17 July 2014 Diffraction operators in paraxial approach
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Proceedings Volume 9289, 12th Education and Training in Optics and Photonics Conference; 92890D (2014) https://doi.org/10.1117/12.2070745
Event: 12th Education and Training in Optics and Photonics Conference, 2013, Porto, Portugal
Nowadays, research in the field of science education points to the creation of alternative ways of teaching contents encouraging the development of more elaborate reasoning, where a high degree of abstraction and generalization of scientific knowledge prevails. On that subject, this research shows a didactic alternative proposal for the construction of Fresnel and Fraunhoffer diffraction concepts applying the Fourier transform technique in the study of electromagnetic waves propagation in free space. Curvature transparency and Fourier sphere operators in paraxial approximation are used in order to make the usual laborious mathematical approach easier. The main result shows that the composition of optic metaxial operators results in the discovery of a simpler way out of the standard electromagnetic wave propagation in free space between a transmitter and a receptor separated from a given distance. This allows to state that the didactic proposal shown encourages the construction of Fresnel and Fraunhoffer diffraction concepts in a more effective and easier way than the traditional teaching.
Lasso, Navas, Añez, Urdaneta, Díaz, and Torres: Diffraction Operators in Paraxial Approach



Current research in science education, show the need to engage in educational practice a number of methods and techniques to ensure the development of critical thinking students predominantly from an increase in the level of reasoning used to understand content, and which are evident forms of scientific knowledge construction involving a higher degree of abstraction and generalization of it. This variation on the traditional way of teaching, the teacher requires finding quality learning and new creative strategies for their achievement in the terms initially exposed. This is to avoid the use of abstractions unmotivated predominantly associated with rote learning and liabilities acquired and decontextualized late in many cases, given the lack of significance for the learner [1]. It should be noted that the attitudes assumed by teachers to enter some content in the classroom in a traditional way, are characterized by the use of epistemological schemes that prevent the development of both creative attitudes in students, and in themselves when they make their educational work [2]. This of course in turn affects the lack of possibilities to form a more complex thinking and meaningful learning achievement. The strategies used to teach physics at this time looking for the transposition of the acquired knowledge to other contexts, thereby facilitating adequate approach to problem situations that might include some experience associated with the solution of these. Most studies involving the search for solutions to these problematic situations are addressed in the context of the teaching of optics, including the use of the Fourier transform, into their applications can stand the theory of diffraction, resonators theory, obtaining optical and digital of the fractional Fourier transform and finally the correlation operation [3].

From this perspective, in this research, we propose an alternative educational way for the construction of the concepts of Fresnel and Fraunhofer diffraction, using the Fourier transform on the study of electromagnetic wave propagation in free space.

The method makes use of the transparency operator of curvature the field and sphere Fourier operator in paraxial approximation, in order to facilitate the mathematical approach the topic by changing the educational approach traditionally used, and encouraging the use of creative attitudes by both the teacher and students, helping them to reason and operate the physical concepts with a higher level than normally obtained in the study of physical phenomena such.



To display the alternative teaching approach since the concept of diffraction, consider Figure 1, which shows an input plane UA(ξ,η) called the diffraction plane and UP(u, v). The output plane is called observation plane. The finite limits of the aperture have been incorporated in the definition of UA(ξ,η), and the input is considered uniformly illuminated by monochromatic plane wave with unit amplitude and normally.

Figure 1.

Free space propagation


Figure 2.

Fresnel diffraction between the spherical surfaces


To demonstrate the use of alternative teaching approach applied to the concept of diffraction, consider Figure 1, which shows an input plane UA(ξ,η) called the diffraction plane and UP(u, v), the output plane; called observation plane. The finite limits of the aperture function have been included in the definition of UA(ξ,η), and the input is considered uniformly illuminated by a monochromatic plane wave normally incident unit amplitude; applying the Fresnel diffraction integral we obtain the following expression:


The origin of time P is exchanged to the origin of time A, then the factor


Has been neglected. Accordingly, the Fresnel diffraction equation can be written as follows:


But the multiplication of the complex amplitude and phase factor can be interpreted in the paraxial approximation equal to the complex amplitude distribution of spherical surfaces with a radius d for the input and radio – d for the output. Therefore can be rewritten as:


The equation (4) and (5) means that the complex amplitude distributions are geometrically spherical in shape for both the diffraction surface 00013_psisdg9289_92890D_page_3_3.jpg and the surface of observation 00013_psisdg9289_92890D_page_3_4.jpg; also shows that the spherical surfaces are tangent to the respective planes. Therefore, the Fresnel diffraction can be written as:


Simultaneously, equation (6) can be transformed into:


Where 00013_psisdg9289_92890D_page_3_7.jpg is a conventional Fourier transform.

