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]


1. Field of the Invention
The present invention relates to an image display apparatus. More specifically, the present invention relates to an image display apparatus using a liquid crystal display panel displaying an image by controlling a polarization state of linearly polarized light.
2. Description of the Related Art
Liquid crystal display panels used in projection-type image display apparatuses are roughly classified into a transmission type and a reflection type.
Referring to FIG. 6, a conventional projection-type image display apparatus using a transmission-type liquid crystal display panel will be described. The image display apparatus includes an illumination unit having a lamp 22 and a concave mirror 23, a transmission-type liquid crystal display panel 27, an input side polarizing plate 28.sub.i, an output side polarizing plate 28.sub.o, a field lens 29, a projection lens 30, and a screen 31.
A light beam from the illumination unit 21 is modulated by the liquid crystal display panel 27 through the polarizing plate 28.sub.i. Then, the light beam passes through the field lens 29 through the polarizing plate 28.sub.o, and is projected on the screen 31 by the projection lens 30. The liquid crystal display panel 27 is provided with the polarizing plate 28.sub.i and the polarizing plate 28.sub.o as polarization selecting elements on a light beam input side (the left side of the drawing) and a light beam output side (the right side of the drawing) in such a manner that their polarization axes are orthogonal to each other. An image is displayed by modulating the linearly polarized light by the liquid crystal display panel 27 so as to control its transmittance. In the case where an electrical field is not applied to a liquid crystal layer in the liquid crystal display panel 27, light transmitted through the polarizing plate 28.sub.i is rotated (i.e., a plane of polarization is rotated) along a twist of liquid crystal molecules by an optical anisotropic effect of the liquid crystal molecules so as to be capable of passing through the polarizing plate 28.sub.o. Consequently, a bright display state (bright state) can be obtained. In the case where an electrical field is applied to the liquid crystal layer, the twist orientation of the liquid crystal molecules is cancelled, so that the liquid crystal layer does not rotate the plane of polarization. As a result, the light transmitted through the polarizing plate 28.sub.i cannot pass through the polarizing plate 28.sub.o, being blocked by the polarizing plate 28.sub.o. Therefore, a dark display state (dark state) is obtained.
The ratio of brightness between a bright state and a dark state in the image display apparatus with the above-mentioned structure is called a contrast ratio. The contrast ratio is an important factor in an image display quality which largely depends upon a degree of polarization P of the two polarizing plates 28.sub.i and 28.sub.o for the following reason: the amount of transmitted light in the case of displaying a dark state largely depends upon the degree of polarization P of the polarizing plates 28.sub.i and 28.sub.o ; therefore, light is transmitted even in the case of displaying a dark state when the degree of polarization P is low; as a result, the contrast ratio decreases.
According to "Introduction to Optics II" (J. Tsujiuchi, Asakura Shoten, p. 203), the degree of polarization P is defined by Formula (1) using a total intensity I.sub.tot and a complete polarized light component I.sub.pol of a light beam transmitted through a polarizing plate: EQU P=I.sub.pol /I.sub.tot (1)
According to "Polarizing-phase difference film for LCD" listed on pp. 53 to 55 in the August, 1995 issue of Monthly Display (Technotimes), the degree of polarization P of a polarizing plate tends to decrease with the increase in transmittance T of the polarizing plate. Therefore, increasing the transmittance T of the polarizing plate so as to realize a bright display in a liquid crystal display apparatus results in decreasing the degree of polarization P of the polarizing plate. This decreases a contrast ratio in the liquid crystal display apparatus. Thus, the transmittance T and the degree of polarization P of two polarizing plates 28.sub.i and 28.sub.o used in the projection-type image display apparatus using the transmission-type liquid crystal display panel is determined by the tradeoff of the brightness and the contrast ratio of a display. In general, the display quality remarkably decreases at a contrast ratio of less than 100:1, so that the degree of polarization P is required to be at least about 99%. In most cases, the polarization degrees of the polarizing plates 28.sub.i and 28.sub.o are at least about 99.9%.
Next, referring to FIG. 7, a conventional projection-type image display apparatus using a reflection-type liquid crystal display panel will be described. The image display apparatus includes an illumination unit 21 having a lamp 22 and a concave mirror 23, a polarizing plate 28.sub.i, a polarization beam splitter 32, a polarizing plate 28.sub.o, a reflection-type liquid crystal display panel 27', a field lens 29, a projection lens 30, and a screen 31. A light beam from the illumination unit 21 is incident upon the polarization beam splitter 32 through the polarizing plate 28.sub.i. A P-polarized light component of the light beam passes through the polarization beam splitter 32, and an S-polarized light component thereof is reflected in a side direction of the polarization beam splitter 32 to be incident upon the liquid crystal panel 27'. In the liquid crystal display panel 27', the S-polarized component has its plane of polarization optically modulated in the same way as in the above-mentioned transmission-type liquid crystal panel and is reflected. Thus, the P-polarized light component included in the reflected light can pass through the polarization beam splitter 32, and is projected on a screen 31 through the field lens 29 and the projection lens 30, whereby an image is displayed.
