$ being the wave frequency. Suppose that the magnetic field ${\mbox{\boldmath $B$}}$ is along the $z$ - axis and both $\upsilon=\omega_{pe}^2/\omega^2$ and $q=\omega/\omega_{ce}$ grow inward the plasma.\
We begin analysis of the EBW behavior assuming validity of the electrostatic approximation. Then the dispersion relation is $$\varepsilon\left(n_\perp,n_\parallel,x\right)=0,$$ where $\varepsilon\equiv\left(n_i\varepsilon_{ik}n_k\right)/n^2$ is the longitudinal dielectric function, $\varepsilon_{ik}$ - the plasma dielectric tensor elements, $n_\perp=k_xc/\omega$ and $n_\parallel=k_\parallel c/\omega$. Since the UHR is known to be the cutoff for electrostatic EBWs, one expand the dielectric tensor components $\varepsilon_{ij}, i,j=(x,y,z)$ in power of $\lambda=k_\perp^2\rho_e^2=\left(n_\perp q\beta\right)^2/2$, where $\rho_e$ is the electron Larmor radius. As it will be seen, $n_\perp$ remains within the limits $1\ll n_\perp\leq \beta^{-1}$, where $\beta=\nu_{te}/c$, $\nu_{te}=\sqrt{2T_e/m_e}$, in the whole region between the UHR and the $q=2$ resonance, so that $\lambda\leq 1$ here. For qualitative investigation we keep only zero- and first-order terms. Then $$\varepsilon=\varepsilon^{(c)}-\frac{\upsilon\beta^2}{2n^2}\left(3n_\parallel^4+\frac{6q^6-3q^4+q^2}{\left(q^2-1\right)^3}
n_\parallel^2n_\perp^2+\frac{3q^4}{\left(q^2-1\right)\left(q^2-4\right)}n_\perp^4\right),$$ where $\varepsilon^{(c)}=\left(\varepsilon_{xx}^{(c)}n_\perp^2+\varepsilon_{zz}^{(c)}n_\parallel^2\right)/n^2$ and $$\varepsilon_{xx}^{(c)}=1-\frac{\upsilon q^2}{q^2-1},
\varepsilon_{zz}^{(c)}=1-\upsilon\nonumber$$ are the dielectric tensor elements in the cold plasma. The electron Bernstein waves are produced via the linear conversion of incident electromagnetic waves with $n_\parallel\leq 1$, while applicability of the electrostatic approximation requires that $n_\perp\gg 1$. This permits one to omit terms proportional to $n_\parallel^2$ and $n_\parallel^4$ in Eq.(2) reducing the dispersion relation (1) to $$\varepsilon_{xx}=\varepsilon_{xx}^{(c)}-\frac{1}{2}\upsilon\beta^2\frac{3q^4}{\left(q^2-1\right)\left(q^2-4\right)}n_\perp^2=0.\nonumber$$ The solution to this equation $$n_\perp^2=-\frac{2}{3\beta^2\upsilon}\varepsilon_{xx}^{(c)}\frac{\left(q^2-1\right)\left(4-q^2\right)}{q^4},$$ where $\varepsilon_{xx}^{(c)}<0$ and $1<q<2$ shows that in the electrostatic approximation the EBW is confined between two cut-offs. One of them is the UHR, for which $\varepsilon_{xx}^{(c)}(x)=0$, and the other one is the cyclotron resonance $q=2$. The validity condition for the electrostatic approximation is $n^2\gg\left|\varepsilon_{ik}\right|$. We consider here typical for STs high - density plasmas with $\upsilon\gg 1$, then the validity condition becomes $$n_\perp^2\gg\upsilon.$$ This condition breaks down close to the UHR and ECR. We do not consider first of these regions because analysis of wave behavior there is the subject of the mode coupling theory. In the dense plasma outside immediate UHR vicinity, Eq.(5) becomes $$n_\perp^2=\frac{2}{3\beta^2}\frac{\left(4-q^2\right)}{q^2}\nonumber$$ so that the characteristic value of $n_\perp$ is $\beta^{-1}$. In the $2^{nd}$ ECR vicinity, where the inequality (6) is broken down a full - wave treatment is required.

