JP2008145843A - Incident angle-dependent diffraction grating - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide an incident angle-dependent diffraction grating having high performance by optimizing a duty ratio or the like of one-dimensional relief type diffraction grating. <P>SOLUTION: The incident angle-dependent diffraction grating comprises a relief type one-dimensional diffraction grating 1 made by distributing a high refractive index part 2 and a low refractive index part 3 periodically in one direction and has ± first-order diffraction efficiency monotonically changed in one direction with respect to the incident angle, wherein the normalization pitch, refractive index modulation, duty ratio and normalization groove depth are set such that Bragg angle satisfying Bragg's condition is made positive, incident angle 0° is included between the Bragg angle and the incident angle which gives the minimum diffraction efficiency appearing first from the Bragg angle, the following relation (A) is satisfied; ä(diffraction efficiency at the Bragg angle-minimum diffraction efficiency appearing first)×0.2 + minimum diffraction efficiency appearing first} ≤ diffraction efficiency at incident angle 0° (A), and it is satisfied that the diffraction efficiency at the Bragg angle is two times or more of the diffraction efficiency at the incident angle 0°. <P>COPYRIGHT: (C)2008,JPO&INPIT

Description

  The present invention relates to an incident angle dependent diffraction grating, and more particularly to a diffraction grating that can be used for an inclination sensor whose diffraction efficiency changes monotonously depending on the incident angle.

Conventionally, using a relief type diffraction grating for a tilt sensor (tilt sensor) using the fact that the diffraction efficiency of ± 1st order diffracted light of a relief type diffraction grating that performs Bragg diffraction changes according to the incident angle is disclosed in Patent Document 1, It is proposed in Patent Document 2 and the like.
Japanese Patent Laid-Open No. 2005-61981 JP 2005-106627 A “Holography” written by Junpei Uchiuchi (November 5, 1997, Hanakabo Co., Ltd.) pp. 65-68 “SPIE”, Vol. 883 (1988), p. 8-11 “JOSA”, Vol. 72 (1982), pp. 1385-1392 “JOSA”, Vol. 73 (1983), pp. 1105-1112

  In Patent Document 1 and Patent Document 2, the Q value of the relief type diffraction grating suitable for the tilt sensor, the groove depth range of the diffraction grating, and the refractive index difference between the material of the diffraction grating and the material filled in the grating groove are described. There is a description.

  It is also disclosed that a one-dimensional diffraction grating is used to detect a tilt in one direction with a single diffraction grating.

  However, in these conventional techniques, in the one-dimensional diffraction grating used to detect the inclination in one direction, the ratio of the high refractive index portion and the low refractive index portion, more precisely, the high refractive index portion in one repetitive pitch. No consideration has been given to the duty ratio, which is the ratio of the width of the.

  The present invention has been made in view of such a situation in the prior art, and an object thereof is to provide a higher performance incident angle dependent diffraction grating by optimizing the duty ratio of a one-dimensional relief type diffraction grating. It is.

The incident angle-dependent diffraction grating of the present invention that achieves the above object has a high refractive index portion and a low refractive index portion that are periodically distributed in one direction, and each high refractive index portion is arranged perpendicular to the plane of the diffraction grating. Diffracted light is emitted in the periodic direction, and the ± 1st-order diffraction efficiency in the periodic direction changes monotonously depending on the incident angle of the incident light with respect to the surface of the diffraction grating. In the lattice,
An angle of incidence of 0 ° is included between the Bragg angle and the incident angle that gives the minimum diffraction efficiency that first appears from the Bragg angle, with the Bragg angle satisfying the Bragg condition being positive; and
{(Diffraction efficiency at Bragg angle-Minimal diffraction efficiency first appearing) x 0.2
+ Minimum diffraction efficiency that appears first} ≦ Diffraction efficiency at an incident angle of 0 ° (A)
The condition (A) is satisfied, and
The diffraction efficiency at the Bragg angle is more than twice the diffraction efficiency at an incident angle of 0 °.
In order to satisfy the above, a standardized pitch Λ, a refractive index modulation Δn, a duty ratio f, and a standardized groove depth d are set. However, the wavelength of the incident light is λ, the repetition pitch of the high refractive index portion and the low refractive index portion in the periodic direction is p, the height of the high refractive index portion is h, the refractive index of the high refractive index portion is n ₂ , and the low refractive index The refractive index of the index part is n ₁ , p / λ is the normalized pitch Λ, h / λ is the normalized groove depth d, and (n ₂ −n ₁ ) / 2 is the refractive index modulation Δn. Further, the ratio of the width w of the high refractive index portion in one repetitive pitch p is defined as a duty ratio f.

