Adjustable optical element

Abstract

An optical element (100) comprises a substrate (110) and a conductive portion (120) configured to, upon application of an electrical current therethrough, generate heat through joule heating. The substrate is configured to receive heat generated in the conductive portion and, in response, undergo thermal expansion so as to change the curvature of a surface of the optical element. A method of changing the curvature of an optical element is also provided. The method comprises generating heat within the optical element so as to cause thermal expansion within the optical element.

Description

[0001] ADJUSTABLE OPTICAL ELEMENT

[0002] This present invention relates to an optical element with an adjustable focal length. More particularly, the present invention relates an optical element with a focal length which is adjustable in response to the generation of heat through the application of an electrical current.

[0003] Background

[0004] In modern optics, there is an increasing demand for lenses with a wide range of focal lengths. Further, modern optical systems often require real-time fine adjustments to the focal length when in use.

[0005] For example, zoom objective lenses, such as those used in digital cameras, comprise a plurality of lenses which form a complex system. Such lens systems typically utilize moving (sliding) lenses for zooming and focussing actions, which requires movement of substantial (heavy) lenses over macroscopic distances.

[0006] There is a high level of energy input required to move the lenses, and the adjustments can be time consuming. Further, the space needed for a well corrected combination of lenses, as required for modern optical systems, is substantial. This means that systems have to be simplified if they are to be integrated into the space available in devices such as integrated cameras in smartphones, USB cameras, and other devices.

[0007] Further, there is demand for fully aberration corrected lenses, which require non- spherical elements with a high-fidelity shape. These are difficult and expensive to produce.

[0008] An optical element that addresses at least some of these issues is desired.

[0009] Summary of Invention

[0010] According to a first aspect of the invention, there is provided an optical element. The optical element comprises a substrate. The optical element comprises a conductive portion configured to, upon application of an electrical current therethrough, generate heat through joule heating. The substrate is configured to receive heat generated in the conductive portion and, in response, undergo thermal expansion so as to change the curvature of a surface of the optical element.

[0011] Changing the curvature of the surface may modify the focal length of the optical element.

[0012] Having an optical element with a focal length that is adjustable through the application of a current enables dynamic optical systems to be formed without requiring multiple moving lenses. This greatly simplifies the design of dynamic optical systems.

[0013] Using thermal expansion to change the curvature, and therefore the focal length, of the optical element enables fine control of the focal length. A user can determine the required temperature change required to create a specific curvature, and so the focal length can be controlled with fine detail.

[0014] Using joule heating to cause thermal expansion to change the focal length of the optical element ensures that the focal length can be adjusted rapidly. The focal length adjustment may take place on microsecond timescales. The focal length adjustment may take place on millisecond timescales. Using such rapid adjustments enables the optical element to be used in situations requiring rapid focal length adjustments.

[0015] The conductance throughout the conductive portion may be non-uniform. The conductance throughout the conductive portion may be distributed such that, upon application of an electrical current through the conductive portion, joule heating causes thermal expansion of the substrate so as to modify the curvature of a surface of the substrate.

[0016] Having a non-uniform conductive portion may ensure that the amount of heat transferred to the substrate varies across the substrate. Varying the amount of heat transferred into the substrate results in different amounts of local thermal expansion across the substrate, meaning specific curvatures and shapes can be created through the application of current through the conductive portion. The thickness of the conductive portion may be non-uniform. Having a conductive portion with non-uniform thickness causes the conductive portion to have non-uniform conductance.

[0017] The conductive portion may comprise a plurality of dopants. The distribution of dopants throughout the conductive portion may be non-uniform.

[0018] Dopants may increase the conductivity of the conductive region. Dopants may decrease the conductivity of the conductive region. Having a non-uniform distribution of dopants creates a non-uniform conductance throughout the conductive portion.

[0019] The conductive portion may comprise a plurality of conductive elements. The conductance of each conductive element may be non-uniform. The distribution of conductive elements throughout the conductive portion may be non-uniform.

[0020] The conductive elements may be conductive tracks or traces. The conductive elements may be individually operable.

[0021] Having individually operable conductive elements enables generation of heat in specific areas, thus enabling the creation of arbitrary bespoke surface curvatures.

[0022] The conductive portion may be, or comprise, a coating applied to the substrate.

