Diffraction patterns of optical discs under the far-ﬁ eld condition

When optical discs are illuminated, bright-colored lines can often be observed on the surface of them. Starting from the oblique incident grating diffraction model, this paper analyzes the physical mechanism behind the appearance of these colored lines and provides the coordinate expression of the location of the colored lines on the optical disc under the far-ﬁ eld condition. The wave-length distribution of the colored lines on the optical disc under white light illumination is also given by the wave vector relationship of the diffraction process. To verify the theoretical analysis results, an experimental apparatus was designed and constructed to measure the position and color of the colored lines. The experimental data, analyzed through Gaussian process regression and direct comparison, demonstrates a good consistency with the theoretical analysis results.


Introduction
Since its emergence in the 1980s, optical discs have been an important digital information storage medium.Due to the periodic tracks on the surface of the optical disc, the optical disc is also often used as an experimental teaching tool that can reproduce the phenomenon of optical diffraction and interference in demonstration experiments or introductory design experimental teaching [1][2][3][4].Even simple spectrometers can be made using optical discs [5].
In these experiments, some interesting optical phenomena, such as rainbow rings [3] and colored lines [6], can be observed somewhere in space after the light is incident on the surface of the disc.
When a light source is used to illuminate the surface of a CD disc, bright and distinct colored lines as illustrated in figure 1 are often observed [6].The colored lines provide tangible experiences for students to understand how the positions and colors of these lines change with variations in the light source and the observer's positions.
Luca et al [6] conducted an analysis of colored lines on optical discs using a reflective diffraction grating model.They concluded that a line with a specific color would appear when the light source is incident at a grazing angle and the observation direction is approximately parallel to the incident light.However, their conclusion was limited to high-angle observations under skimming light illumination.Additionally, their model relied on a prior assumption that the colored line would always align with the angle bisector of the incident light and the line of sight.While this assumption simplifies calculations, it is unrealistic when observing the disc from a lower angle, which is more common in everyday life.
In this paper, we extend the work of Luca et al to provide a more comprehensive and internally consistent analysis of the mechanism and properties of colored lines observed on optical discs under far-field conditions.Our analysis is complemented by experimental investigations.Utilizing the fan-shaped slanted diffraction grating model, we first derive a quantitative expression for the azimuthal position of the bright region on the optical disc under monochromatic light illumination, as a function of the light source direction and observation direction.Subsequently, by examining the phase relationship of the diffraction process, we calculate the color characteristics of the colored lines on the optical disc under white light illumination, establishing the fundamental principles governing the appearance of different spectral colors.Furthermore, we design and construct a simple yet effective experimental setup to measure and validate the predicted positions and color characteristics of the colored lines.Notably, this experiment can be easily replicated in a classroom setting, making it a valuable teaching example for an undergraduate optics course.

Structure of optical discs
Some literatures have explored the structural characteristics of optical discs [7][8][9].Here, we briefly introduce a commonly used and well-supported model.Figure 2 depicts the layered structure typically found in CD and DVD discs [9].The disc surface is comprised of a transparent polycarbonate substrate with a refractive index of 1.585, forming the glossy exterior.Below the substrate lies the data recording layer, followed by a highly reflective layer, a protective layer, and a printing layer.The recording layer contains a spiral track etched with data, extending from the innermost spiral to the outer edge.
Atomic force microscopy (AFM) images reveal disorder along the angular direction but relative orderliness along the radial direction.Lutz et al [8] found that the overall radial structure exhibits periodicity, suggesting a periodic grating structure (as assumed in equation 2 in section 2.2).Consequently, in our analysis, we consider the optical disc as a circular grating structure consisting of numerous small fan-shaped sections.For CDs and DVDs, the common track spacing corresponds to grating constants of 1.5 μm and 0.738 μm, respectively.The recording area measures 60 mm in width and has an approximate thickness of 1 mm.The angular direction of the disc aligns with the grating lines, while the radial direction aligns with the normal direction of the grating lines.It is worth noting that these simplified assumptions yield satisfactory results in our experimental investigation (section 3.3).A more realistic model incorporating the detailed radial structure would necessitate AFM cross-sectional analysis, which exceeds the scope of typical classroom teaching and is therefore not addressed in this study.