  Since the incident angle dependent diffraction grating of the present invention satisfies these three conditions, a one-dimensional tilt angle in one direction can be detected with high accuracy and high sensitivity.

  The incident angle dependent diffraction grating of the present invention will be described below with reference to the drawings.

As schematically shown in FIG. 1, the incident angle dependent diffraction grating according to the present invention has a plane shape parallel to the xy plane, and periodically has a high refractive index portion and a low refractive index portion in the x direction. Is a one-dimensional diffraction grating 1 formed by distribution. When the incident light 10 is incident from the z direction perpendicular to the one-dimensional diffraction grating 1 as described above, + 1st order and −1st order diffracted lights 11 ₊₁ and 11 ₋₁ are generated in the transmission direction. Here, + 1st order diffracted light 11 ₊₁ means first order diffracted light in the + x direction, and −1st order diffracted light 11 ₋₁ means first order diffracted light in the −x direction. Then, each of the diffracted light 11 _{+ 1,} 11 of the two positions corresponding to the _-1 photodetector 12 _+1, place 12 _-1, respectively diffracted light 11 _+1, detects the intensity of 11 _-1 Thus, the tilt angle of the one-dimensional diffraction grating 1 around the y-axis can be detected. That is, the inclination angle around the y-axis can be detected by the difference value between the detection value of the photodetector 12 _{+1 and} the detection value of the photodetector 12 _-1 .

  Such a one-dimensional diffraction grating 1 includes a high-refractive index portion having a rectangular cross section with a pitch p in the x direction and a height h as shown in a plan view in FIG. 2A and a cross-sectional view in FIG. 2 is periodically arranged in the low refractive index portion 3 perpendicular to the plane of the diffraction grating (xy plane). The duty ratio f defined as the ratio of the width w of the high refractive index portion 2 in one repetition pitch p can be arbitrarily set.

For the following description, in addition to the duty ratio f described above, the wavelength of the incident light 10 is λ, and the value p / λ = Λ obtained by normalizing the pitch p with the use wavelength λ is the normalized pitch, and the high refractive index portion 2 The value h / λ = d obtained by normalizing the height h at the use wavelength λ is normalized groove depth, and is half the difference between the refractive index n ₂ of the high refractive index portion ₂ and the refractive index n ₁ of the low refractive index portion 3 ( Let n ₂ −n ₁ ) / 2 = Δn be the refractive index modulation.

  In order to detect the tilt angle with high sensitivity and accuracy using such a one-dimensional diffraction grating 1, diffraction at an angle deviating from the Bragg condition (Bragg angle) is performed as described above. Since the angular dependence of efficiency is used, the duty ratio f, normalized pitch Λ, normalized groove depth d, and refractive index modulation Δn of the one-dimensional diffraction grating 1 are set so as to satisfy the following conditions. It is necessary to set.

(1) When the Bragg angle is positive, the incident angle that gives the minimum diffraction efficiency that appears first from the Bragg angle is negative, or the incident angle that gives the minimum diffraction efficiency that appears first from the Bragg angle and Bragg angle Including an incident angle of 0 ° between
(2)
{(Diffraction efficiency at Bragg angle-Minimal diffraction efficiency first appearing) x 0.2
+ Minimum diffraction efficiency that appears first} ≦ Diffraction efficiency at an incident angle of 0 ° (A)
(3) The diffraction efficiency at the Bragg angle is at least twice the diffraction efficiency at an incident angle of 0 °.

  In the diffraction efficiency theoretical curve of a holographic volume diffraction grating (Non-Patent Document 1), linear approximation is relatively good in the angle range from the Bragg angle to the first minimum on the minus side (the error from the straight line is about 1% or less) ) Is a portion of 20% to 80% in the range of minimum to maximum (Bragg angle) in terms of diffraction efficiency. In order for the portion of the linear approximation to cross the intercept (diffraction efficiency at an incident angle of 0 °), the above (1) and (2) are necessary conditions. Note that the relief type diffraction grating as shown in FIG. 2 has a different refractive index distribution from the volume type diffraction grating, so the peak value is different, but the refractive index distribution is a combination of the sinusoidal distribution of the volume type diffraction grating. It can be said that the diffraction efficiency curves have almost the same shape.