[0023] The conductive portion may be, or comprise, a portion of the substrate. The conductive portion may be, or comprise, a surface of the substrate.

[0024] The optical element may comprise two conductive portions, each conductive portion being associated with a surface of the optical element. The conductance throughout each conductive portion may be distributed such that, upon application of an electrical current through one conductive portion, joule heating causes thermal expansion of the substrate so as to modify the curvature of the surface associated with that conductive portion.

[0025] The optical element may be a lens. The optical element may be a mirror.

[0026] The optical element may have a periodic structure. The optical element may be an optical grating.

[0027] According to a second aspect of the invention, there is provided a method for changing the curvature of an optical element. The optical element may comprise a substrate and a conductive region. The optical element may be the optical element of the first aspect of the invention.

[0028] The method may comprise generating heat within the optical element so as to cause thermal expansion within the optical element. Generating heat may comprise applying a current to the optical element to generate heat through joule heating. Generating heat may comprise generating heat in the conductive region.

[0029] Optional features of any of the above aspects may be combined with the features of any other aspect, in any combination. For example, features described in connection with the optical element of the first aspect may have corresponding features definable with respect to the method of the second aspect, and vice versa, and these embodiments are specifically envisaged. Features which are described in the context or separate aspects and embodiments of the invention may be used together and / or be interchangeable wherever possible. Similarly, where features are, for brevity, described in the context of a single embodiment, those features may also be provided separately or in any suitable sub-combination.

[0030] Brief description of the drawings

[0031] Embodiments of the invention will now be described, by way of example only, with reference to the accompanying drawings, in which:

[0032] Figures la and lb respectively show schematic side views of an optical element according to the present invention before and after the application of a current therethrough; Figures 2a and 2b show side views of optical elements according to the present invention comprising two convex surfaces and two concave surfaces respectively;

[0033] Figures 3a and 3b show schematic side views of optical elements doped with dopants according to the present invention;

[0034] Figures 4a and 4b show schematic side views of optical elements having conductive portions with non-uniform thickness according to the present invention;

[0035] Figures 5a and 5b show schematic side views of optical elements comprising conductive tracks according to the present invention;

[0036] Figure 6 shows a top down view of a temperature distribution for an optical element according to the present invention;

[0037] Figure 7 shows a schematic side view of an optical element comprising two conductive portions; and

[0038] Figures 8(a)-(c) show schematic side views of an optical element comprising a plurality of individually operable conductive elements.

[0039] Detailed description

[0040] Figure la shows a side view of an optical element 100. The optical element 100 comprises a substrate 110 and a conductive portion 120.

[0041] The conductive portion 120 is configured, upon application of an electrical current therethrough, to generate heat through joule heating. In use, heat generated in the conductive portion 120 transfers to the substrate 110, causing the substrate 110 to undergo thermal expansion.

[0042] The conductance of the conductive portion 120 is non-uniform, meaning the electrical resistance varies throughout the conductive portion 120. In this example, the resistance is higher toward the centre of the conductive portion 120 and lower toward the edges of the conductive portion 120. Figure lb shows the optical element 100 of Figure la whilst an electrical current is passing through the conductive portion 120. Because the resistance is greatest towards the centre of the conductive portion 120, there is a greater amount of heat generated via joule heating at the centre of the conductive portion 120 than at the edges of the conductive portion 120. As a result, there is more heat transferred to the centre of the substrate 110 than to the edges of the substrate 110, meaning the centre of the substrate 110 undergoes greater thermal expansion. This creates a change in the curvature of the surface of the optical element 100, which affects the focal length of the optical element.

[0043] The optical element 100 could be a lens, and application of a current through the conductive portion 120 changes the focal length of that lens. The optical element 100 could be a mirror, and application of a current through the conductive portion 120 changes the focal length of that mirror.

[0044] The optical element 100 of Figures la and lb forms a convex surface upon application of an electrical current. In other examples, the surface of the optical element 100 is already convex, but the curvature increases to make the optical element 100 more convex upon the application of an electrical current.

[0045] In other examples, the resistance is greatest at the edge of the optical element 100 and lowest towards the centre of the optical element 100. In such examples, the application of an electrical current creates a concave surface, or increases the curvature of a concave surface, so as to make the surface more concave.