Full-wave equation near $q=2$ resonance
=======================================

To obtain a traceable full - wave hot plasma dispersion relation for EBWs with $n_\perp\sim n_\parallel$ and $n_\perp\ll
\beta^{-1}$ near the $q=2$ resonance we make some simplifications. Consider first the dielectric tensor elements $\varepsilon_{ik}$. It is well known that the elements can be presented as an infinite sums over cyclotron harmonic number $s,-\infty\ll s\ll \infty$ with each term of the sum related to the ECR $q=s$. Since parameter $\lambda$ is small, we calculate the resonance ($s=2$) terms of $\varepsilon_{ik}$ up to the first order in $\lambda$ using the zero order (cold plasma) approximation for non - resonant terms. In this approximation, the elements $\varepsilon_{xz},\varepsilon_{zx},\varepsilon_{yz},\varepsilon_{zy}$ vanish, $\varepsilon_{xx}=\varepsilon_{yy}=\varepsilon_\perp$, $\varepsilon_{xy}=-\varepsilon_{yx}=-i g$ and $$\begin{aligned}
\varepsilon_\perp=\varepsilon_\perp^{(c)}+\frac{1}{2}\frac{\lambda\upsilon}{n_\parallel\beta}Z\left(\xi\right),\,\,
g=g^{(c)}+\frac{1}{2}\frac{\lambda\upsilon}{n_\parallel\beta}Z\left(\xi\right),\end{aligned}$$ here $g^{(c)}=-\upsilon |q|/\left(q^2-1\right)$, $Z$ is the plasma dispersion function defined according to  [@Sw] with the argument $\xi=(q-2)/qn_\parallel\beta$. Consider now the wave equations $$\begin{aligned}
\left(n_\parallel^2-\varepsilon_\perp\right)E_x-igE_y-n_\parallel
n_\perp
E_z\nonumber=0\\
igE_x+\left(n^2-\varepsilon_\perp\right)E_y=0\\
-n_\parallel n_\perp
E_x+\left(n_\parallel^2-\varepsilon_{zz}\right)E_z=0\nonumber\end{aligned}$$ Absolute values of dielectric tensor elements here are of order $\upsilon\gg 1$ and, therefore $E_z\ll E_x$. Omitting $E_z$ in the first equation, that means neglecting terms of order $n_\perp/\upsilon$ and $1/\upsilon$ compared to unity, obtain the dispersion relation $$\begin{aligned}
n_\perp^2\varepsilon_\perp-\left(\varepsilon_\perp-g\right)\left
(\varepsilon_\perp+g\right)=0.\nonumber\end{aligned}$$ Using Eq.(8), we obtain at $\upsilon\gg 1$ $$n_\perp^2\varepsilon_\perp-\frac{\upsilon^2q^2}{q^2-1}\left(1-\frac{n_\perp^2\beta^2(q-1)}{2q}\frac{Z}{n_\parallel\beta}\right)=0.$$ Close to UHR, at large $n_\perp$, $n_\perp^2\gg\upsilon$, the electrostatic approach is valid, the first term in (10) dominates and we returns to the electrostatic equation (5). In the ECR vicinity, at small $n_\perp$, $n_\perp\ll\upsilon$, this term can be omitted and the solution is found explicitly: $$n_\perp=\frac{2}{\beta}\sqrt{\frac{n_\parallel\beta}{Z\left((q-2)/qn_\parallel\beta\right)}}.$$ Thus, in the vicinity of the $q=2$ resonance, where $n_\perp(x)$ goes down with $x$, there is a spatial region $\upsilon^2>n_\perp^2>\upsilon$ where Eqs.(5) and (11) are valid simultaneously.\
The dispersion relation (10) can be also used in the “relativistic” case $n_\parallel\leq\beta$ if the “non-relativistic” plasma dispersion function $Z$ is replaced by a proper relativistic dispersion function.