  And in order to have a sensitivity of a certain level or more, it is necessary to satisfy the above (3).

  FIG. 3 shows the transmission first-order diffraction efficiency of three examples of the one-dimensional diffraction grating 1 as shown in FIG. 2, and the Bragg angle is set to about 5.7 °. Examples 1 and 2 both satisfy the above conditions (1) to (3), and can detect an inclination near an inclination angle of 0 ° with high accuracy and high sensitivity. On the other hand, in the case of a one-dot chain line, the conditions (1) to (2) are not satisfied, and there is no linearity and this is inappropriate.

Therefore, in the present invention, the transmission first-order diffraction efficiency is as described above using the normalized pitch Λ, refractive index modulation Δn, duty ratio f, and normalized groove depth d of the one-dimensional diffraction grating 1 as shown in FIG. 2 as parameters. The range satisfying the conditions (1) to (3) was calculated by applying the RCWA method (strict electromagnetic wave analysis: Non-Patent Documents 2 to 4). The one-dimensional diffraction grating 1 in FIG. 2 has substantially the same diffraction efficiency characteristics regardless of the polarization of incident light. Further, the diffraction efficiency characteristics are almost the same regardless of the average refractive index ((n ₂ + n ₁ ) / 2).

  As a result, it was found that there are at least 6 series as a range satisfying the conditions (1) to (3) within the range of the normalized pitch Λ of 2 to 12. It should be noted that when the standardized pitch Λ is larger than 14, the measurement angle range cannot be taken sufficiently and is omitted.

For the first series, FIG. 4 shows the range of the refractive index modulation Δn when the normalized pitch Λ is changed. FIGS. 5 and 6 show an upper limit f _max and a lower limit f _{min of} the duty ratio f that satisfy the above conditions (1) to (3) when the normalized pitch Λ and the refractive index modulation Δn are changed. And calculated curves (original data) and curves (mathematical approximation) obtained by approximating the limits of those ranges with polynomials. Further, FIGS. 7 and 8 show the upper limit d _max of the normalized groove depth d that satisfies the above conditions (1) to (3) when the normalized pitch Λ and the refractive index modulation Δn are changed. A calculated value (original data) of the lower limit d _min and a curve (mathematical approximation) obtained by approximating the limits of those ranges with a polynomial are shown.

For the second series, FIG. 9 shows the range of the refractive index modulation Δn when the normalized pitch Λ similar to FIG. 4 is changed. 10 and 11, the duty ratio f satisfying the above conditions (1) to (3) when the normalized pitch Λ and the refractive index modulation Δn similar to those in FIGS. 5 and 6 are changed, respectively. The calculated values (original data) of the upper limit f _max and the lower limit f _min and the curve (mathematical approximation) obtained by approximating the limits of these ranges with a polynomial are shown. Further, in FIGS. 12 and 13, normalized grooves satisfying the above conditions (1) to (3) when the normalized pitch Λ and the refractive index modulation Δn similar to those in FIGS. 7 and 8 are changed, respectively. A calculated value (original data) of an upper limit d _max and a lower limit d _min of the depth d and a curve (mathematical approximation) obtained by approximating the limits of those ranges with a polynomial are shown.

  For the third series, FIG. 14 is similar to FIG. 4, FIGS. 15 and 16 are similar to FIGS. 5 and 6, respectively, and FIGS. 17 and 18 are FIGS. A similar diagram is shown.

  For the fourth series, FIG. 19 shows the same diagram as FIG. 4, FIGS. 20 and 21 show the same diagram as FIG. 5 and FIG. 6, respectively, and FIG. 22 and FIG. The figure is shown.

  For the fifth series, FIG. 24 shows the same diagram as FIG. 4, FIGS. 25 and 26 show the same diagrams as FIG. 5 and FIG. 6, respectively, and FIG. 27 and FIG. The figure is shown.

  For the sixth series, FIG. 29 shows the same figure as FIG. 4, FIGS. 30 and 31 show the same figures as FIG. 5 and FIG. 6, respectively, and FIG. 32 and FIG. The figure is shown.