[0046] In other examples, the optical element may have a convex surface and the application of an electrical current creates a change in curvature to produce a concave surface. In other examples, the optical element may have a concave surface and the application of an electrical current creates a change in curvature to produce a convex surface.

[0047] The change in curvature is dependent on the current applied to the conductive portion 120. The power generated through joule heating is given by the equation P=I2R, where P is power, I is the current, and R is the resistance of the region where the heat is being generated. As such, if a greater current is applied, the amount of heat generated increases, meaning the substrate 110 undergoes a greater level of thermal expansion. Therefore, in the optical element of Figure 1, application of a greater current generates more heat and causes greater thermal expansion, meaning the curvature undergoes a greater change.

[0048] Accordingly, the focal length of the optical element 100 can be controlled with high accuracy, as a specific current can be applied to create a specific curvature and hence a specific focal length. In order to maintain a stable focal length, thermal equilibrium must be reached - the heat input into the substrate 110 must be equal to the heat dissipating from the substrate 110.

[0049] Both AC and DC current can be applied to the conductive portion 120. A pulsed current could also be used. Using a DC current may provide the simplest way to maintain a constant focal length. Using a pulsed, or AC, current enables modulation of light travelling through the optical element 100.

[0050] To provide rapid focal length adjustments, a large current can be applied which is quickly reduced as the focal length approaches a target focal length. The large current may provide a large amount of heat input to cause the expansion, and then a smaller current can be used to maintain the level of curvature.

[0051] In the example of Figure 1, the conductive portion 120 is a coating applied onto the substrate 110. In other examples, the conductive portion 120 is part of the substrate 110. In some examples, the optical element 100 comprises an adhesive layer disposed between the coating and the substrate, and, in such examples, the adhesive layer is thermally conductive. In some examples, the conductive portion 120 is embedded within the substrate 110.

[0052] The substrate needs to be, or comprise, a material with a high thermal expansion coefficient. In some embodiments, the substrate has a thermal expansion coefficient exceeding 10'6K1. However, in other embodiments, the substrate can have a thermal expansion coefficient less than 10-6K_1.

[0053] In some embodiments, the substrate has a thermal expansion coefficient exceeding 50X10-6K_ 1. In some embodiments, the substrate has a thermal expansion coefficient exceeding lOOxlO-6K1. In some embodiments, the substrate has a thermal expansion coefficient exceeding 200xl0'6K_1.

[0054] In some examples, the optical element is a lens. In such examples, the substrate and the conductive portion are transparent, or at least partially transparent.

[0055] Substrates for optical elements which are lenses can be, or comprise, one or more of: an amorphous crystal structure (e.g., glass), a single crystalline structure, a metal oxide (e.g., Sapphire, Zirconia), an alkali, and a halogenate (e.g., Sodium chloride crystals, Caesium Iodide, Barium Fluoride, Lithium Fluoride). This list is not exhaustive, and there are other suitable materials for the substrate.

[0056] In some examples, the optical element is a mirror. In such examples, there is no requirement for the substrate to be transparent. In such examples, the surface of the substrate, or a coating applied to the substrate, must be reflective.

[0057] In examples where the optical element is a mirror, the substrate can be, or comprise any of the materials listed above, as well as ceramics with a high thermal expansion coefficient, such as Alumina and Macor. Again, this list is not exhaustive, and there are other suitable materials for the substrate.

[0058] In examples wherein the conductive portion is provided by a coating, and the optical element is a lens, the coating must be transparent, or at least partially transparent. In such examples, the coating may be, or comprise, one or more of: ITO, FTO, NTO, and doped Zinc oxide (Indium Tin Oxide, Fluorine Tin Oxide, Niobium doped anatase). This list is not exhaustive.

[0059] In examples wherein the conductive portion is provided by a coating, and the optical element is a mirror, the coating must be reflective. Example materials which are suitable for forming the coating include pure metals, alloys, or any other conductive materials with a reflectivity of 90% or more.

[0060] In some examples, the coating is deposited onto the substrate using chemical vapour deposition. In some examples, the coating is deposited onto the substrate using Physical vapour deposition. In other examples, alternative methods are used to form the coating. In some examples, there is no coating. In examples of an optical element without a coating, the substrate comprises a surface with non-uniform conductance. In one example, the substrate is glass which is implanted with conductive atoms, such as gallium. In another example of an optical element without a coating, a plurality of conductive tracks are disposed on (e.g., deposited on), or embedded within, a substrate of the optical element.