Oblique incident grating diffraction model of discs
When a beam of light shines onto a disc, it undergoes refraction at the interface between air and the disc base, causing it to enter the disc base.The interaction between the incident light and the recorded layer in the disc can be compared to the diffraction process of light on a reflective plane grating surface, as depicted in figure 4. Following diffraction, the diffracted light exits the disc, undergoes refraction again at the disc base and air interface, and ultimately enters the observer's field of view.
Consider a small, fixed area on the surface of the disc.According to the human eye's visual characteristics, if the diffracted light emitted from that area is at a local maximum of the diffraction light intensity distribution in the observer's direction, then the observer perceives that area as visually brighter than its surroundings (the 'dim regions'), and it may be considered a 'bright region'.To calculate the positions of the bright regions, we need to find all the light paths that allow the light to reach a local maximum intensity.When the positions of light source and observer are fixed, all the bright regions would concatenate to form a larger region (a straight line that passes through the center of the disc under far-field condition, as shall be demonstrated in section 2.5).A toy model and real-life example of the 'bright regions' are shown in figures 3(a) and (b).Similarly, when the positions of the light source and a region of interest (A) on the disc are fixed, and the observer is free to move in the space instead, that specific region of interest would only appear bright when the observer observes at certain positions.We refer to those observing positions as 'best-observing-positions of the region A'.
Let us now narrow our focus to a small region on the disc's radius, which can be considered as a one-dimensional diffraction grating.In the case of a vertically incident parallel beam of light, the surface connecting different best-observing positions of the region would be a flat surface perpendicular to the grating lines.However, when the incident light is obliquely incident, especially when the light waves incident on the same slit has different phases, the best-observing positions of the region no longer form a flat surface but instead form a cone [10].A schematic of the diffraction process of incident light on a disc.For the 'bright regions', the incident light and the diffracted light are situated on the surface of a symmetrical cone with the same vertex angle.The vertex of this cone is located at the region (small enough to be considered as a point) on which the incident light shines, and its axis of symmetry is the direction of the grating lines of that region.Suppose the incident light is monochromatic.Since the region is small enough, the incident light can be approximately regarded as a plane monochromatic wave.The complex amplitude of the light wave which shines on the region can be written as: where A is the complex amplitude, k is the wave vector, and f is the phase.By establishing a Cartesian coordinate system with the origin at the incident point of the region, we can define the x-axis as the radial direction of the disc in this small angle area, the y-axis as the tangential direction of the disc orbit (which is equivalent to the grating groove direction), and the z-axis direction as perpendicular to the grating plane, as shown in figure 4.
The screen function of the one-dimensional grating can be expressed as: where g(x) is the screen function of a single slit, and d is the grating constant.Note that in equation (2), we only assumed a periodic structure for the grating, as explained in section 2.1, so our conclusion is not limited by the specific topology within a single period.After diffraction on the disc, the spatial frequency spectrum function in the diffraction space can be expressed as [11] å where F represents the Fourier transform, and δ represents the Dirac function.f x and f y are the spatial frequencies of the diffraction light in the x and y directions, respectively.Also notes that is the Fourier transform of g(x).
For the best-observing-positions of region , A the spatial frequency spectrum function should take the maximum value (non-zero in the case of delta function).Therefore, a constraint on f x and f y could be established: , , .5 It can be seen that the y-component of the wave vectors of the incident light and the diffracted light with maximum intensity are identical, which indicates that the incident light and the diffracted light are situated on the surface of a symmetrical cone with the same vertex angle, as depicted in figure 4. The vertex of this cone is located at the region A (small enough to be considered as a point) on which the incident light shines, and its axis of symmetry is the direction of the grating lines of that region.From the analysis above, it can be concluded that the bright regions will only appear at specific points where the connecting lines with both the light source and the observing position lie on the same cone.In other words, the projection of the wave vector k of the incident light and the diffracted light in the direction of the grating lines on the disc is equal.Detailed calculations are carried out in the next section.

