Flip-flop coating synthesis revisited William H. Southwell Table Mountain Optics, 509 Marin Street, Suite 125, Thousand Oaks, California 91360, USA ([email protected]) Received 4 September 2013; accepted 7 October 2013; posted 15 November 2013 (Doc. ID 196888); published 16 December 2013

The flip-flop synthesis method is robust with rapid convergence, but the solutions are often not the best. Modifications to the flip-flop algorithm have been developed, one of which puts it on par with needle synthesis. © 2013 Optical Society of America OCIS codes: (310.4165) Multilayer design; (310.5696) Refinement and synthesis methods; (310.6805) Theory and design. http://dx.doi.org/10.1364/AO.53.00A179

1. Introduction

Since the flip-flop synthesis method was introduced in 1985 [1], it has seen a handful of modifications and literature [2–9] and book references (see [10–12], but this list may not be complete), including versions of it in commercial thin-film design software [13]. However, it has not had widespread use. Its virtue is its robustness and speed of convergence. The flip-flop method needs no starting design and it uses no refinement. Its drawback is that while the solutions are sometimes satisfactory, they are not global minima. Presented here are some new approaches to the flip-flop method that enhance its usefulness. It is assumed that a merit function for the desired spectral performance has been established. The flip-flop method begins with a structure of some total physical (or optical) thickness. This structure is equally divided into many very thin layers. The merit function is evaluated for the starting configuration and its value noted. One by one each of the thin layers is flipped from its current index to an alternative index and the merit function is again evaluated. If the merit function has improved then the new index state is retained and the next layer is considered. A single pass consists of a single evaluation of all the layers. Multiple passes are made until no more improvements are seen. Although there may be hundreds of these thin layers, computation time is 1559-128X/14/04A179-07$15.00/0 © 2014 Optical Society of America

greatly reduced through the use of front and base matrices. The method allows for a variety of starting configurations, such as all layers having low index or high index. It is customary to use two coating materials, a low and a high index, but the method is readily modified to allow multiple materials [7], where the material with the most improved merit function is retained at each sublayer. There are also a variety of directions through the thin-layer stack, such as starting from the substrate or starting from the incidence medium. For comparison purposes we have put together some choices, shown in Fig. 1. With this set of options there are 25 implementations of flip-flop, but each one will result in a slightly different solution. While one may work better for a given type of problem, we are not prepared to say that one is best for all problems (although we have not done an exhaustive study of this). An example with a sublayer thickness of 5 nm (and Fig. 1 options) is shown in Fig. 2. For this example we use an edge filter merit function that has 50% reflectance at the wavelength 550 nm and 0% reflectance from 400 to 449 nm and 100% reflectance from 551 to 700 nm. The substrate has index 1.52, the incidence medium is air, and nL  1.47 and nH  2.1. 2. Speed Advantages of the Flip-Flop Method

The spectral features of an assemblage of thin films may be computed from the 2 by 2 matrix M, which is the matrix product of all the characteristic matrices in the stack: 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS

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Fig. 1. Some flip-flop strategies.

M

NY layers j1

 Vj 

M 11 iM 21

 iM 12 ; M 22

(1)

where each layer is characterized by its matrix:  V

cos φ sin φ

 i sin φ∕n ; cos φ

(2)

where for the jth the layer the refractive index is nj and phase thickness φ  2πnj tj ∕λ. Each matrix multiply requires eight double precision multiplication. Knowing the matrix M, the reflectance R, for example, is computed from 

B C



 

M 11 iM 21

iM 12 M 22



1

nsub

 ;

r  ninc B − C∕ninc B  C; 

R  r r:

(3)

