CN102385091B - Combined type phase plates for expanding field depth of optical imaging system - Google Patents

Abstract

The present invention relates to the composite phase plates that three kinds are used for expanding field depth of optical imaging system, when the phase distribution function of one of phase-plate is formed by power function three times, logarithmic function A, exponential function and high math power function superposition, a dimensional expression of phase distribution function are as follows:

--- -- logarithmic function A

In formula: 1~α of α 7 is the parameter of the phase distribution function of the phase-plate; X is normalization aperture coordinate, and value range is [- 1,1]; Sgn is sign function, its value is -1 when as x ＞ 0, its value is 1, x ＜ 0. The composite phase plates for expanding field depth of optical imaging system that system can be made more insensitive to defocus and sharp keen clearly big depth image can be obtained the present invention provides three kinds.

Description

The composite phase plates that is used for expanding field depth of optical imaging system

Technical field

The invention belongs to optical field, relate to a kind of phase mask plate, relate in particular to a kind of composite phase plates for expanding field depth of optical imaging system.

Background technology

The depth of field of expanding optical imaging system always is the focus of academia's research.Since 20th century the mid-80, although panoramic method is proposed for the expansion depth of field, but until doctor Dowski of Univ Colorado-Boulder USA and Cathey teach after the wavefront coded concept of nineteen ninety-five proposition, depth of field continuation has just had breakthrough truly.

Add the phase mask plate through particular design by aperture plane, entrance pupil face or emergent pupil face in traditional imaging system, system will and cause that to out of focus the factor of out of focus is insensitive so.Take the one dimensional optical system as example, its out of focus optical transfer function OTF (Optical transfer function) can obtain by the auto-correlation computation of generalized pupil function, and is as follows:

$H(u,W_{20})=\frac{1}{2}\&CenterDot;\underset{-(1-|u|/2)}{\overset{1-\left|u\right|/2}{\&Integral;}}\exp (j\&CenterDot;(2{\mathrm{<figure-callout\; id="20"\; label="kW"\; filenames="HDA0000098485570000041.png,HDA0000098485570000051.png"\; state="\{\{state\}\}">kW</figure-callout>}}_{20}\mathrm{ux}+f(x+u/2)-f(x-u/2)))\mathrm{dx}$

Wherein, u and x are respectively normalized spatial frequency and aperture plane lateral coordinates; W ₂₀It is maximum defocus wave aberration coefficient; K is wave number; F represents the universal expression formula of employed phase mask plate.

For traditional imaging system, the f item in the following formula does not exist, and the expression that therefore can obtain out of focus OTF is:

$H(u,W_{20})=\frac{\sin (2\mathrm{<figure-callout\; id="20"\; label="k\; W"\; filenames="HDA0000098485570000041.png,HDA0000098485570000051.png"\; state="\{\{state\}\}">k</figure-callout>}{W}_{}{}_{20}\&CenterDot;(1-\frac{\left|u\right|}{2}))}{2{\mathrm{kW}}_{20}u}u\&NotEqual;0$

Can see that when system did not introduce phase-plate, its OTF was highstrung to out of focus, and zero point can periodically occur in the frequency space, thereby cause irreversible information loss.But the cube phase-plate (f (x)=α x that in a single day doctor E.R.Dowski is invented ³) be incorporated on the entrance pupil face of optical system after, we just can obtain a diverse out of focus OTF by the static phase method of approximation, and are as follows:

$H(u,W_{20})\&ap;\frac{1}{2}\&CenterDot;\sqrt{\frac{\mathrm{\&pi;}}{\left|3\mathrm{\&alpha;u}\right|}}\&CenterDot;\exp (j\&CenterDot;(\frac{{\mathrm{\&alpha;u}}^3}{4}-\frac{k^{2}u{{\mathrm{<figure-callout\; id="2"\; label="W"\; filenames="HDA0000098485570000041.png"\; state="\{\{state\}\}">W</figure-callout>}}^2}_{20}}{3\mathrm{\&alpha;}})+j\&CenterDot;\mathrm{sgn}\left(u\right)\&CenterDot;\frac{\mathrm{\&pi;}}{4}),u\&NotEqual;0$