  From the above results, the ranges to be satisfied by Λ, Δn, f, d in the first to sixth sequences are shown below.

<First series>
The range of Λ and Δn is
Λ> 1 and
-0.01Λ + 0.04 <Δn <-0.0003Λ ³ + 0.0089Λ ² -0.0864Λ + 0.299
Because
0.4515-0.1605Λ-3.601Δn + 0.01841Λ ² + 0.5181ΛΔn + 0.9487Δn ^{2
-0.0007005Λ 3 - 0.001691Λ 2 Δn + 2.928ΛΔn} 2 + 0.1092Δn 3
≦ f ≦ 0.3516 + 0.06176Λ + 0.2Δn− 0.008532Λ ² + 0.7653ΛΔn− 0.8939Δn ²
+0.0003957 Λ ³ -0.2844Λ ² Δn-4.389 ΛΔn ² -0.3368Δn ³
,And,
-10.09 + 11.81Λ + 74.36Δn-1.321Λ ² -136.9ΛΔn-35.73Δn ²
+0.0792 Λ ³ + 7.72Λ ² Δn + 517.8ΛΔn ² -34.84Δn ³
≦ d ≦ -7.715-0.9232Λ-562.6Δn + 3.89Λ ² + 562.5ΛΔn-190.7Δn ^{2
-0.1926 Λ 3 - 122.6Λ 2 Δn-} 163.2ΛΔn 2 - 30.25Δn 3
.

<Second series>
The range of Λ and Δn is
Λ> 1 and
-0.01Λ + 0.04 <Δn <0.0001Λ ³ -0.0033Λ ² + 0.0101Λ + 0.1155
Because
0.4134 + 0.05121Λ + 1.146Δn-0.00665Λ ² -1.112ΛΔn-1.124Δn ²
+ 0.0002231Λ ³ -0.006226Λ ² Δn + 4.092ΛΔn ² -0.5992Δn ³
≦ f ≦ 1.023-0.004624Λ-2.723Δn-0.00612Λ ² -0.2689ΛΔn- 0.7496Δn ²
+ 0.0003977Λ ³ -0.06761Λ ² Δn + 3.449ΛΔn ² -0.1509Δn ³
,And,
-25.94 + 13.47Λ + 773.4Δn- 0.4121Λ ² -315.4ΛΔn- 4890Δn ²
-0.0003746Λ ³ + 7.305Λ ² Δn + 1853 ΛΔn ² -1340Δn ³
≦ d ≦ -19.23 + 13.27Λ + 566.1Δn− 0.1128Λ ² -240.8ΛΔn− 4803Δn ^{2
-0.001412Λ 3 - 7.37Λ 2 Δn + 2067ΛΔn} 2 -1502Δn 3
.

<Third series>
The range of Λ and Δn is
Δn> 0.06, Δn <0.05Λ−0.07, and Δn <−0.03Λ + 0.28
Because
4.836-2.451Λ-0.03076Δn + 0.5076Λ ² -4.371ΛΔn + 4.231Δn ²
-0.0338Λ ³ + 0.1792Λ ² Δn + 16.78ΛΔn ² + 1.153Δn ³
≦ f ≦ 2.438-0.6043Λ-9.628Δn + 0.07705Λ ² + 1.281ΛΔn + 0.2567Δn ²
-0.005385 Λ ³ +0.03465 Λ ² Δn + 0.7961 ΛΔn ² + 0.05974Δn ³
,And,
^{75.32-25.21Λ-770.5Δn + 3.284Λ 2 + 292.9ΛΔn} + 29.71Δn 2 - 0.02543Λ 3
-35.75Λ ² Δn + 85.55ΛΔn ² + 4.982Δn ³
≦ d ≦ 87.95-18.82Λ-1498Δn-0.4301Λ ² + 601.4ΛΔn + 36.55Δn ²
+0.3967 Λ ³ -68.54Λ ² Δn + 63.85ΛΔn ² + 3.567Δn ³
<4th series>
The range of Λ and Δn is
Λ> 3 and 0.01Λ + 0.02 <Δn <−0.02Λ + 0.22
Because
16.92-7.218Λ-17.6Δn + 0.7945Λ ² + 0.777ΛΔn + 67.32Δn ²
≦ f ≦ 11.6-7.742Λ-30.42Δn + 1.978Λ ² + 3.243ΛΔn + 5.462Δn ²
-0.1714Λ ³ -0.171Λ ² Δn + 21.22ΛΔn ² + 1.582Δn ³
,And,
-344.9 + 1.952Λ- 1923Δn + 53.92Λ ² + 878.5ΛΔn- 0.9743Δn ²
-7.931 Λ ³ -98.75Λ ² Δn-135.6ΛΔn ² -11.17Δn ³
≦ d ≦ 809.2-449.3Λ-3286Δn + 82.91Λ ² +1470 ΛΔn-41.64Δn ²
-4.973Λ ³ -153.2Λ ² Δn-376.5ΛΔn ² -28.56Δn ³
.