Thus, one needs to compute only two cosine functions and two sine functions for the whole stack. Matrices are precomputed and saved for the low-index layer and the high-index layer, one for each wavelength in the merit function. Furthermore, the number of matrix multiplication is reduced by using the base and forward matrices, as shown in Fig. 3. The algorithm for the flip-flop method using the front and base matrices is given in Fig. 4. In the loop between Start and Done there are three or four matrix multiplication, depending on whether the current layer is being flipped. There is also one merit function evaluation, which takes about the same number of computations as a matrix multiply. Thus, each pass is accomplished with essentially four or five equivalent matrix multiples for each wavelength in the merit function. Notice the use of the inverse matrix V −1 in the algorithm in Fig. 4. Multiplying the forward matrix F from the left with the inverse of the first matrix in F will remove the first matrix from it. This is simply because V −1 V is the identity matrix. The components of V −1 are the same as V in Eq. (2) except the offdiagonal elements have a minus sign. The matrix V flipped is the current layer but with the opposite index. The merit function is a sum over all the wavelengths: N Wavelengths

(4) MF  (5)

X

Rk − Rtk 2 ;

(6)

k1

For air incidence, there are nine multiplication and division operations to get R from the four matrix elements in M. Generally for each wavelength in the merit function there will be 8N layers  9 multiplication and division to determine R. In addition, in general, there will be N layers number of cosine functions and N layers number of sine functions needed to calculate the reflectance R. The biggest time consumption is in computing the cosine and sine for each layer. But for the flip-flop stack all low-index layers have the same thickness and index, as do all high-index layers.

where Rtk is the reflectance target at the kth wavelength. Thus, after the initial merit function evaluation, each additional merit function evaluation requires the computer time of four or five matrix multiplication of 2 by 2 matrices (for each wavelength) regardless of the number of layers in the stack. This compares to N layers as the number of matrix multiplication without the front and base matrices. There is one more trick to increase the speed of the flip-flop method, which is illustrated in Fig. 5. Traditional thin-film software requires the storage of a thickness array and an index array, each of double

Fig. 2. Edge filter using flip-flop from the incident side with all low index. This converged after 10 passes. The merit function is 6.335.

Fig. 3. Only three or four matrix multiplication are needed to compute the spectra from M  BVF, including the matrix updates to the base and front matrices.

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value of n between nLO and nHI and updating the stack with this value before going on to the next layer. We call this the Vary_n method. There are various ways to implement the Vary_n method. One could implement a one-dimensional optimizer, which should be easy because the index variable is strictly bounded between the nL and nH levels. However, in the spirit of the flip-flop method we want to avoid taking derivatives. The flip-flop method gives us the value of the merit function at the two bounds of the index variable (for the layer being varied). We have found that the merit function between these values usually behaves as a convex function. This means the merit function will be a convex surface or a concave surface, which we can model as a quadratic function. If it is convex upward, then there is no new minimum and we use the lowest end point, either nL or nH as the solution for that layer, as shown in Fig. 6. This figure also shows that the quadratic is a good model when either the high or low index are favored (which will usually be the case). But when it is concave then we may have an intermediate minimum, as shown in Fig. 7. One needs three points to fit a quadratic. We already have two; we choose to generate the third by evaluating the merit function at the midpoint between nL and nH . The merit function F is modeled as Fig. 4. Algorithm for the flip-flop method.

F  a  bx  cx2 ; precision, and the array being as long as the number of layers. But in the flip-flop method all layers are of equal thickness so the thickness array is not needed. And a single array of Boolean variables (each either true or false) is used to designate where the layer is the high index or the low index. This saves considerable computer memory and is faster when flipping a layer (changing it from a low or high to a high or low). 3. Vary_n an Alternative Approach

Referring to Fig. 3, the flip-flop method as described above varies the V layer by changing its index from low or high to high or low with constant thickness. There are two other variations that can be made while still exploiting the advantages of the front-base matrix approach. One of them is to find the optimum

Fig. 5. Use of Boolean array to describe the flip-flop stack.