Analysis of the positions of bright regions on the disc
We establish a Cartesian coordinate system with the center of the disc as the origin and the disc plane as the x-y plane.The incident light emitted by the light source located at r , s diffracts upon passing through the recording layer of the light disc and exits as a reflectiontype diffraction process.This process can be equivalently represented as the light source located at the mirror position ¢ r , s which is symmetrical to the light disc, emitting incident light and diffracting through the recording layer as a transmission-type diffraction process.Taking into account the small thickness of the disc, the effect of the medium thickness can be neglected.Thus, it is assumed that upon entering the disc, the incident light diffracts immediately and exits the disc immediately thereafter.In table 1, the coordinates of the light source, mirror image light source, observer, and small grating area on the disc are presented.
As shown in figure 5(a), in this Cartesian coordinate system, the unit vector of the propagation direction from the light source mirrored onto a certain region on the grating, as well as the unit vector of propagation direction from that region to the observer, can be expressed as follow:  Let n be the refractive index of the disc substrate, then the above equations can be rewritten as: The direction vector of the grating lines on the disc is.
According to the previous discussion, the projection of the wave vector k of the incident light and the diffracted light on the direction of the grating lines on the disc is equal, as depicted in figure 5(c).Therefore: Substituting equations (8), (9), and (10) into equation (11) and simplifying, we get: Equation ( 12) represents the necessary condition that must be satisfied by the positions of the bright regions on the disc when observed by an observer located at (x o , y o , z o ) under the illumination of incident light emitted from a light source located at (x s , y s , z s ).

The color of bright regions on the optical disc
When the incident light is not monochromatic but polychromatic, the diffraction of different wavelength incident light on the grating of the optical disc may overlap and display a much more complex pattern in the bright areas.As shown in the light path in figure 5(b), for a small bright region on the grating of the optical disc, the incident light and the diffracted light can be approximated as parallel light.Using the phase difference relationship in the diffraction process of the grating, the wave number k of incident light should satisfy:

ˆ(
) represents the radial normal vector of the grating lines in that area, and m is a positive integer.
If the positions of the light source and observer are given, the coordinates of the bright regions and therefore the value of ¢ ¢ r r , 1 2 can be calculated via equation (12).The wavelength of the light at those bright regions can be subsequently calculated by substituting into equation (13) the value of ¢ ¢ r r , .
1 2 Thus the color of the observed bright bright region can be obtained.
It should be noted that there may be multiple k and m that satisfy equation (13), which means that the diffracted light observed by the observer at a point may actually be a noncoherent superposition of multiple monochromatic waves with different wavelengths.The specific visual color depends on the spectral distribution of the incident light, the coordinates of the diffraction position, and the optical transmittance of the material of the optical disc, which is beyond the scope of this work.

Far-field condition
The equations ( 12) and (13) derived in the previous section give a complete solution to the positions and wavelength distribution of the bright regions on the disc.However, when the distance between the light source and disc, as well as that between the observer and the disc are sufficiently far (  r r r , ( ) It is not difficult to observe that the bright region characterized by equation (14) reduces to a straight line passing through the center of the disc, which is commonly known as the 'colored line'.Under far-field conditions, the position of the colored line can be better illustrated by introducing the spherical coordinate system.We designate the center of the disc as the coordinate origin and introduce the radius r, zenith angle θ, and azimuth angle f.The radius r represents the distance between a point on the disc and the origin, the zenith angle θ represents the angle between the vector pointing from the origin to the point and the positive z-axis direction, and the azimuth angle f represents the angle between the projection of the vector onto the xy plane and the positive x-axis direction.Due to the rotational symmetry of the disc, we may assume that the azimuth angle of the light source is 0. The Cartesian coordinates of r , s r , o and r c of the colored line represented in spherical coordinates are as follows: q q = r r r sin , 0, cos 15 Substituting equations (15), ( 16), and (17) into equations (13) and (14), one can obtain the azimuth angle and wavelength of the color line under the far-field condition: sin cos cos sin sin sin , 19 where m is the diffraction order.If the positions of the light source and observer are determined, the azimuth angle of the color line and the wavelength of the colored line under polychromatic light can be easily calculated, employing equations (18) and (19).

Measurement of the equivalent grating constant of the discs
Two types of discs, namely CD and DVD, are adopted in the experiment.The equivalent grating constant of these optical discs are measured in a similar way as Nöldeke 1990 [12], using a 532 nm laser.The results are shown in table 2, which are relatively close to the data obtained from product specifications.