<5th series>
The range of Λ and Δn is
4 <Λ <9 and -0.003Λ + 0.072 <Δn <-0.003Λ + 0.097
Because
7.756-1.799Λ-48.55Δn + 0.1032Λ ² + 7.421ΛΔn-5.782Δn ²
≦ f ≦ 2.665-0.5912Λ + 0.05526Δn + 0.04807Λ ² -0.5177ΛΔn + 0.001381Δn ²
,And,
58.79-10.53Λ-194.6Δn + 0.7923Λ ² + 10.87ΛΔn + 0.5599 Δn ²
≦ d ≦ -211.8 + 48.96Λ + 2337Δn-1.809Λ ² −383.5ΛΔn + 330.7Δn ²
.

<6th series>
The range of Λ and Δn is
3 <Λ <8 and −0.004Λ + 0.037 <Δn <−0.004Λ + 0.057
Because
-0.8123 + 0.812Λ- 0.5347Δn-0.08148Λ ² -2.631ΛΔn-0.02602Δn ²
≦ f ≦ -0.6139 + 0.8126Λ- 0.5629Δn-0.08154Λ ² -2.623ΛΔn-0.02701Δn ²
,And,
-11.62+ 16.94Λ-413.2Δn- 0.9013Λ ² -61.19ΛΔn- 0.6286Δn ²
≦ d ≦ -23.84 + 33.66Λ-1354Δn-2.742Λ ² -2.268ΛΔn- 0.1423Δn ²
.

  As described above, the normalized pitch Λ, the refractive index modulation Δn, the duty ratio f, and the normalized groove depth d of the one-dimensional diffraction grating 1 of the present invention capable of accurately detecting the tilt angle in a one-dimensional manner with high sensitivity. A relationship was sought.

Specific examples are shown below.
(A) Λ = 2, Δn = 0.06, f = 0.3, d = 5, λ = 0.633 μm, p = 1.27 μm, h = 3.2 μm (first series),
(B) Λ = 6, Δn = 0.02, f = 0.2, d = 24, λ = 0.441 μm, p = 2.65 μm, h = 10.6 μm (second series),
(C) Λ = 5, Δn = 0.10, f = 0.3, d = 14, λ = 0.543 μm, p = 2.72 μm, h = 7.6 μm (third series),
(D) Λ = 5, Δn = 0.08, f = 0.1, d = 19, λ = 0.543 μm, p = 2.72 μm, h = 10.3 μm (fourth series),
(E) Λ = 7, Δn = 0.07, f = 0.5, d = 18, λ = 0.441 μm, p = 0.09 μm, h = 7.9 μm (fifth series),
FIG. 34 shows the change in the transmission first-order diffraction efficiency with respect to the incident angle in each case.

As another example, Λ = 3.4 is selected, and the refractive index n ₁ = 1.48 of the low refractive index portion of the one-dimensional diffraction grating and the refractive index n ₂ = 1.52 of the high refractive index portion, When Δn = (n ₂ −n ₁ ) /2=0.02, from the relationship of the first series, the range of the normalized groove depth d and the duty ratio f is
d = 12.5-25.0
f = 0.058-0.462
Is obtained as the range of the solution. FIGS. 35A and 35B show changes in the transmission first-order diffraction efficiency with respect to the incident angle when f is changed with d = 15 and when d is changed with f = 0.3. Shown in b). 35 (a) and 35 (b) are examples in which the solid line curve is included in the scope of the present invention, the dotted line curve is a comparative example that is out of the scope of the present invention, and the solid line example has an incident angle of 0 °. It can be said that the linearity is relatively good, and the slope related to sensitivity is high. From among them, the conditions may be selected according to specifications such as high sensitivity.

