Total internal reflection is the phenomenon which occurs when a propagated wave strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary and the incident angle is greater than the critical angle, the wave cannot pass through and is entirely reflected. The critical angle is the angle of incidence above which the total internal reflection occurs. This is particularly common as an optical phenomenon, where light waves are involved, but it occurs with many types of waves, such as electromagnetic waves in general or sound waves. When a wave reaches a boundary between different materials with different refractive indices, the wave will in general be partially refracted at the boundary surface, and partially reflected. However, if the angle of incidence is greater (i.e. the direction of propagation is closer to being parallel to the boundary) than the critical angle – the angle of incidence at which light is refracted such that it travels along the boundary – then the wave will not cross the boundary, but will instead be totally reflected back internally. This can only occur when the wave in a medium with a higher refractive index (n1) reaches a boundary with a medium of lower refractive index (n2). For example, it will occur with light reaching air from glass, but not when reaching glass from air.
Total internal reflection of light can be demonstrated using a semi-circular block of glass or plastic. A "ray box" shines a narrow beam of light (a "ray") onto the glass medium. The semi-circular shape ensures that a ray pointing towards the centre of the flat face will hit the curved surface at a right angle; this will prevent refraction at the air/glass boundary of the curved surface. At the glass/air boundary of the flat surface, what happens will depend on the angle. If θc is the critical angle, then the following scenarios depict what will happen according to the size of the incident angle.
- If θ ≤ θc, the ray will split; some of the ray will reflect off the boundary, and some will refract as it passes through. This is not total internal reflection.
- If θ > θc, the entire ray reflects from the boundary. None passes through. This is called total internal reflection. TIR is the abbreviation.
This physical property makes optical fibers useful and prismatic binoculars possible. It is also what gives diamonds their distinctive sparkle, as diamond has an unusually high refractive index.
The critical angle is the angle of incidence for which the angle of refraction is 90°. The angle of incidence is measured with respect to the normal at the refractive boundary (see diagram illustrating Snell's law). Consider a light ray passing from glass into air. The light emanating from the interface is bent towards the glass. When the incident angle is increased sufficiently, the transmitted angle (in air) reaches 90 degrees. It is at this point no light is transmitted into air. The critical angle is given by Snell's law,
Rearranging Snell's Law, we get incidence
To find the critical angle, we find the value for when 90° and thus . The resulting value of is equal to the critical angle .
Now, we can solve for , and we get the equation for the critical angle:
If the incident ray is precisely at the critical angle, the refracted ray is tangent to the boundary at the point of incidence. If for example, visible light were traveling through acrylic glass (with an index of refraction of approximately 1.50) into air (with an index of refraction of 1.00), the calculation would give the critical angle for light from acrylic into air, which is
Light incident on the border with an angle less than 41.8° would be partially transmitted, while light incident on the border at larger angles with respect to normal would be totally internally reflected.
If the fraction is greater than 1, then arcsine is not defined—meaning that total internal reflection does not occur even at very shallow or grazing incident angles.
So the critical angle is only defined when is less than or equal to 1.
Derivation of evanescent wave
An important side effect of total internal reflection is the appearance of an evanescent wave beyond the boundary surface. Essentially, even though the entire incident wave is reflected back into the originating medium, there is some penetration into the second medium at the boundary. The evanescent wave appears to travel along the boundary between the two materials, leading to the Goos-Hänchen shift.
If a plane wave, confined to the xz plane, is incident on a dielectric with an angle and wavevector then a transmitted ray will be created with a corresponding angle of transmittance as shown in the figure above. The transmitted wavevector is given by:
If , then since in the relation obtained from Snell's law, is greater than one for
As a result of this becomes complex:
The electric field of the transmitted plane wave is given by and so evaluating this further one obtains:
Using the fact that and Snell's law, one finally obtains
where and .
This wave in the optically less dense medium is known as the evanescent wave. It is characterized by its propagation in the x direction and its exponential attenuation in the z direction. Although there is a field in the second medium, it can be shown that no energy flows across the boundary. The component of Poynting vector in the direction normal to the boundary is finite, but its time average vanishes. Whereas the other two components of Poynting vector (here x-component only) and their time averaged values are in general found to be finite (i.e. nonzero).