Obviously, this moment out of focus OTF mould, namely MTF (Modulation transfer function) has nothing to do with out of focus wave aberration coefficient, that is to say that the cube phase-plate can make the MTF of system insensitive to out of focus; Although the phase bit position of OTF and degree of blur W ₂₀Relevant, but as long as modulation factor α increases, it is to W ₂₀Dependency degree will significantly reduce.Simultaneously of paramount importancely be, after having added phase-plate, MTF just has to a certain degree decline in amplitude within effective frequency range, and do not have zero point or a nearly zero point, be that system is when out of focus occurring, the information that exceeds the primal system field depth is not lost, and just can effectively be recovered by the digital image restoration algorithm afterwards.Simultaneously because phase-plate can not impact logical light quantity and the resolution of system, so wavefront coded be a kind of novel field depth extension imaging technology that is different from very much reduced bore method, central obscuration method or apodization.

Although it is of a great variety to be used for the phase-plate of depth of field continuation, whether having the phase-plate with better performance still is the focus of research.If the consistance of the corresponding out of focus MTF of phase-plate under different defocusing amounts is higher, and zero point or nearly zero point do not occur, so just can think that this phase-plate has strong depth of field expansion capability.

Summary of the invention

The invention provides three kinds and can make the composite phase plates that is used for expanding field depth of optical imaging system that system is more insensitive to out of focus and can obtain sharp keen clearly large depth image.

Technical solution of the present invention is: the invention provides three kinds of composite phase plates that are used for expanding field depth of optical imaging system, its special character is: when the PHASE DISTRIBUTION function of phase-plate was formed by stacking by third power function, logarithmic function A, exponential function and high order power function, a dimensional expression of its PHASE DISTRIBUTION function was:

$f_1\left(x\right)=({\mathrm{\&alpha;}}_1\&CenterDot;x^3)$

$+\{{\mathrm{\&alpha;}}_2\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\log \left(\right|x\left|\right)+{\mathrm{\&alpha;}}_3\left]\right\}$ -----logarithmic function A

$+[{\mathrm{\&alpha;}}_4\&CenterDot;x\&CenterDot;\exp ({\mathrm{\&alpha;}}_5\&CenterDot;x^2)]$

$+[{\mathrm{\&alpha;}}_6\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;{\left|x\right|}^{{\mathrm{\&alpha;}}_7}]$

In the formula:

α 1～α 7 is parameters of the PHASE DISTRIBUTION function of described phase-plate;

X is normalization aperture coordinate, and its span is [1,1];

Sgn is-symbol function, its value is that 1, x＜0 o'clock its value is-1 when x＞0.

When the PHASE DISTRIBUTION function of phase-plate is when being formed by stacking by third power function, logarithmic function A, exponential function, high order power function and logarithmic function B, a dimensional expression of its PHASE DISTRIBUTION function is:

$f_2\left(x\right)=({\mathrm{\&alpha;}}_1\&CenterDot;x^3)$

$+\{{\mathrm{\&alpha;}}_2\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|x\left|\right)+{\mathrm{\&alpha;}}_3\left]\right\}$ -----logarithmic function A

$+[{\mathrm{\&alpha;}}_4\&CenterDot;x\&CenterDot;\exp ({\mathrm{\&alpha;}}_5\&CenterDot;x^2)]$

$+[{\mathrm{\&alpha;}}_6\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;{\left|x\right|}^{{\mathrm{\&alpha;}}_7}]$

$+\{{\mathrm{\&alpha;}}_8\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|\left|x\right|+{\mathrm{\&alpha;}}_9\left|\right)\left]\right\}$ -----logarithmic function B

In the formula:

α 1～α 9 is parameters of the PHASE DISTRIBUTION function of described phase-plate;

X is normalization aperture coordinate, and its span is [1,1];

Sgn is-symbol function, x＞0 o'clock its value are that 1, x＜0 o'clock its value is for-1.

