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Click on Run to execute the code this also automatically saves the latest version of the code and the plot appearing in the Figure 2 window Fig. The correct interpretation for this sampled rectangle function is a width of 0. In fact, because the rect function coding generates an odd number of samples, if a half width of 0.

This can be visualized by adding the following code: 15 figure 3 16 plot f,'-o' ; 17 axis [80 0 1. The resulting plot Fig. Centering the rectangle in the middle of the vector allows for easy viewing, but as described in Section 2. Add the following the code to the program and run the script to get the plot in Fig. More care needs to be taken when shifting vectors that contain an odd number of samples.

For example, if the fftshift function is used with an odd number of samples, then the ifftshift function should be used to undo the shift. For an even number of samples, the fftshift function works both forward and backward. A second comment is that without the shift operation the FFT algorithm generates a transform for a function that is translated from the zero position, which means a linear phase term will be present in the result shift theorem!

A capital letter is used for the frequency domain vector. Multiplying the result by the sample spacing dx is necessary to correctly approximate the analytic Fourier transform integral. Since each sample in F0 contains two pieces of information the real part and the imaginary part ; or alternatively, the magnitude and the phase, two plots can be used to display this result.

Plot titles and x-axis labels have been included in this code. Run the script and the plots should look like those in Fig. The sinc function nature of the magnitude result in Fig. Thus, the valleys shown in the curve do not necessarily appear to reach zero.

Combining all of this information, the magnitude and phase plots of Fig. In this case the real-valued sinc function could have simply been displayed on one plot; but, in general, Fourier transform results are complex. Once again for display reasons, it is helpful to center the FFT result in the vector. In addition, the spatial frequency coordinates need to be determined.

Running the script generates the plots shown in Fig. This helps diagnose problems and lets you build on previous code with confidence. For the example given in Section 3. The resulting plots Fig. The magnitude results are nearly identical, but the FFT result has slightly higher values than the analytic curve at the edges.

This is a result of the periodic extension property of the FFT Section 2. The example presented here involves the convolution of two Gaussian functions of different widths. In the Editor Window, select New M- file and enter the following code that defines and generates sample values for the two functions fa and fb: magnitude phase 4 discrete discrete 0.

Running this code produces the plot shown in Fig. The functions fa and fb need not be shifted prior to the FFTs since the convolution only depends on their relative positions. This operation is indicated with a period placed before the product operator. Running the code produces the result shown in Fig. As introduced in Section 2. From Fig. Exercise 3. In this book convolutions are coded directly by applying the convolution theorem, but MATLAB has the built-in function conv and for two dimensions, conv2.

For computational speed and efficiency we tend to work with fixed array sizes and pay heed to the support of the signals, as was done in the example above. The side length L is the physical length along one edge of the array where it is assumed that the x and y dimensions are the same. Two identical sample coordinate vectors x and y are defined for the two dimensions. X and Y are used to produce the sampled version of the 2D rect function in the array g.

By using X and Y, the coded version of g appears much like an analytic expression. An image is a common way to visualize optics simulation results. This is helpful in this case where the side length is the same in the x and y directions. The first row of a conventional image file corresponds to the top of the picture.

The axis xy command arranges the y axis to be displayed with increasing values from bottom to top. Running the script produces the image shown in Fig. Add the following code to generate a 1D profile of the x-axis through the center of the array [Fig.

Enter the following code to display the magnitude of the transform results as a surface plot along with a profile slice through the center Fig. The lighting, shading, and colormap commands can be used to change the display. The magnitude could also be displayed in other ways, such as an image or a contour plot. The phase of the result is not shown here, but it could also be displayed in a variety of ways.

Typing a variable name in the Command Window and hitting enter displays the current value, which can be useful for analyzing your code. However, sometimes this gets confusing as code is being edited. The clear all command clears the variable memory. The real or abs functions can be applied to the array to allow the plot to display. For printing purposes the images are displayed in grayscale. However, it is easier to see low-value features with different colormaps—so try some other maps.

To stretch the contrast of a grayscale image to more easily see dim features, a quick trick is to display the nth root of the image values. For example, nthroot g,3 takes the third root of g. The higher the root, the more the contrast is stretched. Just be sure to remember that you are looking at a peak-scaled, contrast-stretched image. The most common programming error is to forget the period when a vectorized operation is needed. Make a habit of checking the vector operations when things are going wrong.