It is important to note that the Goos-Hänchen effect only occurs for linearly polarized light. If the light is circularly or elliptically polarized, it will undergo the analogous Imbert–Fedorov effect.
Frustrated total internal reflection
See also: Attenuated total reflectance
Under ordinary conditions, the evanescent wave transmits zero net energy across the interface. However, if a third medium with a higher refractive index than the low-index second medium is placed within less than several wavelengths distance from the interface between the first medium and the second medium, the evanescent wave will be different from the usual one, and it will pass energy across the second into the third medium. (See evanescent wave coupling.) This process is called "frustrated" total internal reflection (FTIR) and is very similar to quantum tunneling. The quantum tunneling model is mathematically analogous if one thinks of the electromagnetic field as being the wave function of the photon. The low index medium can be thought of as a potential barrier through which photons can tunnel.
The transmission coefficient for FTIR is highly sensitive to the spacing between the third medium and the second medium (the function is approximately exponential until the gap is almost closed), so this effect has often been used to modulate optical transmission and reflection with a large dynamic range.
Phase shift upon total internal reflection
A lesser-known aspect of total internal reflection is that the reflected light has an angle dependent phase shift between the reflected and incident light. Mathematically this means that the Fresnel reflection coefficient becomes a complex rather than a real number. This phase shift is polarization dependent and grows as the incidence angle deviates further from the critical angle toward grazing incidence.
The polarization dependent phase shift is long known and was used by Fresnel to design the Fresnel rhomb which allows one to transform circular polarization to linear polarization and vice versa for a wide range of wavelengths (colors), in contrast to the quarter wave plate. The polarization dependent phase shift is also the reason why TE and TM guided modes have different dispersion relations.
- Total internal reflection is the operating principle of optical fibers, which are used in endoscopes and telecommunications.
- Total internal reflection is the operating principle of automotive rain sensors, which control automatic windscreen/windshield wipers.
- Another application of total internal reflection is the spatial filtering of light.
- Prisms in binoculars use total internal reflection, rather than reflective coatings, to fold optical paths and show erect images.
- Some multi-touch screens use frustrated total internal reflection in combination with a camera and appropriate software to pick up multiple targets.
- Gonioscopy employs total internal reflection to view the anatomical angle formed between the eye's cornea and iris.
- A gait analysis instrument, CatWalk XT, uses frustrated total internal reflection in combination with a high speed camera to capture and analyze footprints of laboratory rodents.
- Optical fingerprinting devices use frustrated total internal reflection in order to record an image of a person's fingerprint without the use of ink.
- A total internal reflection fluorescence microscope uses the evanescent wave produced by TIR to excite fluorophores close to a surface. This is useful for the study of surface properties of biological samples.
- Total internal reflection is the operating principle of LED Light Panels. This technology utilizes LGPs (Light Guide Plates) as the vehicle for transmitting light over large areas. By etching a grid pattern on the second surface of the LGP, frustrated total internal reflection occurs allowing the light to escape the LGP as visible illumination.
- Diamonds shine brightly due to total internal reflection.
- Total internal reflection (TIR) are used on lighting industry to manage narrow distribution (NEMA classification) for floodlight.
Examples in everyday life
Total internal reflection can be observed while swimming, when one opens one's eyes just under the water's surface. If the water is calm, the surface appears mirror-like (except directly overhead, in Snell's window).
Diamonds used for jewelry are normally shaped in such a way as to maximize the amount of light that is totally internally reflected from the back side of the diamond. The very high refractive index of diamond gives a small critical angle allowing almost all the light entering the diamond to be reflected back for optimized diamond cuts such as the brilliant.
- This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C" (in support of MIL-STD-188).
- ^Ehrlich, Robert (1997). Why toast lands jelly-side down: zen and the art of physics demonstrations. Princeton, New Jersey, USA: Princeton University Press. p. 182. ISBN 0-691-02891-5. Retrieved 9 February 2012.
- ^Xu, Yadong; Chan, C. T.; Chen, Huanyang (2015-03-03). "Goos-Hänchen effect in epsilon-near-zero metamaterials". Scientific Reports. 5 (1). Bibcode:2015NatSR...5E8681X. doi:10.1038/srep08681. ISSN 2045-2322.