  As is clear from the above embodiments, the one-dimensional diffraction grating 1 of the embodiments (a) to (e) according to the present invention and the one-dimensional diffraction grating 1 indicated by the solid line of another embodiment are the above (1) to ( It can be seen that the condition 3) is satisfied, and the tilt angle near 0 ° in one direction can be detected with high accuracy and high sensitivity.

  In addition, as a manufacturing method of the incident angle dependent diffraction grating according to the present invention, a diffraction grating having any refractive index (high refractive index or low refractive index) is first formed by an embossing method such as 2P method or etching on a substrate. An example is a method in which the diffraction grating is covered with, for example, a liquid curable material having another refractive index (low refractive index or high refractive index).

  Although the incident angle dependent diffraction grating of the present invention has been described based on the embodiments and the like, the present invention is not limited to these embodiments and can be variously modified.

It is a figure which shows typically the effect | action of the incident angle dependence diffraction grating by this invention. It is the top view and sectional drawing of a one-dimensional diffraction grating used for the incident angle dependence diffraction grating of this invention. It is a figure which shows the transmission 1st-order diffraction efficiency change with respect to the incident angle of the one-dimensional diffraction grating which satisfy | fills the conditions of this invention, and the one-dimensional diffraction grating which is not satisfy | filled. It is a figure which shows the range of refractive index modulation (DELTA) n when the normalization pitch (LAMBDA) is changed about a 1st series. It is a figure which shows the calculated value of the upper limit _fmax of the duty ratio f which satisfies the conditions of this invention when the normalization pitch (LAMBDA) and refractive index modulation (DELTA) n are changed about a 1st series, and the curve which approximated it. It is a figure which shows the calculated value of the minimum _fmin of the duty ratio f which satisfies the conditions of this invention when the normalization pitch (LAMBDA) and refractive index modulation (DELTA) n are changed about a 1st series, and the curve which approximated it. A diagram showing the calculated values and the curve which approximates that of the upper limit d _max condition Standard Kamizo depth d which satisfies the present invention when changing the normalized pitch Λ and the refractive index modulation Δn for the first sequence is there. It is a figure which shows the calculated value of the minimum _dmin of the normalization groove | channel depth d which satisfies the conditions of this invention when the normalization pitch (LAMBDA) and refractive index modulation (DELTA) n are changed about a 1st series, and the curve which approximated it. is there. It is a figure similar to FIG. 4 about the 2nd series. It is a figure similar to FIG. 5 about the 2nd series. It is a figure similar to FIG. 6 about the 2nd series. It is a figure similar to FIG. 7 about the 2nd series. It is a figure similar to FIG. 8 about the 2nd series. It is a figure similar to FIG. 4 about the 3rd series. It is a figure similar to FIG. 5 about the 3rd series. It is a figure similar to FIG. 6 about the 3rd series. It is a figure similar to FIG. 7 about the 3rd series. It is a figure similar to FIG. 8 about the 3rd series. It is a figure similar to FIG. 4 about the 4th series. It is a figure similar to FIG. 5 about the 4th series. It is a figure similar to FIG. 6 about the 4th series. It is a figure similar to FIG. 7 about the 4th series. It is a figure similar to FIG. 8 about the 4th series. It is a figure similar to FIG. 4 about the 5th series. It is a figure similar to FIG. 5 about the 5th series. It is a figure similar to FIG. 6 about the 5th series. It is the same figure as FIG. 7 about the 5th series. It is a figure similar to FIG. 8 about the 5th series. It is a figure similar to FIG. 4 about the 6th series. It is a figure similar to FIG. 5 about the 6th series. It is a figure similar to FIG. 6 about the 6th series. It is a figure similar to FIG. 7 about the 6th series. It is a figure similar to FIG. 8 about the 6th series. It is a figure which shows the transmission 1st-order diffraction efficiency change with respect to the incident angle of the one-dimensional diffraction grating of Example (a)-(e) of this invention. It is a figure which shows the transmission 1st-order diffraction efficiency change with respect to the incident angle of the one-dimensional diffraction grating of another Example of this invention, and a comparative example.