Find the analytic convolution of these functions and compare this result with the discrete result in a plot. Chapter 4 Scalar Diffraction and Propagation Solutions Perhaps the most fundamental task associated with Fourier optics is describing the evolution of an optical field as it propagates from one location to another.

The phenomenon of diffraction underlies the behavior of propagating waves. Extensive theory developed for diffraction provides the basis for modeling optical propagation on the computer. This chapter is essentially a summary of scalar diffraction theory with a listing of the expressions commonly used today to describe optical diffraction of monochromatic light. The presentation closely follows the diffraction development by Goodman.

It accounts for the fact that light rays do not follow strictly rectilinear paths when the wave is disturbed on its boundaries. In our everyday experience we rarely notice diffractive effects of light. The effects of reflection from a mirror , or refraction due to a lens are much more obvious. In fact, the effects of diffraction become most apparent when the confinement size is on the order of the wavelength of the radiation.

Nevertheless, diffraction plays a role in many optical applications and it is a critical consideration for applications involving high resolution, such as astronomical imaging, or long propagation distances such as laser radar, and in applications involving small structures such as photolithographic processes. There is also coupling between the individual components of the electric field, as well as between the magnetic components.

Scalar diffraction refers to the propagation behavior of light under this ideal situation. The long list of assumptions for the medium suggests a rather limited application regime for scalar diffraction theory. However, scalar diffraction can clearly be used for describing free-space optical FSO propagation, which refers to transmission through space or the atmosphere and encompasses a huge number of interesting applications such as lidar, imaging, and laser communications.

Furthermore, for many problems involving less benign propagation media, scalar solutions can provide a reasonable approximation of the principle effects of the propagation and establish a basis for comparison with full vector results.

All of the developments and applications in this book assume scalar diffraction. This expression models a propagating transverse optical electric field of a single polarization. Monochromatic light provides the basis for our analytic and computer simulation approaches to diffraction theory. A truly monochromatic light source is also coherent.

Although some lasers can produce near-monochromatic radiation, true monochromatic light is unachievable. But, as will be discussed in Chapters 7 and 9, the extension of monochromatic results to polychromatic radiation, as well as partially coherent and incoherent radiation, can be straightforward in many useful cases …fortunately! To give an example, a specific form of Eq. This wave has no dependence on x and y and, therefore, is interpreted as extending infinitely in these directions.

If the field in Eq. Furthermore, substituting a complex phasor form for the cosine function provides a valid propagation result and aids in mathematical manipulation. As an example, the phasor form of Eq. To further refine Eq.

Instead, optical detectors respond to the time-averaged squared magnitude of the field. Since we are most interested in the relative spatial form of the field, this constant is usually dropped in our discussions. For example, a typical glass used for visible light might have an index of about 1. For example, in the plane-wave expression of Eq. The term kz gives the number of radians the sinusoid phase of the field has progressed over this distance.

If the plane wave propagates a distance d through a piece of glass with index n, then the OPL is as indicated in Eq. There are other variations of this theme; for example, exp jkr , where r is a radial distance in vacuum. An example 1D profile of the phase of Eq. An important concept is leading and lagging phase. Therefore, we say the phase in the center of the profile in Fig. The further away from the center, the more the phase lags. Interpreting the phase as a representation of an optical wavefront, the center of the wave crest in Fig.

Further physical interpretation of the optical phase is discussed in Section 5. The contributions of these spherical waves are summed at the observation position x, y , allowing for interference. The extension of Eqs. Expression 4. Scalar Diffraction and Propagation Solutions 53 simply re-labeled as x and y. An equivalent expression for Eq. An angular spectrum analysis is often used to derive Eq. The Rayleigh—Sommerfeld expression is the most accurate diffraction solution considered in this book.

By introducing approximations for these terms, a more convenient scalar diffraction form is developed. The criterion of Eq. A looser criterion is the Fresnel number, which is commonly used for determining when the Fresnel expression can be applied. However, a form of the Fraunhofer pattern also appears in the propagation analysis involving lenses.

