Christian Hafner, Cui Xudong, Jasmin Smajic, and Ruediger Vahldieck, "Efficient procedures for the optimization of defects in photonic crystal structures," J. Opt. Soc. Am. A 24, 1177-1188 (2007)

Seven different stochastic binary optimizers—based on the concepts of genetic algorithms and evolutionary strategies—are developed, applied to determine defect locations in several photonic crystal structures that serve as test cases, and compared by extensive statistical analysis. In addition to the stochastic optimizers, a quasi-deterministic optimizer based on an algorithm inspired by hill-climbing algorithms was implemented. The test cases include the prominent 90° photonic crystal waveguide bend and a photonic crystal power divider. The analysis of the results shows that many different photonic crystal structures with high transmission may be found for any operating frequency. All of the eight optimizers outperform standard codes—because they maintain an incomplete fitness table—and find the global optima with a high probability even when the number of fitness evaluations is much smaller than the number of potential solutions contained in the discrete search space. Based on the incomplete fitness table, an algorithm to estimate bit-fitness values is presented. The bit-fitness values are then used to improve the performance of some algorithms. The four best algorithms—an extended microgenetic algorithm, two mutation-based algorithms, and the quasi-deterministic algorithm inspired by hill-climbing algorithms—are considered to be of high value for the optimization of defects in photonic crystals and for similar binary optimization problems.

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Population Size for the Eight Different Runs for the Seven Stochastic Optimizers (STAT, MGA0, MGA1, MGA2, MUT0, MUT1, MUT2) and for the HC Algorithm RHC1^{
a
}

${N}^{\prime}$

Stochastic

RHC1

1

4

1

2

5

2

3

7

4

4

10

7

5

14

11

6

19

16

7

25

22

8

32

29

${N}^{\prime}$ is used in the following tables.

Table 2

Probabilities of Finding Global Optimum in Percent, Averaged over All Test Cases with 100, 200, and 400 Fitness Evaluations, for All Eight Optimizers^{
a
}

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

33

36

39

43

22

46

49

64

2

34

38

40

46

29

50

52

62

3

34

39

42

46

30

55

56

58

4

34

39

43

45

31

57

55

45

5

36

41

44

49

31

53

49

45

6

36

42

45

50

32

44

40

45

7

35

40

45

50

32

34

33

43

8

36

39

43

50

32

29

29

42

av

35

39

43

47

30

46

46

51

${N}^{\prime}$ indicates the population size according to Table 1; av is the average of the values in the corresponding column.

Table 3

Average Relative Fitness (Fitness Found by the Algorithm or Fitness of the Global Optimum) in Percent, Averaged over All Test Cases with 100, 200, and 400 Fitness Evaluations, for All Eight Optimizers^{
a
}

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

99

99

99

99

96

99

99

100

2

99

99

99

99

98

99

99

100

3

99

99

99

99

99

99

99

100

4

99

99

99

99

99

99

99

99

5

99

99

99

99

99

99

99

99

6

99

99

99

100

99

99

99

99

7

99

99

99

100

99

99

99

99

8

99

99

99

100

99

99

99

99

av

99

99

99

99

98

99

99

100

${N}^{\prime}$ indicates the population size according to Table 1.

Table 4

Average Number of Fitness Calls When an IFT Is Used and the Algorithm Is Stopped As Soon As It Finds the Global Optimum, Averaged over All Test Case 100, 200, and 400 Fitness Evaluations, for All Eight Optimizers^{
a
}

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

184

180

175

167

172

163

158

111

2

183

178

173

163

191

157

151

118

3

185

175

171

164

192

149

146

131

4

186

176

171

166

190

144

141

166

5

186

173

167

159

190

126

119

168

6

187

173

167

157

191

110

102

169

7

193

176

165

158

191

99

91

174

8

194

178

168

160

191

89

83

179

av

187

176

170

162

188

130

124

152

${N}^{\prime}$ indicates the population size according to Table 1.

