Mongo查询另一个嵌套对象内的嵌套对象

Question

MongoDb数据库的Employee，Department和Subject课程很少，关系就像这样

public class Employee {
 @Id
 private long id;
 private String name;
 @DBRef
 private Subject subject;
} 

public class subject{
 @Id 
 private long id;
 private String name;
 @DBRef
 private Department department;
}

public class Department{
 @Id     
 private long id;
 private String name;
}

我可以使用主题ID

查询员工

db.employee.find("subject.$id":<subjectId>)

现在我正在尝试使用departmentId获取部门下的员工但是没有得到任何结果

db.employee.find("subject.department.$id":<departmentId>)

请告诉我使用MongoDB中的departmentId查询员工的任何选项

MongoDB中的示例对象

Employee ::

{"_id": 1,"name":"student1","subject":{"$ref":"subject","$id":1}}
{"_id": 2,"name":"student2","subject":{"$ref":"subject","$id":2}}
{"_id": 3,"name":"student3","subject":{"$ref":"subject","$id":3}}

Subject ::
{"_id": 1,"name":"subject1","department":{"$ref":"department","$id":1}}
{"_id": 2,"name":"subject2","department":{"$ref":"department","$id":1}}
{"_id": 3,"name":"subject3","department":{"$ref":"department","$id":2}}

Department ::

{"_id": 1,"name":"department1"}
{"_id": 2,"name":"department2"}

提前致谢