Experimental setup
In order to investigate the colored lines on the disc, a custom experimental apparatus was designed and constructed, as illustrated in figure 6.The apparatus comprises an organic glass plate with semicircular grooves serving as the base.Two cylindrical supports are mounted on the grooves and can move along the semicircular rails.One support holds a mobile phone camera for capturing the diffraction pattern of the disc, while the other support holds an LED light source.We chose a white LED as the light source, which covers the green light band and is readily available in the market.Figure 7 depicts the actual experimental setup, where the disc is placed at the center of the semicircular groove, and a rubber band is fastened to the disc to indicate the orientation of the colored lines.A graduated disc with a minimum scale of 1°is engraved around the disc.The semicircular groove has a radius of 70 cm, which is large enough to be assumed to satisfy the far-field condition.In order to obtain the position angle of the colored line, we rotate the disc around its center and observe it through the camera of the phone, until the rubber band and colored line align.We then read the angular position of the rubber band from the graduated disc engraved on the plastic platform, which exactly indicates the position angle of the colored line observed by the camera.In the experiment, the position of the fixed light source was set to f =  0 , S while the zenith angle q S of the light source was varied by adjusting the height of the LED.The observer's zenith angle was also adjustable by changing the height of the phone.To determine the azimuth angle f C of the colored lines, the azimuth angle of the rubber band fixed on the disc was measured after rotating the disc around its center until the rubber band aligned with the colored lines.
Figure 8 shows the relationship between the observer's azimuth angle f o and the colored lines' azimuth angle f c when the zenith angle of the light source q S is 61.9°and 74.6°, and the observer's zenith angle q o is 72.6°, 66.1°, and 57.3°, respectively.The observational data points and corresponding error bars are marked with crosses in the figure.The shaded area represents the 95% confidence interval determined by fitting the experimental data using Gaussian Process Regression, which represents the range of true function values describing the experimental law: the narrower the shaded area, the higher the probability density of the true function distribution, and vice versa.The regions with sparser data points have less constraint on the true function, so the shaded area is wider.The dashed line is the best-fit curve determined by Gaussian Process Regression, and the solid line is the theoretical curve calculated according to equation (18).
Based on figure 8, it can be observed that when f o is smaller than 130°, the theoretical results agree well with the experimental observations.The theoretical curve falls within the 95% confidence interval determined by the experimental data and almost coincides with the best-fit curve.However, when f o exceeds this value, the deviation between the theoretical curve and the experimental data begins to increase.There may be several reasons for the deviation between the experimental and theoretical models when the azimuth angle is large.Firstly, when f o exceeds a certain value, the theoretical calculation indicates that some bands of colored lines fall outside the visible light range, resulting in a decrease in observable data points, and therefore enlarging the best-fit curve uncertainties.Secondly, when f o is greater than 130°, the experimental observations show that the colored lines begin to curve noticeably as shown in figure 9. Finally, due to the impossibility of achieving perfect far-field conditions in the actual experiment, coupled with the fact that the camera (or human eye) has a certain aperture size, the observer can see multiple colored lines in a small interval of f .For larger f , o such estimation would give a poorer prediction than our theory.

The color of the lines
Under far-field conditions, the position and wavelength distribution of colored diffraction lines can be calculated using equations (18) and (19) for a given light source and observer position.Different wavelengths and diffraction orders can produce different colored lines in the same region of the disc so that with a fixed light source and observer position, colored lines of different wavelengths will overlap.Detailed calculations based on equations (18) and (19) are carried out on Python, and the wavelength and position distribution of colored lines can be derived to further analyze the overlap of colored lines of different wavelengths and orders.In the following discussion, we define the red band as 600-780 nm, the green band as 500-570 nm, and the blue band as 415-485 nm.We calculated the wavelength distribution of colored lines when the incident light fall in these 3 bands separately.
In figures 10 and 11, we present the wavelength and order of colored lines observed with the incident light falling in the red, green, and blue bands, respectively, for a fixed light source at q =  78.7 s and f =  0 , s and an observer at q f , .
o o ( ) It is observed that for a given observing direction q f , , o o ( ) if the incident light comes from a white light source instead of the three band filters, lines from different bands and different orders would be observed in the same    place, and significantly overlapping with each other.The true colors of the observed colored lines by humans would therefore be quite complex, which is related to the spectral characteristics of the actual irradiation light source, the visual sensitivity of the human eye to different light wavelengths, and the spectral characteristics of the reflection process of the discs.
Taking into account the higher sensitivity of the human eye to the green light at 555 nm, we opted to document the perception of the green-colored line during the experiment to examine the distribution pattern of the color lines.By keeping the zenith angle q o constant and adjusting the observer's azimuth angle f , o we marked the site as 1 at each location where the green-colored line was observed and 0 elsewhere.The light source was positioned at q =  78.7 o The experimental results are in good agreement with the theoretical calculations, taking into account the overlapping effects of different wavelengths.Formulas (18) and (19) obtained from the theoretical model accurately describe the wavelength distribution pattern of the color lines under far-field conditions.However, for CD, when observer zenith angles are greater than 68.9°and f o is small, a significant deviation between the theoretical and experimental results is observed due to the overlapping of color lines, which results in greater bias.