The Fraunhofer diffraction expression is a powerful tool and finds use in many applications such as laser beam propagation, image analysis, and spectroscopy. But, since it is a scaled version of the Fourier transform of the initial field, it can be relatively easy to calculate, and as with the Fresnel expression, the Fraunhofer approximation is often used with success in situations where Eq.

For simple source structures such as a plane-wave illuminated aperture, the Fraunhofer result can be useful even when Eq. The Fresnel expression is more tractable, but solutions are still complicated even for simple cases such as a rectangular aperture illuminated by a plane wave. Analytic Fraunhofer diffraction analysis is easier and, for our purposes, serves as a check on some of the computer results. Consider a circular aperture illuminated by a unit amplitude plane wave. Now to choose some mesh parameters.

The Bessel function J1 has a first zero when the argument is equal to 1. Now for some code. It is helpful to first make a function that handles the Bessel function ratio. The masking code may appear to be a roundabout way of doing things, but it allows the input x to be a vector or an array. Then, logical indexing is applied—out mask and x mask —to evaluate the function for all elements where mask is 1.

So beware, not all jinc functions are the same. Now for the Fraunhofer pattern. This is known as the Airy pattern. Running the script produces the results in Fig. The Fraunhofer pattern of a circular aperture is commonly known as the Airy pattern. The central core of this pattern, whose width is given in Eq. Find an expression for the optical path length difference OPD for the two parts of the beam between planes a and b. Plot the analytic Fraunhofer irradiance pattern images and profiles for the above apertures on the computer.

Choose suitable propagation distances z and side lengths L in the observation plane. Figure 4. The field exiting one hole has a magnitude of A1 and the field exiting the other hole has a magnitude of A2. Find an analytic expression for the Fraunhofer irradiance for this aperture.

Hecht, Optics, 4th Ed. Chapter 5 Propagation Simulation Now we look at several implementations of the diffraction expressions of Chapter 4 to simulate optical propagation. Although the material is presented as a teaching exercise, these propagation methods are used extensively in research and industry for modeling laser beam propagation. The concentration is on methods that use the fast Fourier transform FFT and only monochromatic light will be considered here.

When designing a simulation there are a variety of issues related to discrete sampling that need to be considered. A common propagation routine is based on Eq. Here are a few remarks on propTF with associated line numbers: a Line The size function finds the sample dimensions for the input field matrix u1 only M is used.

This helps reduce the number of parameters passed to the propTF function. Note that lower case u is used for the spatial field and upper case U is used for Fourier domain quantities, which is not consistent with the use of upper case for the analytic spatial fields; for example, in Eq. But what can you do? Both are established notations, so we live with a little notational mixing. For making the impulse response propagator, some typing can be saved by starting with a copy of propTF.

Again, the source and observation planes in this approach have the same side length. Due to computational artifacts, the IR approach turns out to be more limited in terms of the situations where it should be used than the TF approach, however, it provides a way to simulate propagation over longer distances and is useful for the discussion of simulation limitations and artifacts. Consider a source plane with dimensions 0.

The source and observation plane side lengths are the same for the TF and IR propagators, i. Assume a square aperture with a half width of 0. The simulation, therefore, places 51 samples across the aperture, which provides a good representation of the square opening see Section 2. The source field is defined in the array u1.

The irradiance is found by squaring the absolute value of the field. Executing this script generates Fig. The next part of the script is where the propagation takes place. Lines 28—47 contain code to display an image of the observation plane irradiance as well as irradiance and field magnitude and phase profiles. The num2str function line 47 is introduced to display the propagation distance in the plot title.

The use of the unwrap function for the phase profile display line 45 is discussed in the next few paragraphs. The irradiance results are shown in Fig. The peaks are offset by valleys with less irradiance. The field magnitude and phase are shown in Fig. The phase is in units of radians. In other words, Fig. The important physical interpretation is that it represents the shape of the optical wavefront at the observation plane.

Therefore, the wavefront profile in Fig. Furthermore, imagine rays projecting normal from the wavefront surface to get an idea of where the energy along the wavefront is headed. The magnitude plot in Fig. Now try the impulse response IR propagator. The results in this case should be identical to those in Figs. Discrete sampling of the source field, sampling of the transfer function or impulse response, and the periodic nature of the FFT can lead to a variety of artifacts in the propagation result.