When the PHASE DISTRIBUTION function of phase-plate is when being formed by stacking by third power function, logarithmic function A, exponential function, high order power function, logarithmic function B and sine function, a dimensional expression of its PHASE DISTRIBUTION function is:

$f_3\left(x\right)=({\mathrm{\&alpha;}}_1\&CenterDot;x^3)$

$+\{{\mathrm{\&alpha;}}_2\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|x\left|\right)+{\mathrm{\&alpha;}}_3\left]\right\}$ -----logarithmic function A

$+[{\mathrm{\&alpha;}}_4\&CenterDot;x\&CenterDot;\exp ({\mathrm{\&alpha;}}_5\&CenterDot;x^2)]$

$+[{\mathrm{\&alpha;}}_6\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;{\left|x\right|}^{{\mathrm{\&alpha;}}_7}]$

$+\{{\mathrm{\&alpha;}}_8\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|\left|x\right|+{\mathrm{\&alpha;}}_9\left|\right)\left]\right\}$ -----logarithmic function B

$+[{\mathrm{\&alpha;}}_{10}\&CenterDot;\sin ({\mathrm{\&alpha;}}_{11}\&CenterDot;x)]$

In the formula:

α 1～α 11 is parameters of the PHASE DISTRIBUTION function of described phase-plate;

X is normalization aperture coordinate, and its span is [1,1];

Sgn is-symbol function, x＞0 o'clock its value are that 1, x＜0 o'clock its value is for-1.

Advantage of the present invention is:

The designed novel combined phase mask plate of the present invention can reduce system more significantly to the susceptibility of out of focus, great variation can not occur with the change of degree of blur in the modulation transfer function (MTF) that is whole optical system, and zero point or nearly zero point can not appear, so can not cause losing of image detail information in the passband scope.After middle blurred picture process deconvolution processing, just can obtain sharp keen clearly large depth image.

Description of drawings

Fig. 1 is the face type curve family of the first phase-plate provided by the present invention;

Fig. 2 is the face type curve family of the second phase-plate provided by the present invention;

Fig. 3 is the face type curve family of third phase provided by the present invention position plate;

Fig. 4 uses three kinds of composite phase plates proposed by the invention under the different Th conditions of Fisher information evaluation and other two kinds of existing composite phase plates to the synoptic diagram of the inhibition ability of out of focus;

Fig. 5 uses three kinds of composite phase plates proposed by the invention under the different Th conditions of Fisher information evaluation and six kinds of typical phase-plates to the synoptic diagram of the inhibition ability of out of focus.

Embodiment

The present invention by the mode of stack, has proposed three kinds of novel combined phase mask plates that are used for expanding field depth of optical imaging system tightly around study hotspot.Analysis result shows that new phase-plate has better out of focus unchangeability, but before depth of field extension capability new, old phase-plate is estimated, is the first step based on the face type optimization under the uniform principles.Because the decline meeting of the MTF that the introducing of phase-plate causes produces adverse influence to follow-up image restoration, so the optimization of face type should be carried out take minimum acceptable MTF (i.e. Th in table 1, table 2 and the table 3) as constraint.

The phase-plate that is used for expanding field depth of optical imaging system involved in the present invention, its PHASE DISTRIBUTION function is mutually superposeed by existing dissimilar phase mask plate and forms.Composite phase plates should be near aperture plane, entrance pupil plane or the emergent pupil plane of imaging system when arranging.Phase-plate face type can also be arranged on the lens surface of close system aperture plane, entrance pupil face or the emergent pupil face of imaging lens.