(7)

where the variable x is the index between its bounds. Let F 1 be the merit function at nL and F 3 be the merit function at nH . The calculated merit function at the midpoint will be F 2 . The coefficients a, b, and c are determined from the three merit function values at the three values of x. The extrema of this function will occur at the value of x that makes the derivative F 0  b  2cx  0. Thus, xmin  −b∕2c;

(8)

Fig. 6. Merit function as a function of layer refractive index and its quadratic fit where the layer favors the high index. The R in the figure is a measure of the goodness of fit to the quadratic function. 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS

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Fig. 7. Merit function as a function of layer refractive index and its quadratic fit where the layer favors an intermediate index. The R’s in the figure are a measure of the goodness of fit to the quadratic function. Note the actual merit function is not a perfect quadratic, but hopefully close enough to locate its minimum.

provided that the constant c is positive. (The second derivative of F is 2c, so the sign of c being positive insures that the extrema is a minimum.) So for the price of one additional merit function evaluation we are able to obtain an estimate for the intermediate layer index. We refer to this as the three fixed-point quadratic method. But the three fixed-point quadratic method does not use the previous best solution. Another way is to select a delta n value, ndif , to evaluate the merit function around the current value of the layer index. This gives three values for the merit function, one at the current value n0 and the other two at n0  ndif . These three merit functions are then fit to a quadratic as above and the minima is found. This ndif method is a little more difficult to implement because it may call for an evaluation point which is outside the bounds (nL , nH ). This means the sample points are not necessarily uniformly spaced, which requires more computation. With the example merit function cited above we performed a comparison for various Vary_n method implementations starting at the substrate with all low index. We found that selecting ndif  0.5nH − nL  gives rapid initial convergence, comparable to the three fixed-point version, as seen in Fig. 8. Smaller values of ndif result in slower convergence but ultimately lower merit functions, as seen in Fig. 9. This plot suggests that a more optimum strategy might be to reduce ndif as the convergence progresses. We have found that when ndif is one or higher times nH − nL , then the results are identical to the flip-flop method. The results with ndif  0.125nH − nL  are shown in Fig. 10. For comparison we show the results of the flip-flop method in Fig. 11. We see that for the same fixed total thickness, the Vary_n method gives lower ripple, a wider high reflection region, and a steeper edge than the flip-flop method. Both the flip-flop and Vary_n methods A182

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Fig. 8. Convergence history for implementations of the Vary_n method. The flip-flop method is given for comparison, but it is complete at six passes.

maintain the initial 2000 nm total physical thickness. No doubt other thicknesses will give better results, but we want to compare the methods on an equal basis. The Vary_n method does not terminate quickly, thus allowing the passes to continue, which

Fig. 9. Convergence history showing that the Vary_n method can result in better solutions than the flip-flop method.

Fig. 10. Results after 100 passes of the Vary_n method with ndif  0.125nH − nL , starting from the substrate with all low index. The merit function is 5.898.

Fig. 11. Results of the flip-flop method, starting from the substrate with all low index. The merit function is 6.911. This converged in seven passes.

Fig. 13. AR with Vary_n with ndiff  0.05nH − nL  from incidence and using all low index to start. MF  0.241%.

4. Vary_t

may result in better solutions. Although the Vary_n solution may eventually result in a two-state index solution, we note, as in Fig. 10, that the solution may contain thin layers surrounding the thicker high- or low-index layers. To speed the convergence, these layers may be converted to additional thicknesses of the adjacent high- and low-index layers without loss of spectral performance. The use of the thin layer equivalence principle [1] is used in this process. Besides being better spectrally, the Vary_n method is seen to remove the troublesome thin layers often associated with the flip-flop method. A.

Antireflection Coating Designs

Consider now the filter that requires the lowest reflection over the visible region 400–700 nm. In Figs. 12 and 13, we compare the results using flipflop and Vary_n methods for a 500 nm total physical thickness. In this case for designs with 500 nm total physical thickness, the Vary_n method produces a filter design with half the reflection as the flip-flop method. A smaller ndiff was used because much of the coating uses a gradient index region. There are more internal index values used than with designs requiring high reflectance.