Much of the trouble comes because the chirp functions on the right side of Eqs. This issue is introduced here with some example results. On the other hand, the IR result exhibits periodic copies of the pattern. The IR result is smooth. At longer distances the irradiance pattern is predicted by Fraunhofer theory to take on a sinc2 form.

This generally appears to be the case for Fig. Corresponding irradiance patterns are shown in Fig. In this section we examine criteria used to predict when there will be problems. For more details on these and other criteria, see Appendix A and References 1 and 2 in this chapter and A. This helps reduce artifacts at the edges of the array after propagation due to the periodic extension properties of the FFT.

Further criteria are derived by considering the effects of sampling the chirp functions in the Fresnel transfer function H and impulse response h expressions Appendix A. Oversampling is a good thing, in general. If Eq. Table 5. For each regime, a criterion is described that involves the source field bandwidth B1.

In practice, the source bandwidth criteria of Table 5. So, an effective bandwidth B1 can be used when considering the criteria. Here, the support size available in the observation plane is limited. Thus, the TF Table 5. The undersampled IR phase function has an aliased, periodic phase representation, and using this approach produces periodic copies of the field.

The source bandwidth B1 is only limited in the usual way by the sampling theorem in the source plane. Here, the bandwidth available for the source field becomes limited. This was illustrated in Fig. To consider the criteria in Table 5. Referring to Section 2. Referring to Table 5. However, the observation plane size limitation has a negligable effect on the TF result [Fig.

Thus, most of the significant source spectrum obeys the criterion. But, small ticks still creep into the TF approach result [Fig. On the other hand, artifacts are not apparent in the IR result [Fig. Thus, the TF approach causes significant stair-step artifacts [Fig. The IR approach actually suppresses source frequency components that lie beyond the available bandwidth. This gives a smoother result, but with the small, spurious sidelobes near the array edge [Fig.

The higher sample rate shows whether any spatial details are lost in the propagator results. A Fresnel integral routine was used to compute C and S. The TF and IR propagator profiles are displayed with solid lines, and the analytic results are displayed with dashed lines. There is generally good consistence between the curves. Note the magnitude results are plotted on a log scale to emphasize the small differences in the wings of the profiles.

The phase profiles [Fig. Overall, the propagator result appears quite accurate. The primary deviation is in the wings and is of little consequence. In this case critical sampling leads to an extremely close match with the analytic result. The propagator phase in Fig. However, the criteria can still be used to help find reasonable simulation parameters, although, it often becomes something of an art form to juggle sampling and field parameters to get a satisfactory propagation result.

Critical sampling helps minimize artifacts by allowing full use of the array side length and sampling bandwidth. It seems prudent to try and use critical sampling, but maintaining this condition can be inconvenient. For a given situation, the critical condition may dictate either too many samples for a practical FFT calculation or too few to adequately sample the source or observation planes. Other requirements can be at odds with the critical criterion. For example, phase screens used to simulate propagation through atmospheric turbulence have their own set of sample interval and array size conditions.

In practice, Step 3 in Table 5. If there are signs of artifacts such as the stair-step or sidelobe features illustrated in Figs. This is because a succession of TF propagations is the same as applying the product of the transfer functions to the initial field.

So, even if the shorter propagations are critically sampled, the final result is the same as a single propagation! It is the total propagation distance that is important; however, split- step simulations are applied in many situations for reasons such as propagating between a series of atmospheric turbulence phase screens. Previously, it was noted that the reason the IR approach behaved better for the long propagation example is that it effectively suppresses source frequency content where the frequency chirp function is going bad.

In fact, the IR approach is mainly introduced to give a quick and relatively easy way to approach longer propagation distances. But there are other ways to handle this issue. Researchers working with laser beam propagation simulations also apply window functions to either suppress the source spectrum or remove energy in the wings of the source field. This, combined with multi-step propagation, can give good results. This subject is covered in more detail by Schmidt in reference 2.

Suppose a simulation involves some fixed parameters in the source or observation planes such that a single side length and sample interval will not serve for modeling both planes. In this situation the ability to independently select the physical side lengths of the source and observation planes is helpful.

The two-step method allows the source and observation plane side lengths to be different. This is described and analyzed in Appendix B. While it still suffers from some of the same sampling limitations described for the TF approach, it affords flexibility in the simulation design. When using the FFT to compute the Fraunhofer field, the source and observation plane side lengths are not generally the same. From Eq.