Thus, the Fresnel diffraction is a Fourier transform between two spherical surfaces where the vertices are located in the same axis and separated by a distance similar to the radius of the sphere. This result allows to define the Fourier sphere operator. Consequently 00013_psisdg9289_92890D_page_3_9.jpg is the Fourier sphere [4-6] of 00013_psisdg9289_92890D_page_3_10.jpg. Obviously, this interpretation complies with the Huygens principle, ie the spherical surface of the radio transmitter that propagates d be observed at a distance d, but on a spherical surface of the radio receiver -d.

Thus, "If ASphe is the emitter spherical with radius R = d, and PSphe is a receiver spherical with radius R =-d and both form a concentric surface. The field that is transferred from ASphe to PSphe corresponds to Fraunhofer diffraction phenomenon and is mathematically expressed as a Fourier transform."



Considering the geometry of Figure 3, assume that a spherical monochromatic wave is illuminating the left side of a positive lens.

Figure 3.

Curvature transparency operator


To calculate the complex amplitude distribution of the field on the right side of the lens UB(ξ,η), considering Figure 3, and using properties of the Fourier transform of a lens, one can show that:


And assuming that R1 = RB and R2 = RA the amplitude distribution on the right side of the lens becomes


In equation (9), the factor 00013_psisdg9289_92890D_page_4_4.jpg is a quadratic phase factor, represented by a quadratic approximation to a spherical wave converging to a point of light at a distance RB.

Similarly, the factor 00013_psisdg9289_92890D_page_4_5.jpg is a quadratic phase factor, represented by a quadratic approximation to a spherical wave diverging light toward a midpoint in a distance RA.

Then equation (9) is given as:


If Δo ≠ 0, it is assumed that the lens has a thickness and the spherical surfaces coincide with the surfaces of the faces of the lens. Where UBSph is the complex amplitude distribution of radio RB and UASph is the amplitude distribution is the spherical surface with radius RA.

In the case where Δo → 0 transparency of curvature operator is obtained. Then:


Considering two spherical segments A and B with radii RA and RB shown in Figure 3, states that the transfer from the emitter field spherical ASph tangential to a spherical surface BSph is expressed by a quadratic phase factor dependent and radii of curvature ASph and BSph.



Using the described operators, equation electromagnetic field transfer between a spherical emitter A and spherical receiver B separated by a distance d can be obtained with complete generality (see Figure 4).

Figure 4.

Emitter and receiver system.


With successive applications of two operators, one can achieve the electromagnetic transfer, with complete generality between transmitter RA and receiver RB separated by a distance d (see Figure 4.). Therefore:


In the special case of diffraction from a plane screen located at a finite distance, when RA →∞ and RB →∞, the Fresnel diffraction formula is achieved; for the case RA →∞ or RB →∞ a finite distance, ie spherical transmitter or receiver, an appropriate scaling the coordinates allows obtaining a fractional Fourier transforn; finally, when RA →∞, RB →∞ and d →∞, the resulting expression corresponds to the known phenomenon of Fraunhofer diffraction[7].


The analysis proposed in this work can prove, that the passage of a spherical wave with a radius of curvature well known through a thin lens and the spherical wave study, corresponds mathematically transparent curvature operator. Furthermore, it was found that the Fresnel diffraction metaxial approximation corresponds mathematically to the Fourier transform between spherical surfaces, reaching the sphere of Fourier operator. Furthermore, it is important to note that the composition of the optical operators allows a more simple way to study the propagation of electromagnetic waves in free space, that is, with the application of the two operators, it is possible to obtain the propagation of the electromagnetic field more generally from sender to receiver, when they are separated by an arbitrary distance given. This last aspect, suggests that the proposal submitted, using the transparency of curvature operators and Fourier sphere to study wave propagation in free space, facilitates the construction of the concepts of Fresnel diffraction and Fraunhofer of more effectively and easily than traditional teaching, emphasizing its educational role.



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© (2014) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
William Lasso, Marianela Navas, Liz Añez, Romer Urdaneta, Leonardo Díaz, César O. Torres, "Diffraction operators in paraxial approach", Proc. SPIE 9289, 12th Education and Training in Optics and Photonics Conference, 92890D (17 July 2014); doi: 10.1117/12.2070745; https://doi.org/10.1117/12.2070745

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