Same as in Table 2, When All Algorithms Are Stopped After 100 Fitness Evaluations

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

16

17

17

20

11

21

22

44

2

15

17

17

20

12

23

22

37

3

16

17

19

21

12

24

25

30

4

16

17

19

20

13

25

25

20

5

16

19

20

23

12

30

29

22

6

17

17

21

23

14

29

27

23

7

16

16

20

24

14

26

25

21

8

16

16

19

22

14

24

25

20

av

16

17

19

22

13

25

25

27

Table 7

Same as in Table 2, When All Algorithms Are Stopped After 200 Fitness Evaluations

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

29

31

35

39

21

42

46

67

2

29

35

35

43

25

47

50

65

3

30

36

38

43

25

54

54

56

4

30

36

38

41

27

59

58

40

5

32

37

39

46

27

56

53

41

6

31

39

40

48

28

47

44

41

7

32

38

42

49

28

36

34

39

8

32

36

39

48

28

30

30

39

av

31

36

38

44

26

46

46

48

Table 8

Same as in Table 2, When All Algorithms Are Stopped After 400 Fitness Evaluations

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

55

60

65

71

34

75

80

81

2

57

62

68

75

51

81

85

85

3

55

64

68

74

53

86

88

89

4

56

65

70

75

54

88

83

75

5

58

68

73

77

53

73

66

72

6

60

69

74

78

54

56

50

71

7

58

67

74

78

54

42

39

70

8

59

66

72

79

54

34

32

68

av

57

65

70

76

50

67

65

76

Table 9

Probabilities of Finding the Global Optimum in Percent for the RHC1 Algorithm, for the 14 Bit Bend Optimization Stopped after 100, 200, 400 Fitness Evaluations (Columns B1, B2, B4, Respectively) and for the Power Divider Optimization Stopped After 100, 200, 400 Fitness Evaluations (Columns D1, D2, D4, Respectively)^{
a
}

${N}_{\mathrm{pop}}$

B1

B2

B4

D1

D2

D4

1

7

17

27

51

81

89

2

7

17

40

43

74

86

4

39

64

86

18

35

68

7

6

16

38

16

31

59

11

7

13

27

16

30

55

16

7

14

28

16

30

53

22

6

13

27

15

27

50

29

6

10

23

14

28

47

${N}_{\mathrm{pop}}$ indicates the number of individuals in the initialization step.

Table 10

Bit-Fitness Values for the Ten Bits of the HF Solution of the 90° Bend Structure^{
a
}

Bit

Bit Fitness

Best

1

0.5020626

1

2

0.4509298

0

3

0.3468390

$\underset{\u0331}{1}$

4

0.0569637

0

5

0.3744641

0

6

0.3250102

$\underset{\u0331}{1}$

7

0.2519560

0

8

0.3823080

0

9

0.3907066

0

10

0.4766179

0

The “Best” column shows the bits of the global optimum solution. For the underlined values, the bit-fitness value is misleading; i.e., the corresponding bit of the best solution is 1, although the bit-fitness value is $<0.5$. Note that 80% of the bit-fitness values provides a correct estimate. Bit 4 has a very small value, indicating very high probability for a defect at position 4. According to Fig. 1, this is the cell at the input and output ports, where one also intuitively expects a defect.

Tables (10)

Table 1

Population Size for the Eight Different Runs for the Seven Stochastic Optimizers (STAT, MGA0, MGA1, MGA2, MUT0, MUT1, MUT2) and for the HC Algorithm RHC1^{
a
}

${N}^{\prime}$

Stochastic

RHC1

1

4

1

2

5

2

3

7

4

4

10

7

5

14

11

6

19

16

7

25

22

8

32

29

${N}^{\prime}$ is used in the following tables.

Table 2

Probabilities of Finding Global Optimum in Percent, Averaged over All Test Cases with 100, 200, and 400 Fitness Evaluations, for All Eight Optimizers^{
a
}

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

33

36

39

43

22

46

49

64

2

34

38

40

46

29

50

52

62

3

34

39

42

46

30

55

56

58

4

34

39

43

45

31

57

55

45

5

36

41

44

49

31

53

49

45

6

36

42

45

50

32

44

40

45

7

35

40

45

50

32

34

33

43

8

36

39

43

50

32

29

29

42

av

35

39

43

47

30

46

46

51

${N}^{\prime}$ indicates the population size according to Table 1; av is the average of the values in the corresponding column.

Table 3

Average Relative Fitness (Fitness Found by the Algorithm or Fitness of the Global Optimum) in Percent, Averaged over All Test Cases with 100, 200, and 400 Fitness Evaluations, for All Eight Optimizers^{
a
}

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

99

99

99

99

96

99

99

100

2

99

99

99

99

98

99

99

100

3

99

99

99

99

99

99

99

100

4

99

99

99

99

99

99

99

99

5

99

99

99

99

99

99

99

99

6

99

99

99

100

99

99

99

99

7

99

99

99

100

99

99

99

99

8

99

99

99

100

99

99

99

99

av

99

99

99

99

98

99

99

100

${N}^{\prime}$ indicates the population size according to Table 1.