In the projection-type image display apparatus with the above-mentioned structure, the degree of polarization of the polarization beam splitter 32 with respect to a light beam reflected therefrom or transmitted therethrough largely influences the contrast ratio. However, even if the polarization beam splitter 32 has a degree of polarization of at least about 99% with respect to incident light (e.g., laser light) whose scattering angle is negligible, in an optical system of the projection-type image display apparatus, illumination light generally has a scattering angle of .+-. several degrees; therefore, the degree of polarization of the polarization beam splitter 32 is lower, compared with those of the polarizing plates 28.sub.i and 28.sub.o. More specifically, since the transmittance of the P-polarized light component with respect to the polarization beam splitter 32 is about 95%, and that of the S-polarized light component is about 5%, the degree of polarization of the polarization beam splitter 32 is very low, i.e., about 95% in accordance with the above-mentioned Formula (1). As a result, in the projection-type image display apparatus with the above-mentioned structure, the contrast ratio is very low, i.e., 20:1. In order to solve this problem, the following attempt has been made: the polarizing plates 28.sub.i and 28.sub.o are provided as polarization selecting elements on an illumination unit side and a projection element side of the polarization beam splitter 32, and the degree of polarization of the combination of the polarizing plate 28.sub.i and the polarization beam splitter 32 and that of the combination of the polarizing plate 28.sub.o and the polarization beam splitter 32 are prescribed to be at least about 99%, whereby a contrast ratio of at least 100:1 is achieved.
However, in the projection-type image display apparatus using the transmission-type liquid crystal display panel as shown in FIG. 6, an effective component of the illumination light which can be used by the transmission-type liquid crystal display panel 27 is limited to that along a polarization axis of the polarizing plate 28.sub.i. The component which cannot pass through the transmission-type liquid crystal display panel 27 is absorbed by the polarizing plate 28.sub.i and converted into heat. Therefore, the use efficiency of light becomes 1/2 or less. As a result, a bright display cannot be obtained in such a projection-type image display apparatus. Furthermore, due to the heat absorption by the polarizing plate 28.sub.i, the characteristics of the polarizing plate 28.sub.i are degraded and the temperature of the liquid crystal layer in the liquid crystal display panel 27 is increased to phase-change a liquid crystal material included therein. Thus, display characteristics are degraded.
Likewise, in the projection-type image display apparatus using the reflection-type liquid crystal display panel as shown in FIG. 7, the polarizing plate 28.sub.i on the illumination unit side of the polarization beam splitter 32 transmits only linearly polarized light along its polarization axis and converts the other component into heat which it absorbs. Therefore, the use efficiency of light becomes 1/2 or less. Furthermore, due to the heat absorption by the polarizing plate 28.sub.i, the characteristics of the polarizing plate 28.sub.i are degraded and the temperature of the liquid crystal layer in the liquid crystal panel 27' is increased to phase-change a liquid crystal material included therein. Thus, display characteristics are degraded.
In order to solve the above-mentioned problems, the following techniques have been proposed. For example, Japanese Laid-open Publication No. 4-178683 describes a liquid crystal display apparatus (projection-type image display apparatus) in which a component of illumination light from an illumination unit which cannot pass through an input side polarizing plate is converted into polarized light which can pass through the input side polarizing plate by a polarization separating element and a polarization rotating element. A polarization converting optical system of the liquid crystal display apparatus described in this publication is shown in FIG. 8. As shown in FIG. 8, the polarization converting optical system includes an illumination unit 41 having a lamp 42 and a concave mirror 43, a cube-shaped polarization beam splitter 44, a polarization rotating element 45, a reflector 46, a prism 47, and a transmission-type liquid crystal display panel 48.
The polarization beam splitter 44 transmits a particular polarized light component (P-polarized light a.sub.1 in FIG. 8) and reflects a polarized component orthogonal thereto (S-polarized light a.sub.2 in FIG. 8) in a side direction of the polarization beam splitter 44, thereby splitting the light beam. Thus, white random light emitted from the illumination unit 41 is split into the P-polarized light a.sub.1 and the S-polarized light a.sub.2 by the polarization beam splitter 44. The P-polarized light a.sub.1 transmitted through the polarization beam splitter 44 is incident upon the prism 47 as it is. The S-polarized light a.sub.2 reflected in the side direction passes through the polarization rotating element 45, thereby having its plane of polarization rotated by 90.degree..
Thus, the plane of polarization of the S-polarized light a.sub.2 is aligned with the same direction as that of the P-polarized light a.sub.1 (i.e., direction parallel to the drawing surface) to become P-polarized light a.sub.2 '. Thereafter, the P-polarized light a.sub.2 ' is reflected by the reflector 46 and is incident upon the prism 47. Then, two light beams of the P-polarized light a.sub.1 and the P-polarized light a.sub.2 ' are combined by the prism 47 and radiated to the transmission-type liquid crystal display panel 48. In this way, according to the structure described in the above-mentioned publication, two light beams of the P-polarized light a.sub.1 and the S-polarized light a.sub.2 are radiated to the transmission-type liquid crystal display panel 48 with their planes of polarization being alighted, whereby the use efficiency of light is improved, and the increase in temperature and the decrease in display characteristics of the transmission-type liquid crystal display panel 48 are minimized.