- ^Boundless (2016-05-26). "Total Internal Reflection and Fiber Optics". Boundless.
- ^"BBC - GCSE Bitesize: Total internal reflection". Retrieved 2017-09-04.
- ^Moreno, Ivan; J. Jesus Araiza; Maximino Avendano-Alejo (2005). "Thin-film spatial filters"(PDF). Optics Letters. 30 (8): 914–916. Bibcode:2005OptL...30..914M. doi:10.1364/OL.30.000914. PMID 15865397.
- ^OD, By Emily Bruce, OD, Rodney Bendure, OD, Sarah Krein, OD, and Nathan Lighthizer,. "Zoom in on Gonioscopy". Retrieved 2017-09-04.
- ^"Gonioscopy - Glaucoma Associates of Texas". Glaucoma Associates of Texas. Retrieved 2017-09-04.
- ^"Gait analysis system and software for rodents | CatWalk". Noldus.com. Retrieved 26 August 2010.
- ^Axelrod, D. (1 April 1981). "Cell-substrate contacts illuminated by total internal reflection fluorescence". The Journal of Cell Biology. 89 (1): 141–145. doi:10.1083/jcb.89.1.141. PMC 2111781. PMID 7014571.
- ^"Total Internal Reflection: Diamonds & Fiber Optics". laser.physics.sunysb.edu. Retrieved 2017-09-04.
Learning Through Reflection
We learn by experiences that allow us to (Wertenbroch, Nabeth, 2000):
- Absorb (see, hear, feel, taste, smell)
- Do (perform an activity)
- Interact (socialize)
In addition, we also learn by reflecting on such experiences (Dewey 1933). Reflection is thinking for an extended period by linking recent experiences to earlier ones in order to promote a more complex and interrelated mental schema or patterns. The thinking involves looking for commonalities, differences, and interrelations beyond their superficial elements. The goal is to develop higher order thinking skills.
Educators often consider Dewey the modern day originator of the concept of reflection, although he drew on the ideas of earlier educators, such as Aristotle, Plato, and Confucius . He thought of reflection as a form of problem solving that chained several ideas together by linking each idea with its predecessor in order to resolve an issue.
Essentials of Reflection
Hatton and Smith (1995) identified four essential issues concerning reflection:
We should learn to frame and reframe complex or ambiguous problems, test out various interpretations, and then modify our actions consequently.
Our thoughts should be extended and systematic by looking back upon our actions shortly after they have taken place.
Certain activities labeled as reflective, such as the use of journals or group discussions following practical experiences, are often not directed towards the solution of specific problems.
We should consciously account for the wider historic, cultural, and political values or beliefs in framing practical problems to arrive at a solution. This is often identified as critical reflection. However, the term critical reflection, like reflection itself, appears to be used loosely, some taking it to mean no more than constructive self-criticism of one's actions with a view to improvement.
Taking it a step further is Critical Reflection — the process of analyzing, reconsidering and questioning experiences within a broad context of issues (Murray, Kujundzic, 2005). Four activities are central to critical reflection (Brookfield 1988):
Assumption analysis - This is the first step in the critical reflection process. It involves thinking in such a manner that it challenges our beliefs, values, cultural practices, and social structures in order to assess their impact on our daily proceedings. Assumptions are our way of seeing reality and to aid us in describing how the order of relationships.
Contextual awareness - Realizing that our assumptions are socially and personally created in a specific historical and cultural context.
Imaginative speculation - Imagining alternative ways of thinking about phenomena in order to provide an opportunity to challenge our prevailing ways of knowing and acting.
Reflective skepticism - Questioning of universal truth claims or unexamined patterns of interaction through the prior three activities - assumption analysis, contextual awareness, and imaginative speculation. It is the ability to think about a subject so that the available evidence from that subject's field is suspended or temporarily rejected in order to establish the truth or viability of a proposition or action.
Most educators believe that reflection is useful in the learning process, even without the supporting research data. However, it is often difficult to encourage reflection among the learners. Gustafson and Bennett (1999) found that promoting reflection among military cadets by means of written responses in diaries or journals was difficult. Cadets across three different years generally did not produce responses indicating any deep reflection. Although the results were disappointing, they are consistent with the research literature that suggests promoting reflection is difficult to accomplish (Stamper, 1996).