[0061] Figure 2a shows an optical element 200a comprising two convex surfaces. The optical element 100 comprises two conductive portions (not shown), each conductive portion being associated with one of the convex surfaces. Similar to the optical element 100 of Figure 1, each conductive portion has non-uniform conductance such that application of an electrical current through a conductive portion associated with a surface causes a change in the curvature of that surface via joule heating and thermal expansion.

[0062] Figure 2b shows an optical element 200b which comprises two concave surfaces. As with the optical element 200a of Figure 2a, each of the concave surfaces has an associated conductive portion (not shown). Each conductive portion has non-uniform conductance such that application of an electrical current through a conductive portion associated with a surface causes a change in the curvature of that surface via joule heating and thermal expansion.

[0063] In other examples, an optical element can have one convex surface and / or one concave surface.

[0064] For optical elements comprising multiple surfaces with multiple associated conductive portions, each conductive portion may be individually operable. The curvature of one surface can be modified by applying an electrical current through the associated conductive portion without modifying the other surface.

[0065] Figures 3a and 3b show doped concave and convex optical elements 300a, 300b, respectively.

[0066] In both optical elements 300a, b, the conductive portion 320a, b is a coating applied onto the substrate 310a, b. The conductive portion 320a, b is doped with ions to increase the conductance. In other examples, ion implantation is used instead of doping. In other examples, the optical element 300a, b does not comprise a coating, and the surface of the substrate is doped directly.

[0067] In Figure 3a, the conductive portion 320a has a greater density of dopants in the centre, meaning the resistance is low and the conductance is high at the centre. As such, when an electrical current is passed through the conductive portion 320a, there is more joule heating at the edges of the conductive portion 320a than at the centre, and so the substrate 310a undergoes a greater level of thermal expansion at its edges to increase the curvature of the concave surface.

[0068] The conductive portion 320a comprises electrodes 301 at opposing edges. The electrodes 301 are configured to enable the application of a current through the conductive portion 320a. The electrodes 301 can be any suitable means for connecting external electrical circuitry to the optical element 300a to enable the application of a current through the conductive portion 320a.

[0069] In Figure 3b, the conductive portion 320b has a greater density of dopants at the edges, away from the centre, meaning the resistance is low and the conductance is high at the edges. As such, when an electrical current is passed through the conductive portion 320b, there is more joule heating at the centre of the conductive portion 320b than at the edges, and so the substrate undergoes a greater level of thermal expansion at its centre to increase the curvature of the convex surface.

[0070] Figures 4a and 4b show concave and convex optical elements 400a, 400b, respectively.

[0071] In both optical elements 400a, b, the conductive portion 420a, b is a coating applied onto the substrate 410a, b. The conductive portion 420a, b is a layer of electrically conductive material with non-uniform thickness.

[0072] In Figure 4a, the conductive portion 420a has a greater thickness at its centre than at its edges. Therefore, the resistance is lowest at the centre of the conductive portion 420a and is greatest at the edges of the conductive portion 420a. Upon application of an electrical current, there is more joule heating at the edges of the conductive portion 420a and so there is greater thermal expansion at the edges of the substrate 410a. In Figure 4b, the conductive portion 420b has a greater thickness at its edges than at its centre. Therefore, the resistance is lowest at the edges of the conductive portion 420b and is greatest at the centre of the conductive portion 420b. Upon application of an electrical current, there is more joule heating at the centre of the conductive portion 420b and so there is greater thermal expansion at the centre of the substrate 410a.

[0073] The conductive portion 420a comprises electrodes 401 at opposing edges. The electrodes 401 are configured to enable the application of a current through the conductive portion 420a.

[0074] Figures 5a and 5b show convex optical elements 500a, 500b, respectively.

[0075] In both optical elements 500a, b, the conductive portion 520 is part of the substrate 510. The conductive portion 520 is a portion of the substrate 510 in which there are a plurality of electrically conductive traces 525.

[0076] The conductive portion 520 comprises electrodes 501 at opposing edges. The electrodes 501 are configured to enable the application of a current through the conductive traces 525. The conductive traces 525 can be individually operable, meaning current can be applied through one or more of the conductive traces individually.

