# 模拟波动光学的非近轴高斯光束公式

2018年 6月 26日

### 平面波的角谱

\left ( \frac

{\partial^2}{\partial x^2} +\frac{\partial^2} {\partial y^2}

+ k^2 \right ) E_z = 0,

E_z(x,y) = \int_

{k_x^2+k_y^2=k^2}

A(k_y)e^

{i(k_x x +k_y y)}

dk_y,

E_z(x,y) = \int_{-\pi/2}^

{\pi/2} A(\varphi)e^{ik(x \cos \varphi +y \sin \varphi)}d \varphi.

E_z(r,\theta) = \int_{-\pi/2}^{\pi/2}

A(\varphi) e^

{ikr \cos (\theta-\varphi)}

d \varphi.

1. 波矢数
2. 最大横波数

### 参考文献

1. P. Vaveliuk, “Limits of the paraxial approximation in laser beams”, Optics Letters, vol. 32, no. 8, 2007.
2. M. Born and E. Wolf, Principles of Optics, ed. 7, Cambridge University Press, 1999.
3. J. W. Goodman, Fourier Optics.
4. G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation”, Phys. Rev. a. 27, pp. 1693–1695, 1983.

#### 评论 (13)

##### 妙 彭
2020-03-11

Thanks Yosuke for such detailed explanation. The model I am currently working on is a Gaussian beam focused by a high NA (=1.25) objective lens. But I don’t know how to produce the tightly focused Gaussian beam with comsol. Can you help me ?

##### wei bao
2020-03-12 COMSOL 员工

2020-03-13

##### wei bao
2020-03-13 COMSOL 员工

2020-03-13

##### Yosuke Mizuyama
2020-03-13 COMSOL 员工

Hi,
Thank you for reading my blog.
Are you talking about the liquid immersion lens?
If so, it’s a little bit tricky to set it up correctly.
1. Calculate the NA in the vacuum instead of your material. NA_vac = 1.25/n. (I assume your liquid’s index is 1.51 or something. Then the NA_vac is about 0.83.)
2. Calculate a rough spot radius from the above result, r = 0.61*lambda/NA_vac (= 0.73*lambda if n=1.51).
3. From the above calculation, determine if your beam is paraxial or non-paraxial. If it’s a liquid immersion lens, it is most likely highly non-paraxial. So you will have to use the non-paraxial Gaussian beam option with the plane wave expansion.
4. In a liquid, Gaussian beams change the focal position. You will have to calculate the focal position shift in your liquid. Probably you will have to use the Gaussian ABCD method. Refer to Shojiro Nemoto, “Waist shift of a Gaussian beam by plane dielectric interfaces”, Applied Optics, Vol. 27, No. 9 (1988) for this focus shift physics.
5. Back-calculate the focal position in the vacuum and enter the focal position in the “Beam radius” setting in COMSOL.

Since your beam is highly non-paraxial, you will need more settings.
6. Select Plane wave expansion
7. Select User defined for Wave vector distribution type
8. Enter a certain large number for Wave vector count, say 40.
9. Enter a certain number larger than ewfd.k0, say 1.2*ewfd.k0
10. Enter the beam radius you calculated above
11. Enter the adjusted focal position in the vacuum that you calculated above

Then it should work.
This is a highly difficult and advanced simulation. You may need to contact support@comsol.com for your successful settings of this since there is limitation for teaching in this comment space.

Best regards,

Yosuke

##### 妙 彭
2020-03-14

Hi, Dr. Yosuke,
Thank you very much for your kindness help and patience. But I still have some confusion and need your help.
1. Yes, I am talking about the liquid immersion lens, and the liquid’s index is 1.51.
2. If the focus position is in water (n = 1.33) rather than in vacuum or air, do I need to consider the refractive index of water according to the formula you provided: r = 0.61*lambda/NA_vac?
3. As you mentioned in point 5, should it be enter the spot radius in the “Beam radius”, and the focal position in the “focal plane along the axis” in COMSOL?
4. As you mentioned in point 8 and 9, How to calculate the wave vector count and the maximum transverse wave number? Can I set any number I want? And iust make sure that this number is greater than ewfd.k0?
5. Actually, I am trying to simulate the motion of nanosphere in an optical tweezer system using COMSOL. After the Gaussian beam passes through the oil-immersed objective lens (NA = 1.25), it will illuminate on the nanospheres, where the environment is water (n = 1.33), and then focus behind the nanospheres. So, when I calculate the focal position, whether the refractive index of microsphere and water should be considered? And when the position of the microsphere changes, does the focal position need to be recalculated every time?
Best wishes
Miao Peng

##### Yosuke Mizuyama
2020-03-15 COMSOL 员工

Hi Miao,
The settings for the wave vector count and the max transverse wave number are not something that can be uniquely calculated. It’s a trade-off between the accuracy and the aliasing issue. So you will have to tweak it around. But please try 40 and 1.2*ewfd.k0 as I mentioned as starting points.
Once you have fixed the focal position for the initial nanosphere position, you don’t have to change it even if the sphere moves around unless the beam moves around as well.
I hope this helps.
Best regards,
Yosuke

##### 妙 彭
2020-03-15

Hi, Dr. Yosuke,
Thank you very much for your detailed explanation. It will help me a lot. And I am looking forward to more such a brilliant blogs in the future, maybe more about the simulation of optical tweezers.
Best wishes
Miao Peng

##### Yosuke Mizuyama
2020-03-17 COMSOL 员工

You are welcome!
Best regards,
Yosuke

##### 妙 彭
2020-03-19

Hi，Dr. Yosuke
I am sorry to bothering you again.
1. In COMSOL 5.5, does the plane wave expansion formula of Gaussian beam is transmitted in vacuum by default? So when we think about transmission in other media, we need to consider the refractive index of the media, right?
2. You said, “Calculate a rough spot radius from the above result, r = 0.61*lambda/NA_vac.” If my focus is in a medium such as water or air, do I need to consider the effect of the refractive index of the medium on the spot size?
3. In “ewfd” setting, the default value of the “Focal plane along the axis” of the beam is zero, right? If so, when the focus position of the tightly focused Gaussian beam is in the water, is the calculated “focal position shift” should be directly enter in the text box of “Focal plane along the axis”?
Best wishes
Sincerely,
Miao Peng

##### Yosuke Mizuyama
2020-03-19 COMSOL 员工

Hi Miao,
I think you have to consider if the background field feature is appropriate or not for your application. We’ve been discussing under an assumption that you can somehow use the background field formulation by doing some pre-calculations. But it may be more appropriate to use the Gaussian beam boundary condition, rather than the (domain) background. We introduced the Gaussian beam feature in the scattering boundary condition at version 5.5 but it’s only the paraxial formulation. For your high NA case, you can use the integral formula that I wrote in this blog.
Best regards,
Yosuke

##### 妙 彭
2020-03-24

Hi, Dr. Yosuke
Thank you for your kindness help again. I see what you mean.
Best wishes
Sincerely,
Miao Peng