In their work, Gustafson and Bennett (1999) identified variables that affected the cadets' lack of reflective behavior. These eleven variables are grouped into three main characteristics:
1. Learner's skill and experience in reflective thinking
The ability to reflect is a learned behavior that is cultivated by the individual over a period of time. How reflective an individual can become is probably a personality trait. However, designing appropriate learning experiences can develop reflecting skills.
2. Breadth of learner's knowledge of the content area
The ability to reflect on a specific topic is directly proportional to how much one already knows. If a learner's schema for a topic is limited, then there is less ability to relate new information to it.
3. Learner's motivation to complete the reflection task
Both internal and external sources of motivation affect the quality of reflection. Internal motivation by nature is difficult to elevate and even more difficult to accurately estimate or measure. External strategies, such as creating a mental challenge, organizing the learners into pairs, or forming competitive teams enhance motivation, but the effectiveness of these and other strategies for promoting reflection awaits verification.
4. Mental preparation (mental set) for reflecting
Although the mental set of the individual might be considered a motivational variable, it is described separately to highlight its probable importance to promoting reflection.
5. Degree of security felt in reporting actual reflections versus perceived desired responses
When there is confidence in the professionalism and integrity of reviewers, the amount and quality of responses are enhanced. This is particularly true when items call for making judgments about the worth of an activity or the quality of the instruction. This type of reflection can be used to promote thinking about what was and was not included that the learner wanted or needed to learn, what the designer of the instruction may have incorrectly assumed about the learner's entering knowledge or skill, or why the instruction was or was not effective.
6. Physical environment in which reflection occurs
The opportunity for the learner to establish an appropriate mental set for reflecting is related to the nature of the physical environment in which reflection is expected to take place. Other factors may contribute to a poor physical environment, such as competing stimuli (e.g. televisions, personal conversations, ambient noise, poor ventilation, high or low temperature, uncomfortable furniture).
7. Interpersonal environment in which reflection occurs
Environments that promote interpersonal interaction may result in greater reflection (Bandura, 1977). Social interaction may enhance motivation and prolong engagement with the task. Social interaction would almost certainly bring forth more information and ideas that could be shared and perhaps result in deeper thinking about the subject. This interaction might take place during the learning activity or it may occur later in formal or informal group discussions.
Reflection Task Characteristics
8. Nature of the stimulus questions, directions, or probes
The nature of the stimulus to reflect will impact the quality of the reflection. Surbeck, Han, and Moyer (1991) identified three levels of reflection:
9. Format required for reporting reflections
Yinger and Clark (1981) believe that reflection results written down are more powerful than reporting them orally. However, handwriting is slow, requires a writing surface, and revisions or extensions of what has been recorded are less likely than for products produced on a word processor. Word processing has the advantage of easy revision, but requires that equipment be readily available.
10. Quality of the feedback provided following reflection
Feedback takes several forms, ranging from no feedback, to acknowledging that the work was done, to commenting on how well it was done, to extending beyond or elaborating on what was submitted.
11. Consequences of Reflecting
Zeichner and Liston (1996) posited a five-part taxonomy of reflection, of which reflection upon completion of the action is only one type:
Of the eleven variables listed above, number 7 - Interpersonal Environment, may hold the most promise for encouraging reflection.
Hatton and Smith (1995) observed students undertaking a four-year secondary Bachelor of Education degree. They were required to complete several activities designed to encourage reflection. The activities included peer interviews in "critical friend" dyads and written reports where they reflected upon the factors that had influenced their thinking and action.
Their research indicated that engaging with another person in a way that encourages talking with, questioning, or confronting, helps the reflective process by placing the learner in a safe environment so that self-revelation may take place.
In addition, students were able to distance themselves from their actions, ideas, and beliefs, by holding them up for scrutiny in the company of a peer with whom they are willing to take such risks.
The study also identified a framework for four types of writing, the first one is non-reflective, while the other three are characterized as different kinds of reflection.
Descriptive writing for reporting events. Its main purpose is to provide a support or a starting point for the framework.
Descriptive reflection attempts to provide reasons based upon personal judgment. For example, "I choose this problem solving activity because I believe the learners should be active rather than passive."