Conclusion
This study investigated the phenomenon of colored lines on CD and DVD discs under farfield illumination conditions.We discovered that the azimuth angle of the colored line on the discs, when observed from various angles, does not simply align with the angle bisector of the incident light and the line of sight, but rather follows a more complicated functional relationship with the zenith angle of light source and the observer, in addition to the azimuth angle of the observer.We also reported a useful relationship to estimate the color of the colored line, under far-field condition.Our results may be interesting to undergraduate students who are currently taking optics courses, and may serve as an excellent teaching example.Further research is required to analyze and investigate the bending colored lines and color overlap under non-far-field conditions or observation azimuth angles greater than 130°.

Figure 1 .
Figure 1.A distinct 'colored line' would appear when observed from certain angles.

Figure 2 .
Figure 2. The spiral track and the layered structure of optical disc.

Figure 4 .
Figure 4.A schematic of the diffraction process of incident light on a disc.For the 'bright regions', the incident light and the diffracted light are situated on the surface of a symmetrical cone with the same vertex angle.The vertex of this cone is located at the region (small enough to be considered as a point) on which the incident light shines, and its axis of symmetry is the direction of the grating lines of that region.

Figure 3 .
Figure 3. (a) Diagram of bright and dim regions on the surface of a disc.When the positions of light source and observer are fixed, all the bright regions seen by the observer would concatenate to form a larger region.(b) A real-life example of the 'bright regions' and 'dim regions'.

Figure 5 .
Figure 5.A schematic of the light path.(a) The coordinate system and the position of light source, light source mirror, as well as the observer (not marked in the figure).(b) The zoom-in view of the 'bright region' of the disc.(c) The zoom-in view of the middle plate in subplot (b).

Figure 6 .
Figure 6.A sketch of the experimental setup.

oalso marked the f f = 2 co 2 co
We / curve, which is the assumption of Luca in deriving his conclusions, with dash-dot line in figure8.It can be seen that f f = / can only give a crude

Figure 7 .
Figure 7. (a) The actual experimental setup and (b) the 'colored line' pattern on the disc obtained by the mobile phone.

Figure 9 .
Figure 9.The colored line begins to curve noticeably when f o is greater than 130°.

Figure 8 .
Figure 8.The relation between f c and f o when q S and q o are given.The experimental data points and their corresponding error bars are denoted by crosses, and the shaded region represents the 95% confidence interval obtained by fitting the experimental data using Gaussian Process Regression.The dashed lines represent the best-fit curve, while the solid line represents the theoretical curve calculated using equation (18), in comparison to the bisector of the azimuth angle between the light source and observer (as assumed by Luca et al) marked by dash-dot lines.The theoretical predictions are in good agreement with the experimental observations, except for cases where f o is relatively large.Further analysis and discussion on this deviation can be found in section 3.3. o

Figure 10 .
Figure 10.The distribution of observation zenith and azimuth angle for the red, green and blue bands of colored lines on the CD disc.

Figure 11 .
Figure 11.The distribution of observation zenith and azimuth angle for the red, green and blue bands of colored lines on the DVD disc.

Figure 12 .Figure 13 .
Figure 12.The comparison between the experimentally observed and theoretically calculated result of the green-colored line on a CD, for different observer zenith angles (q o ) and azimuth angles (f o ).The red dashed lines indicate the observations, while the black solid lines represent the theoretically calculated results of the green-colored line at different angles.The zenith angle of light source is fixed at q =  78.7 .s s fixed throughout the experiment.Figures12(a)-(h) and 13(a)-(h) present a comparison between the experimentally observed and theoretically calculated appearance of the green-colored line for CD and DVD, respectively, for different observer zenith angles (q o ) and azimuth angles (f o ).The red dashed lines indicate the observations, while the black solid lines represent the theoretically calculated results of the green-colored line at different f .

Table 1 .
Symbolic conventions.The coordinate notations of each object is listed.The corresponding direction vector is marked in bold.
(12)two equations above can be further simplified.In this scenario, equation(12)becomes.

Table 2 .
The equivalent grating constant of the CD and DVD used in the following experiment.Measurement results of the azimuth angle of the colored lines Category Product specification d nm ( ) Experimental measurement d nm ( )