When above-mentioned phase-plate PHASE DISTRIBUTION function was formed by stacking by third power function, logarithmic function A, exponential function and high order power function, its one dimension function expression was:

$f_1\left(x\right)=({\mathrm{\&alpha;}}_1\&CenterDot;x^3)$

$+\{{\mathrm{\&alpha;}}_2\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\log \left(\right|x\left|\right)+{\mathrm{\&alpha;}}_3\left]\right\}$ -----logarithmic function A

$+[{\mathrm{\&alpha;}}_4\&CenterDot;x\&CenterDot;\exp ({\mathrm{\&alpha;}}_5\&CenterDot;x^2)]$

$+[{\mathrm{\&alpha;}}_6\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;{\left|x\right|}^{{\mathrm{\&alpha;}}_7}]$

In the formula, α 1～α 7 is parameters of described phase-plate PHASE DISTRIBUTION function, and x is normalization aperture coordinate, and its span is [1,1].Sgn is-symbol function, x＞0 o'clock its value are that 1, x＜0 o'clock its value is for-1.

After use simulated annealing is carried out multi-parameters optimization to above-mentioned phase-plate, just can obtain the optimum face shape parameter under the corresponding various boundary conditions of phase-plate, as shown in table 1, and with the corresponding one dimension PHASE DISTRIBUTION of these parameters as shown in Figure 1, wherein each bar curve (equals 0.21 with Th corresponding to the parameter shown in one group of table 1,0.25,0.29 0.33 is example, Fig. 2 and 3 together), horizontal ordinate is the aperture coordinate after the normalization, and ordinate is phase-modulation intensity f ₁

Table 1

Th	α1	α2	α3	α4	α5	α6	α7
								0.21	63.04	86.06	0.47	71.86	1.06	55.25	1.27
0.23	56.56	62.07	0.27	61.75	1.19	99.82	2.01
								0.25	22.82	96.75	0.26	74.67	1.07	18.46	1.94
0.27	48.46	63.52	0.15	68.93	0.83	53.30	2.75
								0.29	50.57	57.95	0.007	14.52	2.24	48.60	1.65
0.31	49.53	54.50	0.12	5.46	2.76	70.45	2.56
								0.33	43.75	65.96	0.17	6.51	2.01	98.88	2.76
0.35	6.85	43.06	0.21	17.28	1.78	99.24	2.30

When above-mentioned phase-plate PHASE DISTRIBUTION function is when being formed by stacking by third power function, logarithmic function A, exponential function, high order power function and logarithmic function B, its one dimension function expression is:

$f_2\left(x\right)=({\mathrm{\&alpha;}}_1\&CenterDot;x^3)$

$+\{{\mathrm{\&alpha;}}_2\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|x\left|\right)+{\mathrm{\&alpha;}}_3\left]\right\}$ -----logarithmic function A

$+[{\mathrm{\&alpha;}}_4\&CenterDot;x\&CenterDot;\exp ({\mathrm{\&alpha;}}_5\&CenterDot;x^2)]$

$+[{\mathrm{\&alpha;}}_6\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;{\left|x\right|}^{{\mathrm{\&alpha;}}_7}]$

$+\{{\mathrm{\&alpha;}}_8\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|\left|x\right|+{\mathrm{\&alpha;}}_9\left|\right)\left]\right\}$ -----logarithmic function B

In the formula, α 1～α 9 is parameters of described phase-plate PHASE DISTRIBUTION function, and x is normalization aperture coordinate, and its span is [1,1].Sgn is-symbol function, x＞0 o'clock its value are that 1, x＜0 o'clock its value is for-1.

After use simulated annealing is carried out multi-parameters optimization to above-mentioned phase-plate, just can obtain the optimum face shape parameter under the corresponding various boundary conditions of phase-plate, as shown in table 2, and with the corresponding one dimension PHASE DISTRIBUTION of these parameters as shown in Figure 2, wherein each bar curve is corresponding to the parameter shown in one group of table 2, horizontal ordinate is the aperture coordinate after the normalization, and ordinate is phase-modulation intensity f ₂