Otherwise, the side lengths are different. Stretch the contrast of the irradiance pattern with the nthroot function to bring out the sidelobes. The simulation result can be checked against the analytic Fraunhofer result. Points in b are analytic values. Now it is your turn: insert Eq. Usually, the irradiance is of interest when calculating the Fraunhofer pattern, so the complex exponentials out front disappear. But, suppose the Fraunhofer field is of interest, including the chirp term. Based on Eq.

This implies a large M. Fortunately, the Fraunhofer phase is not often required. This makes the functions easier to use but it is redundant. Speed and efficiency are not a big problem for the examples in this book, but they can be an important issue when running many iterations of a propagation code. M-Lint is an analyzer that checks the code in the Editor for possible problems. The Profiler tracks the execution time of the various statements and function calls in your code.

It can help find problems and improve the efficiency of your code. Assume critical sampling for a Fresnel propagation. How many samples span the diameter of the circle function? Is the propagation distance within the Fresnel region? Try both TF and IR simulations. What are the distances z that result in critical sampling?

Assuming no absorption or scatter of the light, which is true for the simulations presented in this book, the power proportional to watts should be conserved. In other words, the source and observation planes should contain the same optical power. If not, there may be a code error or a sampling problem.

Maybe two of these? What about dx and dy? You can remove the semicolon from the end of the line with the power calculation so that the value displays in the Command Window when the script is executed. Are there differences between the Fresnel and Rayleigh—Sommerfeld results?

What can you say about applying Fresnel versus Rayleigh—Sommerfeld propagation in this case? Fourier methods are well suited for simulating laser beam propagation. Typically, a laser beam obeys the paraxial ray angle approximation, which is valid for the Fresnel expression. Also, the Gaussian function used to describe the beam profile is more forgiving in terms of sampling artifacts than a square or circular aperture beam of similar support.

Create the Gaussian beam of Eq. Compare irradiance results with the analytic result of Eq. Test the source bandwidth criterion for a m propagation distance. Show that your result is consistent with the analytic expression in Eq. Compare the split-step result with a single TF propagation of 20, m. Are the results the same? Compare discrete and analytic results in an x-axis irradiance profile. Note that there is no attempt in this exercise to model the Fraunhofer field such that the phase is adequately sampled.

Find a criterion for the number of linear samples M necessary for the simulation array in order to adequately sample the Fraunhofer field phase. The result should contain no variables— just a number. Code up the two-step propagator function described in Appendix B. The 0. What are the apparent artifacts for the other distances? Propagation Simulation 87 5. Voelz and M. Goodman, Introduction to Fourier Optics, 3rd ed. Roggemann, and B. Chapter 6 Transmittance Functions, Lenses, and Gratings The beam sources implemented in Chapter 5 are for the most part simple apertures illuminated by a plane wave.

They are modeled with real functions and, in effect, have a zero phase component. In general, these transmittance functions can be thought of as multiplying an incident field to create a desired effect; however, some represent well-known optical components such as a diffraction grating or a lens. The functions discussed in this chapter provide considerable utility in their own right, but like the basic functions they can be combined to create more elaborate fields.

As a matter of convenience these functions are described as part of the source, or as applied in the source plane. However, they can be applied in other planes; for example, the pupil of an imaging system, which is coming up in Chapter 7. An expression for the dashed line in Fig.

This essentially requires replacing the position z with a phase quantity. As time progresses the wave moves in the positive z direction, but as noted previously , the phase representation becomes more negative. This reverses the sign of the expression. Sampling limitations also exist for this technique. As one might guess, if the tilt is large enough to translate the beam in the observation plane beyond the grid boundary, there will be trouble see Exercise 6.

To study this limitation, consider that tilt is a linear phase exponential applied to the source function U1. Using Eq. Some comments about this criterion include the following: a The result is approximate as the specific interaction of the source and propagator phase is not accounted for in Eq. For example, one may cancel some of the effects of the other. Try this tilt angle and see what happens. The resulting beam should appear close to the array edge. In general, it is a good idea to work with tilt angles that are well within the limit set by Eq.