Dialogic reflection forms a discourse with one's self through the exploration of possible reasons. For example, "I became aware that a number of students did not respond to written text materials. Thinking about this, there may have been several reasons. A number of students may still have lacked confidence in handling the level of language in the text.
Critical reflection involves giving reasons for decisions or events, which takes into account the broader historical, social and/or political contexts.
Strategies for Fostering Reflection
Hatton and Smith (1995) reported four activities that in in the process of reflection:
Action Learning (or action research) projects
Case and cultural studies
Structured curriculum tasks:
Reading fiction and non-fiction
Writing tasks such as narratives, biographies, reflective essays, and keeping journals.
However, although these strategies have the potential to encourage reflection, there is little research evidence to show that this is actually being achieved.
Fact questions that are obvious do not promote reflection (e.g., what are the functional areas of a manufacturing plant?). In addition, posing hypothetical situations produced similarly disappointing results (e.g., Assume you have inherited a significant sum of money and wish to buy land in an environmentally sensitive area on which to build. What factors will go into your decision and why?).
In contrast, the most successful probe is asking learners to write a one page letter to a parent, sibling or other significant person in their lives describing a recent experience or event.
Extending evaluative feedback might have even more powerful effects. Providing probes may cause the learner to continue to think about the topic, such as:
- "Have you thought about how a skilled operator might do this?"
- "How much does safety get compromised when you don't use safety shoes?"
Pointing out other possibilities may also result in additional thinking about relationships among factors not previously considered, such as:
- "Another factor you might consider is how many different tools will be required if you use different size bolts in the design?"
- "But what if the rate of water flow is doubled?"
Although such feedback may be provided via written comments, they are normally the most powerful when used in interpersonal dialogue. Carrying on a dialogue with one or more learners about the work they have submitted is probably the ultimate in promoting reflection via feedback. But the logistics of doing so and having discussion leaders who are skilled in the content and possess good interpersonal skills may be beyond the capacity of the system to provide; unless it is computer mediated in some way.
Other hints for encouraging reflection include:
- Seek alternatives
- View from various perspectives
- Seek the framework, theoretical basis, underlying rationale (of behaviors, methods, techniques, programs)
- Compare and contrast
- Put into different/varied contexts
- Ask, "What if . . . ?"
- Consider consequences
Bandura, A. (1977). Social Learning Theory. Englewood Cliffs, NJ: Prentice-Hall.
Brookfield, S. (1988). Developing Critically Reflective Practitioners: A Rationale for Training Educators of Adults. Training Educators of Adults: The Theory and Practice of Graduate Adult Education. Brookfield (Ed). New York: Routledge.
Dewey, J. (1933). How We Think: A Restatement of the Relation of Reflective Thinking to the Educative Process. Boston: D.C. Heath.
Gustafson, K., Bennett, W. (1999). Issues and Difficulties in Promoting Learner Reflection: Results from a Three-Year Study. http://it.coe.uga.edu/~kgustafs/document/promoting.html
Hatton, N., Smith, D. (1995). Reflection in Teacher Education: Towards Definition and Implementation. The University of Sydney: School of Teaching and Curriculum Studies. http://www2.edfac.usyd.edu.au/LocalResource/Study1/hattonart.html
Zeichner, K., Liston, D. (eds.) (1996). Reflective Teaching: An Introduction. Mahwah, NJ: Lawrence Erlbaum Associates.
Murray, M., Kujundzic, N., (2005). Critical Reflection: A Textbook For Critical Thinking. Québec, Canada: McGill-Queen's University Press.
Stamper, C. (1996). Fostering Reflective Thinking Through Computer Mediated Journaling. Unpublished doctoral dissertation. Tempe: Arizona State University.
Surbeck, E., Park Han, E., Moyer, J. (1991). Assessing Reflective Responses in Journals. Educational Leadership. March, 25-27.
Yinger, R., Clark, M. (1981). Reflective Journal Writing: Theory and Practice. East Lansing, MI: Michigan State University. Institute for Research on Teaching (Occasional Paper No. 50). .
Wertenbroch, A., Nabeth, T. (2000). Advanced Learning Approaches and Technologies: The CALT Perspective. http://www.insead.fr/CALT/Publication/CALTReport/calt-perspective.pdf