[0077] In Figure 5a, the conductive portion 520a has a greater density of conductive traces 525a at its centre than at its edges. Joule heating only occurs within the conductive traces 525a, and so having a greater density of conductive traces 525a at the centre of the conductive portion 520a causes a greater level of thermal expansion at the centre of the substrate 510a than at its edges. In other examples, the density of conductive traces 525a may be greater at the edges of the conductive portion 520b than at the centre.

[0078] In Figure 5b, the conductive portion 520b has a greater density of conductive traces 525b at its edges than at its centre. Therefore, there is more joule heating, and a greater level of thermal expansion, at the edges of the optical element 500b than at the centre. In other examples, the resistance of each conductive trace varies instead of, or in addition to, the density of the conductive traces. Fore example, the thickness and / or conductivity of each conductive traces may vary.

[0079] In other examples, an optical element can comprise two or more of the concepts from the examples of Figures 3-5 in order to provide non-uniform conductance. For example, the thickness of the conductive portion can be non-uniform and it can also be doped with ions and / or contain electrically conductive traces.

[0080] Figure 6 shows a top-down view of the temperature distribution of an optical element 600. The resistivity (and temperature) is indicated according to colour. The brighter areas have a greater resistivity and therefore provide more joule heating upon application of a current.

[0081] In Figure 6, a ring portion 601 is shown as being brighter than the rest of the optical element 600. This indicates that the ring 601 has the highest temperature and hence undergoes the greatest level of thermal expansion. A central region 602 within the ring 601 is darker, meaning it has a lower temperature and hence undergoes less thermal expansion. This creates a concave surface wherein the central region 602 is recessed relative to the ring 601.

[0082] In some embodiments, the optical element comprises a plurality of conductive portions, each conductive portion being associated with the same surface. Applying current to different conductive portions may create different changes in curvature. For example, a first conductive portion may reduce how convex the surface is, and a second conductive portion may increase how convex the surface is. Such an embodiment is shown in Figure 7.

[0083] In Figure 7, optical element 700 comprises a first conductive portion 720a and a second conductive portion 720b. The conductive portions 720a, b are arranged to form layers. The layers are electrically insulated from each other such that current may be applied to each conductive portion 720a, b individually.

[0084] Conductive portion 720a has a greater density of dopants at the edges, away from the centre, meaning the resistance is low and the conductance is high at the edges. As such, when an electrical current is passed through the conductive portion 720a, there is more joule heating at the centre of the conductive portion 720a than at the edges, and so the substrate undergoes a greater level of thermal expansion at its centre to increase the curvature of the convex surface.

[0085] Conductive portion 720b has a greater density of dopants in the centre, meaning the resistance is low and the conductance is high at the centre. As such, when an electrical current is passed through the conductive portion 720b, there is more joule heating at the edges of the conductive portion 720b than at the centre, and so the substrate 710 undergoes a greater level of thermal expansion at its edges to create a concave surface.

[0086] In other embodiments, the optical element may comprise a plurality of conductive portions, each conductive portion being associated with a specific region of the surface of the optical element.

[0087] While concave / convex lenses and mirrors (i.e., objects with point symmetry, rotational symmetry, or even centrosymmetry) are common optical elements falling within the scope of the invention, the invention also encompasses other types of optical element with other types of symmetry (e.g., a periodic arrangement like in a grating or array), or no symmetry at all (as would be required for adaptive optics).

[0088] In standard lenses and mirrors, which can be described using simple mathematical equations (e.g., spherical, parabolic), aberrations occur. In existing optical systems, multi-lens systems are required to overcome the aberration effects by creating an inverse of the aberration. However, in the present invention, any arbitrary surface can be created through the application of a specific current value in order to create an optical element with the exact required curvature to overcome aberration effects.

[0089] Figures 8(a)-(c) show an optical element 800 comprising a plurality of conducive traces 825. Each conductive trace 825 is individually operable, meaning each conductive trace 825 acts as a separate conductive portion.

[0090] Figure 8(b) shows the optical element 800 with alternative conductive traces 825 activated (carrying current). Alternate conductive traces 825 generate heat through joule heating and transfer that heat to the substrate 810. As a result, only the regions of substrate 810 surrounding the active conductive traces 825 undergo thermal expansion, thus creating a sinusoidal surface curvature. In such an example, the optical element 800 acts as a grating. In other examples, different gratings can be formed by activating every third conductive trace, or by activating the conductive traces in any desired periodic pattern.