Table 4

Average Number of Fitness Calls When an IFT Is Used and the Algorithm Is Stopped As Soon As It Finds the Global Optimum, Averaged over All Test Case 100, 200, and 400 Fitness Evaluations, for All Eight Optimizers^{
a
}

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

184

180

175

167

172

163

158

111

2

183

178

173

163

191

157

151

118

3

185

175

171

164

192

149

146

131

4

186

176

171

166

190

144

141

166

5

186

173

167

159

190

126

119

168

6

187

173

167

157

191

110

102

169

7

193

176

165

158

191

99

91

174

8

194

178

168

160

191

89

83

179

av

187

176

170

162

188

130

124

152

${N}^{\prime}$ indicates the population size according to Table 1.

Same as in Table 2, When All Algorithms Are Stopped After 100 Fitness Evaluations

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

16

17

17

20

11

21

22

44

2

15

17

17

20

12

23

22

37

3

16

17

19

21

12

24

25

30

4

16

17

19

20

13

25

25

20

5

16

19

20

23

12

30

29

22

6

17

17

21

23

14

29

27

23

7

16

16

20

24

14

26

25

21

8

16

16

19

22

14

24

25

20

av

16

17

19

22

13

25

25

27

Table 7

Same as in Table 2, When All Algorithms Are Stopped After 200 Fitness Evaluations

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

29

31

35

39

21

42

46

67

2

29

35

35

43

25

47

50

65

3

30

36

38

43

25

54

54

56

4

30

36

38

41

27

59

58

40

5

32

37

39

46

27

56

53

41

6

31

39

40

48

28

47

44

41

7

32

38

42

49

28

36

34

39

8

32

36

39

48

28

30

30

39

av

31

36

38

44

26

46

46

48

Table 8

Same as in Table 2, When All Algorithms Are Stopped After 400 Fitness Evaluations

${N}^{\prime}$

STAT

MGA0

MGA1

MGA2

MUT0

MUT1

MUT2

RHC1

1

55

60

65

71

34

75

80

81

2

57

62

68

75

51

81

85

85

3

55

64

68

74

53

86

88

89

4

56

65

70

75

54

88

83

75

5

58

68

73

77

53

73

66

72

6

60

69

74

78

54

56

50

71

7

58

67

74

78

54

42

39

70

8

59

66

72

79

54

34

32

68

av

57

65

70

76

50

67

65

76

Table 9

Probabilities of Finding the Global Optimum in Percent for the RHC1 Algorithm, for the 14 Bit Bend Optimization Stopped after 100, 200, 400 Fitness Evaluations (Columns B1, B2, B4, Respectively) and for the Power Divider Optimization Stopped After 100, 200, 400 Fitness Evaluations (Columns D1, D2, D4, Respectively)^{
a
}

${N}_{\mathrm{pop}}$

B1

B2

B4

D1

D2

D4

1

7

17

27

51

81

89

2

7

17

40

43

74

86

4

39

64

86

18

35

68

7

6

16

38

16

31

59

11

7

13

27

16

30

55

16

7

14

28

16

30

53

22

6

13

27

15

27

50

29

6

10

23

14

28

47

${N}_{\mathrm{pop}}$ indicates the number of individuals in the initialization step.

Table 10

Bit-Fitness Values for the Ten Bits of the HF Solution of the 90° Bend Structure^{
a
}

Bit

Bit Fitness

Best

1

0.5020626

1

2

0.4509298

0

3

0.3468390

$\underset{\u0331}{1}$

4

0.0569637

0

5

0.3744641

0

6

0.3250102

$\underset{\u0331}{1}$

7

0.2519560

0

8

0.3823080

0

9

0.3907066

0

10

0.4766179

0

The “Best” column shows the bits of the global optimum solution. For the underlined values, the bit-fitness value is misleading; i.e., the corresponding bit of the best solution is 1, although the bit-fitness value is $<0.5$. Note that 80% of the bit-fitness values provides a correct estimate. Bit 4 has a very small value, indicating very high probability for a defect at position 4. According to Fig. 1, this is the cell at the input and output ports, where one also intuitively expects a defect.