In typical simulations the maximum available tilt angle is quite small, which is consistent with the paraxial nature of the Fresnel propagator. Transmittance Functions, Lenses, and Gratings 93 6. A beam with a spherical wavefront, as shown in Fig.

Thus, the negative sign in Eq. A positive sign corresponds to a diverging wavefront. Borrowing from the discussion of the Fresnel diffraction in Section 4. That looks like a pretty good focus! In this example the focal distance is the same as the propagation distance, so a small spot is expected. The pattern is, in fact, a scaled Fraunhofer pattern. Check out Section 6. Try some other focal distances—see what happens.

Can you get the pattern to expand to fill the observation plane grid? A negative focus value puts the focal point in a virtual position behind the plane and causes a diverging wave Fig. Try it! Multiplying a source field by Eq.

The pattern looks reasonable; however, phase aliasing errors are just starting to creep in on the array edges. This can be seen in the unwrapped phase profile of the observation plane field. This is essentially the same complex exponential defined for focus with zf replaced by f. A positive focal length produces a converging wavefront from a plane-wave input and a negative focal length produces a diverging wavefront.

The pupil function accounts for the physical size of the lens—the opening available to collect light. It is not always practical to implement the transmittance function of Eq. This is because the focal length f is governed by the same criterion as zf given in Eq. If the field incident on the lens is U1 x1, y1 , then the field exiting the lens is U1 x1, y1 tA x1, y1.

Insert this into Eq. The irradiance pattern in Fig. The focused irradiance pattern formed with an ideal circular-shaped lens, such as shown in Fig. A special case of interest is when the source field is located in the front focal plane of a positive lens, a distance f from the lens Fig.

The fourth root is applied for a. The large peak irradiance value in b is because all of the power in the unit amplitude field incident on the lens is being focused to a very small area. U1 x1, y1 U2 x2, y2 f f Figure 6. The chirp phase factor out front is now gone, so the focal plane field is a scaled Fourier transform of the input field.

The arguments in the pupil function account for vignetting, which is a loss of light for off-axis points in the input field due to the finite pupil size. The effect of vignetting is reduced if the lens pupil is oversized compared to the support of the input field. Incident light diffracts either in transmission or reflection from the structure, and the colors wavelength components of the light become spatially separated some distance from the grating.

Gratings are commonly used in spectrometers for examining the wavelength spectrum of an optical signal or in spectrophotometers that measure the spectral characteristics of an optical component. The diffraction pattern from a grating is usually observed in the Fraunhofer region. Transmittance Functions, Lenses, and Gratings 99 6. In Eq. The cosine pattern is only a function of x and has a period P. The Fraunhofer pattern is created using a lens or mirror of focal length f.

At least two are required to satisfy the sampling theorem. Figure 6. To make sure the simulation is working properly, the results can be compared with the analytic expression for the Fraunhofer pattern. Try this and see if the discrete and analytic plots come out the same. The main application for a grating is wavelength separation. The first-order peaks are clearly separated. You can test this by making D1 larger. In this example the ucomb function is used to create a 1D periodic sequence of unit sample delta functions defined in Appendix C.

This delta is defined as a unit value at the coordinate of interest. The ucomb function truncates the input values at the sixth decimal position, so small round-off error will not cause problems in placing the unit samples. For the ucomb function to work properly, the vector coordinates must be such that a sample is found at each position where the delta function is needed. The repmat function is a quick way to fill the rows of the 2D array u1 with the vector ux.

The field created by the grating is propagated using propFF, and the profile for I2 is shown in Fig. It is similar to the cosine grating but with some additional higher-order peaks. Again, analytic theory can be used to verify the simulation. Compare the results with the numerical simulation. Can you get them to match? To do this, the y-dependent terms are removed, meshgrid is no longer required, and the function repmat is not required.

Also, a 1D Fraunhofer calculation is required. The advantage of 1D modeling is that larger vectors can be used; so, more overall width and cycles across the grating can be modeled. The greater number of samples results in an increase of the side length of the Fraunhofer pattern. The expanded view in Fig. More samples across the periodic features in the 1D model produce fewer artifacts small ripples in the result compared to the 2D result.

Although not a practical device, it is not uncommon for a beam of light incident on a grating to cover hundreds or thousands of the periodic cycles—more than anything modeled here so far. But, more importantly, for an infinite-support grating the diffracted orders for an infinite analytic grating are delta functions and the multipliers for these delta functions indicate the relative amount of optical power that is directed to each order.