Explanation of symbols

1 ... 1-dimensional diffraction grating 2 ... high refractive index portion 3 ... low refractive index portion 10 ... incident light 11 ₊₁ ... + 1-order diffracted light 11 _-1 ... -1 order diffracted light 12 _+1, 12 _-1 ... photodetector

Claims (7)

A high-refractive index part and a low-refractive index part are periodically distributed in one direction, and each high-refractive index part is composed of a relief type one-dimensional diffraction grating arranged perpendicularly to the surface of the diffraction grating. In the incident angle-dependent diffraction grating in which the ± 1st-order diffraction efficiency in the period direction monotonously changes depending on the incident angle of the incident light with respect to the surface of the diffraction grating,
An angle of incidence of 0 ° is included between the Bragg angle and the incident angle that gives the minimum diffraction efficiency that first appears from the Bragg angle, with the Bragg angle satisfying the Bragg condition being positive; and
{(Diffraction efficiency at Bragg angle-Minimal diffraction efficiency first appearing) x 0.2
+ Minimum diffraction efficiency that appears first} ≦ Diffraction efficiency at an incident angle of 0 ° (A)
The condition (A) is satisfied, and
The diffraction efficiency at the Bragg angle is more than twice the diffraction efficiency at an incident angle of 0 °.
An incident angle dependent diffraction grating characterized in that a normalized pitch Λ, a refractive index modulation Δn, a duty ratio f, and a normalized groove depth d are set so as to satisfy the above. However, the wavelength of the incident light is λ, the repetition pitch of the high refractive index portion and the low refractive index portion in the periodic direction is p, the height of the high refractive index portion is h, the refractive index of the high refractive index portion is n ₂ , and the low refractive index The refractive index of the index part is n ₁ , p / λ is the normalized pitch Λ, h / λ is the normalized groove depth d, and (n ₂ −n ₁ ) / 2 is the refractive index modulation Δn. Further, the ratio of the width w of the high refractive index portion in one repetition pitch p is defined as a duty ratio f. The range of Λ and Δn is
Λ> 1 and
-0.01Λ + 0.04 <Δn <-0.0003Λ ³ + 0.0089Λ ² -0.0864Λ + 0.299
Because
0.4515-0.1605Λ-3.601Δn + 0.01841Λ ² + 0.5181ΛΔn + 0.9487Δn ^{2
-0.0007005Λ 3 - 0.001691Λ 2 Δn + 2.928ΛΔn} 2 + 0.1092Δn 3
≦ f ≦ 0.3516 + 0.06176Λ + 0.2Δn− 0.008532Λ ² + 0.7653ΛΔn− 0.8939Δn ²
+0.0003957 Λ ³ -0.2844Λ ² Δn-4.389 ΛΔn ² -0.3368Δn ³
,And,
-10.09 + 11.81Λ + 74.36Δn-1.321Λ ² -136.9ΛΔn-35.73Δn ²
+0.0792 Λ ³ + 7.72Λ ² Δn + 517.8ΛΔn ² -34.84Δn ³
≦ d ≦ -7.715-0.9232Λ-562.6Δn + 3.89Λ ² + 562.5ΛΔn-190.7Δn ^{2
-0.1926 Λ 3 - 122.6Λ 2 Δn-} 163.2ΛΔn 2 - 30.25Δn 3