Japanese Laid-open Publication No. 2-93580 describes a projection-type image display apparatus as shown in FIG. 9. The projection-type image display apparatus includes a reflector 56 on a surface of a polarization beam splitter 55 orthogonal to a transmission-type liquid crystal display panel 57, and a quarter-wave plate 54 which is a polarization rotating element disposed between an illumination unit 51 irradiating a parallel light beam and the polarization beam splitter 55. With this structure, the technique described in the publication contemplates that all the components of the light beam from the illumination unit 51 are radiated to the transmission-type liquid crystal display panel 57. In the optical system shown in FIG. 9, the polarization beam splitter 55 transmits P-polarized light a.sub.1 and reflects S-polarized light a.sub.2 in a side direction, thereby splitting white random light emitted from the illumination unit 51. The reflected S-polarized light a.sub.2 is reflected by the reflector 56 and reflected again by the polarization beam splitter 55. Then, the S-polarized light a.sub.2 travels back to the illumination unit 51, is further reflected by a concave mirror 53 of the illumination unit 51 twice, and is incident upon the polarization beam splitter 55.
During the above-mentioned process, the polarization state of the S-polarized light a.sub.2 changes as follows. When returning to the concave mirror 53 of the illumination unit 51 from the polarization beam splitter 55, the S-polarized light a.sub.2 passes through the quarter-wave plate 54 to become circularly polarized light. Then, the circularly polarized light passes through the quarter-wave plate 54 to become P-polarized light a.sub.2 ', thereby having its plane of polarization aligned with the P-polarized light a.sub.1. Accordingly, the S-polarized light a.sub.2 is re-used, so that the amount of light which can be used by the transmission-type liquid crystal display panel 57 increases, and the increase in temperature of the liquid crystal layer and the decrease in display characteristics can be minimized.
Japanese Laid-open Publication No. 4-127120 describes a polarization converting optical system not using a polarization rotating element. The structure of the optical system is shown in FIG. 10. The optical system includes a polarization beam splitter 64 between an illumination unit 61 irradiating a parallel light beam and a transmission-type liquid crystal display panel 65. The polarization beam splitter 64 transmits P-polarized light a.sub.1 of white random light from the illumination unit 61 and reflects S-polarized light a.sub.2 thereof to the illumination unit 61. The S-polarized light a.sub.2 reflected from the polarization beam splitter 64 is reflected by a concave mirror 63 twice and is incident upon the polarization beam splitter 64. At this time, depending upon the positions at which the S-polarized light a.sub.2 is reflected in a cross-section of the concave mirror 63, linearly polarized light having a different polarization axis (i.e., P-polarized light a.sub.2 ') is generated. Thus, the P-polarized light a.sub.1 and a.sub.2 ' pass through the polarization beam splitter 64 to be radiated to the transmission-type liquid crystal display panel 65. Therefore, the use efficiency of light is improved and the heat generation of the liquid crystal layer and the degradation of the display characteristics are minimized.
However, the above-mentioned techniques have respective problems which will be described below.
According to the technique (FIG. 8) described in Japanese Laid-open Publication No. 4-178683, the liquid crystal display panel 48 is irradiated with light beams from two directions, so that the-incident angle of illumination light becomes large and the optical system itself becomes large. When the incident angle of the illumination light becomes large, the aperture of the projection lens of the projection unit should be enlarged. Furthermore, the number of components such as the polarization beam splitter 44, the polarization rotating element 45, and the like increases. This causes the apparatus to be enlarged and the cost to be increased.
According to the technique (FIG. 9) described in Japanese Laid-open Publication No. 2-93580, the polarization beam splitter 55, the reflector 56, and the quarter-wave plate 54 are required to be provided. Therefore, the technique shown in FIG. 9 also has the problem that the apparatus is enlarged, and the cost is increased in the same way as in the technique described in Japanese Laid-open Publication No. 4-178683. In addition, because of the absorption and reflection of illumination light by the quarter-wave plate 54, the loss of the illumination-light is caused. Furthermore, since the quarter-wave plate 54 is disposed close to the illumination unit 51, the characteristics of the quarter-wave plate 54 are degraded by the heat of the illumination unit 51.
The technique (FIG. 10) described in Japanese Laid-open Publication No. 4-127120 also has the problem that the apparatus is enlarged and the cost is increased because of the presence of the polarization beam splitter 64. Furthermore, a discontinuous portion is present in a polarization separating film of the polarization beam splitter 64. It is very difficult to uniformly form the polarization separating film in the discontinuous portion. This changes the polarization separation characteristics, resulting in display inconsistency.
As described above, there has been a demand for a miniaturized and light-weight projection-type image display apparatus which has outstanding use efficiency of light from an illumination unit, enables a bright image with a high contrast to be obtained, has an outstanding display quality, and is produced at low cost.