For optical spectral analysis, high diffraction efficiency into the first order is usually desired. The discrete Fourier transform, applied to obtain the Fraunhofer pattern, produces a result that is associated with repeating copies of the input array periodic extension. Since the source is arranged to be perfectly continuous at the array boundaries, the result is the transform of an infinite periodic structure.

Return to Section 6. Executing the script yields sample delta functions at the diffractive order positions. To find the percentage of optical power in each order, compute the optical power at each sample in the observation 1 0. A display of Ppct is shown in Fig. Use this arrangement to derive the same tilt criterion as defined in Eq. Assume critical sampling and Fraunhofer propagation. Transmittance Functions, Lenses, and Gratings c Is it generally possible to describe the irradiance in the focal plane of a cylindrical lens in terms of a Fraunhofer pattern as was done in Eq.

Derive the analytic result and compare with the simulation result in a profile plot. The transmittance of this plate is illustrated in Fig. If a plane wave illuminates this plate, how do you expect the transmitted field to behave? Create a Fresnel propagation simulation for this plate in the source plane. Assume the following: unit amplitude Figure 6.

The choice of an extremely large value of f relative to the plate radius is necessary for sampling, and also to provide a magnified pattern at the observation plane. Should the transfer function or impulse response approach be used? Display the patterns and profiles. Which is more efficient? Do some testing! What happens? Try some other values make sure M is still even so that other numerical issues are not also happening. Diffraction occurs because of periodic optical path length changes across the grating.

A reflection grating can be modeled in the computer as a phase grating. Adjust the factor m; for example, 1, 2, and 4, and notice the effect on the Fraunhofer pattern. Choose the vector size and other sampling parameters. Compare this result with the numerical result of part a.

Plot the Fraunhofer pattern profile. Plot the power percentage result. Chapter 7 Imaging and Diffraction- Limited Imaging Simulation Imaging is about reproducing the field, or more often the irradiance pattern of an object or scene, at an image plane. Geometrical optics, where optical rays are assumed to travel in rectilinear fashion without diffraction, is used extensively in lens and optical system design. Geometrical optics provides useful relationships between the object and image locations and sizes and is also applied in the analysis of the pupils of an imaging system.

A proficient approach for image modeling draws on both geometrical optics and diffraction theory. This chapter begins with a review of geometrical imaging concepts and relationships that are helpful for the imaging simulations that follow. However, our concern is with imaging, and in order to form a real image, light from an arbitrary object point must be collected and focused at the image plane. For the imaging situation shown in Fig. Principal planes are a virtual concept for geometrical lens analysis.

They are normal to the optical axis. A ray incident on the front principal plane at some height from the optical axis will exit the back principal plane at the same height. In other words, principal planes are planes of unit magnification. A cone of rays from the base or tip of the object are collected by the lens and directed to the corresponding image points. To form a real image, z1 and z2 are positive and the lens focal length f is positive. Practical imaging systems often use combinations of lenses to control aberrations or for packaging reasons, but imaging still requires a positive focal length for the combined lens group.

An imaging system is also characterized by its pupils. The pupils are images of the physical element in the system, known as the aperture stop, which limits the collection of light. The lens is the stop for the system in Fig. There are other system issues, such as aberrations, that further disrupt the image, but the diffractive effects due to the stop represent the fundamental performance limit of an imaging system.

The stop and other system effects can all be incorporated in the pupils, so this concept is utilized for diffraction analysis. Figure 7. Although Fig. In a single thin lens system Fig. This is a useful case to fall back on when thinking about the examples in this chapter. But, to provide some food for thought, refer to Fig.

The iris is the stop, and the exit pupil a virtual aperture is co-located with the iris and has the same diameter as the iris. In this case the pupil and principal plane distances are different. This parameter was briefly introduced in Chapter 6 in the discussion of lenses. A summary of the key points of the geometrical optics discussion is as follows: a The principal plane distances z1, z2 define the transverse magnification of the image.

We refer the reader to other references for further discussions of principal planes, pupils, and geometrical optical imaging. The general imaging arrangement considered is shown in Fig. Imaging with coherent illumination, such as with a coherent laser, is described in its simplest form as a convolution operation involving the optical field.