The incident angle dependent diffraction grating according to claim 1, wherein the following condition is satisfied. The range of Λ and Δn is
Λ> 1 and
-0.01Λ + 0.04 <Δn <0.0001Λ ³ -0.0033Λ ² + 0.0101Λ + 0.1155
Because
0.4134 + 0.05121Λ + 1.146Δn-0.00665Λ ² -1.112ΛΔn-1.124Δn ²
+ 0.0002231Λ ³ -0.006226Λ ² Δn + 4.092ΛΔn ² -0.5992Δn ³
≦ f ≦ 1.023-0.004624Λ-2.723Δn-0.00612Λ ² -0.2689ΛΔn- 0.7496Δn ²
+ 0.0003977Λ ³ -0.06761Λ ² Δn + 3.449ΛΔn ² -0.1509Δn ³
,And,
-25.94 + 13.47Λ + 773.4Δn- 0.4121Λ ² -315.4ΛΔn- 4890Δn ²
-0.0003746Λ ³ + 7.305Λ ² Δn + 1853 ΛΔn ² -1340Δn ³
≦ d ≦ -19.23 + 13.27Λ + 566.1Δn− 0.1128Λ ² -240.8ΛΔn− 4803Δn ^{2
-0.001412Λ 3 - 7.37Λ 2 Δn + 2067ΛΔn} 2 -1502Δn 3
The incident angle dependent diffraction grating according to claim 1, wherein the following condition is satisfied. The range of Λ and Δn is
Δn> 0.06, Δn <0.05Λ−0.07, and Δn <−0.03Λ + 0.28
Because
4.836-2.451Λ-0.03076Δn + 0.5076Λ ² -4.371ΛΔn + 4.231Δn ²
-0.0338Λ ³ + 0.1792Λ ² Δn + 16.78ΛΔn ² + 1.153Δn ³
≦ f ≦ 2.438-0.6043Λ-9.628Δn + 0.07705Λ ² + 1.281ΛΔn + 0.2567Δn ²
-0.005385 Λ ³ +0.03465 Λ ² Δn + 0.7961 ΛΔn ² + 0.05974Δn ³
,And,
^{75.32-25.21Λ-770.5Δn + 3.284Λ 2 + 292.9ΛΔn} + 29.71Δn 2 - 0.02543Λ 3
-35.75Λ ² Δn + 85.55ΛΔn ² + 4.982Δn ³
≦ d ≦ 87.95-18.82Λ-1498Δn-0.4301Λ ² + 601.4ΛΔn + 36.55Δn ²
+0.3967 Λ ³ -68.54Λ ² Δn + 63.85ΛΔn ² + 3.567Δn ³
The incident angle dependent diffraction grating according to claim 1, wherein the following condition is satisfied. The range of Λ and Δn is
Λ> 3 and 0.01Λ + 0.02 <Δn <−0.02Λ + 0.22
Because
16.92-7.218Λ-17.6Δn + 0.7945Λ ² + 0.777ΛΔn + 67.32Δn ²
≦ f ≦ 11.6-7.742Λ-30.42Δn + 1.978Λ ² + 3.243ΛΔn + 5.462Δn ²
-0.1714Λ ³ -0.171Λ ² Δn + 21.22ΛΔn ² + 1.582Δn ³
,And,
-344.9 + 1.952Λ- 1923Δn + 53.92Λ ² + 878.5ΛΔn- 0.9743Δn ²
-7.931 Λ ³ -98.75Λ ² Δn-135.6ΛΔn ² -11.17Δn ³
≦ d ≦ 809.2-449.3Λ-3286Δn + 82.91Λ ² +1470 ΛΔn-41.64Δn ²
-4.973Λ ³ -153.2Λ ² Δn-376.5ΛΔn ² -28.56Δn ³
The incident angle dependent diffraction grating according to claim 1, wherein the following condition is satisfied. The range of Λ and Δn is
4 <Λ <9 and -0.003Λ + 0.072 <Δn <-0.003Λ + 0.097
Because
7.756-1.799Λ-48.55Δn + 0.1032Λ ² + 7.421ΛΔn-5.782Δn ²
≦ f ≦ 2.665-0.5912Λ + 0.05526Δn + 0.04807Λ ² -0.5177ΛΔn + 0.001381Δn ²
,And,
58.79-10.53Λ-194.6Δn + 0.7923Λ ² + 10.87ΛΔn + 0.5599 Δn ²
≦ d ≦ -211.8 + 48.96Λ + 2337Δn-1.809Λ ² −383.5ΛΔn + 330.7Δn ²
The incident angle dependent diffraction grating according to claim 1, wherein the following condition is satisfied. The range of Λ and Δn is
3 <Λ <8 and −0.004Λ + 0.037 <Δn <−0.004Λ + 0.057
Because
-0.8123 + 0.812Λ- 0.5347Δn-0.08148Λ ² -2.631ΛΔn-0.02602Δn ²
≦ f ≦ -0.6139 + 0.8126Λ- 0.5629Δn-0.08154Λ ² -2.623ΛΔn-0.02701Δn ²
,And,
-11.62+ 16.94Λ-413.2Δn- 0.9013Λ ² -61.19ΛΔn- 0.6286Δn ²
≦ d ≦ -23.84 + 33.66Λ-1354Δn-2.742Λ ² -2.268ΛΔn- 0.1423Δn ²
The incident angle dependent diffraction grating according to claim 1, wherein the following condition is satisfied.

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