[0091] In addition to the examples given above, wherein the optical elements are described as being convex / concave lenses, mirrors, and gratings, the optical element may also provide an arbitrarily shaped surface. For example, in Figure 8(c), the three leftmost conductive traces 825 are active whilst the three rightmost conductive traces are inactive, thus creating a unique surface curvature having greater thermal expansion at the left of the figure than at the right. In other embodiments, any combination of conductive traces can be activated in order to create any desired bespoke surface.

[0092] In some embodiments, different levels of current can be applied to each conductive trace. Applying more current results in greater generation and transfer of heat to the substrate, meaning there is more thermal expansion. Different levels of current can be applied to each conductive traces to create bespoke surfaces.

[0093] From reading the present disclosure, other variations and modifications will be apparent to the skilled person. Such variations and modifications may involve equivalent and other features which are already known in the art of adjustable optics and optical systems, and which may be used instead of, or in addition to, features already described herein.

[0094] Although the appended claims are directed to particular combinations of features, it should be understood that the scope of the disclosure of the present invention also includes any novel feature or any novel combination of features disclosed herein either explicitly or implicitly or an generalisation thereof, whether or not it relates to the same invention as presently claimed in any claim and whether or not it mitigates any or all of the same technical problems as does the present invention.

[0095] Features which are described in the context of separate embodiments may also be provided in combination in a single embodiment. Conversely, various features which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable sub-combination. The applicant hereby gives notice that new claims may be formulated to such features and / or combinations of such features during the prosecution of the present application or of any further application derived therefrom.

[0096] For the sake of completeness, it is also stated that the term "comprising" does not exclude other elements or steps, the term "a" or "an" does not exclude a plurality, a single processor or other unit may fulfil the functions of several means recited in the claims and any reference signs in the claims shall not be construed as limiting the scope of the claims.

Claims

Claims1. An optical element comprising: a substrate; and a conductive portion configured to, upon application of an electrical current therethrough, generate heat through joule heating; wherein the substrate is configured to receive heat generated in the conductive portion and, in response, undergo thermal expansion so as to change the curvature of a surface of the optical element.

2. The optical element of claim 1, wherein the conductance throughout the conductive portion is non-uniform.

3. The optical element of claim 1 or claim 2, wherein the change in curvature of the optical element modifies the focal length of the optical element.

4. The optical element of any preceding claim, wherein the thickness of the conductive portion is non-uniform.

5. The optical element of any preceding claim, wherein the conductive portion comprises a plurality of dopants.

6. The optical element of claim 5, wherein the distribution of dopants throughout the conductive portion is non-uniform.

7. The optical element of any preceding claim, wherein the conductive portion comprises a plurality of conductive elements.

8. The optical element of claim 7, wherein the conductance of each conductive element is non-uniform.

9. The optical element of claim 7 or claim 8, wherein the distribution of conductive elements throughout the conductive portion is non-uniform.

10. The optical element of any of claims 7-9, wherein the conductive elements are individually operable.

11. The optical element of any preceding claim, wherein the conductive portion is, or comprises, a coating applied to the substrate.

12. The optical element of any preceding claim, wherein the conductive portion is, or comprises, a portion of the substrate.

13. The optical element of claim 12, wherein the conductive portion is, or comprises, a surface of the substrate.

14. The optical element of any of claims 3-13, wherein the optical element comprises two conductive portions, each conductive portion being associated a different surface of the optical element; and wherein the conductance throughout each conductive portion is distributed such that, upon application of an electrical current through one conductive portion, joule heating causes thermal expansion of the substrate so as to modify the curvature of the surface associated with that conductive portion.

15. The optical element of any preceding claim, wherein the optical element is a lens.

16. The optical element of any of claims 1-14, wherein the optical element is a mirror.

17. A method of changing the curvature of an optical element, the method comprising: generating heat within the optical element so as to cause thermal expansion within the optical element.

18. The method of claim 17, wherein the step of generating heat comprises applying a current to the optical element to generate heat through joule heating.

19. The method of claim 17 or claim 18, wherein the optical element comprises a substrate and a conductive region, and wherein generating heat comprises generating heat in the conductive region.

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