In the frequency domain the corresponding spectra for Eq. Thus, the coherent transfer function takes on the attributes of the XP. A few comments about Eq. This inversion is associated with the inversion of the ideal geometrical image indicated by Mt in Eq. Thus, the pupil function is defined relative to an ideal spherical wavefront.

Complex exponential terms are included in the pupil function to describe wavefront deviations from a sphere covered in Chapter 8. Unlike the square aperture, the cutoff frequency in this case is the same radially in all directions in the frequency plane.

Applying Eq. Consider a thin lens with a diameter of Working with a larger array increases the image size. This illustrates that the Fourier optics-based simulation described here examines a small part of the image plane for near- diffraction-limited performance.

However, a very large array is needed to model a modest field of view, which might correspond to an image size of, say, 10 or 20 mm in this case. The first thing we need is the ideal image. Actual test charts are printed on a glass substrate, and the USAF is still used today for testing lenses and optical systems.

Similar images of the chart can also be found on the internet, or you can use another image file of your choosing, although preferably something with a variety of feature sizes. As coded here, the image file needs to be resident in the current directory. If you use your own image with a different format, the format may need to be converted or the script set up to use it appropriately. Changes such as defining separate u and v coordinate vectors are needed if the image is not square.

Since an image is conventionally stored with the top row first, the flipud function is used to reverse the order of the rows of A so the bottom of the image corresponds to the negative v coordinates. The single command converts the. The max command, applied twice to find the maximum value of a 2D array, is used to set the peak image value to unity for reference. Since the image file represents an irradiance image, take the square root to get the magnitude of the field. This actually has no effect on this particular test chart image since after normalization it only contains zeros and ones.

Beyond that, a phase component can be included to simulate a complex coherent field, but that comes in the next section. For now, zero phase is assumed across the ideal image. The ideal test chart image is shown in Fig. Thus, Eq. The coherent transfer function is displayed in Fig. The combination of surface plotting and lighting Figure 7. Imaging and Diffraction-Limited Imaging Simulation commands in the script help improve the display of H. The resulting image is displayed in Fig.

The features are blurred and some of the three-bar groups are unresolved. A physical v-axis coordinate value vvalue is selected, and the nearest array index value is found. The round function rounds a floating point value to the nearest integer. The plot compares the ideal and simulated image profiles.

Element 6 vertical bars are resolved but with much less contrast. Even though the square magnitude of the field is taken for the irradiance, the apparent resolution in this kind of simulated coherent irradiance image, typically, appears close to the coherent cutoff. The log of the spectra magnitude is displayed to bring out low contrast details. Furthermore, the phase of the source field across the chart was constant equal to zero.

However, when a Group -2 Group -2 1. Image scale is the same as Fig. In other words, the random dips and bumps of a surface, unless it is machined like a mirror, are large enough that an incident plane wave reflecting from one dip will have a significantly different phase from the portion of the wave that reflects off a bump. A simple way to model this effect is to apply a random complex exponential phase term to the object ideal image field. Therefore, every sample point has a phase that is independent and uncorrelated from every other point.

The code now produces the irradiance image of Fig. Coherent speckle is a well-known phenomenon. The sparkling is a speckle effect. Imaging and Diffraction-Limited Imaging Simulation 7. The object may actually be illuminated by a source, like the sun, but ultimately the field exiting the object surface involves a spectrum of wavelengths and randomly changing phase in time.

Perfectly incoherent light refers to the situation where the complex field phasors from the radiating point sources are stochastically independent; where there is no correlation between the field phasors at different points or times. To visualize this idea, imagine the speckles in the image of Fig.

With enough averaging the image texture will tend to become smooth. In contrast to coherent imaging, which is linear with the field, incoherent imaging is linear with irradiance. The impulse response h u, v 2 is commonly known as the point spread function PSF. The corresponding spectra of the functions in Eq. This means the OTF is not assumed to affect the total optical power associated with the ideal geometrical image.

The incoherent cutoff is the same as the cutoff suggested in Eq. What about the circular pupil function for the coherent example in Section 7. For its OTF, the normalized autocorrelation of the coherent transfer function in Eq. Note that the OTF is left in the shifted arrangement. But, numerical precision can produce small imaginary and negative values in the results.

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