Fig. 12. AR coating designed with flip-flop from incidence and using all low index to start. MF  0.518%, which is the average reflectance.

A second alternative approach is called Vary_t, which varies the layer thickness instead of the index of each layer. In this case we cannot start with an all low-index or an all high-index configuration. We start with an alternating high-low thin layer configuration and let each thickness vary with zero as a strict lower bound. When the optimization selects a layer with a near-zero thickness, that layer is removed and the surrounding layers are combined. This reduces the total layer count and the result often ends up as a standard quarter-wave type design. An example of this Vary_t approach is shown in Fig. 14, which ended with 27 layers. As one might expect, the Vary_t method does not preserve total thickness. Performance of the example shown in Fig. 14 may be better than by using other methods, but that may be due to its greater thickness. We should note that when using Vary_t we can no longer use the Boolean array to characterize the stack. We must revert to a thickness array and a corresponding index array. When combining adjacent like layers the thicknesses are no longer equal. However, the base and forward matrices are still used. For the single layer thicknesses optimization we use a derivative method, which is a one-dimensional least-squares method. We use zero as a lower bound and twice the current layer thickness as the upper

Fig. 14. Design generated with the Vary_t method starting with 400 layers of alternating high and low index each 5 nm physical thickness. Near-zero thickness layers were removed and the adjacent layers combined. 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS

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bound. Very thin layers are banished when they become lower than a specified thickness, which we usually set at 1 nm. Other one-dimensional optimization methods may be considered, including the quadratic model described above. We observe that when combining like adjacent layers, the total layer count is reduced and the Vary_t passes are thus sped up. The example given in Fig. 14 starts with 400 layers but ends up with 27 layers. A.

Combined Methods

Faster convergence may be achieved by combining various methods. For example, one could start with regular flip-flop, then combine adjacent layers of similar index, and then implement Vary_t passes—all using front and base matrices. This approach uses the Vary_t passes as a refinement processes to follow the flip-flop synthesis. This approach was followed in the example shown in Fig. 15. The merit function in this example was reduced to 5.837 with 23 layers. This is very close to the OptiLayer [14] solution 5.834, which has 24 layers, although the flip-flop/Vary_t method was much slower converging. This is because it operates on only one layer at a time, whereas OptiLayer uses a powerful modified damped least-squares method, which can handle all layers at once. 5. Delayed Updates

In the flip-flop method, if instead of retaining the best flip-flop layer index at each layer in the pass (instant gratification), consider waiting until the entire stack is scanned and then only flip the sublayer, which has the most merit function improvement (delayed update). Consider a plot of the value of the merit function as a function of the sublayer position as each layer is flipped but without updating it. When the value of the merit function of the starting configuration is subtracted from this plot, then one can easily see the regions where the merit function can be improved by flipping. We call this the flip-flop Q-function. An example Q-function is shown plotted as the solid line in Fig. 16. This is compared to the P-function of the needle method computed with the OptiLayer software. They appear to be the same. The most negative point on this flip-flop Q-function will be the position of the sublayer that will produce

Fig. 15. Design using the flip-flop method followed by the Vary_t method on the combined layers. The merit function is 5.837. A184

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Fig. 16. Flip-flop Q-function (solid line) for the first pass of the edge filter example. The needle P-function (dashed line) for the same case is shown for comparison.