Table 2

Th	α1	α2	α3	α4	α5	α6	α7	α8	α9
										0.21	57.36	86.15	0.41	93.29	0.98	85.25	2.03	36.96	0.29
0.23	57.61	56.70	0.17	67.20	1.18	76.36	1.33	20.24	1.64
										0.25	4.96	65.27	0.21	91.64	1.09	2.34	1.01	66.50	1.34
0.27	9.23	63.85	0.33	38.33	1.58	56.69	1.29	72.14	0.31
										0.29	61.38	93.08	0.05	7.15	1.78	93.50	2.47	0.58	0.72
0.31	57.23	71.44	0.39	12.83	0.10	91.93	2.68	51.19	1.46
										0.33	31.65	69.07	0.29	23.97	0.81	97.87	2.70	52.37	0.63
0.35	40.64	69.28	0.05	6.68	2.04	86.77	2.69	43.11	0.68

When above-mentioned phase-plate PHASE DISTRIBUTION function is when being formed by stacking by third power function, logarithmic function A, exponential function, high order power function, logarithmic function B and sine function, its one dimension function expression is:

$f_3\left(x\right)=({\mathrm{\&alpha;}}_1\&CenterDot;x^3)$

$+\{{\mathrm{\&alpha;}}_2\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|x\left|\right)+{\mathrm{\&alpha;}}_3\left]\right\}$ -----logarithmic function A

$+[{\mathrm{\&alpha;}}_4\&CenterDot;x\&CenterDot;\exp ({\mathrm{\&alpha;}}_5\&CenterDot;x^2)]$

$+[{\mathrm{\&alpha;}}_6\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;{\left|x\right|}^{{\mathrm{\&alpha;}}_7}]$

$+\{{\mathrm{\&alpha;}}_8\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|\left|x\right|+{\mathrm{\&alpha;}}_9\left|\right)\left]\right\}$ -----logarithmic function B

$+[{\mathrm{\&alpha;}}_{10}\&CenterDot;\sin ({\mathrm{\&alpha;}}_{11}\&CenterDot;x)]$

In the formula, α 1～α 11 is parameters of described phase-plate PHASE DISTRIBUTION function, and x is normalization aperture coordinate, and its span is [1,1].Sgn is-symbol function, x＞0 o'clock its value are that 1, x＜0 o'clock its value is for-1.

After use simulated annealing is carried out multi-parameters optimization to above-mentioned phase-plate, just can obtain the optimum face shape parameter under the corresponding various boundary conditions of phase-plate, as shown in table 3, and with the corresponding one dimension PHASE DISTRIBUTION of these parameters as shown in Figure 3, wherein each bar curve is corresponding to the parameter shown in one group of table 3, horizontal ordinate is the aperture coordinate after the normalization, and ordinate is phase-modulation intensity f ₃

Table 3

Th	α1	α2	α3	α4	α5	α6	α7	α8	α9	α10	α11
												0.21	43.33	91.83	0.55	69.74	1.16	26.11	1.39	69.76	1.98	35.58	0.08
0.23	58.42	62.17	0.12	78.15	1.09	79.76	1.34	39.94	0.54	33.11	1.08
												0.25	1.03	51.07	0.15	89.70	1.16	37.23	1.91	9.82	1.87	59.51	1.38
0.27	50.30	62.09	0.24	19.80	1.79	92.63	2.06	91.63	0.15	63.93	1.32

0.29	47.28	36.86	0.10	37.53	1.05	57.46	2.34	88.81	1.00	39.05	1.89
												0.31	49.10	65.50	0.03	13.94	1.83	42.59	2.04	32.63	1.45	39.84	0.56
0.33	15.18	71.98	0.16	11.64	2.34	25.93	2.03	89.66	0.90	3.40	0.93
												0.35	12.27	61.20	0.05	30.80	1.27	57.80	1.47	70.87	1.80	31.00	0.33

In above-mentioned table 1, table 2 and the table 3, symbol Th is a threshold value, and satisfies ∫ MTF (u, W ₂₀=0, P) du 〉=Th, wherein MTF is the modulation transfer function of wavefront coded system; U, W ₂₀And P is respectively the normalization spatial frequency, out of focus wave aberration coefficient and phase-plate parameter.Th is the degree that reduces at phase-plate face type optimizing process for constrained system MTF, and namely the Th value is larger, and the corresponding MTF curve of system after optimizing is just higher, otherwise lower.