Wave behavior in the $2^{nd}$ ECR layer
=======================================

Close to the $2^{nd}$ ECR layer we consider, for simplicity, as $x$ dependent only the resonant factor $(q-2)/q$ in the argument of the $Z$ function, putting $(q-2)/q=x/l$, where $l$ is the dimensionless characteristic scale-length of the magnetic field variation. Using the asymptotic expression $Z\sim-1/\xi$, where $\xi=x/L$ and $L=l\beta n_\parallel$, at $\left|\xi\right|\gg 1$, one find that at negative $x$ outside the resonance layer Eq.(11) differs from the electrostatic equation (5) only by a constant factor in the right-hand side. Inside the EC layer at $\left|\xi\right|\sim 1$ the function $Z$ is a complex function with $|Z|\sim 1$ and $Re(Z)\sim Im{(Z)}$. Solution to Eq.(11) is also complex with $\left|n_\perp\right|\sim
\left(n_{\parallel}/\beta\right)^{1/2}$, so that the applicability condition for the WKB theory in the resonance region takes the form $$n_\perp L=l n_\parallel^{3/2}\beta^{1/2}\gg 1,$$ The estimation for the relativistic case can be obtained from this equation by putting $n_\parallel\sim\beta$. We assume that the WKB approximation is applicable in the region outside the resonance layer where $\left|n_\perp\right|$ grows. Suppose now that inequality (12) is satisfied and consider solution to Eq.(11) inside the resonance layer. This solution in a parameter free form $$N=\pm Z^{-1/2}, N=n_\perp\left(\beta/(4n_\parallel)\right)^{1/2}$$ is shown in fig.3 on the plane $\left(Re(N), Im(N)\right)$.

Brunches located in the upper and lower complex $N$-plane correspond to the signs $(-)$ and $(+)$ in Eq.(13), respectively. Consider first the brunch $N= - Z^{-1/2}$. The fragment of the curve with $Re(N)<0$, $Re(N)\gg Im(N)$ belongs to the region $x<0$, $|x|\gg L$ when the asymptotic expression for $Z$ can be used and $n_\perp\rightarrow - \left(4|x|n_\parallel/(\beta
l)\right)^{1/2}$. Since $\left(Im\left(Z\right)\right)^{-1/2}<0$, the whole curve lies in the upper complex $N$ plane. At $|x|\gg L$ we have $n_\perp\rightarrow i \left(4|x|n_\parallel/(\beta
l)\right)^{1/2}$. The second curve describes transition from $n_\perp\rightarrow \left(4|x|n_\parallel/(\beta l)\right)^{1/2}$ to $n_\perp\rightarrow -i \left(4|x|n_\parallel/(\beta
l)\right)^{1/2}$ via the lower complex $N$ plane. Note that the waves under consideration have their phase and group velocities directed oppositely. Therefore, the incident wave has asymptotically a negative $n_\perp$ in the propagation region $x<0$. This wave is described by the $N=-Z^{-1/2}$ dispersion curve. At large positive $x$ this mode vanishes exponentially. The second brunch describes the wave reflected from the ECR. This mode grows exponentially in the non propagation region $|x|\gg L$.Three important conclusions can be drawn from this analysis. First, incoming waves incident on the ECR layer from the high-field side are not converted in the resonance region into outgoing EBWs with large $n_\perp$ propagating on the low field side of the ECR. Instead, the incident waves become non-propagating beyond the resonance layer. Second, two effects are simultaneously responsible for $Im\left(n_\perp\right)$: the ECR damping and the wave transition into the non-propagation region. These two contribution can not be separated. Finally, since the function $|Z(\xi)^{-1}|$ has no zeros at finite $|\xi|$, two brunches of the dispersion curves are separated in the whole complex $x$ plane. As a result, reflection from the ECR layer can only be due to approximate nature of the WKB theory.