the most reduction in the merit function. This is then identical to the needle approach for inserting a new layer. Thus, the flip-flop with delayed updates method begins with the same premise for inserting a new layer as the needle method. The pass then concludes with the addition of two layers when the one best layer is flipped. The needle method then refines the thicknesses of all layers before computing a new P-function. Here, we want to consider ways to proceed without refining the layer thicknesses. In the spirit of flip-flop, one way to do this is to apply the flip-flop method (with immediate updates) to the layers surrounding the newly flipped layer, alternating sublayers on each side around it until the merit function no longer improves. This, in effect, finds the approximate best thickness for the inserted (flipped) layer. At that point a new Q-function is computed and the process is repeated. This approach retains the total thickness and the fixed number of equal-thickness sublayers. While this method is successful, its convergence is not as good as the needle method. The difference is that this flip-flop approach only “refines” the new flipped layer while holding all other layers fixed, whereas, the needle method refines all layer thicknesses with the insertion of each new layer. Another approach is to insert the one best flipped layer from the Q-function and then widen the flipped layer by applying the standard flip-flop method on the sublayers surrounding the flipped layer until the merit function no longer improves, as in the previous approach. At that point, combine adjacent like layers to reduce the layer count and use the Vary_t method as a refining method for all the layer thicknesses. This can be done while continuing to use the base and forward matrices. As mentioned above, this requires the use of the thickness and index arrays. When that is done, subdivide all layers of the design into equal thicknesses again and compute the Qfunction to find the best layer to flip and thus repeat the process.

6. Summary

The flip-flop method has been described and reviewed to exhibit, but not necessarily explain, its fast convergence. It is still remarkable that it can rapidly provide reasonable solutions. We have presented some alternative ideas to complement or enhance the basic flip-flop method. One significant result is the introduction of the flip-flop Q-function, which turns out to be proportional to the needle P-function, enabling the insertion of a new layer at an optimum position in the stack. Another idea of practical utility is that the flip-flop method can be used to find a suitable starting point for more advanced optimization techniques. Appreciation is given to Joseph Peeples whose programming skills enabled the evaluation of a great variety of methods in a most timely manner. References and Note 1. W. H. Southwell, “Coating design using very thin high- and low-index layers,” Appl. Opt. 24, 457–460 (1985). 2. J. A. Dobrowolski, “Comparison of the Fourier transform and flip-flop synthesis methods,” Appl. Opt. 25, 1966–1972 (1986).

3. T. Skettrup, “Three-layer approximation of dielectric thin films systems,” Appl. Opt. 28, 2860–2863 (1989). 4. J. A. Dobrowolski and R. A. Kemp, “Flip-flop thin-film design program with enhanced capabilities,” Appl. Opt. 31, 3807–3812 (1992). 5. L. Li and J. A. Dobrowolski, “Computational speeds of different optical thin-film synthesis methods,” Appl. Opt. 31, 3790–3799 (1992). 6. J. A. Dobrowolski and R. A. Kemp, “Interface design methods for two-material optical multilayer coatings,” Appl. Opt. 31, 6747–6756 (1992). 7. J. Hrdina, “A technologically acceptable coating synthesis based on the flip-flop synthesis method,” J. Mod. Opt. 36, 111–118 (1989). 8. H. Arabshahi and M. Asmari, “Optimum designing of thin film filter layers of SiO2 and SnTe based on optical particle swarm optimizer,” Int. J. Phys. Sci. 5, 57–61 (2010). 9. J. Baedi, H. Arabshahi, M. Gordi Armak, and E. Hosseini, “Optical design of multilayer filter by using PSO algorithm,” Res. J. Appl. Sci. Eng. Technol. 2, 56–59 (2010). 10. P. W. Baumeister, Optical Coating Technology (SPIE, 2004). 11. H. A. Macleod, ed., Thin-Film Optical Filters, 4th ed. (CRC Press, 2010). 12. R. R. Willey, Practical Design of Optical Thin Films (Willey Optical, 2011). 13. FilmStar and Film Wizard list Flip-Flop as an option but this list may be incomplete. 14. A. V. Tikhonravov and M. K. Trubetskov, Optilayer Thin Film Software, http://www.optilayer.com.

1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS

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Flip-flop coating synthesis revisited.

The flip-flop synthesis method is robust with rapid convergence, but the solutions are often not the best. Modifications to the flip-flop algorithm ha...
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