Curve comparison by Fig. 1, Fig. 2 and Fig. 3 can see that there is some difference between the corresponding optimum face type curve of composite phase plates, similar but not identical, and this has just determined that its out of focus suppresses the size of ability.

Fisher information represents system to the susceptibility of out of focus, and its value is less, and system is just more insensitive to out of focus so.According to the Fisher information definition, when the degree of blur span was 0-9 π, the corresponding Fisher information of which kind of phase-plate curve was lower, and so this phase-plate is better to the insensitivity of out of focus, that is to say that it has stronger depth of field expansion capability.

As shown in Figure 4 and Figure 5, three kinds of composite phase plates proposed by the invention carry out after the multiparameter combined optimization, its insensitivity to out of focus has had raising clearly, especially when the modulation transfer function that requires wavefront coded system is higher, be that the Th value is when larger, the raising of this performance is especially obvious because new composite phase plates corresponding Fisher curve in the position of large defocusing amount obvious reduction is arranged.

In Fig. 4, the corresponding Fisher information of three kinds of novel combined phase-plates proposed by the invention is all less than the Fisher information of existing two kinds of composite phase plates, show that namely it has stronger out of focus and suppresses ability, six width of cloth images of Fig. 4 correspond respectively to Th and equal 0.23,0.25,0.27,0.29,0.31 and 0.33, and the out of focus scope is 0～9 π.Black, powder, blue or green corresponding to new composite phase plates f ₁, f ₂And f ₃, red and indigo plant is then corresponding to other two kinds of existing composite phase plates, and the phase-plate that is formed by stacking by cube function and logarithmic function A of red expression wherein is blue then represent the phase-plate that is formed by stacking by cube function, logarithmic function A and exponential function.Wherein, horizontal ordinate is degree of blur, and ordinate is Fisher information.

Fig. 5 has then provided the performance comparison between novel combined phase-plate and the six kinds of single phase-plates, and six width of cloth images correspond respectively to Th and equal 0.21,0.23, and 0.25,0.27,0.31 and 0.33, and the out of focus scope is 0～9 π.Wherein not the curve of zone circle corresponding to new composite phase plates f ₁, f ₂, f ₃, and above-mentioned two kinds of existing composite phase plates; The curve of zone circle then represents single phase-plate: cube type, logarithm A type, logarithm Type B, exponential type, high order square and sinusoidal pattern.Wherein, horizontal ordinate is degree of blur, and ordinate is Fisher information.

Can see that novel phase-plate proposed by the invention can make system more insensitive to out of focus, when especially the Th value is larger, more remarkable to the inhibition of out of focus.In addition, three kinds of novel combined phase-plates are when the Th value is larger, and its inhibition ability to out of focus almost can't be distinguished, because its corresponding Fisher information curve coincides together basically.

With employed phase-plate in the prior art comparatively speaking, use novel combined phase-plate involved in the present invention can make wavefront coded system that out of focus is become more insensitive, great variation can not occur with the change of degree of blur in the modulation transfer function (MTF) that is whole imaging system, and zero point or nearly zero point can not appear, so can not cause losing of image detail information in the passband scope.After middle blurred picture process deconvolution processing, we just can obtain sharp keen clearly depth image.Because the MTF of system has fabulous stability, only use a digital filter just can the middle blurred picture of all positions to be restored simultaneously.

Follow-on phase-plate involved in the present invention is for wave-front coding imaging technology, preferably is arranged near the aperture of imaging system, near the entrance pupil or near the emergent pupil.If because imaging system has certain restriction to its operating distance, volume, quality or number of elements etc., so also can directly the face type of phase-plate be processed on certain surface of existing imaging lens, but in order to bring into play to greatest extent the performance of phase-plate, this surface is preferably near system aperture plane, entrance pupil face or emergent pupil face.