Reflection from ECR layer
=========================

We analyze reflection of EBWs from the $q=2$ cyclotron resonance with the use of the model wave equation obtained from the dispersion relation (13) by replacement $n_\perp\rightarrow-id/dx$: $$U^{''}(x)+n_\perp(x)^2U(x)=0.$$ Here $n_\perp^2$ is given by the right - hand side of Eq.(11) and we assume that function $U$ is related linearly to the wave field components. More definite interpretation of this function is not required. In the WKB approximation Eq.(14) has two linearly independent solutions $$\begin{aligned}
U_1=\frac{1}{n_\perp^{1/2}}\exp{\left(-i\int_a^x{n_\perp
dx}\right)},\,\,
U_2=\frac{1}{n_\perp^{1/2}}\exp{\left(i\int_a^x{n_\perp
dx}\right)}\end{aligned}$$ where $a$ is an arbitrary constant. As it has been mentioned before, $U_1$ and $U_2$ represent incoming and outgoing waves, respectively. The solution to Eq.(14) describing the ECR damping and reflection of the wave incident from the high-field side vanishes at $|x|\gg L$ and has the asymptotic form $$U=U_1+R U_2$$ at $x<0$,$|x|\gg L$. Here $R$ is a constant and the quantity $|R|^2$ is the energy reflection coefficient. Presenting Eq.(14) in the form $$U^{''}(x)+\left(V_0+V_1\right)U(x)=0,$$ where $V_0=n_\perp^2-V_1$, $V_1=\left(n_\perp^{-1/2}\right)^{''}n_\perp^{1/2}$. One can easily see that functions $U_{1,2}$ are exact solutions to the equation $U_{1,2}^{''}+V_0U_{1,2}=0$ and $V_1$ is small, of order $\left(\left|n_\perp L\right|\right)^{-2}$ compared to $V_0$, if the WKB method applicability condition $\left|n_\perp\right|L\gg
1$ is satisfied. In this case the reflection coefficient $R$ can be calculating by treating the term in Eq.(17) as a small perturbation and using $U_1$ as a zero order solution: $$R=\frac{1}{2}\int_{-\infty}^{\infty}{V_1U_0^2dx}$$ or explicitly, $$R=\frac{1}{\Gamma_0}\int_{-\infty}^\infty{\left(Z^{1/4}\right)^{''}Z^{1/4}\exp{\left(-i\Gamma_0\int_{-\infty}^\xi{Z^{-1/2}\left(\zeta\right)}d\zeta\right)}d\xi},$$ where the prime denotes differentiation with respect to $\xi$ and $\Gamma_0=2l\beta^{1/2}n_\parallel^{3/2}$. For Eq.(19) validity, $\Gamma_0$ must be large compared to unity. Since the integrand has no saddle points at finite $|\xi|$, and contribution to the integral is spread over the whole $|\xi|\leq 1$, the possibility of obtaining an accurate analytical estimation of the integral seems rather doubtful. Numerical evaluation shows that $\left|R\right|^2$ is negligibly small at $\Gamma_0\geq 1$.\
In the limiting case $\left|k_\perp\right|L<1$ opposite to the WKB one, the reflection coefficient can be found by presenting $Z^{-1}(\xi)\simeq -\xi+V$ and treating $V$ as a perturbation. Substituting these expressions into Eq.(14) and introducing the new independent variable $\tau=k_0x\left(\beta^2l/4\right)^{-1/3}$, obtain $$U^{''}-\left(\tau-\gamma V\left(\tau/\gamma\right)\right)U=0,$$ where $\gamma=\Gamma_0^{2/3}$ is a small parameter. In the present case the reflection coefficient is close to unity. Calculating the correction to it in the lowest order in $\gamma$ yields $$\begin{aligned}
|R|\simeq 1-\frac{\Gamma_0^2}{\xi^2},\end{aligned}$$ where $\xi^{-1}=\pi Ai(0)=1.12$.

![Absolute value of the reflection coefficient $|R|$ for the non-relativistic ($n_\parallel\gg\beta$) case.](fig5.eps){height="8"}

The comparison of the reflection coefficient, $R$, calculated numerically and its analytical estimation (21) is presented in fig.4. The reasonable agreement is demonstrated. Note in conclusion that standard ray tracing method using the full wave dispersion relation or the approximate dispersion relation (11) can be used for high-field side propagation only if the wave is damped in the periphery of the ECR, where $Im(n_\perp)\ll Re(n_\perp)$. Otherwise the wave penetrates into the region of strong cyclotron damping where the ray tracing method is inapplicable.

Summary
=======

Existence of EBWs in the region between the UHR and ECR requires inhomogeneous plasma density.\
Waves in this region are adequately described by the approximate full-wave dispersion relation (10). Incoming waves incident on the ECR layer from the high-field side are not converted in the resonance region into outgoing EBWs propagating on the low field side of the ECR. Instead, the incident waves become non-propagating beyond the resonance layer.\
Decreasing of the wave amplitude within ECR layer is due to combine effect of the ECR damping and non-propagation.\
In the WKB approximation, the waves are fully damped in the ECR layer. Reflection from the ECR layer is only due to approximate nature of the WKB theory.\
Standard ray tracing method can be used for high-field side propagation only if the wave is damped in the periphery of the ECR layer. Otherwise the wave penetrates into the region of strong cyclotron damping where the ray tracing method is inapplicable.\
\
[*Acknowledgment:*]{} This work has been supported by RFBR 04-02-16404, 02-02-17683, Scientific School grant 2159.2003.2.\
\

[99]{} , [*Plasma Phys. Control. Fusion*]{} [**45**]{} (2003) 1309 , [*Physics of Plasmas*]{} [**11**]{} (2004) 1028 , [*Mod. Rev. Phys.*]{} [**67**]{} (1995) 837