Claims (3)

1. composite phase plates that is used for expanding field depth of optical imaging system, it is characterized in that: the PHASE DISTRIBUTION function of described composite phase plates is formed by stacking by third power function, logarithmic function A, exponential function and high order power function; The a dimensional expression of the PHASE DISTRIBUTION function of described composite phase plates is:

$f_1\left(x\right)=({\mathrm{\&alpha;}}_1\&CenterDot;x^3)$

$+\{{\mathrm{\&alpha;}}_2\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\log \left(\right|x\left|\right)+{\mathrm{\&alpha;}}_3\left]\right\}$ -----logarithmic function A

$+[{\mathrm{\&alpha;}}_4\&CenterDot;x\&CenterDot;\exp ({\mathrm{\&alpha;}}_5\&CenterDot;x^2)]$

$+[{\mathrm{\&alpha;}}_6\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;{\left|x\right|}^{{\mathrm{\&alpha;}}_7}]$

In the formula:

α 1～α 7 is parameters of the PHASE DISTRIBUTION function of described phase-plate;

X is normalization aperture coordinate, and its span is [1,1];

Sgn is-symbol function, its value is that 1, x＜0 o'clock its value is-1 when x＞0.

2. composite phase plates that is used for expanding field depth of optical imaging system, it is characterized in that: the PHASE DISTRIBUTION function of described composite phase plates is to be formed by stacking by third power function, logarithmic function A, exponential function, high order power function and logarithmic function B; The a dimensional expression of the PHASE DISTRIBUTION function of described composite phase plates is:

$f_2\left(x\right)=({\mathrm{\&alpha;}}_1\&CenterDot;x^3)$

$+\{{\mathrm{\&alpha;}}_2\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|x\left|\right)+{\mathrm{\&alpha;}}_3\left]\right\}$ -----logarithmic function A

$+[{\mathrm{\&alpha;}}_4\&CenterDot;x\&CenterDot;\exp ({\mathrm{\&alpha;}}_5\&CenterDot;x^2)]$

$+[{\mathrm{\&alpha;}}_6\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;{\left|x\right|}^{{\mathrm{\&alpha;}}_7}]$

$+\{{\mathrm{\&alpha;}}_8\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|\left|x\right|+{\mathrm{\&alpha;}}_9\left|\right)\left]\right\}$ -----logarithmic function B

In the formula:

α 1～α 9 is parameters of the PHASE DISTRIBUTION function of described phase-plate;

X is normalization aperture coordinate, and its span is [1,1];

Sgn is-symbol function, x＞0 o'clock its value are that 1, x＜0 o'clock its value is for-1.

3. composite phase plates that is used for expanding field depth of optical imaging system, it is characterized in that: the PHASE DISTRIBUTION function of described composite phase plates is to be formed by stacking by third power function, logarithmic function A, exponential function, high order power function, logarithmic function B and sine function; The a dimensional expression of the PHASE DISTRIBUTION function of described composite phase plates is:

$f_3\left(x\right)=({\mathrm{\&alpha;}}_1\&CenterDot;x^3)$

$+\{{\mathrm{\&alpha;}}_2\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|x\left|\right)+{\mathrm{\&alpha;}}_3\left]\right\}$ -----logarithmic function A

$+[{\mathrm{\&alpha;}}_4\&CenterDot;x\&CenterDot;\exp ({\mathrm{\&alpha;}}_5\&CenterDot;x^2)]$

$+[{\mathrm{\&alpha;}}_6\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;{\left|x\right|}^{{\mathrm{\&alpha;}}_7}]$

$+\{{\mathrm{\&alpha;}}_8\&CenterDot;\mathrm{sgn}\left(x\right)\&CenterDot;x^2\&CenterDot;[\ln \left(\right|\left|x\right|+{\mathrm{\&alpha;}}_9\left|\right)\left]\right\}$ -----logarithmic function B

$+[{\mathrm{\&alpha;}}_{10}\&CenterDot;\sin ({\mathrm{\&alpha;}}_{11}\&CenterDot;x)]$

In the formula:

α 1～α 11 is parameters of the PHASE DISTRIBUTION function of described phase-plate;

X is normalization aperture coordinate, and its span is [1,1];

Sgn is-symbol function, x＞0 o'clock its value are that 1, x＜0 o'clock its value is for-1.

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